Daily Papers

Transformers have been recently explored for 3D point cloud understanding with impressive progress achieved. A large number of points, over 0.1 million, make the global self-attention infeasible for point cloud data. Thus, most methods propose to apply the transformer in a local region, e.g., spherical or cubic window. However, it still contains a large number of Query-Key pairs, which requires high computational costs. In addition, previous methods usually learn the query, key, and value using a linear projection without modeling the local 3D geometric structure. In this paper, we attempt to reduce the costs and model the local geometry prior by developing a new transformer block, named ConDaFormer. Technically, ConDaFormer disassembles the cubic window into three orthogonal 2D planes, leading to fewer points when modeling the attention in a similar range. The disassembling operation is beneficial to enlarging the range of attention without increasing the computational complexity, but ignores some contexts. To provide a remedy, we develop a local structure enhancement strategy that introduces a depth-wise convolution before and after the attention. This scheme can also capture the local geometric information. Taking advantage of these designs, ConDaFormer captures both long-range contextual information and local priors. The effectiveness is demonstrated by experimental results on several 3D point cloud understanding benchmarks. Code is available at https://github.com/LHDuan/ConDaFormer .

3D shape generation aims to produce innovative 3D content adhering to specific conditions and constraints. Existing methods often decompose 3D shapes into a sequence of localized components, treating each element in isolation without considering spatial consistency. As a result, these approaches exhibit limited versatility in 3D data representation and shape generation, hindering their ability to generate highly diverse 3D shapes that comply with the specified constraints. In this paper, we introduce a novel spatial-aware 3D shape generation framework that leverages 2D plane representations for enhanced 3D shape modeling. To ensure spatial coherence and reduce memory usage, we incorporate a hybrid shape representation technique that directly learns a continuous signed distance field representation of the 3D shape using orthogonal 2D planes. Additionally, we meticulously enforce spatial correspondences across distinct planes using a transformer-based autoencoder structure, promoting the preservation of spatial relationships in the generated 3D shapes. This yields an algorithm that consistently outperforms state-of-the-art 3D shape generation methods on various tasks, including unconditional shape generation, multi-modal shape completion, single-view reconstruction, and text-to-shape synthesis.

Recent trends in semi-supervised learning have significantly boosted the performance of 3D semi-supervised medical image segmentation. Compared with 2D images, 3D medical volumes involve information from different directions, e.g., transverse, sagittal, and coronal planes, so as to naturally provide complementary views. These complementary views and the intrinsic similarity among adjacent 3D slices inspire us to develop a novel annotation way and its corresponding semi-supervised model for effective segmentation. Specifically, we firstly propose the orthogonal annotation by only labeling two orthogonal slices in a labeled volume, which significantly relieves the burden of annotation. Then, we perform registration to obtain the initial pseudo labels for sparsely labeled volumes. Subsequently, by introducing unlabeled volumes, we propose a dual-network paradigm named Dense-Sparse Co-training (DeSCO) that exploits dense pseudo labels in early stage and sparse labels in later stage and meanwhile forces consistent output of two networks. Experimental results on three benchmark datasets validated our effectiveness in performance and efficiency in annotation. For example, with only 10 annotated slices, our method reaches a Dice up to 86.93% on KiTS19 dataset.

Reconstructing detailed 3D objects from single-view images remains a challenging task due to the limited information available. In this paper, we introduce FDGaussian, a novel two-stage framework for single-image 3D reconstruction. Recent methods typically utilize pre-trained 2D diffusion models to generate plausible novel views from the input image, yet they encounter issues with either multi-view inconsistency or lack of geometric fidelity. To overcome these challenges, we propose an orthogonal plane decomposition mechanism to extract 3D geometric features from the 2D input, enabling the generation of consistent multi-view images. Moreover, we further accelerate the state-of-the-art Gaussian Splatting incorporating epipolar attention to fuse images from different viewpoints. We demonstrate that FDGaussian generates images with high consistency across different views and reconstructs high-quality 3D objects, both qualitatively and quantitatively. More examples can be found at our website https://qjfeng.net/FDGaussian/.

Single-image piece-wise planar 3D reconstruction aims to simultaneously segment plane instances and recover 3D plane parameters from an image. Most recent approaches leverage convolutional neural networks (CNNs) and achieve promising results. However, these methods are limited to detecting a fixed number of planes with certain learned order. To tackle this problem, we propose a novel two-stage method based on associative embedding, inspired by its recent success in instance segmentation. In the first stage, we train a CNN to map each pixel to an embedding space where pixels from the same plane instance have similar embeddings. Then, the plane instances are obtained by grouping the embedding vectors in planar regions via an efficient mean shift clustering algorithm. In the second stage, we estimate the parameter for each plane instance by considering both pixel-level and instance-level consistencies. With the proposed method, we are able to detect an arbitrary number of planes. Extensive experiments on public datasets validate the effectiveness and efficiency of our method. Furthermore, our method runs at 30 fps at the testing time, thus could facilitate many real-time applications such as visual SLAM and human-robot interaction. Code is available at https://github.com/svip-lab/PlanarReconstruction.

We characterize the cut patterns that can be produced by "orthogonal fold & cut": folding an axis-aligned rectangular sheet of paper along horizontal and vertical creases, and then making a single straight cut (at any angle). Along the way, we solve a handful of related problems: orthogonal fold & punch, 1D fold & cut, signed 1D fold & cut, and 1D interval fold & cut.

Extracting planes from a 3D scene is useful for downstream tasks in robotics and augmented reality. In this paper we tackle the problem of estimating the planar surfaces in a scene from posed images. Our first finding is that a surprisingly competitive baseline results from combining popular clustering algorithms with recent improvements in 3D geometry estimation. However, such purely geometric methods are understandably oblivious to plane semantics, which are crucial to discerning distinct planes. To overcome this limitation, we propose a method that predicts multi-view consistent plane embeddings that complement geometry when clustering points into planes. We show through extensive evaluation on the ScanNetV2 dataset that our new method outperforms existing approaches and our strong geometric baseline for the task of plane estimation.

The task of synthesizing novel views from a single image has useful applications in virtual reality and mobile computing, and a number of approaches to the problem have been proposed in recent years. A Multiplane Image (MPI) estimates the scene as a stack of RGBA layers, and can model complex appearance effects, anti-alias depth errors and synthesize soft edges better than methods that use textured meshes or layered depth images. And unlike neural radiance fields, an MPI can be efficiently rendered on graphics hardware. However, MPIs are highly redundant and require a large number of depth layers to achieve plausible results. Based on the observation that the depth complexity in local image regions is lower than that over the entire image, we split an MPI into many small, tiled regions, each with only a few depth planes. We call this representation a Tiled Multiplane Image (TMPI). We propose a method for generating a TMPI with adaptive depth planes for single-view 3D photography in the wild. Our synthesized results are comparable to state-of-the-art single-view MPI methods while having lower computational overhead.

Modeling and re-rendering dynamic 3D scenes is a challenging task in 3D vision. Prior approaches build on NeRF and rely on implicit representations. This is slow since it requires many MLP evaluations, constraining real-world applications. We show that dynamic 3D scenes can be explicitly represented by six planes of learned features, leading to an elegant solution we call HexPlane. A HexPlane computes features for points in spacetime by fusing vectors extracted from each plane, which is highly efficient. Pairing a HexPlane with a tiny MLP to regress output colors and training via volume rendering gives impressive results for novel view synthesis on dynamic scenes, matching the image quality of prior work but reducing training time by more than 100times. Extensive ablations confirm our HexPlane design and show that it is robust to different feature fusion mechanisms, coordinate systems, and decoding mechanisms. HexPlane is a simple and effective solution for representing 4D volumes, and we hope they can broadly contribute to modeling spacetime for dynamic 3D scenes.

3D plane recovery from a single image can usually be divided into several subtasks of plane detection, segmentation, parameter estimation and possibly depth estimation. Previous works tend to solve this task by either extending the RCNN-based segmentation network or the dense pixel embedding-based clustering framework. However, none of them tried to integrate above related subtasks into a unified framework but treat them separately and sequentially, which we suspect is potentially a main source of performance limitation for existing approaches. Motivated by this finding and the success of query-based learning in enriching reasoning among semantic entities, in this paper, we propose PlaneRecTR, a Transformer-based architecture, which for the first time unifies all subtasks related to single-view plane recovery with a single compact model. Extensive quantitative and qualitative experiments demonstrate that our proposed unified learning achieves mutual benefits across subtasks, obtaining a new state-of-the-art performance on public ScanNet and NYUv2-Plane datasets. Codes are available at https://github.com/SJingjia/PlaneRecTR.

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.

Based on the Manhattan World assumption, most existing indoor layout estimation schemes focus on recovering layouts from vertically compressed 1D sequences. However, the compression procedure confuses the semantics of different planes, yielding inferior performance with ambiguous interpretability. To address this issue, we propose to disentangle this 1D representation by pre-segmenting orthogonal (vertical and horizontal) planes from a complex scene, explicitly capturing the geometric cues for indoor layout estimation. Considering the symmetry between the floor boundary and ceiling boundary, we also design a soft-flipping fusion strategy to assist the pre-segmentation. Besides, we present a feature assembling mechanism to effectively integrate shallow and deep features with distortion distribution awareness. To compensate for the potential errors in pre-segmentation, we further leverage triple attention to reconstruct the disentangled sequences for better performance. Experiments on four popular benchmarks demonstrate our superiority over existing SoTA solutions, especially on the 3DIoU metric. The code is available at https://github.com/zhijieshen-bjtu/DOPNet.

The Multiplane Image (MPI), containing a set of fronto-parallel RGBA layers, is an effective and efficient representation for view synthesis from sparse inputs. Yet, its fixed structure limits the performance, especially for surfaces imaged at oblique angles. We introduce the Structural MPI (S-MPI), where the plane structure approximates 3D scenes concisely. Conveying RGBA contexts with geometrically-faithful structures, the S-MPI directly bridges view synthesis and 3D reconstruction. It can not only overcome the critical limitations of MPI, i.e., discretization artifacts from sloped surfaces and abuse of redundant layers, and can also acquire planar 3D reconstruction. Despite the intuition and demand of applying S-MPI, great challenges are introduced, e.g., high-fidelity approximation for both RGBA layers and plane poses, multi-view consistency, non-planar regions modeling, and efficient rendering with intersected planes. Accordingly, we propose a transformer-based network based on a segmentation model. It predicts compact and expressive S-MPI layers with their corresponding masks, poses, and RGBA contexts. Non-planar regions are inclusively handled as a special case in our unified framework. Multi-view consistency is ensured by sharing global proxy embeddings, which encode plane-level features covering the complete 3D scenes with aligned coordinates. Intensive experiments show that our method outperforms both previous state-of-the-art MPI-based view synthesis methods and planar reconstruction methods.

Polygonal meshes are ubiquitous, but have only played a relatively minor role in the deep learning revolution. State-of-the-art neural generative models for 3D shapes learn implicit functions and generate meshes via expensive iso-surfacing. We overcome these challenges by employing a classical spatial data structure from computer graphics, Binary Space Partitioning (BSP), to facilitate 3D learning. The core operation of BSP involves recursive subdivision of 3D space to obtain convex sets. By exploiting this property, we devise BSP-Net, a network that learns to represent a 3D shape via convex decomposition without supervision. The network is trained to reconstruct a shape using a set of convexes obtained from a BSP-tree built over a set of planes, where the planes and convexes are both defined by learned network weights. BSP-Net directly outputs polygonal meshes from the inferred convexes. The generated meshes are watertight, compact (i.e., low-poly), and well suited to represent sharp geometry. We show that the reconstruction quality by BSP-Net is competitive with those from state-of-the-art methods while using much fewer primitives. We also explore variations to BSP-Net including using a more generic decoder for reconstruction, more general primitives than planes, as well as training a generative model with variational auto-encoders. Code is available at https://github.com/czq142857/BSP-NET-original.

Grid-based structures are commonly used to encode explicit features for graphics primitives such as images, signed distance functions (SDF), and neural radiance fields (NeRF) due to their simple implementation. However, in n-dimensional space, calculating the value of a sampled point requires interpolating the values of its 2^n neighboring vertices. The exponential scaling with dimension leads to significant computational overheads. To address this issue, we propose a simplex-based approach for encoding graphics primitives. The number of vertices in a simplex-based structure increases linearly with dimension, making it a more efficient and generalizable alternative to grid-based representations. Using the non-axis-aligned simplicial structure property, we derive and prove a coordinate transformation, simplicial subdivision, and barycentric interpolation scheme for efficient sampling, which resembles transformation procedures in the simplex noise algorithm. Finally, we use hash tables to store multiresolution features of all interest points in the simplicial grid, which are passed into a tiny fully connected neural network to parameterize graphics primitives. We implemented a detailed simplex-based structure encoding algorithm in C++ and CUDA using the methods outlined in our approach. In the 2D image fitting task, the proposed method is capable of fitting a giga-pixel image with 9.4% less time compared to the baseline method proposed by instant-ngp, while maintaining the same quality and compression rate. In the volumetric rendering setup, we observe a maximum 41.2% speedup when the samples are dense enough.

Recent novel view synthesis methods obtain promising results for relatively small scenes, e.g., indoor environments and scenes with a few objects, but tend to fail for unbounded outdoor scenes with a single image as input. In this paper, we introduce SAMPLING, a Scene-adaptive Hierarchical Multiplane Images Representation for Novel View Synthesis from a Single Image based on improved multiplane images (MPI). Observing that depth distribution varies significantly for unbounded outdoor scenes, we employ an adaptive-bins strategy for MPI to arrange planes in accordance with each scene image. To represent intricate geometry and multi-scale details, we further introduce a hierarchical refinement branch, which results in high-quality synthesized novel views. Our method demonstrates considerable performance gains in synthesizing large-scale unbounded outdoor scenes using a single image on the KITTI dataset and generalizes well to the unseen Tanks and Temples dataset.The code and models will soon be made available.

We present UniPlane, a novel method that unifies plane detection and reconstruction from posed monocular videos. Unlike existing methods that detect planes from local observations and associate them across the video for the final reconstruction, UniPlane unifies both the detection and the reconstruction tasks in a single network, which allows us to directly optimize final reconstruction quality and fully leverage temporal information. Specifically, we build a Transformers-based deep neural network that jointly constructs a 3D feature volume for the environment and estimates a set of per-plane embeddings as queries. UniPlane directly reconstructs the 3D planes by taking dot products between voxel embeddings and the plane embeddings followed by binary thresholding. Extensive experiments on real-world datasets demonstrate that UniPlane outperforms state-of-the-art methods in both plane detection and reconstruction tasks, achieving +4.6 in F-score in geometry as well as consistent improvements in other geometry and segmentation metrics.

We introduce AlphaTablets, a novel and generic representation of 3D planes that features continuous 3D surface and precise boundary delineation. By representing 3D planes as rectangles with alpha channels, AlphaTablets combine the advantages of current 2D and 3D plane representations, enabling accurate, consistent and flexible modeling of 3D planes. We derive differentiable rasterization on top of AlphaTablets to efficiently render 3D planes into images, and propose a novel bottom-up pipeline for 3D planar reconstruction from monocular videos. Starting with 2D superpixels and geometric cues from pre-trained models, we initialize 3D planes as AlphaTablets and optimize them via differentiable rendering. An effective merging scheme is introduced to facilitate the growth and refinement of AlphaTablets. Through iterative optimization and merging, we reconstruct complete and accurate 3D planes with solid surfaces and clear boundaries. Extensive experiments on the ScanNet dataset demonstrate state-of-the-art performance in 3D planar reconstruction, underscoring the great potential of AlphaTablets as a generic 3D plane representation for various applications. Project page is available at: https://hyzcluster.github.io/alphatablets

Monocular depth estimation aims to infer a dense depth map from a single image, which is a fundamental and prevalent task in computer vision. Many previous works have shown impressive depth estimation results through carefully designed network structures, but they usually ignore the planar information and therefore perform poorly in low-texture areas of indoor scenes. In this paper, we propose Plane2Depth, which adaptively utilizes plane information to improve depth prediction within a hierarchical framework. Specifically, in the proposed plane guided depth generator (PGDG), we design a set of plane queries as prototypes to softly model planes in the scene and predict per-pixel plane coefficients. Then the predicted plane coefficients can be converted into metric depth values with the pinhole camera model. In the proposed adaptive plane query aggregation (APGA) module, we introduce a novel feature interaction approach to improve the aggregation of multi-scale plane features in a top-down manner. Extensive experiments show that our method can achieve outstanding performance, especially in low-texture or repetitive areas. Furthermore, under the same backbone network, our method outperforms the state-of-the-art methods on the NYU-Depth-v2 dataset, achieves competitive results with state-of-the-art methods KITTI dataset and can be generalized to unseen scenes effectively.

We present a computational framework that transforms single images into 3D physical objects. The visual geometry of a physical object in an image is determined by three orthogonal attributes: mechanical properties, external forces, and rest-shape geometry. Existing single-view 3D reconstruction methods often overlook this underlying composition, presuming rigidity or neglecting external forces. Consequently, the reconstructed objects fail to withstand real-world physical forces, resulting in instability or undesirable deformation -- diverging from their intended designs as depicted in the image. Our optimization framework addresses this by embedding physical compatibility into the reconstruction process. We explicitly decompose the three physical attributes and link them through static equilibrium, which serves as a hard constraint, ensuring that the optimized physical shapes exhibit desired physical behaviors. Evaluations on a dataset collected from Objaverse demonstrate that our framework consistently enhances the physical realism of 3D models over existing methods. The utility of our framework extends to practical applications in dynamic simulations and 3D printing, where adherence to physical compatibility is paramount.

Although polygon meshes have been a standard representation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning-based applications. In this work, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neural shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the boundary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target surface alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.

The aerodynamic coefficients of aircrafts are significantly impacted by its geometry, especially when the angle of attack (AoA) is large. In the field of aerodynamics, traditional polynomial-based parameterization uses as few parameters as possible to describe the geometry of an airfoil. However, because the 3D geometry of a wing is more complicated than the 2D airfoil, polynomial-based parameterizations have difficulty in accurately representing the entire shape of a wing in 3D space. Existing deep learning-based methods can extract massive latent neural representations for the shape of 2D airfoils or 2D slices of wings. Recent studies highlight that directly taking geometric features as inputs to the neural networks can improve the accuracy of predicted aerodynamic coefficients. Motivated by geometry theory, we propose to incorporate Riemannian geometric features for learning Coefficient of Pressure (CP) distributions on wing surfaces. Our method calculates geometric features (Riemannian metric, connection, and curvature) and further inputs the geometric features, coordinates and flight conditions into a deep learning model to predict the CP distribution. Experimental results show that our method, compared to state-of-the-art Deep Attention Network (DAN), reduces the predicted mean square error (MSE) of CP by an average of 8.41% for the DLR-F11 aircraft test set.

Driven by the appealing properties of neural fields for storing and communicating 3D data, the problem of directly processing them to address tasks such as classification and part segmentation has emerged and has been investigated in recent works. Early approaches employ neural fields parameterized by shared networks trained on the whole dataset, achieving good task performance but sacrificing reconstruction quality. To improve the latter, later methods focus on individual neural fields parameterized as large Multi-Layer Perceptrons (MLPs), which are, however, challenging to process due to the high dimensionality of the weight space, intrinsic weight space symmetries, and sensitivity to random initialization. Hence, results turn out significantly inferior to those achieved by processing explicit representations, e.g., point clouds or meshes. In the meantime, hybrid representations, in particular based on tri-planes, have emerged as a more effective and efficient alternative to realize neural fields, but their direct processing has not been investigated yet. In this paper, we show that the tri-plane discrete data structure encodes rich information, which can be effectively processed by standard deep-learning machinery. We define an extensive benchmark covering a diverse set of fields such as occupancy, signed/unsigned distance, and, for the first time, radiance fields. While processing a field with the same reconstruction quality, we achieve task performance far superior to frameworks that process large MLPs and, for the first time, almost on par with architectures handling explicit representations.

Single-image room layout reconstruction aims to reconstruct the enclosed 3D structure of a room from a single image. Most previous work relies on the cuboid-shape prior. This paper considers a more general indoor assumption, i.e., the room layout consists of a single ceiling, a single floor, and several vertical walls. To this end, we first employ Convolutional Neural Networks to detect planes and vertical lines between adjacent walls. Meanwhile, estimating the 3D parameters for each plane. Then, a simple yet effective geometric reasoning method is adopted to achieve room layout reconstruction. Furthermore, we optimize the 3D plane parameters to reconstruct a geometrically consistent room layout between planes and lines. The experimental results on public datasets validate the effectiveness and efficiency of our method.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

The paper presents a novel approach for vector-floorplan generation via a diffusion model, which denoises 2D coordinates of room/door corners with two inference objectives: 1) a single-step noise as the continuous quantity to precisely invert the continuous forward process; and 2) the final 2D coordinate as the discrete quantity to establish geometric incident relationships such as parallelism, orthogonality, and corner-sharing. Our task is graph-conditioned floorplan generation, a common workflow in floorplan design. We represent a floorplan as 1D polygonal loops, each of which corresponds to a room or a door. Our diffusion model employs a Transformer architecture at the core, which controls the attention masks based on the input graph-constraint and directly generates vector-graphics floorplans via a discrete and continuous denoising process. We have evaluated our approach on RPLAN dataset. The proposed approach makes significant improvements in all the metrics against the state-of-the-art with significant margins, while being capable of generating non-Manhattan structures and controlling the exact number of corners per room. A project website with supplementary video and document is here https://aminshabani.github.io/housediffusion.

The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenkovi\'c, Solomon, and Than [CCLMST23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that for every two vertices u and v in the graph, there exists a path between u and v that intersects only a few clusters. They proved that any planar graph admits a shortcut partition and gave several applications, including a construction of tree cover for arbitrary planar graphs with stretch 1+varepsilon and O(1) many trees for any fixed varepsilon in (0,1). However, the construction heavily exploits planarity in multiple steps, and is thus inherently limited to planar graphs. In this work, we breach the "planarity barrier" to construct a shortcut partition for K_r-minor-free graphs for any r. To this end, we take a completely different approach -- our key contribution is a novel deterministic variant of the cop decomposition in minor-free graphs [And86, AGG14]. Our shortcut partition for K_r-minor-free graphs yields several direct applications. Most notably, we construct the first optimal distance oracle for K_r-minor-free graphs, with 1+varepsilon stretch, linear space, and constant query time for any fixed varepsilon in (0,1). The previous best distance oracle [AG06] uses O(nlog n) space and O(log n) query time, and its construction relies on Robertson-Seymour structural theorem and other sophisticated tools. We also obtain the first tree cover of O(1) size for minor-free graphs with stretch 1+varepsilon, while the previous best (1+varepsilon)-tree cover has size O(log^2 n) [BFN19].

We address 2D floorplan reconstruction from 3D scans. Existing approaches typically employ heuristically designed multi-stage pipelines. Instead, we formulate floorplan reconstruction as a single-stage structured prediction task: find a variable-size set of polygons, which in turn are variable-length sequences of ordered vertices. To solve it we develop a novel Transformer architecture that generates polygons of multiple rooms in parallel, in a holistic manner without hand-crafted intermediate stages. The model features two-level queries for polygons and corners, and includes polygon matching to make the network end-to-end trainable. Our method achieves a new state-of-the-art for two challenging datasets, Structured3D and SceneCAD, along with significantly faster inference than previous methods. Moreover, it can readily be extended to predict additional information, i.e., semantic room types and architectural elements like doors and windows. Our code and models are available at: https://github.com/ywyue/RoomFormer.

2D keypoints are commonly used as an additional cue to refine estimated 3D human meshes. Current methods optimize the pose and shape parameters with a reprojection loss on the provided 2D keypoints. Such an approach, while simple and intuitive, has limited effectiveness because the optimal solution is hard to find in ambiguous parameter space and may sacrifice depth. Additionally, divergent gradients from distal joints complicate and deviate the refinement of proximal joints in the kinematic chain. To address these, we introduce Kinematic-Tree Rotation (KITRO), a novel mesh refinement strategy that explicitly models depth and human kinematic-tree structure. KITRO treats refinement from a bone-wise perspective. Unlike previous methods which perform gradient-based optimizations, our method calculates bone directions in closed form. By accounting for the 2D pose, bone length, and parent joint's depth, the calculation results in two possible directions for each child joint. We then use a decision tree to trace binary choices for all bones along the human skeleton's kinematic-tree to select the most probable hypothesis. Our experiments across various datasets and baseline models demonstrate that KITRO significantly improves 3D joint estimation accuracy and achieves an ideal 2D fit simultaneously. Our code available at: https://github.com/MartaYang/KITRO.

One-class learning is the classic problem of fitting a model to data for which annotations are available only for a single class. In this paper, we propose a novel objective for one-class learning. Our key idea is to use a pair of orthonormal frames -- as subspaces -- to "sandwich" the labeled data via optimizing for two objectives jointly: i) minimize the distance between the origins of the two subspaces, and ii) to maximize the margin between the hyperplanes and the data, either subspace demanding the data to be in its positive and negative orthant respectively. Our proposed objective however leads to a non-convex optimization problem, to which we resort to Riemannian optimization schemes and derive an efficient conjugate gradient scheme on the Stiefel manifold. To study the effectiveness of our scheme, we propose a new dataset~Dash-Cam-Pose, consisting of clips with skeleton poses of humans seated in a car, the task being to classify the clips as normal or abnormal; the latter is when any human pose is out-of-position with regard to say an airbag deployment. Our experiments on the proposed Dash-Cam-Pose dataset, as well as several other standard anomaly/novelty detection benchmarks demonstrate the benefits of our scheme, achieving state-of-the-art one-class accuracy.

We propose a method, named DualMesh-UDF, to extract a surface from unsigned distance functions (UDFs), encoded by neural networks, or neural UDFs. Neural UDFs are becoming increasingly popular for surface representation because of their versatility in presenting surfaces with arbitrary topologies, as opposed to the signed distance function that is limited to representing a closed surface. However, the applications of neural UDFs are hindered by the notorious difficulty in extracting the target surfaces they represent. Recent methods for surface extraction from a neural UDF suffer from significant geometric errors or topological artifacts due to two main difficulties: (1) A UDF does not exhibit sign changes; and (2) A neural UDF typically has substantial approximation errors. DualMesh-UDF addresses these two difficulties. Specifically, given a neural UDF encoding a target surface S to be recovered, we first estimate the tangent planes of S at a set of sample points close to S. Next, we organize these sample points into local clusters, and for each local cluster, solve a linear least squares problem to determine a final surface point. These surface points are then connected to create the output mesh surface, which approximates the target surface. The robust estimation of the tangent planes of the target surface and the subsequent minimization problem constitute our core strategy, which contributes to the favorable performance of DualMesh-UDF over other competing methods. To efficiently implement this strategy, we employ an adaptive Octree. Within this framework, we estimate the location of a surface point in each of the octree cells identified as containing part of the target surface. Extensive experiments show that our method outperforms existing methods in terms of surface reconstruction quality while maintaining comparable computational efficiency.

There are few known exponential speedups for quantum algorithms and these tend to fall into even fewer families. One speedup that has mostly resisted generalization is the use of quantum walks to traverse the welded-tree graph, due to Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman. We show how to generalize this to a large class of hierarchical graphs in which the vertices are grouped into "supervertices" which are arranged according to a d-dimensional lattice. Supervertices can have different sizes, and edges between supervertices correspond to random connections between their constituent vertices. The hitting times of quantum walks on these graphs are related to the localization properties of zero modes in certain disordered tight binding Hamiltonians. The speedups range from superpolynomial to exponential, depending on the underlying dimension and the random graph model. We also provide concrete realizations of these hierarchical graphs, and introduce a general method for constructing graphs with efficient quantum traversal times using graph sparsification.

This paper studies the problem of recovering cameras from a set of fundamental matrices. A set of fundamental matrices is said to be compatible if a set of cameras exists for which they are the fundamental matrices. We focus on the complete graph, where fundamental matrices for each pair of cameras are given. Previous work has established necessary and sufficient conditions for compatibility as rank and eigenvalue conditions on the n-view fundamental matrix obtained by concatenating the individual fundamental matrices. In this work, we show that the eigenvalue condition is redundant. We provide explicit homogeneous polynomials that describe necessary and sufficient conditions for compatibility in terms of the fundamental matrices and their epipoles. In this direction, we find that quadruple-wise compatibility is enough to ensure global compatibility for any number of cameras. We demonstrate that for four cameras, compatibility is generically described by triple-wise conditions and one additional equation involving all fundamental matrices.

Learning about the three-dimensional world from two-dimensional images is a fundamental problem in computer vision. An ideal neural network architecture for such tasks would leverage the fact that objects can be rotated and translated in three dimensions to make predictions about novel images. However, imposing SO(3)-equivariance on two-dimensional inputs is difficult because the group of three-dimensional rotations does not have a natural action on the two-dimensional plane. Specifically, it is possible that an element of SO(3) will rotate an image out of plane. We show that an algorithm that learns a three-dimensional representation of the world from two dimensional images must satisfy certain geometric consistency properties which we formulate as SO(2)-equivariance constraints. We use the induced and restricted representations of SO(2) on SO(3) to construct and classify architectures which satisfy these geometric consistency constraints. We prove that any architecture which respects said consistency constraints can be realized as an instance of our construction. We show that three previously proposed neural architectures for 3D pose prediction are special cases of our construction. We propose a new algorithm that is a learnable generalization of previously considered methods. We test our architecture on three pose predictions task and achieve SOTA results on both the PASCAL3D+ and SYMSOL pose estimation tasks.

While following different technical routes, both low-rank and orthogonal adaptation techniques can efficiently adapt large-scale pre-training models in specific tasks or domains based on a small piece of trainable parameters. In this study, we bridge the gap between these two techniques, proposing a simple but effective adaptation method based on Householder reflections. Given a pre-trained model, our method fine-tunes its layers by multiplying each frozen weight matrix with an orthogonal matrix constructed by a chain of learnable Householder reflections (HRs). This HR-based orthogonal fine-tuning is equivalent to an adaptive low-rank adaptation. Moreover, we show that the orthogonality of the reflection planes corresponding to the HRs impacts the model capacity and regularity. The analysis motivates us to regularize the orthogonality of the HRs, leading to different implementations of the proposed Householder reflection adaptation (HRA) method. Compared with state-of-the-art methods, HRA achieves superior performance with fewer learnable parameters when adapting large language models and conditional image generators. The code is available at https://github.com/DaShenZi721/HRA

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a spectral loss. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a 10times increase in performance speed.

Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning incorporate non-linear positional encoding, addressing issues like spectral bias; however, this poses challenges in applying mesh extraction techniques based on CPWA functions. Focusing on trilinear interpolating methods as positional encoding, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint. Moreover, we introduce a method for approximating intersecting points among three hypersurfaces contributing to broader applications. We empirically validate correctness and parsimony through chamfer distance and efficiency, and angular distance, while examining the correlation between the eikonal loss and the planarity of the hypersurfaces.

3D Gaussian Splatting (3D-GS) has recently attracted great attention with real-time and photo-realistic renderings. This technique typically takes perspective images as input and optimizes a set of 3D elliptical Gaussians by splatting them onto the image planes, resulting in 2D Gaussians. However, applying 3D-GS to panoramic inputs presents challenges in effectively modeling the projection onto the spherical surface of {360^circ} images using 2D Gaussians. In practical applications, input panoramas are often sparse, leading to unreliable initialization of 3D Gaussians and subsequent degradation of 3D-GS quality. In addition, due to the under-constrained geometry of texture-less planes (e.g., walls and floors), 3D-GS struggles to model these flat regions with elliptical Gaussians, resulting in significant floaters in novel views. To address these issues, we propose 360-GS, a novel 360^{circ} Gaussian splatting for a limited set of panoramic inputs. Instead of splatting 3D Gaussians directly onto the spherical surface, 360-GS projects them onto the tangent plane of the unit sphere and then maps them to the spherical projections. This adaptation enables the representation of the projection using Gaussians. We guide the optimization of 360-GS by exploiting layout priors within panoramas, which are simple to obtain and contain strong structural information about the indoor scene. Our experimental results demonstrate that 360-GS allows panoramic rendering and outperforms state-of-the-art methods with fewer artifacts in novel view synthesis, thus providing immersive roaming in indoor scenarios.

The hull of a linear code C is the intersection of C with its dual code. We present and analyze the number of linear q-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension k and length nge 2k the number of all [n,k]_q linear codes with hull dimension l decreases as l increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length n and dimension k, comparing the obtained numbers with the number of all linear codes for the given n and k.

Micro-Aerial Vehicles (MAVs) have the advantage of moving freely in 3D space. However, creating compact and sparse map representations that can be efficiently used for planning for such robots is still an open problem. In this paper, we take maps built from noisy sensor data and construct a sparse graph containing topological information that can be used for 3D planning. We use a Euclidean Signed Distance Field, extract a 3D Generalized Voronoi Diagram (GVD), and obtain a thin skeleton diagram representing the topological structure of the environment. We then convert this skeleton diagram into a sparse graph, which we show is resistant to noise and changes in resolution. We demonstrate global planning over this graph, and the orders of magnitude speed-up it offers over other common planning methods. We validate our planning algorithm in real maps built onboard an MAV, using RGB-D sensing.

We propose a novel method, LoLep, which regresses Locally-Learned planes from a single RGB image to represent scenes accurately, thus generating better novel views. Without the depth information, regressing appropriate plane locations is a challenging problem. To solve this issue, we pre-partition the disparity space into bins and design a disparity sampler to regress local offsets for multiple planes in each bin. However, only using such a sampler makes the network not convergent; we further propose two optimizing strategies that combine with different disparity distributions of datasets and propose an occlusion-aware reprojection loss as a simple yet effective geometric supervision technique. We also introduce a self-attention mechanism to improve occlusion inference and present a Block-Sampling Self-Attention (BS-SA) module to address the problem of applying self-attention to large feature maps. We demonstrate the effectiveness of our approach and generate state-of-the-art results on different datasets. Compared to MINE, our approach has an LPIPS reduction of 4.8%-9.0% and an RV reduction of 73.9%-83.5%. We also evaluate the performance on real-world images and demonstrate the benefits.

Given a K-vertex simplex in a d-dimensional space, suppose we measure n points on the simplex with noise (hence, some of the observed points fall outside the simplex). Vertex hunting is the problem of estimating the K vertices of the simplex. A popular vertex hunting algorithm is successive projection algorithm (SPA). However, SPA is observed to perform unsatisfactorily under strong noise or outliers. We propose pseudo-point SPA (pp-SPA). It uses a projection step and a denoise step to generate pseudo-points and feed them into SPA for vertex hunting. We derive error bounds for pp-SPA, leveraging on extreme value theory of (possibly) high-dimensional random vectors. The results suggest that pp-SPA has faster rates and better numerical performances than SPA. Our analysis includes an improved non-asymptotic bound for the original SPA, which is of independent interest.

In stark contrast to the case of images, finding a concise, learnable discrete representation of 3D surfaces remains a challenge. In particular, while polygon meshes are arguably the most common surface representation used in geometry processing, their irregular and combinatorial structure often make them unsuitable for learning-based applications. In this work, we present VoroMesh, a novel and differentiable Voronoi-based representation of watertight 3D shape surfaces. From a set of 3D points (called generators) and their associated occupancy, we define our boundary representation through the Voronoi diagram of the generators as the subset of Voronoi faces whose two associated (equidistant) generators are of opposite occupancy: the resulting polygon mesh forms a watertight approximation of the target shape's boundary. To learn the position of the generators, we propose a novel loss function, dubbed VoroLoss, that minimizes the distance from ground truth surface samples to the closest faces of the Voronoi diagram which does not require an explicit construction of the entire Voronoi diagram. A direct optimization of the Voroloss to obtain generators on the Thingi32 dataset demonstrates the geometric efficiency of our representation compared to axiomatic meshing algorithms and recent learning-based mesh representations. We further use VoroMesh in a learning-based mesh prediction task from input SDF grids on the ABC dataset, and show comparable performance to state-of-the-art methods while guaranteeing closed output surfaces free of self-intersections.

Coarse architectural models are often generated at scales ranging from individual buildings to scenes for downstream applications such as Digital Twin City, Metaverse, LODs, etc. Such piece-wise planar models can be abstracted as twins from 3D dense reconstructions. However, these models typically lack realistic texture relative to the real building or scene, making them unsuitable for vivid display or direct reference. In this paper, we present TwinTex, the first automatic texture mapping framework to generate a photo-realistic texture for a piece-wise planar proxy. Our method addresses most challenges occurring in such twin texture generation. Specifically, for each primitive plane, we first select a small set of photos with greedy heuristics considering photometric quality, perspective quality and facade texture completeness. Then, different levels of line features (LoLs) are extracted from the set of selected photos to generate guidance for later steps. With LoLs, we employ optimization algorithms to align texture with geometry from local to global. Finally, we fine-tune a diffusion model with a multi-mask initialization component and a new dataset to inpaint the missing region. Experimental results on many buildings, indoor scenes and man-made objects of varying complexity demonstrate the generalization ability of our algorithm. Our approach surpasses state-of-the-art texture mapping methods in terms of high-fidelity quality and reaches a human-expert production level with much less effort. Project page: https://vcc.tech/research/2023/TwinTex.

We tackle the problem of estimating a Manhattan frame, i.e. three orthogonal vanishing points, and the unknown focal length of the camera, leveraging a prior vertical direction. The direction can come from an Inertial Measurement Unit that is a standard component of recent consumer devices, e.g., smartphones. We provide an exhaustive analysis of minimal line configurations and derive two new 2-line solvers, one of which does not suffer from singularities affecting existing solvers. Additionally, we design a new non-minimal method, running on an arbitrary number of lines, to boost the performance in local optimization. Combining all solvers in a hybrid robust estimator, our method achieves increased accuracy even with a rough prior. Experiments on synthetic and real-world datasets demonstrate the superior accuracy of our method compared to the state of the art, while having comparable runtimes. We further demonstrate the applicability of our solvers for relative rotation estimation. The code is available at https://github.com/cvg/VP-Estimation-with-Prior-Gravity.

Dictionary learning is a widely used unsupervised learning method in signal processing and machine learning. Most existing works of dictionary learning are in an offline manner. There are mainly two offline ways for dictionary learning. One is to do an alternative optimization of both the dictionary and the sparse code; the other way is to optimize the dictionary by restricting it over the orthogonal group. The latter one is called orthogonal dictionary learning which has a lower complexity implementation, hence, it is more favorable for lowcost devices. However, existing schemes on orthogonal dictionary learning only work with batch data and can not be implemented online, which is not applicable for real-time applications. This paper proposes a novel online orthogonal dictionary scheme to dynamically learn the dictionary from streaming data without storing the historical data. The proposed scheme includes a novel problem formulation and an efficient online algorithm design with convergence analysis. In the problem formulation, we relax the orthogonal constraint to enable an efficient online algorithm. In the algorithm design, we propose a new Frank-Wolfe-based online algorithm with a convergence rate of O(ln t/t^(1/4)). The convergence rate in terms of key system parameters is also derived. Experiments with synthetic data and real-world sensor readings demonstrate the effectiveness and efficiency of the proposed online orthogonal dictionary learning scheme.

Imagine two high-dimensional clusters and a hyperplane separating them. Consider in particular the angle between: the direction joining the two clusters' centroids and the normal to the hyperplane. In linear classification, this angle depends on the level of L2 regularization used. Can you explain why?

Reconstructing 3D shapes from planar cross-sections is a challenge inspired by downstream applications like medical imaging and geographic informatics. The input is an in/out indicator function fully defined on a sparse collection of planes in space, and the output is an interpolation of the indicator function to the entire volume. Previous works addressing this sparse and ill-posed problem either produce low quality results, or rely on additional priors such as target topology, appearance information, or input normal directions. In this paper, we present OReX, a method for 3D shape reconstruction from slices alone, featuring a Neural Field as the interpolation prior. A modest neural network is trained on the input planes to return an inside/outside estimate for a given 3D coordinate, yielding a powerful prior that induces smoothness and self-similarities. The main challenge for this approach is high-frequency details, as the neural prior is overly smoothing. To alleviate this, we offer an iterative estimation architecture and a hierarchical input sampling scheme that encourage coarse-to-fine training, allowing the training process to focus on high frequencies at later stages. In addition, we identify and analyze a ripple-like effect stemming from the mesh extraction step. We mitigate it by regularizing the spatial gradients of the indicator function around input in/out boundaries during network training, tackling the problem at the root. Through extensive qualitative and quantitative experimentation, we demonstrate our method is robust, accurate, and scales well with the size of the input. We report state-of-the-art results compared to previous approaches and recent potential solutions, and demonstrate the benefit of our individual contributions through analysis and ablation studies.

A spanning tree T of graph G is a rho-approximate universal Steiner tree (UST) for root vertex r if, for any subset of vertices S containing r, the cost of the minimal subgraph of T connecting S is within a rho factor of the minimum cost tree connecting S in G. Busch et al. (FOCS 2012) showed that every graph admits 2^{O(log n)}-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions. We settle these open questions by giving polynomial-time algorithms for computing both O(log ^ 7 n)-approximate USTs and poly-logarithmic strong sparse partition hierarchies. For graphs with constant doubling dimension or constant pathwidth we improve this to O(log n)-approximate USTs and O(1) strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms. We reduce the existence of these objects to the previously studied cluster aggregation problem and what we call dangling nets.

Building Footprint Extraction (BFE) from off-nadir aerial images often involves roof segmentation and offset prediction to adjust roof boundaries to the building footprint. However, this multi-stage approach typically produces low-quality results, limiting its applicability in real-world data production. To address this issue, we present OBMv2, an end-to-end and promptable model for polygonal footprint prediction. Unlike its predecessor OBM, OBMv2 introduces a novel Self Offset Attention (SOFA) mechanism that improves performance across diverse building types, from bungalows to skyscrapers, enabling end-to-end footprint prediction without post-processing. Additionally, we propose a Multi-level Information System (MISS) to effectively leverage roof masks, building masks, and offsets for accurate footprint prediction. We evaluate OBMv2 on the BONAI and OmniCity-view3 datasets and demonstrate its generalization on the Huizhou test set. The code will be available at https://github.com/likaiucas/OBMv2.

As the most common representation for 3D shapes, mesh is often stored discretely with arrays of vertices and faces. However, 3D shapes in the real world are presented continuously. In this paper, we propose to learn a continuous representation for meshes with fixed topology, a common and practical setting in many faces-, hand-, and body-related applications. First, we split the template into multiple closed manifold genus-0 meshes so that each genus-0 mesh can be parameterized onto the unit sphere. Then we learn spherical implicit surface (SIS), which takes a spherical coordinate and a global feature or a set of local features around the coordinate as inputs, predicting the vertex corresponding to the coordinate as an output. Since the spherical coordinates are continuous, SIS can depict a mesh in an arbitrary resolution. SIS representation builds a bridge between discrete and continuous representation in 3D shapes. Specifically, we train SIS networks in a self-supervised manner for two tasks: a reconstruction task and a super-resolution task. Experiments show that our SIS representation is comparable with state-of-the-art methods that are specifically designed for meshes with a fixed resolution and significantly outperforms methods that work in arbitrary resolutions.

We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.

Open-world 3D reconstruction models have recently garnered significant attention. However, without sufficient 3D inductive bias, existing methods typically entail expensive training costs and struggle to extract high-quality 3D meshes. In this work, we introduce MeshFormer, a sparse-view reconstruction model that explicitly leverages 3D native structure, input guidance, and training supervision. Specifically, instead of using a triplane representation, we store features in 3D sparse voxels and combine transformers with 3D convolutions to leverage an explicit 3D structure and projective bias. In addition to sparse-view RGB input, we require the network to take input and generate corresponding normal maps. The input normal maps can be predicted by 2D diffusion models, significantly aiding in the guidance and refinement of the geometry's learning. Moreover, by combining Signed Distance Function (SDF) supervision with surface rendering, we directly learn to generate high-quality meshes without the need for complex multi-stage training processes. By incorporating these explicit 3D biases, MeshFormer can be trained efficiently and deliver high-quality textured meshes with fine-grained geometric details. It can also be integrated with 2D diffusion models to enable fast single-image-to-3D and text-to-3D tasks. Project page: https://meshformer3d.github.io

Polynomial filters, a kind of Graph Neural Networks, typically use a predetermined polynomial basis and learn the coefficients from the training data. It has been observed that the effectiveness of the model is highly dependent on the property of the polynomial basis. Consequently, two natural and fundamental questions arise: Can we learn a suitable polynomial basis from the training data? Can we determine the optimal polynomial basis for a given graph and node features? In this paper, we propose two spectral GNN models that provide positive answers to the questions posed above. First, inspired by Favard's Theorem, we propose the FavardGNN model, which learns a polynomial basis from the space of all possible orthonormal bases. Second, we examine the supposedly unsolvable definition of optimal polynomial basis from Wang & Zhang (2022) and propose a simple model, OptBasisGNN, which computes the optimal basis for a given graph structure and graph signal. Extensive experiments are conducted to demonstrate the effectiveness of our proposed models.

When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. While classical regression techniques prove effective for parameterized geometries, practical scenarios often involve the absence of shape parametrization during the inference stage, leaving us with only mesh discretizations as available data. Learning simulations from such mesh-based representations poses significant challenges, with recent advances relying heavily on deep graph neural networks to overcome the limitations of conventional machine learning approaches. Despite their promising results, graph neural networks exhibit certain drawbacks, including their dependency on extensive datasets and limitations in providing built-in predictive uncertainties or handling large meshes. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making. In the considered numerical experiments, the proposed method is competitive with respect to existing graph neural networks, regarding training efficiency and accuracy of the predictions.

Accurate reconstruction of both the geometric and topological details of a 3D object from a single 2D image embodies a fundamental challenge in computer vision. Existing explicit/implicit solutions to this problem struggle to recover self-occluded geometry and/or faithfully reconstruct topological shape structures. To resolve this dilemma, we introduce LIST, a novel neural architecture that leverages local and global image features to accurately reconstruct the geometric and topological structure of a 3D object from a single image. We utilize global 2D features to predict a coarse shape of the target object and then use it as a base for higher-resolution reconstruction. By leveraging both local 2D features from the image and 3D features from the coarse prediction, we can predict the signed distance between an arbitrary point and the target surface via an implicit predictor with great accuracy. Furthermore, our model does not require camera estimation or pixel alignment. It provides an uninfluenced reconstruction from the input-view direction. Through qualitative and quantitative analysis, we show the superiority of our model in reconstructing 3D objects from both synthetic and real-world images against the state of the art.

The irregularity and permutation invariance of point cloud data pose challenges for effective learning. Conventional methods for addressing this issue involve converting raw point clouds to intermediate representations such as 3D voxel grids or range images. While such intermediate representations solve the problem of permutation invariance, they can result in significant loss of information. Approaches that do learn on raw point clouds either have trouble in resolving neighborhood relationships between points or are too complicated in their formulation. In this paper, we propose a novel approach to representing point clouds as a locality preserving 1D ordering induced by the Hilbert space-filling curve. We also introduce Point2Point, a neural architecture that can effectively learn on Hilbert-sorted point clouds. We show that Point2Point shows competitive performance on point cloud segmentation and generation tasks. Finally, we show the performance of Point2Point on Spatio-temporal Occupancy prediction from Point clouds.

Sublinear time complexity is required by the massively parallel computation (MPC) model. Breaking dynamic programs into a set of sparse dynamic programs that can be divided, solved, and merged in sublinear time. The rectangle escape problem (REP) is defined as follows: For n axis-aligned rectangles inside an axis-aligned bounding box B, extend each rectangle in only one of the four directions: up, down, left, or right until it reaches B and the density k is minimized, where k is the maximum number of extensions of rectangles to the boundary that pass through a point inside bounding box B. REP is NP-hard for k>1. If the rectangles are points of a grid (or unit squares of a grid), the problem is called the square escape problem (SEP) and it is still NP-hard. We give a 2-approximation algorithm for SEP with kgeq2 with time complexity O(n^{3/2}k^2). This improves the time complexity of existing algorithms which are at least quadratic. Also, the approximation ratio of our algorithm for kgeq 3 is 3/2 which is tight. We also give a 8-approximation algorithm for REP with time complexity O(nlog n+nk) and give a MPC version of this algorithm for k=O(1) which is the first parallel algorithm for this problem.

Recent progress in neural rendering has brought forth pioneering methods, such as NeRF and Gaussian Splatting, which revolutionize view rendering across various domains like AR/VR, gaming, and content creation. While these methods excel at interpolating {\em within the training data}, the challenge of generalizing to new scenes and objects from very sparse views persists. Specifically, modeling 3D humans from sparse views presents formidable hurdles due to the inherent complexity of human geometry, resulting in inaccurate reconstructions of geometry and textures. To tackle this challenge, this paper leverages recent advancements in Gaussian Splatting and introduces a new method to learn generalizable human Gaussians that allows photorealistic and accurate view-rendering of a new human subject from a limited set of sparse views in a feed-forward manner. A pivotal innovation of our approach involves reformulating the learning of 3D Gaussian parameters into a regression process defined on the 2D UV space of a human template, which allows leveraging the strong geometry prior and the advantages of 2D convolutions. In addition, a multi-scaffold is proposed to effectively represent the offset details. Our method outperforms recent methods on both within-dataset generalization as well as cross-dataset generalization settings.

We introduce a novel neural network-based algorithm to compute optimal transport (OT) plans for general cost functionals. In contrast to common Euclidean costs, i.e., ell^1 or ell^2, such functionals provide more flexibility and allow using auxiliary information, such as class labels, to construct the required transport map. Existing methods for general costs are discrete and have limitations in practice, i.e. they do not provide an out-of-sample estimation. We address the challenge of designing a continuous OT approach for general costs that generalizes to new data points in high-dimensional spaces, such as images. Additionally, we provide the theoretical error analysis for our recovered transport plans. As an application, we construct a cost functional to map data distributions while preserving the class-wise structure.

We study the joint image of lines incident to points, meaning the set of image tuples obtained from fixed cameras observing a varying 3D point-line incidence. We prove a formula for the number of complex critical points of the triangulation problem that aims to compute a 3D point-line incidence from noisy images. Our formula works for an arbitrary number of images and measures the intrinsic difficulty of this triangulation. Additionally, we conduct numerical experiments using homotopy continuation methods, comparing different approaches of triangulation of such incidences. In our setup, exploiting the incidence relations gives both a faster point reconstruction and in three views more accurate.

We introduce a new approach for generating realistic 3D models with UV maps through a representation termed "Object Images." This approach encapsulates surface geometry, appearance, and patch structures within a 64x64 pixel image, effectively converting complex 3D shapes into a more manageable 2D format. By doing so, we address the challenges of both geometric and semantic irregularity inherent in polygonal meshes. This method allows us to use image generation models, such as Diffusion Transformers, directly for 3D shape generation. Evaluated on the ABO dataset, our generated shapes with patch structures achieve point cloud FID comparable to recent 3D generative models, while naturally supporting PBR material generation.

Trit-plane coding enables deep progressive image compression, but it cannot use autoregressive context models. In this paper, we propose the context-based trit-plane coding (CTC) algorithm to achieve progressive compression more compactly. First, we develop the context-based rate reduction module to estimate trit probabilities of latent elements accurately and thus encode the trit-planes compactly. Second, we develop the context-based distortion reduction module to refine partial latent tensors from the trit-planes and improve the reconstructed image quality. Third, we propose a retraining scheme for the decoder to attain better rate-distortion tradeoffs. Extensive experiments show that CTC outperforms the baseline trit-plane codec significantly in BD-rate on the Kodak lossless dataset, while increasing the time complexity only marginally. Our codes are available at https://github.com/seungminjeon-github/CTC.

In computer-aided design (CAD) systems, 2D line drawings are commonly used to illustrate 3D object designs. To reconstruct the 3D models depicted by a single 2D line drawing, an important key is finding the edge loops in the line drawing which correspond to the actual faces of the 3D object. In this paper, we approach the classical problem of face identification from a novel data-driven point of view. We cast it as a sequence generation problem: starting from an arbitrary edge, we adopt a variant of the popular Transformer model to predict the edges associated with the same face in a natural order. This allows us to avoid searching the space of all possible edge loops with various hand-crafted rules and heuristics as most existing methods do, deal with challenging cases such as curved surfaces and nested edge loops, and leverage additional cues such as face types. We further discuss how possibly imperfect predictions can be used for 3D object reconstruction.

We present a learning-based approach to reconstruct buildings as 3D polygonal meshes from airborne LiDAR point clouds. What makes 3D building reconstruction from airborne LiDAR hard is the large diversity of building designs and especially roof shapes, the low and varying point density across the scene, and the often incomplete coverage of building facades due to occlusions by vegetation or to the viewing angle of the sensor. To cope with the diversity of shapes and inhomogeneous and incomplete object coverage, we introduce a generative model that directly predicts 3D polygonal meshes from input point clouds. Our autoregressive model, called Point2Building, iteratively builds up the mesh by generating sequences of vertices and faces. This approach enables our model to adapt flexibly to diverse geometries and building structures. Unlike many existing methods that rely heavily on pre-processing steps like exhaustive plane detection, our model learns directly from the point cloud data, thereby reducing error propagation and increasing the fidelity of the reconstruction. We experimentally validate our method on a collection of airborne LiDAR data of Zurich, Berlin and Tallinn. Our method shows good generalization to diverse urban styles.

We investigate novel random graph embeddings that can be computed in expected polynomial time and that are able to distinguish all non-isomorphic graphs in expectation. Previous graph embeddings have limited expressiveness and either cannot distinguish all graphs or cannot be computed efficiently for every graph. To be able to approximate arbitrary functions on graphs, we are interested in efficient alternatives that become arbitrarily expressive with increasing resources. Our approach is based on Lov\'asz' characterisation of graph isomorphism through an infinite dimensional vector of homomorphism counts. Our empirical evaluation shows competitive results on several benchmark graph learning tasks.

Diffusion models have emerged as the state-of-the-art for image generation, among other tasks. Here, we present an efficient diffusion-based model for 3D-aware generation of neural fields. Our approach pre-processes training data, such as ShapeNet meshes, by converting them to continuous occupancy fields and factoring them into a set of axis-aligned triplane feature representations. Thus, our 3D training scenes are all represented by 2D feature planes, and we can directly train existing 2D diffusion models on these representations to generate 3D neural fields with high quality and diversity, outperforming alternative approaches to 3D-aware generation. Our approach requires essential modifications to existing triplane factorization pipelines to make the resulting features easy to learn for the diffusion model. We demonstrate state-of-the-art results on 3D generation on several object classes from ShapeNet.

3D Gaussian Splatting (3DGS) has recently revolutionized radiance field reconstruction, achieving high quality novel view synthesis and fast rendering speed without baking. However, 3DGS fails to accurately represent surfaces due to the multi-view inconsistent nature of 3D Gaussians. We present 2D Gaussian Splatting (2DGS), a novel approach to model and reconstruct geometrically accurate radiance fields from multi-view images. Our key idea is to collapse the 3D volume into a set of 2D oriented planar Gaussian disks. Unlike 3D Gaussians, 2D Gaussians provide view-consistent geometry while modeling surfaces intrinsically. To accurately recover thin surfaces and achieve stable optimization, we introduce a perspective-accurate 2D splatting process utilizing ray-splat intersection and rasterization. Additionally, we incorporate depth distortion and normal consistency terms to further enhance the quality of the reconstructions. We demonstrate that our differentiable renderer allows for noise-free and detailed geometry reconstruction while maintaining competitive appearance quality, fast training speed, and real-time rendering. Our code will be made publicly available.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

We present a learning-based method, namely GeoUDF,to tackle the long-standing and challenging problem of reconstructing a discrete surface from a sparse point cloud.To be specific, we propose a geometry-guided learning method for UDF and its gradient estimation that explicitly formulates the unsigned distance of a query point as the learnable affine averaging of its distances to the tangent planes of neighboring points on the surface. Besides,we model the local geometric structure of the input point clouds by explicitly learning a quadratic polynomial for each point. This not only facilitates upsampling the input sparse point cloud but also naturally induces unoriented normal, which further augments UDF estimation. Finally, to extract triangle meshes from the predicted UDF we propose a customized edge-based marching cube module. We conduct extensive experiments and ablation studies to demonstrate the significant advantages of our method over state-of-the-art methods in terms of reconstruction accuracy, efficiency, and generality. The source code is publicly available at https://github.com/rsy6318/GeoUDF.

In this paper, we propose Img2CAD, the first approach to our knowledge that uses 2D image inputs to generate CAD models with editable parameters. Unlike existing AI methods for 3D model generation using text or image inputs often rely on mesh-based representations, which are incompatible with CAD tools and lack editability and fine control, Img2CAD enables seamless integration between AI-based 3D reconstruction and CAD software. We have identified an innovative intermediate representation called Structured Visual Geometry (SVG), characterized by vectorized wireframes extracted from objects. This representation significantly enhances the performance of generating conditioned CAD models. Additionally, we introduce two new datasets to further support research in this area: ABC-mono, the largest known dataset comprising over 200,000 3D CAD models with rendered images, and KOCAD, the first dataset featuring real-world captured objects alongside their ground truth CAD models, supporting further research in conditioned CAD model generation.

Unsupervised learning of depth from indoor monocular videos is challenging as the artificial environment contains many textureless regions. Fortunately, the indoor scenes are full of specific structures, such as planes and lines, which should help guide unsupervised depth learning. This paper proposes PLNet that leverages the plane and line priors to enhance the depth estimation. We first represent the scene geometry using local planar coefficients and impose the smoothness constraint on the representation. Moreover, we enforce the planar and linear consistency by randomly selecting some sets of points that are probably coplanar or collinear to construct simple and effective consistency losses. To verify the proposed method's effectiveness, we further propose to evaluate the flatness and straightness of the predicted point cloud on the reliable planar and linear regions. The regularity of these regions indicates quality indoor reconstruction. Experiments on NYU Depth V2 and ScanNet show that PLNet outperforms existing methods. The code is available at https://github.com/HalleyJiang/PLNet.

A recent trend among generalizable novel view synthesis methods is to learn a rendering operator acting over single camera rays. This approach is promising because it removes the need for explicit volumetric rendering, but it effectively treats target images as collections of independent pixels. Here, we propose to learn a global rendering operator acting over all camera rays jointly. We show that the right representation to enable such rendering is a 5-dimensional plane sweep volume consisting of the projection of the input images on a set of planes facing the target camera. Based on this understanding, we introduce our Convolutional Global Latent Renderer (ConvGLR), an efficient convolutional architecture that performs the rendering operation globally in a low-resolution latent space. Experiments on various datasets under sparse and generalizable setups show that our approach consistently outperforms existing methods by significant margins.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

Implicit neural networks have emerged as a crucial technology in 3D surface reconstruction. To reconstruct continuous surfaces from discrete point clouds, encoding the input points into regular grid features (plane or volume) has been commonly employed in existing approaches. However, these methods typically use the grid as an index for uniformly scattering point features. Compared with the irregular point features, the regular grid features may sacrifice some reconstruction details but improve efficiency. To take full advantage of these two types of features, we introduce a novel and high-efficiency attention mechanism between the grid and point features named Point-Grid Transformer (GridFormer). This mechanism treats the grid as a transfer point connecting the space and point cloud. Our method maximizes the spatial expressiveness of grid features and maintains computational efficiency. Furthermore, optimizing predictions over the entire space could potentially result in blurred boundaries. To address this issue, we further propose a boundary optimization strategy incorporating margin binary cross-entropy loss and boundary sampling. This approach enables us to achieve a more precise representation of the object structure. Our experiments validate that our method is effective and outperforms the state-of-the-art approaches under widely used benchmarks by producing more precise geometry reconstructions. The code is available at https://github.com/list17/GridFormer.

While current multi-frame restoration methods combine information from multiple input images using 2D alignment techniques, recent advances in novel view synthesis are paving the way for a new paradigm relying on volumetric scene representations. In this work, we introduce the first 3D-based multi-frame denoising method that significantly outperforms its 2D-based counterparts with lower computational requirements. Our method extends the multiplane image (MPI) framework for novel view synthesis by introducing a learnable encoder-renderer pair manipulating multiplane representations in feature space. The encoder fuses information across views and operates in a depth-wise manner while the renderer fuses information across depths and operates in a view-wise manner. The two modules are trained end-to-end and learn to separate depths in an unsupervised way, giving rise to Multiplane Feature (MPF) representations. Experiments on the Spaces and Real Forward-Facing datasets as well as on raw burst data validate our approach for view synthesis, multi-frame denoising, and view synthesis under noisy conditions.

We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation through a Dyck tiling on a ribbon. We introduce a new combinatorial object similar to a tree-like tableau, which we call a forest. A forest is shown to give a permutation, and be bijective to a network corresponding to the inverse of the permutation. We show that the poset of networks is a finite graded lattice and admits an EL-labeling. By use of this EL-labeling, we show the lattice is supersolvable and compute the M\"obius function of an interval of the poset.

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.

Semi-transitive graphs, defined in hps98 as examples where ``uniform percolation" holds whenever p>p_c, are a large class of graphs more general than quasi-transitive graphs. Let G be a semi-transitive graph with one end which can be properly embedded into the plane with uniformly bounded face degree for finite faces and minimal vertex degree at least 7. We show that p_u^{site}(G) +p_c^{site}(G_*)=1, where G_* denotes the matching graph of G. This fulfils and extends an observation of Sykes and Essam in 1964 (SE64) to semi-transitive graphs.

Computer-aided design (CAD) tools are utilized in the manufacturing industry for modeling everything from cups to spacecraft. These programs are complex to use and typically require years of training and experience to master. Structured and well-constrained 2D sketches and 3D constructions are crucial components of CAD modeling. A well-executed CAD model can be seamlessly integrated into the manufacturing process, thereby enhancing production efficiency. Deep generative models of 3D shapes and 3D object reconstruction models have garnered significant research interest. However, most of these models produce discrete forms of 3D objects that are not editable. Moreover, the few models based on CAD operations often have substantial input restrictions. In this work, we fine-tuned pre-trained models to create OpenECAD models (0.55B, 0.89B, 2.4B and 3.1B), leveraging the visual, logical, coding, and general capabilities of visual language models. OpenECAD models can process images of 3D designs as input and generate highly structured 2D sketches and 3D construction commands, ensuring that the designs are editable. These outputs can be directly used with existing CAD tools' APIs to generate project files. To train our network, we created a series of OpenECAD datasets. These datasets are derived from existing public CAD datasets, adjusted and augmented to meet the specific requirements of vision language model (VLM) training. Additionally, we have introduced an approach that utilizes dependency relationships to define and generate sketches, further enriching the content and functionality of the datasets.

A central goal of visual recognition is to understand objects and scenes from a single image. 2D recognition has witnessed tremendous progress thanks to large-scale learning and general-purpose representations. Comparatively, 3D poses new challenges stemming from occlusions not depicted in the image. Prior works try to overcome these by inferring from multiple views or rely on scarce CAD models and category-specific priors which hinder scaling to novel settings. In this work, we explore single-view 3D reconstruction by learning generalizable representations inspired by advances in self-supervised learning. We introduce a simple framework that operates on 3D points of single objects or whole scenes coupled with category-agnostic large-scale training from diverse RGB-D videos. Our model, Multiview Compressive Coding (MCC), learns to compress the input appearance and geometry to predict the 3D structure by querying a 3D-aware decoder. MCC's generality and efficiency allow it to learn from large-scale and diverse data sources with strong generalization to novel objects imagined by DALLcdotE 2 or captured in-the-wild with an iPhone.

Learning 3D representations that generalize well to arbitrarily oriented inputs is a challenge of practical importance in applications varying from computer vision to physics and chemistry. We propose a novel multi-resolution convolutional architecture for learning over concentric spherical feature maps, of which the single sphere representation is a special case. Our hierarchical architecture is based on alternatively learning to incorporate both intra-sphere and inter-sphere information. We show the applicability of our method for two different types of 3D inputs, mesh objects, which can be regularly sampled, and point clouds, which are irregularly distributed. We also propose an efficient mapping of point clouds to concentric spherical images, thereby bridging spherical convolutions on grids with general point clouds. We demonstrate the effectiveness of our approach in improving state-of-the-art performance on 3D classification tasks with rotated data.

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of R^{n} for three symmetry groups that are missing from the machine learning literature: O(n), the orthogonal group; SO(n), the special orthogonal group; and Sp(n), the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of R^{n} when the group is O(n) or SO(n), and in the symplectic basis of R^{n} when the group is Sp(n).

In the realm of 3D computer vision, parametric models have emerged as a ground-breaking methodology for the creation of realistic and expressive 3D avatars. Traditionally, they rely on Principal Component Analysis (PCA), given its ability to decompose data to an orthonormal space that maximally captures shape variations. However, due to the orthogonality constraints and the global nature of PCA's decomposition, these models struggle to perform localized and disentangled editing of 3D shapes, which severely affects their use in applications requiring fine control such as face sculpting. In this paper, we leverage diffusion models to enable diverse and fully localized edits on 3D meshes, while completely preserving the un-edited regions. We propose an effective diffusion masking training strategy that, by design, facilitates localized manipulation of any shape region, without being limited to predefined regions or to sparse sets of predefined control vertices. Following our framework, a user can explicitly set their manipulation region of choice and define an arbitrary set of vertices as handles to edit a 3D mesh. Compared to the current state-of-the-art our method leads to more interpretable shape manipulations than methods relying on latent code state, greater localization and generation diversity while offering faster inference than optimization based approaches. Project page: https://rolpotamias.github.io/Shapefusion/

Given a matrix Ain R^{ntimes d} and a vector bin R^n, we consider the regression problem with ell_infty guarantees: finding a vector x'in R^d such that |x'-x^*|_infty leq epsilon{d}cdot |Ax^*-b|_2cdot |A^dagger| where x^*=argmin_{xin R^d}|Ax-b|_2. One popular approach for solving such ell_2 regression problem is via sketching: picking a structured random matrix Sin R^{mtimes n} with mll n and SA can be quickly computed, solve the ``sketched'' regression problem argmin_{xin R^d} |SAx-Sb|_2. In this paper, we show that in order to obtain such ell_infty guarantee for ell_2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m=epsilon^{-2}dlog^3(n/delta) such that solving the sketched regression problem gives the ell_infty guarantee, with probability at least 1-delta. Moreover, the matrix SA can be computed in time O(ndlog n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in d rows, m=Omega(epsilon^{-2}d^{1+gamma}) for gamma=Theta(frac{loglog n{log d}}) is required. We also develop a novel analytical framework for ell_infty guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

We propose to investigate detecting and characterizing the 3D planar articulation of objects from ordinary videos. While seemingly easy for humans, this problem poses many challenges for computers. We propose to approach this problem by combining a top-down detection system that finds planes that can be articulated along with an optimization approach that solves for a 3D plane that can explain a sequence of observed articulations. We show that this system can be trained on a combination of videos and 3D scan datasets. When tested on a dataset of challenging Internet videos and the Charades dataset, our approach obtains strong performance. Project site: https://jasonqsy.github.io/Articulation3D

Generating 3D images of complex objects conditionally from a few 2D views is a difficult synthesis problem, compounded by issues such as domain gap and geometric misalignment. For instance, a unified framework such as Generative Adversarial Networks cannot achieve this unless they explicitly define both a domain-invariant and geometric-invariant joint latent distribution, whereas Neural Radiance Fields are generally unable to handle both issues as they optimize at the pixel level. By contrast, we propose a simple and novel 2D to 3D synthesis approach based on conditional diffusion with vector-quantized codes. Operating in an information-rich code space enables high-resolution 3D synthesis via full-coverage attention across the views. Specifically, we generate the 3D codes (e.g. for CT images) conditional on previously generated 3D codes and the entire codebook of two 2D views (e.g. 2D X-rays). Qualitative and quantitative results demonstrate state-of-the-art performance over specialized methods across varied evaluation criteria, including fidelity metrics such as density, coverage, and distortion metrics for two complex volumetric imagery datasets from in real-world scenarios.

We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small. Especially, it is completely independent of the width of the network if the weight matrices are orthogonal. Our bound does not contradict to the existing bounds but is a complement to the existing bounds. As supported by several existing empirical results, low-rankness is not the only reason for generalization. Furthermore, our bound can be combined with the existing bounds to obtain a tighter bound. Our result sheds new light on understanding generalization of neural networks with full-rank weight matrices, and it provides a connection between operator-theoretic analysis and generalization of neural networks.

Recent years have witnessed considerable achievements in facial avatar reconstruction with neural volume rendering. Despite notable advancements, the reconstruction of complex and dynamic head movements from monocular videos still suffers from capturing and restoring fine-grained details. In this work, we propose a novel approach, named Tri^2-plane, for monocular photo-realistic volumetric head avatar reconstructions. Distinct from the existing works that rely on a single tri-plane deformation field for dynamic facial modeling, the proposed Tri^2-plane leverages the principle of feature pyramids and three top-to-down lateral connections tri-planes for details improvement. It samples and renders facial details at multiple scales, transitioning from the entire face to specific local regions and then to even more refined sub-regions. Moreover, we incorporate a camera-based geometry-aware sliding window method as an augmentation in training, which improves the robustness beyond the canonical space, with a particular improvement in cross-identity generation capabilities. Experimental outcomes indicate that the Tri^2-plane not only surpasses existing methodologies but also achieves superior performance across quantitative and qualitative assessments. The project website is: https://songluchuan.github.io/Tri2Plane.github.io/.

The polygon mesh representation of 3D data exhibits great flexibility, fast rendering speed, and storage efficiency, which is widely preferred in various applications. However, given its unstructured graph representation, the direct generation of high-fidelity 3D meshes is challenging. Fortunately, with a pre-defined ordering strategy, 3D meshes can be represented as sequences, and the generation process can be seamlessly treated as an auto-regressive problem. In this paper, we validate the Neural Coordinate Field (NeurCF), an explicit coordinate representation with implicit neural embeddings, is a simple-yet-effective representation for large-scale sequential mesh modeling. After that, we present MeshXL, a family of generative pre-trained auto-regressive models, which addresses the process of 3D mesh generation with modern large language model approaches. Extensive experiments show that MeshXL is able to generate high-quality 3D meshes, and can also serve as foundation models for various down-stream applications.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

Complex networks provide a means to describe cities through their street mesh, expressing characteristics that refer to the structure and organization of an urban zone. Although other studies have used complex networks to model street meshes, we observed a lack of methods to characterize the relationship between cities by using their topological features. Accordingly, this paper aims to describe interactions between cities by using vectors of topological features extracted from their street meshes represented as complex networks. The methodology of this study is based on the use of digital maps. Over the computational representation of such maps, we extract global complex-network features that embody the characteristics of the cities. These vectors allow for the use of multidimensional projection and clustering techniques, enabling a similarity-based comparison of the street meshes. We experiment with 645 cities from the Brazilian state of Sao Paulo. Our results show how the joint of global features describes urban indicators that are deep-rooted in the network's topology and how they reveal characteristics and similarities among sets of cities that are separated from each other.

In this article, we study the parameterized complexity of the Set Cover problem restricted to semi-ladder-free hypergraphs, a class defined by Fabianski et al. [Proceedings of STACS 2019]. We observe that two algorithms introduced by Langerman and Morin [Discrete & Computational Geometry 2005] in the context of geometric covering problems can be adapted to this setting, yielding simple FPT and kernelization algorithms for Set Cover in semi-ladder-free hypergraphs. We complement our algorithmic results with a compression lower bound for the problem, which proves the tightness of our kernelization under standard complexity-theoretic assumptions.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Voronoi diagrams are essential geometrical structures with numerous applications, particularly astrophysics-driven finite volume methods. While serial algorithms for constructing these entities are well-established, parallel construction remains challenging. This is especially true in distributed memory systems, where each host manages only a subset of the input points. This process requires redistributing points across hosts and accurately computing the corresponding Voronoi cells. In this paper, we introduce a new distributed construction algorithm, which is implemented in our open-source C++ 3-dimensional Voronoi construction framework. Our approach leverages Delaunay triangulation as an intermediate step, which is then transformed into a Voronoi diagram. We introduce the algorithms we implemented for the precise construction and our load-balancing approach and compare the running time with other state-of-the-art frameworks. MadVoro is a versatile tool that can be applied in various scientific domains, such as mesh decomposition, computational physics, chemistry, and machine learning.

Novel view synthesis from an in-the-wild video is difficult due to challenges like scene dynamics and lack of parallax. While existing methods have shown promising results with implicit neural radiance fields, they are slow to train and render. This paper revisits explicit video representations to synthesize high-quality novel views from a monocular video efficiently. We treat static and dynamic video content separately. Specifically, we build a global static scene model using an extended plane-based scene representation to synthesize temporally coherent novel video. Our plane-based scene representation is augmented with spherical harmonics and displacement maps to capture view-dependent effects and model non-planar complex surface geometry. We opt to represent the dynamic content as per-frame point clouds for efficiency. While such representations are inconsistency-prone, minor temporal inconsistencies are perceptually masked due to motion. We develop a method to quickly estimate such a hybrid video representation and render novel views in real time. Our experiments show that our method can render high-quality novel views from an in-the-wild video with comparable quality to state-of-the-art methods while being 100x faster in training and enabling real-time rendering.

We introduce ReMatching, a novel shape correspondence solution based on the functional maps framework. Our method, by exploiting a new and appropriate re-meshing paradigm, can target shape-matching tasks even on meshes counting millions of vertices, where the original functional maps does not apply or requires a massive computational cost. The core of our procedure is a time-efficient remeshing algorithm which constructs a low-resolution geometry while acting conservatively on the original topology and metric. These properties allow translating the functional maps optimization problem on the resulting low-resolution representation, thus enabling efficient computation of correspondences with functional map approaches. Finally, we propose an efficient technique for extending the estimated correspondence to the original meshes. We show that our method is more efficient and effective through quantitative and qualitative comparisons, outperforming state-of-the-art pipelines in quality and computational cost.

3D Gaussian Splatting has recently emerged as a highly promising technique for modeling of static 3D scenes. In contrast to Neural Radiance Fields, it utilizes efficient rasterization allowing for very fast rendering at high-quality. However, the storage size is significantly higher, which hinders practical deployment, e.g.~on resource constrained devices. In this paper, we introduce a compact scene representation organizing the parameters of 3D Gaussian Splatting (3DGS) into a 2D grid with local homogeneity, ensuring a drastic reduction in storage requirements without compromising visual quality during rendering. Central to our idea is the explicit exploitation of perceptual redundancies present in natural scenes. In essence, the inherent nature of a scene allows for numerous permutations of Gaussian parameters to equivalently represent it. To this end, we propose a novel highly parallel algorithm that regularly arranges the high-dimensional Gaussian parameters into a 2D grid while preserving their neighborhood structure. During training, we further enforce local smoothness between the sorted parameters in the grid. The uncompressed Gaussians use the same structure as 3DGS, ensuring a seamless integration with established renderers. Our method achieves a reduction factor of 8x to 26x in size for complex scenes with no increase in training time, marking a substantial leap forward in the domain of 3D scene distribution and consumption. Additional information can be found on our project page: https://fraunhoferhhi.github.io/Self-Organizing-Gaussians/

Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

Recent advances in modeling 3D objects mostly rely on synthetic datasets due to the lack of large-scale realscanned 3D databases. To facilitate the development of 3D perception, reconstruction, and generation in the real world, we propose OmniObject3D, a large vocabulary 3D object dataset with massive high-quality real-scanned 3D objects. OmniObject3D has several appealing properties: 1) Large Vocabulary: It comprises 6,000 scanned objects in 190 daily categories, sharing common classes with popular 2D datasets (e.g., ImageNet and LVIS), benefiting the pursuit of generalizable 3D representations. 2) Rich Annotations: Each 3D object is captured with both 2D and 3D sensors, providing textured meshes, point clouds, multiview rendered images, and multiple real-captured videos. 3) Realistic Scans: The professional scanners support highquality object scans with precise shapes and realistic appearances. With the vast exploration space offered by OmniObject3D, we carefully set up four evaluation tracks: a) robust 3D perception, b) novel-view synthesis, c) neural surface reconstruction, and d) 3D object generation. Extensive studies are performed on these four benchmarks, revealing new observations, challenges, and opportunities for future research in realistic 3D vision.

We present LRM-Zero, a Large Reconstruction Model (LRM) trained entirely on synthesized 3D data, achieving high-quality sparse-view 3D reconstruction. The core of LRM-Zero is our procedural 3D dataset, Zeroverse, which is automatically synthesized from simple primitive shapes with random texturing and augmentations (e.g., height fields, boolean differences, and wireframes). Unlike previous 3D datasets (e.g., Objaverse) which are often captured or crafted by humans to approximate real 3D data, Zeroverse completely ignores realistic global semantics but is rich in complex geometric and texture details that are locally similar to or even more intricate than real objects. We demonstrate that our LRM-Zero, trained with our fully synthesized Zeroverse, can achieve high visual quality in the reconstruction of real-world objects, competitive with models trained on Objaverse. We also analyze several critical design choices of Zeroverse that contribute to LRM-Zero's capability and training stability. Our work demonstrates that 3D reconstruction, one of the core tasks in 3D vision, can potentially be addressed without the semantics of real-world objects. The Zeroverse's procedural synthesis code and interactive visualization are available at: https://desaixie.github.io/lrm-zero/.

We propose a novel mixed-integer programming (MIP) formulation for generating precise sparse correspondences for highly non-rigid shapes. To this end, we introduce a projected Laplace-Beltrami operator (PLBO) which combines intrinsic and extrinsic geometric information to measure the deformation quality induced by predicted correspondences. We integrate the PLBO, together with an orientation-aware regulariser, into a novel MIP formulation that can be solved to global optimality for many practical problems. In contrast to previous methods, our approach is provably invariant to rigid transformations and global scaling, initialisation-free, has optimality guarantees, and scales to high resolution meshes with (empirically observed) linear time. We show state-of-the-art results for sparse non-rigid matching on several challenging 3D datasets, including data with inconsistent meshing, as well as applications in mesh-to-point-cloud matching.

We present NeX, a new approach to novel view synthesis based on enhancements of multiplane image (MPI) that can reproduce next-level view-dependent effects -- in real time. Unlike traditional MPI that uses a set of simple RGBalpha planes, our technique models view-dependent effects by instead parameterizing each pixel as a linear combination of basis functions learned from a neural network. Moreover, we propose a hybrid implicit-explicit modeling strategy that improves upon fine detail and produces state-of-the-art results. Our method is evaluated on benchmark forward-facing datasets as well as our newly-introduced dataset designed to test the limit of view-dependent modeling with significantly more challenging effects such as rainbow reflections on a CD. Our method achieves the best overall scores across all major metrics on these datasets with more than 1000times faster rendering time than the state of the art. For real-time demos, visit https://nex-mpi.github.io/

We present Magic123, a two-stage coarse-to-fine approach for high-quality, textured 3D meshes generation from a single unposed image in the wild using both2D and 3D priors. In the first stage, we optimize a neural radiance field to produce a coarse geometry. In the second stage, we adopt a memory-efficient differentiable mesh representation to yield a high-resolution mesh with a visually appealing texture. In both stages, the 3D content is learned through reference view supervision and novel views guided by a combination of 2D and 3D diffusion priors. We introduce a single trade-off parameter between the 2D and 3D priors to control exploration (more imaginative) and exploitation (more precise) of the generated geometry. Additionally, we employ textual inversion and monocular depth regularization to encourage consistent appearances across views and to prevent degenerate solutions, respectively. Magic123 demonstrates a significant improvement over previous image-to-3D techniques, as validated through extensive experiments on synthetic benchmarks and diverse real-world images. Our code, models, and generated 3D assets are available at https://github.com/guochengqian/Magic123.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

Polar slice sampling (Roberts & Rosenthal, 2002) is a Markov chain approach for approximate sampling of distributions that is difficult, if not impossible, to implement efficiently, but behaves provably well with respect to the dimension. By updating the directional and radial components of chain iterates separately, we obtain a family of samplers that mimic polar slice sampling, and yet can be implemented efficiently. Numerical experiments in a variety of settings indicate that our proposed algorithm outperforms the two most closely related approaches, elliptical slice sampling (Murray et al., 2010) and hit-and-run uniform slice sampling (MacKay, 2003). We prove the well-definedness and convergence of our methods under suitable assumptions on the target distribution.

Inspired by cognitive theories, we introduce AnyHome, a framework that translates any text into well-structured and textured indoor scenes at a house-scale. By prompting Large Language Models (LLMs) with designed templates, our approach converts provided textual narratives into amodal structured representations. These representations guarantee consistent and realistic spatial layouts by directing the synthesis of a geometry mesh within defined constraints. A Score Distillation Sampling process is then employed to refine the geometry, followed by an egocentric inpainting process that adds lifelike textures to it. AnyHome stands out with its editability, customizability, diversity, and realism. The structured representations for scenes allow for extensive editing at varying levels of granularity. Capable of interpreting texts ranging from simple labels to detailed narratives, AnyHome generates detailed geometries and textures that outperform existing methods in both quantitative and qualitative measures.

Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

Modeling and designing urban building layouts is of significant interest in computer vision, computer graphics, and urban applications. A building layout consists of a set of buildings in city blocks defined by a network of roads. We observe that building layouts are discrete structures, consisting of multiple rows of buildings of various shapes, and are amenable to skeletonization for mapping arbitrary city block shapes to a canonical form. Hence, we propose a fully automatic approach to building layout generation using graph attention networks. Our method generates realistic urban layouts given arbitrary road networks, and enables conditional generation based on learned priors. Our results, including user study, demonstrate superior performance as compared to prior layout generation networks, support arbitrary city block and varying building shapes as demonstrated by generating layouts for 28 large cities.

Contemporary 3D research, particularly in reconstruction and generation, heavily relies on 2D images for inputs or supervision. However, current designs for these 2D-3D mapping are memory-intensive, posing a significant bottleneck for existing methods and hindering new applications. In response, we propose a pair of highly scalable components for 3D neural fields: Lightplane Render and Splatter, which significantly reduce memory usage in 2D-3D mapping. These innovations enable the processing of vastly more and higher resolution images with small memory and computational costs. We demonstrate their utility in various applications, from benefiting single-scene optimization with image-level losses to realizing a versatile pipeline for dramatically scaling 3D reconstruction and generation. Code: https://github.com/facebookresearch/lightplane.

We present a new class of fast polylog-linear algorithms based on the theory of structured matrices (in particular low displacement rank) for integrating tensor fields defined on weighted trees. Several applications of the resulting fast tree-field integrators (FTFIs) are presented, including (a) approximation of graph metrics with tree metrics, (b) graph classification, (c) modeling on meshes, and finally (d) Topological Transformers (TTs) (Choromanski et al., 2022) for images. For Topological Transformers, we propose new relative position encoding (RPE) masking mechanisms with as few as three extra learnable parameters per Transformer layer, leading to 1.0-1.5%+ accuracy gains. Importantly, most of FTFIs are exact methods, thus numerically equivalent to their brute-force counterparts. When applied to graphs with thousands of nodes, those exact algorithms provide 5.7-13x speedups. We also provide an extensive theoretical analysis of our methods.

This work proposes a novel representation of injective deformations of 3D space, which overcomes existing limitations of injective methods: inaccuracy, lack of robustness, and incompatibility with general learning and optimization frameworks. The core idea is to reduce the problem to a deep composition of multiple 2D mesh-based piecewise-linear maps. Namely, we build differentiable layers that produce mesh deformations through Tutte's embedding (guaranteed to be injective in 2D), and compose these layers over different planes to create complex 3D injective deformations of the 3D volume. We show our method provides the ability to efficiently and accurately optimize and learn complex deformations, outperforming other injective approaches. As a main application, we produce complex and artifact-free NeRF and SDF deformations.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.

It is inherently ambiguous to lift 2D results from pre-trained diffusion models to a 3D world for text-to-3D generation. 2D diffusion models solely learn view-agnostic priors and thus lack 3D knowledge during the lifting, leading to the multi-view inconsistency problem. We find that this problem primarily stems from geometric inconsistency, and avoiding misplaced geometric structures substantially mitigates the problem in the final outputs. Therefore, we improve the consistency by aligning the 2D geometric priors in diffusion models with well-defined 3D shapes during the lifting, addressing the vast majority of the problem. This is achieved by fine-tuning the 2D diffusion model to be viewpoint-aware and to produce view-specific coordinate maps of canonically oriented 3D objects. In our process, only coarse 3D information is used for aligning. This "coarse" alignment not only resolves the multi-view inconsistency in geometries but also retains the ability in 2D diffusion models to generate detailed and diversified high-quality objects unseen in the 3D datasets. Furthermore, our aligned geometric priors (AGP) are generic and can be seamlessly integrated into various state-of-the-art pipelines, obtaining high generalizability in terms of unseen shapes and visual appearance while greatly alleviating the multi-view inconsistency problem. Our method represents a new state-of-the-art performance with an 85+% consistency rate by human evaluation, while many previous methods are around 30%. Our project page is https://sweetdreamer3d.github.io/

Feed-forward 3D generative models like the Large Reconstruction Model (LRM) have demonstrated exceptional generation speed. However, the transformer-based methods do not leverage the geometric priors of the triplane component in their architecture, often leading to sub-optimal quality given the limited size of 3D data and slow training. In this work, we present the Convolutional Reconstruction Model (CRM), a high-fidelity feed-forward single image-to-3D generative model. Recognizing the limitations posed by sparse 3D data, we highlight the necessity of integrating geometric priors into network design. CRM builds on the key observation that the visualization of triplane exhibits spatial correspondence of six orthographic images. First, it generates six orthographic view images from a single input image, then feeds these images into a convolutional U-Net, leveraging its strong pixel-level alignment capabilities and significant bandwidth to create a high-resolution triplane. CRM further employs Flexicubes as geometric representation, facilitating direct end-to-end optimization on textured meshes. Overall, our model delivers a high-fidelity textured mesh from an image in just 10 seconds, without any test-time optimization.

Neural graphics primitives are faster and achieve higher quality when their neural networks are augmented by spatial data structures that hold trainable features arranged in a grid. However, existing feature grids either come with a large memory footprint (dense or factorized grids, trees, and hash tables) or slow performance (index learning and vector quantization). In this paper, we show that a hash table with learned probes has neither disadvantage, resulting in a favorable combination of size and speed. Inference is faster than unprobed hash tables at equal quality while training is only 1.2-2.6x slower, significantly outperforming prior index learning approaches. We arrive at this formulation by casting all feature grids into a common framework: they each correspond to a lookup function that indexes into a table of feature vectors. In this framework, the lookup functions of existing data structures can be combined by simple arithmetic combinations of their indices, resulting in Pareto optimal compression and speed.

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions R^3, position and orientations R^3 {times} S^2, and the group SE(3) itself. Among these, R^3 {times} S^2 is an optimal choice due to the ability to represent directional information, which R^3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

For any given dimension d, all reflexive d-polytopes can be found (in principle) as subpolytopes of a number of maximal polyhedra that are defined in terms of (d+1)-tuples of integers (weights), or combinations of k-tuples of weights with k<d+1. We present the results of a complete classification of sextuples of weights pertaining to the construction of all reflexive polytopes in five dimensions. We find 322 383 760 930 such weight systems. 185 269 499 015 of them give rise directly to reflexive polytopes and thereby to mirror pairs of Calabi-Yau fourfolds. These lead to 532 600 483 distinct sets of Hodge numbers.

Inverse design, where we seek to design input variables in order to optimize an underlying objective function, is an important problem that arises across fields such as mechanical engineering to aerospace engineering. Inverse design is typically formulated as an optimization problem, with recent works leveraging optimization across learned dynamics models. However, as models are optimized they tend to fall into adversarial modes, preventing effective sampling. We illustrate that by instead optimizing over the learned energy function captured by the diffusion model, we can avoid such adversarial examples and significantly improve design performance. We further illustrate how such a design system is compositional, enabling us to combine multiple different diffusion models representing subcomponents of our desired system to design systems with every specified component. In an N-body interaction task and a challenging 2D multi-airfoil design task, we demonstrate that by composing the learned diffusion model at test time, our method allows us to design initial states and boundary shapes that are more complex than those in the training data. Our method generalizes to more objects for N-body dataset and discovers formation flying to minimize drag in the multi-airfoil design task. Project website and code can be found at https://github.com/AI4Science-WestlakeU/cindm.

We introduce PyTorch Geometric, a library for deep learning on irregularly structured input data such as graphs, point clouds and manifolds, built upon PyTorch. In addition to general graph data structures and processing methods, it contains a variety of recently published methods from the domains of relational learning and 3D data processing. PyTorch Geometric achieves high data throughput by leveraging sparse GPU acceleration, by providing dedicated CUDA kernels and by introducing efficient mini-batch handling for input examples of different size. In this work, we present the library in detail and perform a comprehensive comparative study of the implemented methods in homogeneous evaluation scenarios.

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix multiplications in fully-connected layers with Hamilton products of Quaternions, both enjoy parameter savings with only 1/4 learnable parameters and achieve comparable performance in various applications. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions (4D, 8D, and 16D). This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility using arbitrarily 1/n learnable parameters compared with the fully-connected layer counterpart. Experiments of applications to the LSTM and Transformer models on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.

We study the problem of visualizing large-scale and high-dimensional data in a low-dimensional (typically 2D or 3D) space. Much success has been reported recently by techniques that first compute a similarity structure of the data points and then project them into a low-dimensional space with the structure preserved. These two steps suffer from considerable computational costs, preventing the state-of-the-art methods such as the t-SNE from scaling to large-scale and high-dimensional data (e.g., millions of data points and hundreds of dimensions). We propose the LargeVis, a technique that first constructs an accurately approximated K-nearest neighbor graph from the data and then layouts the graph in the low-dimensional space. Comparing to t-SNE, LargeVis significantly reduces the computational cost of the graph construction step and employs a principled probabilistic model for the visualization step, the objective of which can be effectively optimized through asynchronous stochastic gradient descent with a linear time complexity. The whole procedure thus easily scales to millions of high-dimensional data points. Experimental results on real-world data sets demonstrate that the LargeVis outperforms the state-of-the-art methods in both efficiency and effectiveness. The hyper-parameters of LargeVis are also much more stable over different data sets.

We target a 3D generative model for general natural scenes that are typically unique and intricate. Lacking the necessary volumes of training data, along with the difficulties of having ad hoc designs in presence of varying scene characteristics, renders existing setups intractable. Inspired by classical patch-based image models, we advocate for synthesizing 3D scenes at the patch level, given a single example. At the core of this work lies important algorithmic designs w.r.t the scene representation and generative patch nearest-neighbor module, that address unique challenges arising from lifting classical 2D patch-based framework to 3D generation. These design choices, on a collective level, contribute to a robust, effective, and efficient model that can generate high-quality general natural scenes with both realistic geometric structure and visual appearance, in large quantities and varieties, as demonstrated upon a variety of exemplar scenes.

In this paper, we present CAD2Program, a new method for reconstructing 3D parametric models from 2D CAD drawings. Our proposed method is inspired by recent successes in vision-language models (VLMs), and departs from traditional methods which rely on task-specific data representations and/or algorithms. Specifically, on the input side, we simply treat the 2D CAD drawing as a raster image, regardless of its original format, and encode the image with a standard ViT model. We show that such an encoding scheme achieves competitive performance against existing methods that operate on vector-graphics inputs, while imposing substantially fewer restrictions on the 2D drawings. On the output side, our method auto-regressively predicts a general-purpose language describing 3D parametric models in text form. Compared to other sequence modeling methods for CAD which use domain-specific sequence representations with fixed-size slots, our text-based representation is more flexible, and can be easily extended to arbitrary geometric entities and semantic or functional properties. Experimental results on a large-scale dataset of cabinet models demonstrate the effectiveness of our method.

Novel view synthesis and 3D modeling using implicit neural field representation are shown to be very effective for calibrated multi-view cameras. Such representations are known to benefit from additional geometric and semantic supervision. Most existing methods that exploit additional supervision require dense pixel-wise labels or localized scene priors. These methods cannot benefit from high-level vague scene priors provided in terms of scenes' descriptions. In this work, we aim to leverage the geometric prior of Manhattan scenes to improve the implicit neural radiance field representations. More precisely, we assume that only the knowledge of the indoor scene (under investigation) being Manhattan is known -- with no additional information whatsoever -- with an unknown Manhattan coordinate frame. Such high-level prior is used to self-supervise the surface normals derived explicitly in the implicit neural fields. Our modeling allows us to cluster the derived normals and exploit their orthogonality constraints for self-supervision. Our exhaustive experiments on datasets of diverse indoor scenes demonstrate the significant benefit of the proposed method over the established baselines. The source code will be available at https://github.com/nikola3794/normal-clustering-nerf.

LiDAR scene generation has been developing rapidly recently. However, existing methods primarily focus on generating static and single-frame scenes, overlooking the inherently dynamic nature of real-world driving environments. In this work, we introduce DynamicCity, a novel 4D LiDAR generation framework capable of generating large-scale, high-quality LiDAR scenes that capture the temporal evolution of dynamic environments. DynamicCity mainly consists of two key models. 1) A VAE model for learning HexPlane as the compact 4D representation. Instead of using naive averaging operations, DynamicCity employs a novel Projection Module to effectively compress 4D LiDAR features into six 2D feature maps for HexPlane construction, which significantly enhances HexPlane fitting quality (up to 12.56 mIoU gain). Furthermore, we utilize an Expansion & Squeeze Strategy to reconstruct 3D feature volumes in parallel, which improves both network training efficiency and reconstruction accuracy than naively querying each 3D point (up to 7.05 mIoU gain, 2.06x training speedup, and 70.84% memory reduction). 2) A DiT-based diffusion model for HexPlane generation. To make HexPlane feasible for DiT generation, a Padded Rollout Operation is proposed to reorganize all six feature planes of the HexPlane as a squared 2D feature map. In particular, various conditions could be introduced in the diffusion or sampling process, supporting versatile 4D generation applications, such as trajectory- and command-driven generation, inpainting, and layout-conditioned generation. Extensive experiments on the CarlaSC and Waymo datasets demonstrate that DynamicCity significantly outperforms existing state-of-the-art 4D LiDAR generation methods across multiple metrics. The code will be released to facilitate future research.

Crystal structures are characterized by atomic bases within a primitive unit cell that repeats along a regular lattice throughout 3D space. The periodic and infinite nature of crystals poses unique challenges for geometric graph representation learning. Specifically, constructing graphs that effectively capture the complete geometric information of crystals and handle chiral crystals remains an unsolved and challenging problem. In this paper, we introduce a novel approach that utilizes the periodic patterns of unit cells to establish the lattice-based representation for each atom, enabling efficient and expressive graph representations of crystals. Furthermore, we propose ComFormer, a SE(3) transformer designed specifically for crystalline materials. ComFormer includes two variants; namely, iComFormer that employs invariant geometric descriptors of Euclidean distances and angles, and eComFormer that utilizes equivariant vector representations. Experimental results demonstrate the state-of-the-art predictive accuracy of ComFormer variants on various tasks across three widely-used crystal benchmarks. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

A common approach to localize 3D human joints in a synchronized and calibrated multi-view setup consists of two-steps: (1) apply a 2D detector separately on each view to localize joints in 2D, and (2) perform robust triangulation on 2D detections from each view to acquire the 3D joint locations. However, in step 1, the 2D detector is limited to solving challenging cases which could potentially be better resolved in 3D, such as occlusions and oblique viewing angles, purely in 2D without leveraging any 3D information. Therefore, we propose the differentiable "epipolar transformer", which enables the 2D detector to leverage 3D-aware features to improve 2D pose estimation. The intuition is: given a 2D location p in the current view, we would like to first find its corresponding point p' in a neighboring view, and then combine the features at p' with the features at p, thus leading to a 3D-aware feature at p. Inspired by stereo matching, the epipolar transformer leverages epipolar constraints and feature matching to approximate the features at p'. Experiments on InterHand and Human3.6M show that our approach has consistent improvements over the baselines. Specifically, in the condition where no external data is used, our Human3.6M model trained with ResNet-50 backbone and image size 256 x 256 outperforms state-of-the-art by 4.23 mm and achieves MPJPE 26.9 mm.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

We present a simple yet powerful neural network that implicitly represents and renders 3D objects and scenes only from 2D observations. The network models 3D geometries as a general radiance field, which takes a set of 2D images with camera poses and intrinsics as input, constructs an internal representation for each point of the 3D space, and then renders the corresponding appearance and geometry of that point viewed from an arbitrary position. The key to our approach is to learn local features for each pixel in 2D images and to then project these features to 3D points, thus yielding general and rich point representations. We additionally integrate an attention mechanism to aggregate pixel features from multiple 2D views, such that visual occlusions are implicitly taken into account. Extensive experiments demonstrate that our method can generate high-quality and realistic novel views for novel objects, unseen categories and challenging real-world scenes.

Deep neural representations of 3D shapes as implicit functions have been shown to produce high fidelity models surpassing the resolution-memory trade-off faced by the explicit representations using meshes and point clouds. However, most such approaches focus on representing closed shapes. Unsigned distance function (UDF) based approaches have been proposed recently as a promising alternative to represent both open and closed shapes. However, since the gradients of UDFs vanish on the surface, it is challenging to estimate local (differential) geometric properties like the normals and tangent planes which are needed for many downstream applications in vision and graphics. There are additional challenges in computing these properties efficiently with a low-memory footprint. This paper presents a novel approach that models such surfaces using a new class of implicit representations called the closest surface-point (CSP) representation. We show that CSP allows us to represent complex surfaces of any topology (open or closed) with high fidelity. It also allows for accurate and efficient computation of local geometric properties. We further demonstrate that it leads to efficient implementation of downstream algorithms like sphere-tracing for rendering the 3D surface as well as to create explicit mesh-based representations. Extensive experimental evaluation on the ShapeNet dataset validate the above contributions with results surpassing the state-of-the-art.

In this paper, we present a novel indoor 3D reconstruction method with occluded surface completion, given a sequence of depth readings. Prior state-of-the-art (SOTA) methods only focus on the reconstruction of the visible areas in a scene, neglecting the invisible areas due to the occlusions, e.g., the contact surface between furniture, occluded wall and floor. Our method tackles the task of completing the occluded scene surfaces, resulting in a complete 3D scene mesh. The core idea of our method is learning 3D geometry prior from various complete scenes to infer the occluded geometry of an unseen scene from solely depth measurements. We design a coarse-fine hierarchical octree representation coupled with a dual-decoder architecture, i.e., Geo-decoder and 3D Inpainter, which jointly reconstructs the complete 3D scene geometry. The Geo-decoder with detailed representation at fine levels is optimized online for each scene to reconstruct visible surfaces. The 3D Inpainter with abstract representation at coarse levels is trained offline using various scenes to complete occluded surfaces. As a result, while the Geo-decoder is specialized for an individual scene, the 3D Inpainter can be generally applied across different scenes. We evaluate the proposed method on the 3D Completed Room Scene (3D-CRS) and iTHOR datasets, significantly outperforming the SOTA methods by a gain of 16.8% and 24.2% in terms of the completeness of 3D reconstruction. 3D-CRS dataset including a complete 3D mesh of each scene is provided at project webpage.

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Image-based 3D reconstruction has increasingly stunning results over the past few years with the latest improvements in computer vision and graphics. Geometry and topology are two fundamental concepts when dealing with 3D mesh structures. But the latest often remains a side issue in the 3D mesh-based reconstruction literature. Indeed, performing per-vertex elementary displacements over a 3D sphere mesh only impacts its geometry and leaves the topological structure unchanged and fixed. Whereas few attempts propose to update the geometry and the topology, all need to lean on costly 3D ground-truth to determine the faces/edges to prune. We present in this work a method that aims to refine the topology of any 3D mesh through a face-pruning strategy that extensively relies upon 2D alpha masks and camera pose information. Our solution leverages a differentiable renderer that renders each face as a 2D soft map. Its pixel intensity reflects the probability of being covered during the rendering process by such a face. Based on the 2D soft-masks available, our method is thus able to quickly highlight all the incorrectly rendered faces for a given viewpoint. Because our module is agnostic to the network that produces the 3D mesh, it can be easily plugged into any self-supervised image-based (either synthetic or natural) 3D reconstruction pipeline to get complex meshes with a non-spherical topology.

Given a set of K probability densities, we consider the multimarginal generative modeling problem of learning a joint distribution that recovers these densities as marginals. The structure of this joint distribution should identify multi-way correspondences among the prescribed marginals. We formalize an approach to this task within a generalization of the stochastic interpolant framework, leading to efficient learning algorithms built upon dynamical transport of measure. Our generative models are defined by velocity and score fields that can be characterized as the minimizers of simple quadratic objectives, and they are defined on a simplex that generalizes the time variable in the usual dynamical transport framework. The resulting transport on the simplex is influenced by all marginals, and we show that multi-way correspondences can be extracted. The identification of such correspondences has applications to style transfer, algorithmic fairness, and data decorruption. In addition, the multimarginal perspective enables an efficient algorithm for reducing the dynamical transport cost in the ordinary two-marginal setting. We demonstrate these capacities with several numerical examples.

Representing and rendering dynamic scenes has been an important but challenging task. Especially, to accurately model complex motions, high efficiency is usually hard to maintain. We introduce the 4D Gaussian Splatting (4D-GS) to achieve real-time dynamic scene rendering while also enjoying high training and storage efficiency. An efficient deformation field is constructed to model both Gaussian motions and shape deformations. Different adjacent Gaussians are connected via a HexPlane to produce more accurate position and shape deformations. Our 4D-GS method achieves real-time rendering under high resolutions, 70 FPS at a 800times800 resolution on an RTX 3090 GPU, while maintaining comparable or higher quality than previous state-of-the-art methods. More demos and code are available at https://guanjunwu.github.io/4dgs/.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size n is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in R^K, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a ell_2 distance of at most varepsilon from the true simplex (for any varepsilon>0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have ngeleft(K^2/varepsilon^2right)e^{Omegaleft(K/SNR^2right)} samples, where SNR stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as SNRgeOmegaleft(K^{1/2}right), the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in ashtiani2018nearly, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

Graph neural networks that model 3D data, such as point clouds or atoms, are typically desired to be SO(3) equivariant, i.e., equivariant to 3D rotations. Unfortunately equivariant convolutions, which are a fundamental operation for equivariant networks, increase significantly in computational complexity as higher-order tensors are used. In this paper, we address this issue by reducing the SO(3) convolutions or tensor products to mathematically equivalent convolutions in SO(2) . This is accomplished by aligning the node embeddings' primary axis with the edge vectors, which sparsifies the tensor product and reduces the computational complexity from O(L^6) to O(L^3), where L is the degree of the representation. We demonstrate the potential implications of this improvement by proposing the Equivariant Spherical Channel Network (eSCN), a graph neural network utilizing our novel approach to equivariant convolutions, which achieves state-of-the-art results on the large-scale OC-20 and OC-22 datasets.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

We present Simplex Random Features (SimRFs), a new random feature (RF) mechanism for unbiased approximation of the softmax and Gaussian kernels by geometrical correlation of random projection vectors. We prove that SimRFs provide the smallest possible mean square error (MSE) on unbiased estimates of these kernels among the class of weight-independent geometrically-coupled positive random feature (PRF) mechanisms, substantially outperforming the previously most accurate Orthogonal Random Features at no observable extra cost. We present a more computationally expensive SimRFs+ variant, which we prove is asymptotically optimal in the broader family of weight-dependent geometrical coupling schemes (which permit correlations between random vector directions and norms). In extensive empirical studies, we show consistent gains provided by SimRFs in settings including pointwise kernel estimation, nonparametric classification and scalable Transformers.

In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the d-dimensional advection-diffusion equation, our proposal is a two-level algorithm that merges the solutions obtained from the discretization of the equation over highly anisotropic submeshes to compute an initial approximation of the fine solution. The algorithm then iteratively refines the initial guess by leveraging the structure of the residual. Performing costly calculations on anisotropic submeshes enable us to reduce the dimensionality of the problem by one, and the merging process, which involves the computation of solutions over disjoint domains, allows for parallel implementation.

Triggered by limitations of graph-based deep learning methods in terms of computational expressivity and model flexibility, recent years have seen a surge of interest in computational models that operate on higher-order topological domains such as hypergraphs and simplicial complexes. While the increased expressivity of these models can indeed lead to a better classification performance and a more faithful representation of the underlying system, the computational cost of these higher-order models can increase dramatically. To this end, we here explore a simplicial complex neural network learning architecture based on random walks and fast 1D convolutions (SCRaWl), in which we can adjust the increase in computational cost by varying the length and number of random walks considered while accounting for higher-order relationships. Importantly, due to the random walk-based design, the expressivity of the proposed architecture is provably incomparable to that of existing message-passing simplicial neural networks. We empirically evaluate SCRaWl on real-world datasets and show that it outperforms other simplicial neural networks.

Strategies for the generation of periodic discrete structures with identical two-point correlation are developed. Starting from a pair of root structures, which are not related by translation, phase inversion or axis reflections, child structures of arbitrary resolution (i.e., pixel or voxel numbers) and number of phases (i.e., material phases/species) can be generated by means of trivial embedding based phase extension, application of kernels and/or phase coalescence, such that the generated structures inherit the two-point-correlation equivalence. Proofs of the inheritance property are provided by means of the Discrete Fourier Transform theory. A Python 3 implementation of the results is offered by the authors through the Github repository https://github.com/DataAnalyticsEngineering/EQ2PC in order to make the provided results reproducible and useful for all interested readers. Examples for the generation of structures are demonstrated, together with applications in the homogenization theory of periodic media.

Surface reconstruction from multi-view images is a challenging task, with solutions often requiring a large number of sampled images with high overlap. We seek to develop a method for few-view reconstruction, for the case of the human foot. To solve this task, we must extract rich geometric cues from RGB images, before carefully fusing them into a final 3D object. Our FOUND approach tackles this, with 4 main contributions: (i) SynFoot, a synthetic dataset of 50,000 photorealistic foot images, paired with ground truth surface normals and keypoints; (ii) an uncertainty-aware surface normal predictor trained on our synthetic dataset; (iii) an optimization scheme for fitting a generative foot model to a series of images; and (iv) a benchmark dataset of calibrated images and high resolution ground truth geometry. We show that our normal predictor outperforms all off-the-shelf equivalents significantly on real images, and our optimization scheme outperforms state-of-the-art photogrammetry pipelines, especially for a few-view setting. We release our synthetic dataset and baseline 3D scans to the research community.

We introduce a method for the generation of images from an input scene graph. The method separates between a layout embedding and an appearance embedding. The dual embedding leads to generated images that better match the scene graph, have higher visual quality, and support more complex scene graphs. In addition, the embedding scheme supports multiple and diverse output images per scene graph, which can be further controlled by the user. We demonstrate two modes of per-object control: (i) importing elements from other images, and (ii) navigation in the object space, by selecting an appearance archetype. Our code is publicly available at https://www.github.com/ashual/scene_generation

In this work, we recover the underlying 3D structure of non-geometrically consistent scenes. We focus our analysis on hand-drawn images from cartoons and anime. Many cartoons are created by artists without a 3D rendering engine, which means that any new image of a scene is hand-drawn. The hand-drawn images are usually faithful representations of the world, but only in a qualitative sense, since it is difficult for humans to draw multiple perspectives of an object or scene 3D consistently. Nevertheless, people can easily perceive 3D scenes from inconsistent inputs! In this work, we correct for 2D drawing inconsistencies to recover a plausible 3D structure such that the newly warped drawings are consistent with each other. Our pipeline consists of a user-friendly annotation tool, camera pose estimation, and image deformation to recover a dense structure. Our method warps images to obey a perspective camera model, enabling our aligned results to be plugged into novel-view synthesis reconstruction methods to experience cartoons from viewpoints never drawn before. Our project page is https://toon3d.studio/.

Aerial imagery analysis is critical for many research fields. However, obtaining frequent high-quality aerial images is not always accessible due to its high effort and cost requirements. One solution is to use the Ground-to-Aerial (G2A) technique to synthesize aerial images from easily collectible ground images. However, G2A is rarely studied, because of its challenges, including but not limited to, the drastic view changes, occlusion, and range of visibility. In this paper, we present a novel Geometric Preserving Ground-to-Aerial (G2A) image synthesis (GPG2A) model that can generate realistic aerial images from ground images. GPG2A consists of two stages. The first stage predicts the Bird's Eye View (BEV) segmentation (referred to as the BEV layout map) from the ground image. The second stage synthesizes the aerial image from the predicted BEV layout map and text descriptions of the ground image. To train our model, we present a new multi-modal cross-view dataset, namely VIGORv2 which is built upon VIGOR with newly collected aerial images, maps, and text descriptions. Our extensive experiments illustrate that GPG2A synthesizes better geometry-preserved aerial images than existing models. We also present two applications, data augmentation for cross-view geo-localization and sketch-based region search, to further verify the effectiveness of our GPG2A. The code and data will be publicly available.

In representation learning, uniformity refers to the uniform feature distribution in the latent space (i.e., unit hypersphere). Previous work has shown that improving uniformity contributes to the learning of under-represented classes. However, most of the previous work focused on classification; the representation space of imbalanced regression remains unexplored. Classification-based methods are not suitable for regression tasks because they cluster features into distinct groups without considering the continuous and ordered nature essential for regression. In a geometric aspect, we uniquely focus on ensuring uniformity in the latent space for imbalanced regression through two key losses: enveloping and homogeneity. The enveloping loss encourages the induced trace to uniformly occupy the surface of a hypersphere, while the homogeneity loss ensures smoothness, with representations evenly spaced at consistent intervals. Our method integrates these geometric principles into the data representations via a Surrogate-driven Representation Learning (SRL) framework. Experiments with real-world regression and operator learning tasks highlight the importance of uniformity in imbalanced regression and validate the efficacy of our geometry-based loss functions.

The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at https://github.com/chaitjo/geometric-gnn-dojo

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

Modern methods for vision-centric autonomous driving perception widely adopt the bird's-eye-view (BEV) representation to describe a 3D scene. Despite its better efficiency than voxel representation, it has difficulty describing the fine-grained 3D structure of a scene with a single plane. To address this, we propose a tri-perspective view (TPV) representation which accompanies BEV with two additional perpendicular planes. We model each point in the 3D space by summing its projected features on the three planes. To lift image features to the 3D TPV space, we further propose a transformer-based TPV encoder (TPVFormer) to obtain the TPV features effectively. We employ the attention mechanism to aggregate the image features corresponding to each query in each TPV plane. Experiments show that our model trained with sparse supervision effectively predicts the semantic occupancy for all voxels. We demonstrate for the first time that using only camera inputs can achieve comparable performance with LiDAR-based methods on the LiDAR segmentation task on nuScenes. Code: https://github.com/wzzheng/TPVFormer.

With the recent advancements in single-image-based human mesh recovery, there is a growing interest in enhancing its performance in certain extreme scenarios, such as occlusion, while maintaining overall model accuracy. Although obtaining accurately annotated 3D human poses under occlusion is challenging, there is still a wealth of rich and precise 2D pose annotations that can be leveraged. However, existing works mostly focus on directly leveraging 2D pose coordinates to estimate 3D pose and mesh. In this paper, we present PostoMETRO(Pose token enhanced MEsh TRansfOrmer), which integrates occlusion-resilient 2D pose representation into transformers in a token-wise manner. Utilizing a specialized pose tokenizer, we efficiently condense 2D pose data to a compact sequence of pose tokens and feed them to the transformer together with the image tokens. This process not only ensures a rich depiction of texture from the image but also fosters a robust integration of pose and image information. Subsequently, these combined tokens are queried by vertex and joint tokens to decode 3D coordinates of mesh vertices and human joints. Facilitated by the robust pose token representation and the effective combination, we are able to produce more precise 3D coordinates, even under extreme scenarios like occlusion. Experiments on both standard and occlusion-specific benchmarks demonstrate the effectiveness of PostoMETRO. Qualitative results further illustrate the clarity of how 2D pose can help 3D reconstruction. Code will be made available.

All tables can be represented as grids. Based on this observation, we propose GridFormer, a novel approach for interpreting unconstrained table structures by predicting the vertex and edge of a grid. First, we propose a flexible table representation in the form of an MXN grid. In this representation, the vertexes and edges of the grid store the localization and adjacency information of the table. Then, we introduce a DETR-style table structure recognizer to efficiently predict this multi-objective information of the grid in a single shot. Specifically, given a set of learned row and column queries, the recognizer directly outputs the vertexes and edges information of the corresponding rows and columns. Extensive experiments on five challenging benchmarks which include wired, wireless, multi-merge-cell, oriented, and distorted tables demonstrate the competitive performance of our model over other methods.

Advancements in 3D Gaussian Splatting have significantly accelerated 3D reconstruction and generation. However, it may require a large number of Gaussians, which creates a substantial memory footprint. This paper introduces GES (Generalized Exponential Splatting), a novel representation that employs Generalized Exponential Function (GEF) to model 3D scenes, requiring far fewer particles to represent a scene and thus significantly outperforming Gaussian Splatting methods in efficiency with a plug-and-play replacement ability for Gaussian-based utilities. GES is validated theoretically and empirically in both principled 1D setup and realistic 3D scenes. It is shown to represent signals with sharp edges more accurately, which are typically challenging for Gaussians due to their inherent low-pass characteristics. Our empirical analysis demonstrates that GEF outperforms Gaussians in fitting natural-occurring signals (e.g. squares, triangles, and parabolic signals), thereby reducing the need for extensive splitting operations that increase the memory footprint of Gaussian Splatting. With the aid of a frequency-modulated loss, GES achieves competitive performance in novel-view synthesis benchmarks while requiring less than half the memory storage of Gaussian Splatting and increasing the rendering speed by up to 39%. The code is available on the project website https://abdullahamdi.com/ges .

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

Recent advancements in 3D object reconstruction from single images have primarily focused on improving the accuracy of object shapes. Yet, these techniques often fail to accurately capture the inter-relation between the object, ground, and camera. As a result, the reconstructed objects often appear floating or tilted when placed on flat surfaces. This limitation significantly affects 3D-aware image editing applications like shadow rendering and object pose manipulation. To address this issue, we introduce ORG (Object Reconstruction with Ground), a novel task aimed at reconstructing 3D object geometry in conjunction with the ground surface. Our method uses two compact pixel-level representations to depict the relationship between camera, object, and ground. Experiments show that the proposed ORG model can effectively reconstruct object-ground geometry on unseen data, significantly enhancing the quality of shadow generation and pose manipulation compared to conventional single-image 3D reconstruction techniques.

We propose Gaussian Frosting, a novel mesh-based representation for high-quality rendering and editing of complex 3D effects in real-time. Our approach builds on the recent 3D Gaussian Splatting framework, which optimizes a set of 3D Gaussians to approximate a radiance field from images. We propose first extracting a base mesh from Gaussians during optimization, then building and refining an adaptive layer of Gaussians with a variable thickness around the mesh to better capture the fine details and volumetric effects near the surface, such as hair or grass. We call this layer Gaussian Frosting, as it resembles a coating of frosting on a cake. The fuzzier the material, the thicker the frosting. We also introduce a parameterization of the Gaussians to enforce them to stay inside the frosting layer and automatically adjust their parameters when deforming, rescaling, editing or animating the mesh. Our representation allows for efficient rendering using Gaussian splatting, as well as editing and animation by modifying the base mesh. We demonstrate the effectiveness of our method on various synthetic and real scenes, and show that it outperforms existing surface-based approaches. We will release our code and a web-based viewer as additional contributions. Our project page is the following: https://anttwo.github.io/frosting/

Neural 3D scene representations have shown great potential for 3D reconstruction from 2D images. However, reconstructing real-world captures of complex scenes still remains a challenge. Existing generic 3D reconstruction methods often struggle to represent fine geometric details and do not adequately model reflective surfaces of large-scale scenes. Techniques that explicitly focus on reflective surfaces can model complex and detailed reflections by exploiting better reflection parameterizations. However, we observe that these methods are often not robust in real unbounded scenarios where non-reflective as well as reflective components are present. In this work, we propose UniSDF, a general purpose 3D reconstruction method that can reconstruct large complex scenes with reflections. We investigate both view-based as well as reflection-based color prediction parameterization techniques and find that explicitly blending these representations in 3D space enables reconstruction of surfaces that are more geometrically accurate, especially for reflective surfaces. We further combine this representation with a multi-resolution grid backbone that is trained in a coarse-to-fine manner, enabling faster reconstructions than prior methods. Extensive experiments on object-level datasets DTU, Shiny Blender as well as unbounded datasets Mip-NeRF 360 and Ref-NeRF real demonstrate that our method is able to robustly reconstruct complex large-scale scenes with fine details and reflective surfaces. Please see our project page at https://fangjinhuawang.github.io/UniSDF.

Learning radiance fields (NeRF) with powerful 2D diffusion models has garnered popularity for text-to-3D generation. Nevertheless, the implicit 3D representations of NeRF lack explicit modeling of meshes and textures over surfaces, and such surface-undefined way may suffer from the issues, e.g., noisy surfaces with ambiguous texture details or cross-view inconsistency. To alleviate this, we present DreamMesh, a novel text-to-3D architecture that pivots on well-defined surfaces (triangle meshes) to generate high-fidelity explicit 3D model. Technically, DreamMesh capitalizes on a distinctive coarse-to-fine scheme. In the coarse stage, the mesh is first deformed by text-guided Jacobians and then DreamMesh textures the mesh with an interlaced use of 2D diffusion models in a tuning free manner from multiple viewpoints. In the fine stage, DreamMesh jointly manipulates the mesh and refines the texture map, leading to high-quality triangle meshes with high-fidelity textured materials. Extensive experiments demonstrate that DreamMesh significantly outperforms state-of-the-art text-to-3D methods in faithfully generating 3D content with richer textual details and enhanced geometry. Our project page is available at https://dreammesh.github.io.

We develop a novel theoretical framework for understating OT schemes respecting a class structure. For this purpose, we propose a convex OT program with a sum-of-norms regularization term, which provably recovers the underlying class structure under geometric assumptions. Furthermore, we derive an accelerated proximal algorithm with a closed-form projection and proximal operator scheme, thereby affording a more scalable algorithm for computing optimal transport plans. We provide a novel argument for the uniqueness of the optimum even in the absence of strong convexity. Our experiments show that the new regularizer not only results in a better preservation of the class structure in the data but also yields additional robustness to the data geometry, compared to previous regularizers.

Creating diverse and high-quality 3D assets with an automatic generative model is highly desirable. Despite extensive efforts on 3D generation, most existing works focus on the generation of a single category or a few categories. In this paper, we introduce a diffusion-based feed-forward framework for synthesizing massive categories of real-world 3D objects with a single generative model. Notably, there are three major challenges for this large-vocabulary 3D generation: a) the need for expressive yet efficient 3D representation; b) large diversity in geometry and texture across categories; c) complexity in the appearances of real-world objects. To this end, we propose a novel triplane-based 3D-aware Diffusion model with TransFormer, DiffTF, for handling challenges via three aspects. 1) Considering efficiency and robustness, we adopt a revised triplane representation and improve the fitting speed and accuracy. 2) To handle the drastic variations in geometry and texture, we regard the features of all 3D objects as a combination of generalized 3D knowledge and specialized 3D features. To extract generalized 3D knowledge from diverse categories, we propose a novel 3D-aware transformer with shared cross-plane attention. It learns the cross-plane relations across different planes and aggregates the generalized 3D knowledge with specialized 3D features. 3) In addition, we devise the 3D-aware encoder/decoder to enhance the generalized 3D knowledge in the encoded triplanes for handling categories with complex appearances. Extensive experiments on ShapeNet and OmniObject3D (over 200 diverse real-world categories) convincingly demonstrate that a single DiffTF model achieves state-of-the-art large-vocabulary 3D object generation performance with large diversity, rich semantics, and high quality.

A comprehensive semantic understanding of a scene is important for many applications - but in what space should diverse semantic information (e.g., objects, scene categories, material types, texture, etc.) be grounded and what should be its structure? Aspiring to have one unified structure that hosts diverse types of semantics, we follow the Scene Graph paradigm in 3D, generating a 3D Scene Graph. Given a 3D mesh and registered panoramic images, we construct a graph that spans the entire building and includes semantics on objects (e.g., class, material, and other attributes), rooms (e.g., scene category, volume, etc.) and cameras (e.g., location, etc.), as well as the relationships among these entities. However, this process is prohibitively labor heavy if done manually. To alleviate this we devise a semi-automatic framework that employs existing detection methods and enhances them using two main constraints: I. framing of query images sampled on panoramas to maximize the performance of 2D detectors, and II. multi-view consistency enforcement across 2D detections that originate in different camera locations.

Micro Aerial Vehicles (MAVs) that operate in unstructured, unexplored environments require fast and flexible local planning, which can replan when new parts of the map are explored. Trajectory optimization methods fulfill these needs, but require obstacle distance information, which can be given by Euclidean Signed Distance Fields (ESDFs). We propose a method to incrementally build ESDFs from Truncated Signed Distance Fields (TSDFs), a common implicit surface representation used in computer graphics and vision. TSDFs are fast to build and smooth out sensor noise over many observations, and are designed to produce surface meshes. Meshes allow human operators to get a better assessment of the robot's environment, and set high-level mission goals. We show that we can build TSDFs faster than Octomaps, and that it is more accurate to build ESDFs out of TSDFs than occupancy maps. Our complete system, called voxblox, will be available as open source and runs in real-time on a single CPU core. We validate our approach on-board an MAV, by using our system with a trajectory optimization local planner, entirely on-board and in real-time.

Coordinate-based neural representations have shown significant promise as an alternative to discrete, array-based representations for complex low dimensional signals. However, optimizing a coordinate-based network from randomly initialized weights for each new signal is inefficient. We propose applying standard meta-learning algorithms to learn the initial weight parameters for these fully-connected networks based on the underlying class of signals being represented (e.g., images of faces or 3D models of chairs). Despite requiring only a minor change in implementation, using these learned initial weights enables faster convergence during optimization and can serve as a strong prior over the signal class being modeled, resulting in better generalization when only partial observations of a given signal are available. We explore these benefits across a variety of tasks, including representing 2D images, reconstructing CT scans, and recovering 3D shapes and scenes from 2D image observations.

This work considers gradient-based mesh optimization, where we iteratively optimize for a 3D surface mesh by representing it as the isosurface of a scalar field, an increasingly common paradigm in applications including photogrammetry, generative modeling, and inverse physics. Existing implementations adapt classic isosurface extraction algorithms like Marching Cubes or Dual Contouring; these techniques were designed to extract meshes from fixed, known fields, and in the optimization setting they lack the degrees of freedom to represent high-quality feature-preserving meshes, or suffer from numerical instabilities. We introduce FlexiCubes, an isosurface representation specifically designed for optimizing an unknown mesh with respect to geometric, visual, or even physical objectives. Our main insight is to introduce additional carefully-chosen parameters into the representation, which allow local flexible adjustments to the extracted mesh geometry and connectivity. These parameters are updated along with the underlying scalar field via automatic differentiation when optimizing for a downstream task. We base our extraction scheme on Dual Marching Cubes for improved topological properties, and present extensions to optionally generate tetrahedral and hierarchically-adaptive meshes. Extensive experiments validate FlexiCubes on both synthetic benchmarks and real-world applications, showing that it offers significant improvements in mesh quality and geometric fidelity.

Lattice-based planning techniques simplify the motion planning problem for autonomous vehicles by limiting available motions to a pre-computed set of primitives. These primitives are then combined online to generate more complex maneuvers. A set of motion primitives t-span a lattice if, given a real number t at least 1, any configuration in the lattice can be reached via a sequence of motion primitives whose cost is no more than a factor of t from optimal. Computing a minimal t-spanning set balances a trade-off between computed motion quality and motion planning performance. In this work, we formulate this problem for an arbitrary lattice as a mixed integer linear program. We also propose an A*-based algorithm to solve the motion planning problem using these primitives. Finally, we present an algorithm that removes the excessive oscillations from planned motions -- a common problem in lattice-based planning. Our method is validated for autonomous driving in both parking lot and highway scenarios.

This article presents a comprehensive review of and analysis of state-of-the-art mapping algorithms for UAV (Unmanned Aerial Vehicle) applications, focusing on canopy-level and high-speed scenarios. This article presents a comprehensive exploration of sensor technologies suitable for UAV mapping, assessing their capabilities to provide measurements that meet the requirements of fast UAV mapping. Furthermore, the study conducts extensive experiments in a simulated environment to evaluate the performance of three distinct mapping algorithms: Direct Sparse Odometry (DSO), Stereo DSO (SDSO), and DSO Lite (DSOL). The experiments delve into mapping accuracy and mapping speed, providing valuable insights into the strengths and limitations of each algorithm. The results highlight the versatility and shortcomings of these algorithms in meeting the demands of modern UAV applications. The findings contribute to a nuanced understanding of UAV mapping dynamics, emphasizing their applicability in complex environments and high-speed scenarios. This research not only serves as a benchmark for mapping algorithm comparisons but also offers practical guidance for selecting sensors tailored to specific UAV mapping applications.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Reconstructing interacting hands from a single RGB image is a very challenging task. On the one hand, severe mutual occlusion and similar local appearance between two hands confuse the extraction of visual features, resulting in the misalignment of estimated hand meshes and the image. On the other hand, there are complex spatial relationship between interacting hands, which significantly increases the solution space of hand poses and increases the difficulty of network learning. In this paper, we propose a decoupled iterative refinement framework to achieve pixel-alignment hand reconstruction while efficiently modeling the spatial relationship between hands. Specifically, we define two feature spaces with different characteristics, namely 2D visual feature space and 3D joint feature space. First, we obtain joint-wise features from the visual feature map and utilize a graph convolution network and a transformer to perform intra- and inter-hand information interaction in the 3D joint feature space, respectively. Then, we project the joint features with global information back into the 2D visual feature space in an obfuscation-free manner and utilize the 2D convolution for pixel-wise enhancement. By performing multiple alternate enhancements in the two feature spaces, our method can achieve an accurate and robust reconstruction of interacting hands. Our method outperforms all existing two-hand reconstruction methods by a large margin on the InterHand2.6M dataset.

Neural implicit methods have achieved high-quality 3D object surfaces under slight specular highlights. However, high specular reflections (HSR) often appear in front of target objects when we capture them through glasses. The complex ambiguity in these scenes violates the multi-view consistency, then makes it challenging for recent methods to reconstruct target objects correctly. To remedy this issue, we present a novel surface reconstruction framework, NeuS-HSR, based on implicit neural rendering. In NeuS-HSR, the object surface is parameterized as an implicit signed distance function (SDF). To reduce the interference of HSR, we propose decomposing the rendered image into two appearances: the target object and the auxiliary plane. We design a novel auxiliary plane module by combining physical assumptions and neural networks to generate the auxiliary plane appearance. Extensive experiments on synthetic and real-world datasets demonstrate that NeuS-HSR outperforms state-of-the-art approaches for accurate and robust target surface reconstruction against HSR. Code is available at https://github.com/JiaxiongQ/NeuS-HSR.

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form h = g circ p where p : R^d rightarrow R is a degree k polynomial and g: R rightarrow R is a degree q polynomial. This function class generalizes the single-index model, which corresponds to k=1, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree k polynomials p, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target h up to vanishing test error in mathcal{O}(d^k) samples and polynomial time. This is a strict improvement over kernel methods, which require widetilde Theta(d^{kq}) samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of p being a quadratic. When p is indeed a quadratic, we achieve the information-theoretically optimal sample complexity mathcal{O}(d^2), which is an improvement over prior work~nichani2023provable requiring a sample size of widetildeTheta(d^4). Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature p with mathcal{O}(d^k) samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

Self-supervised learning has emerged as a promising approach for acquiring transferable 3D representations from unlabeled 3D point clouds. Unlike 2D images, which are widely accessible, acquiring 3D assets requires specialized expertise or professional 3D scanning equipment, making it difficult to scale and raising copyright concerns. To address these challenges, we propose learning 3D representations from procedural 3D programs that automatically generate 3D shapes using simple primitives and augmentations. Remarkably, despite lacking semantic content, the 3D representations learned from this synthesized dataset perform on par with state-of-the-art representations learned from semantically recognizable 3D models (e.g., airplanes) across various downstream 3D tasks, including shape classification, part segmentation, and masked point cloud completion. Our analysis further suggests that current self-supervised learning methods primarily capture geometric structures rather than high-level semantics.

For any graph G having n vertices and its automorphism group Aut(G), we provide a full characterisation of all of the possible Aut(G)-equivariant neural networks whose layers are some tensor power of R^{n}. In particular, we find a spanning set of matrices for the learnable, linear, Aut(G)-equivariant layer functions between such tensor power spaces in the standard basis of R^{n}.

Auto-regressive models have achieved impressive results in 2D image generation by modeling joint distributions in grid space. In this paper, we extend auto-regressive models to 3D domains, and seek a stronger ability of 3D shape generation by improving auto-regressive models at capacity and scalability simultaneously. Firstly, we leverage an ensemble of publicly available 3D datasets to facilitate the training of large-scale models. It consists of a comprehensive collection of approximately 900,000 objects, with multiple properties of meshes, points, voxels, rendered images, and text captions. This diverse labeled dataset, termed Objaverse-Mix, empowers our model to learn from a wide range of object variations. However, directly applying 3D auto-regression encounters critical challenges of high computational demands on volumetric grids and ambiguous auto-regressive order along grid dimensions, resulting in inferior quality of 3D shapes. To this end, we then present a novel framework Argus3D in terms of capacity. Concretely, our approach introduces discrete representation learning based on a latent vector instead of volumetric grids, which not only reduces computational costs but also preserves essential geometric details by learning the joint distributions in a more tractable order. The capacity of conditional generation can thus be realized by simply concatenating various conditioning inputs to the latent vector, such as point clouds, categories, images, and texts. In addition, thanks to the simplicity of our model architecture, we naturally scale up our approach to a larger model with an impressive 3.6 billion parameters, further enhancing the quality of versatile 3D generation. Extensive experiments on four generation tasks demonstrate that Argus3D can synthesize diverse and faithful shapes across multiple categories, achieving remarkable performance.

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

Single point-supervised object detection is gaining attention due to its cost-effectiveness. However, existing approaches focus on generating horizontal bounding boxes (HBBs) while ignoring oriented bounding boxes (OBBs) commonly used for objects in aerial images. This paper proposes PointOBB, the first single Point-based OBB generation method, for oriented object detection. PointOBB operates through the collaborative utilization of three distinctive views: an original view, a resized view, and a rotated/flipped (rot/flp) view. Upon the original view, we leverage the resized and rot/flp views to build a scale augmentation module and an angle acquisition module, respectively. In the former module, a Scale-Sensitive Consistency (SSC) loss is designed to enhance the deep network's ability to perceive the object scale. For accurate object angle predictions, the latter module incorporates self-supervised learning to predict angles, which is associated with a scale-guided Dense-to-Sparse (DS) matching strategy for aggregating dense angles corresponding to sparse objects. The resized and rot/flp views are switched using a progressive multi-view switching strategy during training to achieve coupled optimization of scale and angle. Experimental results on the DIOR-R and DOTA-v1.0 datasets demonstrate that PointOBB achieves promising performance, and significantly outperforms potential point-supervised baselines.

Single image 3D reconstruction is an important but challenging task that requires extensive knowledge of our natural world. Many existing methods solve this problem by optimizing a neural radiance field under the guidance of 2D diffusion models but suffer from lengthy optimization time, 3D inconsistency results, and poor geometry. In this work, we propose a novel method that takes a single image of any object as input and generates a full 360-degree 3D textured mesh in a single feed-forward pass. Given a single image, we first use a view-conditioned 2D diffusion model, Zero123, to generate multi-view images for the input view, and then aim to lift them up to 3D space. Since traditional reconstruction methods struggle with inconsistent multi-view predictions, we build our 3D reconstruction module upon an SDF-based generalizable neural surface reconstruction method and propose several critical training strategies to enable the reconstruction of 360-degree meshes. Without costly optimizations, our method reconstructs 3D shapes in significantly less time than existing methods. Moreover, our method favors better geometry, generates more 3D consistent results, and adheres more closely to the input image. We evaluate our approach on both synthetic data and in-the-wild images and demonstrate its superiority in terms of both mesh quality and runtime. In addition, our approach can seamlessly support the text-to-3D task by integrating with off-the-shelf text-to-image diffusion models.

Although various 3D datasets with different functions and scales have been proposed recently, it remains challenging for individuals to complete the whole pipeline of large-scale data collection, sanitization, and annotation. Moreover, the created datasets usually suffer from extremely imbalanced class distribution or partial low-quality data samples. Motivated by this, we explore the procedurally synthetic 3D data generation paradigm to equip individuals with the full capability of creating large-scale annotated photogrammetry point clouds. Specifically, we introduce a synthetic aerial photogrammetry point clouds generation pipeline that takes full advantage of open geospatial data sources and off-the-shelf commercial packages. Unlike generating synthetic data in virtual games, where the simulated data usually have limited gaming environments created by artists, the proposed pipeline simulates the reconstruction process of the real environment by following the same UAV flight pattern on different synthetic terrain shapes and building densities, which ensure similar quality, noise pattern, and diversity with real data. In addition, the precise semantic and instance annotations can be generated fully automatically, avoiding the expensive and time-consuming manual annotation. Based on the proposed pipeline, we present a richly-annotated synthetic 3D aerial photogrammetry point cloud dataset, termed STPLS3D, with more than 16 km^2 of landscapes and up to 18 fine-grained semantic categories. For verification purposes, we also provide a parallel dataset collected from four areas in the real environment. Extensive experiments conducted on our datasets demonstrate the effectiveness and quality of the proposed synthetic dataset.

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

This paper studies implicit surface reconstruction leveraging differentiable ray casting. Previous works such as IDR and NeuS overlook the spatial context in 3D space when predicting and rendering the surface, thereby may fail to capture sharp local topologies such as small holes and structures. To mitigate the limitation, we propose a flexible neural implicit representation leveraging hierarchical voxel grids, namely Neural Deformable Anchor (NeuDA), for high-fidelity surface reconstruction. NeuDA maintains the hierarchical anchor grids where each vertex stores a 3D position (or anchor) instead of the direct embedding (or feature). We optimize the anchor grids such that different local geometry structures can be adaptively encoded. Besides, we dig into the frequency encoding strategies and introduce a simple hierarchical positional encoding method for the hierarchical anchor structure to flexibly exploit the properties of high-frequency and low-frequency geometry and appearance. Experiments on both the DTU and BlendedMVS datasets demonstrate that NeuDA can produce promising mesh surfaces.

Empirically validating new 3D-printing related algorithms and implementations requires testing data representative of inputs encountered in the wild. An ideal benchmarking dataset should not only draw from the same distribution of shapes people print in terms of class (e.g., toys, mechanisms, jewelry), representation type (e.g., triangle soup meshes) and complexity (e.g., number of facets), but should also capture problems and artifacts endemic to 3D printing models (e.g., self-intersections, non-manifoldness). We observe that the contextual and geometric characteristics of 3D printing models differ significantly from those used for computer graphics applications, not to mention standard models (e.g., Stanford bunny, Armadillo, Fertility). We present a new dataset of 10,000 models collected from an online 3D printing model-sharing database. Via analysis of both geometric (e.g., triangle aspect ratios, manifoldness) and contextual (e.g., licenses, tags, classes) characteristics, we demonstrate that this dataset represents a more concise summary of real-world models used for 3D printing compared to existing datasets. To facilitate future research endeavors, we also present an online query interface to select subsets of the dataset according to project-specific characteristics. The complete dataset and per-model statistical data are freely available to the public.

We consider the problem of constructing small coresets for k-Median in Euclidean spaces. Given a large set of data points Psubset R^d, a coreset is a much smaller set Ssubset R^d, so that the k-Median costs of any k centers w.r.t. P and S are close. Existing literature mainly focuses on the high-dimension case and there has been great success in obtaining dimension-independent bounds, whereas the case for small d is largely unexplored. Considering many applications of Euclidean clustering algorithms are in small dimensions and the lack of systematic studies in the current literature, this paper investigates coresets for k-Median in small dimensions. For small d, a natural question is whether existing near-optimal dimension-independent bounds can be significantly improved. We provide affirmative answers to this question for a range of parameters. Moreover, new lower bound results are also proved, which are the highest for small d. In particular, we completely settle the coreset size bound for 1-d k-Median (up to log factors). Interestingly, our results imply a strong separation between 1-d 1-Median and 1-d 2-Median. As far as we know, this is the first such separation between k=1 and k=2 in any dimension.

A key step in any scanning-based asset creation workflow is to convert unordered point clouds to a surface. Classical methods (e.g., Poisson reconstruction) start to degrade in the presence of noisy and partial scans. Hence, deep learning based methods have recently been proposed to produce complete surfaces, even from partial scans. However, such data-driven methods struggle to generalize to new shapes with large geometric and topological variations. We present Points2Surf, a novel patch-based learning framework that produces accurate surfaces directly from raw scans without normals. Learning a prior over a combination of detailed local patches and coarse global information improves generalization performance and reconstruction accuracy. Our extensive comparison on both synthetic and real data demonstrates a clear advantage of our method over state-of-the-art alternatives on previously unseen classes (on average, Points2Surf brings down reconstruction error by 30\% over SPR and by 270\%+ over deep learning based SotA methods) at the cost of longer computation times and a slight increase in small-scale topological noise in some cases. Our source code, pre-trained model, and dataset are available on: https://github.com/ErlerPhilipp/points2surf

The increasing demand for virtual reality applications has highlighted the significance of crafting immersive 3D assets. We present a text-to-3D 360^{circ} scene generation pipeline that facilitates the creation of comprehensive 360^{circ} scenes for in-the-wild environments in a matter of minutes. Our approach utilizes the generative power of a 2D diffusion model and prompt self-refinement to create a high-quality and globally coherent panoramic image. This image acts as a preliminary "flat" (2D) scene representation. Subsequently, it is lifted into 3D Gaussians, employing splatting techniques to enable real-time exploration. To produce consistent 3D geometry, our pipeline constructs a spatially coherent structure by aligning the 2D monocular depth into a globally optimized point cloud. This point cloud serves as the initial state for the centroids of 3D Gaussians. In order to address invisible issues inherent in single-view inputs, we impose semantic and geometric constraints on both synthesized and input camera views as regularizations. These guide the optimization of Gaussians, aiding in the reconstruction of unseen regions. In summary, our method offers a globally consistent 3D scene within a 360^{circ} perspective, providing an enhanced immersive experience over existing techniques. Project website at: http://dreamscene360.github.io/

Recent advancements in 3D reconstruction from single images have been driven by the evolution of generative models. Prominent among these are methods based on Score Distillation Sampling (SDS) and the adaptation of diffusion models in the 3D domain. Despite their progress, these techniques often face limitations due to slow optimization or rendering processes, leading to extensive training and optimization times. In this paper, we introduce a novel approach for single-view reconstruction that efficiently generates a 3D model from a single image via feed-forward inference. Our method utilizes two transformer-based networks, namely a point decoder and a triplane decoder, to reconstruct 3D objects using a hybrid Triplane-Gaussian intermediate representation. This hybrid representation strikes a balance, achieving a faster rendering speed compared to implicit representations while simultaneously delivering superior rendering quality than explicit representations. The point decoder is designed for generating point clouds from single images, offering an explicit representation which is then utilized by the triplane decoder to query Gaussian features for each point. This design choice addresses the challenges associated with directly regressing explicit 3D Gaussian attributes characterized by their non-structural nature. Subsequently, the 3D Gaussians are decoded by an MLP to enable rapid rendering through splatting. Both decoders are built upon a scalable, transformer-based architecture and have been efficiently trained on large-scale 3D datasets. The evaluations conducted on both synthetic datasets and real-world images demonstrate that our method not only achieves higher quality but also ensures a faster runtime in comparison to previous state-of-the-art techniques. Please see our project page at https://zouzx.github.io/TriplaneGaussian/.

We consider using deep neural networks to solve time-dependent partial differential equations (PDEs), where multi-scale processing is crucial for modeling complex, time-evolving dynamics. While the U-Net architecture with skip connections is commonly used by prior studies to enable multi-scale processing, our analysis shows that the need for features to evolve across layers results in temporally misaligned features in skip connections, which limits the model's performance. To address this limitation, we propose SineNet, consisting of multiple sequentially connected U-shaped network blocks, referred to as waves. In SineNet, high-resolution features are evolved progressively through multiple stages, thereby reducing the amount of misalignment within each stage. We furthermore analyze the role of skip connections in enabling both parallel and sequential processing of multi-scale information. Our method is rigorously tested on multiple PDE datasets, including the Navier-Stokes equations and shallow water equations, showcasing the advantages of our proposed approach over conventional U-Nets with a comparable parameter budget. We further demonstrate that increasing the number of waves in SineNet while maintaining the same number of parameters leads to a monotonically improved performance. The results highlight the effectiveness of SineNet and the potential of our approach in advancing the state-of-the-art in neural PDE solver design. Our code is available as part of AIRS (https://github.com/divelab/AIRS).

3D understanding and rendering of moving humans from monocular videos is a challenging task. Despite recent progress, the task remains difficult in real-world scenarios, where obstacles may block the camera view and cause partial occlusions in the captured videos. Existing methods cannot handle such defects due to two reasons. First, the standard rendering strategy relies on point-point mapping, which could lead to dramatic disparities between the visible and occluded areas of the body. Second, the naive direct regression approach does not consider any feasibility criteria (ie, prior information) for rendering under occlusions. To tackle the above drawbacks, we present OccNeRF, a neural rendering method that achieves better rendering of humans in severely occluded scenes. As direct solutions to the two drawbacks, we propose surface-based rendering by integrating geometry and visibility priors. We validate our method on both simulated and real-world occlusions and demonstrate our method's superiority.

Computer-aided design (CAD) significantly enhances the efficiency, accuracy, and innovation of design processes by enabling precise 2D and 3D modeling, extensive analysis, and optimization. Existing methods for creating CAD models rely on latent vectors or point clouds, which are difficult to obtain and costly to store. Recent advances in Multimodal Large Language Models (MLLMs) have inspired researchers to use natural language instructions and images for CAD model construction. However, these models still struggle with inferring accurate 3D spatial location and orientation, leading to inaccuracies in determining the spatial 3D starting points and extrusion directions for constructing geometries. This work introduces CAD-GPT, a CAD synthesis method with spatial reasoning-enhanced MLLM that takes either a single image or a textual description as input. To achieve precise spatial inference, our approach introduces a 3D Modeling Spatial Mechanism. This method maps 3D spatial positions and 3D sketch plane rotation angles into a 1D linguistic feature space using a specialized spatial unfolding mechanism, while discretizing 2D sketch coordinates into an appropriate planar space to enable precise determination of spatial starting position, sketch orientation, and 2D sketch coordinate translations. Extensive experiments demonstrate that CAD-GPT consistently outperforms existing state-of-the-art methods in CAD model synthesis, both quantitatively and qualitatively.

This work presents DAVINCI, a unified architecture for single-stage Computer-Aided Design (CAD) sketch parameterization and constraint inference directly from raster sketch images. By jointly learning both outputs, DAVINCI minimizes error accumulation and enhances the performance of constrained CAD sketch inference. Notably, DAVINCI achieves state-of-the-art results on the large-scale SketchGraphs dataset, demonstrating effectiveness on both precise and hand-drawn raster CAD sketches. To reduce DAVINCI's reliance on large-scale annotated datasets, we explore the efficacy of CAD sketch augmentations. We introduce Constraint-Preserving Transformations (CPTs), i.e. random permutations of the parametric primitives of a CAD sketch that preserve its constraints. This data augmentation strategy allows DAVINCI to achieve reasonable performance when trained with only 0.1% of the SketchGraphs dataset. Furthermore, this work contributes a new version of SketchGraphs, augmented with CPTs. The newly introduced CPTSketchGraphs dataset includes 80 million CPT-augmented sketches, thus providing a rich resource for future research in the CAD sketch domain.

We propose a novel point-based representation, Gaussian surfels, to combine the advantages of the flexible optimization procedure in 3D Gaussian points and the surface alignment property of surfels. This is achieved by directly setting the z-scale of 3D Gaussian points to 0, effectively flattening the original 3D ellipsoid into a 2D ellipse. Such a design provides clear guidance to the optimizer. By treating the local z-axis as the normal direction, it greatly improves optimization stability and surface alignment. While the derivatives to the local z-axis computed from the covariance matrix are zero in this setting, we design a self-supervised normal-depth consistency loss to remedy this issue. Monocular normal priors and foreground masks are incorporated to enhance the quality of the reconstruction, mitigating issues related to highlights and background. We propose a volumetric cutting method to aggregate the information of Gaussian surfels so as to remove erroneous points in depth maps generated by alpha blending. Finally, we apply screened Poisson reconstruction method to the fused depth maps to extract the surface mesh. Experimental results show that our method demonstrates superior performance in surface reconstruction compared to state-of-the-art neural volume rendering and point-based rendering methods.

The large abundance of perspective camera datasets facilitated the emergence of novel learning-based strategies for various tasks, such as camera localization, single image depth estimation, or view synthesis. However, panoramic or omnidirectional image datasets, including essential information, such as pose and depth, are mostly made with synthetic scenes. In this work, we introduce a large scale 360^{circ} videos dataset in the wild. This dataset has been carefully scraped from the Internet and has been captured from various locations worldwide. Hence, this dataset exhibits very diversified environments (e.g., indoor and outdoor) and contexts (e.g., with and without moving objects). Each of the 25K images constituting our dataset is provided with its respective camera's pose and depth map. We illustrate the relevance of our dataset for two main tasks, namely, single image depth estimation and view synthesis.

Geometric Deep Learning has recently made striking progress with the advent of continuous Deep Implicit Fields. They allow for detailed modeling of watertight surfaces of arbitrary topology while not relying on a 3D Euclidean grid, resulting in a learnable parameterization that is not limited in resolution. Unfortunately, these methods are often not suitable for applications that require an explicit mesh-based surface representation because converting an implicit field to such a representation relies on the Marching Cubes algorithm, which cannot be differentiated with respect to the underlying implicit field. In this work, we remove this limitation and introduce a differentiable way to produce explicit surface mesh representations from Deep Signed Distance Functions. Our key insight is that by reasoning on how implicit field perturbations impact local surface geometry, one can ultimately differentiate the 3D location of surface samples with respect to the underlying deep implicit field. We exploit this to define MeshSDF, an end-to-end differentiable mesh representation which can vary its topology. We use two different applications to validate our theoretical insight: Single-View Reconstruction via Differentiable Rendering and Physically-Driven Shape Optimization. In both cases our differentiable parameterization gives us an edge over state-of-the-art algorithms.