MDNF: Multi-Diffusion-Nets for Neural Fields on Meshes
Avigail Cohen Rimon Tal Shnitzer Mirela Ben Chen

*
    Axel Klawonn0000-0003-4765-7387 
 Martin Lanser0000-0002-4232-9395
    
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§ ABSTRACT 
We propose a novel framework for representing neural fields on triangle meshes that is multi-resolution across both spatial and frequency domains.
Inspired by the Neural Fourier Filter Bank (NFFB), our architecture decomposes the spatial and frequency domains by associating finer spatial resolution levels with higher frequency bands, while coarser resolutions are mapped to lower frequencies.
To achieve geometry-aware spatial decomposition we leverage multiple DiffusionNet components, each associated with a different spatial resolution level.
Subsequently, we apply a Fourier feature mapping to encourage finer resolution levels to be associated with higher frequencies.
The final signal is composed in a wavelet-inspired manner using a sine-activated MLP, aggregating higher-frequency signals on top of lower-frequency ones.
Our architecture attains high accuracy in learning complex neural fields and is robust to discontinuities, exponential scale variations of the target field, and mesh modification.
We demonstrate the effectiveness of our approach through its application to diverse neural fields, such as synthetic RGB functions, UV texture coordinates, and vertex normals, illustrating different challenges.
To validate our method, we compare its performance against two alternatives, showcasing the advantages of our multi-resolution architecture.



Keywords: Triangle Meshes · Neural Fields · Multi-Resolution





§ INTRODUCTION

Recent advancements in machine learning have lead to a surge of interest in solving visual computing problems using coordinate-based neural networks, known as Neural fields. These networks parameterize the physical properties of scenes or objects across spatial and temporal dimensions. Neural fields have gained widespread adoption due to their ability to encode continuous signals over arbitrary dimensions at high resolutions, enabling accurate, high-fidelity, and expressive solutions <cit.>.
They have demonstrated remarkable success in a variety of tasks, including animation of human bodies <cit.>, mesh smoothing and deformations <cit.>, novel view synthesis <cit.>, mesh geometry and texture editing <cit.>, 3D reconstruction <cit.>, textured 3D reconstruction from images <cit.>, shape representation and completion <cit.>, and neural stylization of meshes <cit.>. 

Despite their widespread success, these coordinate-based neural architectures remain vulnerable to spectral bias <cit.> and demand significant computational resources.
Among other generalizations, these shortcomings have been addressed through spatial decomposition strategies using grids <cit.>, which support rapid training and level of detail control.
Additionally, techniques that encode input data using high-dimensional features through frequency transformations, such as sinusoidal representations <cit.>, help mitigate the inherent low-frequency bias of neural fields <cit.>.


Wu et al. wu2023neural propose an architecture that bridges these two approaches. They demonstrate that employing different grid resolutions focused on distinct frequency components, combined with proper localization, achieves state-of-the-art performance in terms of model compactness and convergence speed across multiple tasks.
However, the proposed grid-based methods, including <cit.>, are designed for Euclidean spaces and do not account for the unique properties of non-Euclidean, irregular geometric domains like triangle meshes. Although adapting such methods to fit such structures via data modification has shown efficacy, it often overlooks the inherent characteristics of mesh data.
Notably, meshes typically represent smooth manifolds with defined geometry, offering potential for enhanced understanding and representation.
Furthermore, we aim to enhance the architecture's invariance to the multi-representational nature of mesh geometry, accommodating different resolutions and many equivalent vertex connectivities. 




In this work, we introduce a novel geometry-aware framework for representing neural fields on triangle meshes that are multi-resolution across both spatial and frequency domains. Inspired by the Neural Fourier Filter Bank (NFFB) <cit.> and leveraging the geometry-aware DiffusionNet architecture <cit.>, our approach decomposes the spatial and frequency domains using multiple DiffusionNet components representing different spatial resolutions and controlling frequency bands using Fourier feature mappings at different scales.
We associate finer spatial resolution levels with higher frequency bands, while coarser resolutions are mapped to lower frequencies. 
This wavelet-inspired decomposition, combined with a carefully designed network architecture, enables our method to effectively learn and represent complex neural fields, accurately capturing intricate details and frequency variations.
We demonstrate the efficacy of our approach through its application to diverse neural fields, including synthetic RGB functions, UV texture coordinates, and vertex normals, showcasing its robustness to discontinuities, exponential scale variations, and mesh modification.







 §.§ Related Work

We highlight relevant work that is related to the key components of our method: architectures for learning on meshes that leverage mesh geometry and neural fields on non-Euclidean domains.



  
Learning on Meshes 
Several works have proposed unique architectures to leverage mesh geometry and other structural properties for learning on meshes <cit.>. 
Hanocka et al. hanocka2019meshcnn defined MeshCNN, a convolutional layer on meshes by learning edge features and defining pooling operations through edge collapse. Milano et al. milano2020primal captures triangle adjacency in meshes through graphs of mesh edges and dual edges, and Smirnov et al. smirnov2021hodgenet learns spectral geometry elements to construct custom mesh features. DiffusionNet <cit.> leverages the heat equation and learns multiscale diffusion operations to propagate information across the manifold.
These architectures commonly focus on segmentation, classification and correspondence learning tasks. While they form the basic architecture of our work, they are typically incapable of capturing subtle differences in multiple resolutions, as demonstrated in the experimental results, Section <ref>. 


  
Neural Fields
Neural fields have been increasingly used for learning functional representations in arbitrary resolutions, most commonly for Euclidean domains, e.g. <cit.>. The foundational work, NeRF <cit.>, a coordinate-based neural network for view synthesis, demonstrated the importance of positional encodings to facilitate learning of high frequency data by neural networks. 
Subsequent works have used periodic activation functions <cit.>, wavelet-like multi-resolution decomposition <cit.>, and a decomposition to a cascade of band-limited neural fields <cit.>.
These works focus on Euclidean spaces, encoding non-Euclidean 2D manifolds as volumes, resulting in higher computational costs or failure to capture the manifold structure. 


  
Neural Fields on Manifolds
Recently, a few works have proposed methods for learning neural fields on non-Euclidean domains <cit.>.
Bensaïd et al. bensaid2023partial leverages neural fields for learning partial matching of nonrigid shapes. They use intrinsic positional encodings and a neural representation in the spectral domain to interpolate between matched sparse landmarks of partial shapes. 
NeuTex <cit.> represents meshes as 3D volume in a Euclidean space, but encodes texture with a 2D network. To enable texture representation and editing, they train mapping networks between the two spaces, which can be seen as learning a representation of the 2D surface.
Koestler et al. koestler2022intrinsic takes into account the manifold structure by using the eigenfunctions of the Laplace-Beltrami Operator as positional encodings, serving as point embeddings in the input of the trained neural network. 
This approach is conceptually similar to the concatenation of DiffusionNet <cit.> and a Multi-Layer Perceptron (MLP), and we compare our method to such an architecture in Section <ref>.
<cit.> further extends this concept and learns a continuous field, independent of the manifold discretization, by mapping a series of mesh poses to an implicit canonical representation and learning surface deformations fields for each pose. This approach requires a series of related inputs, such as a human mesh in different poses.



 §.§ Contribution

To summarize, our contributions are as follows:


  * We propose a novel geometry-aware framework for neural fields representation on triangle meshes that is multi-resolution across both spatial and frequency domains.


  * We show that our method attains high precision in learning diverse neural fields, such as synthetic RGB functions, UV texture coordinates, and vertex normals, illustrating different challenges.         

  * We show that our method outperforms DiffusionNet in learning high-detail functions. Moreover, we provide an ablation study comparing our model against a single-resolution variant, demonstrating the efficacy of our multi-resolution approach.




§ BACKGROUND

Our architecture draws inspiration from the architecture proposed by Wu et al. wu2023neural, and builds upon two main works: DiffusionNet <cit.> and Fourier feature mapping <cit.>. For completeness, we provide here a brief review of these works.



 §.§ DiffusionNet


DiffusionNet <cit.> is a discretization agnostic architecture for deep learning on surfaces.
The architecture consists of successive identical DiffusionNet blocks. A central feature of each DiffusionNet block is the use of learned diffusion based on the heat equation to propagate information across the surface.
This diffusion is discretized via the Laplacian 𝐋 and mass M matrices of the surface. In our work, we use the cotangent-Laplace matrix, which is ubiquitous in geometry processing applications <cit.>.

For efficient diffusion computation, the authors propose to use a spectral method that utilizes the k smallest eigenpairs ϕ_i, λ_i of the generalized eigenvalue problem Lϕ_i = λ_i Mϕ_i.
The diffusion layer h_t(u) corresponding to time t is implemented by projecting the input feature channel u onto this truncated basis, exponentially scaling the coefficients by -λ_i t, and projecting back:

    h_t(u) := Φ[ e^-λ_1 t; e^-λ_2 t;        ⋮ ]⊙ (Φ^T 𝐌u)

where ⊙ denotes the Hadamard (elementwise) product and Φ, Λ are the matrices of generalized eigenvectors and eigenvalues, respectively.
The learned diffusion times, optimized per feature channel, control the spatial support ranging from purely local to totally global. Here, we have briefly reviewed only the aspects critical for understanding our method; see <cit.> for further details.


 §.§ Fourier Feature Mapping

The work by Tancik et al. tancik2020fourier addresses the problem of "spectral bias" in coordinate-based multi-layer perceptrons (MLPs), which refers to their inherent limitation in accurately modeling high-frequency components due to the rapid decay of eigenvalues in their neural tangent kernels (NTKs) <cit.>.
The authors propose using a Fourier feature mapping that applies a non-linear transformation to the input coordinates before passing them to the MLP.
They report a Gaussian mapping as most effective, where input coordinates are multiplied by random Gaussian matrices to produce high-dimensional Fourier features.
Theoretically, they show that this mapping transforms the NTK into a stationary kernel with a tunable bandwidth. This bandwidth, which determines the width of the kernel’s effective frequency spectrum, is controlled by the scale (standard deviation) of the Gaussian matrices.
A larger scale allows representing higher frequencies, overcoming spectral bias. 



 §.§ Neural Fourier Filter Bank (NFFB)

Wu et al. wu2023neural introduce a novel neural field framework named "Neural Fourier Filter Bank" (NFFB) that decomposes the target signal jointly in the spatial and frequency domains, inspired by wavelet decomposition <cit.>.
The core idea is to utilize multi-layer perceptrons (MLPs) to implement a low-pass filter by leveraging their inherent frequency bias, and to employ grid features at varying spatial resolutions alongside Fourier feature mappings <cit.> at different scales to create a high-pass filter.
A novel aspect of this framework is the correlation of finer spatial resolutions with higher frequency bands, whereas coarser resolutions correspond to lower frequencies. Fourier feature mappings are applied at scales that match the respective spatial resolutions of each grid feature.
The proposed architecture feeds the high-frequency components into sine-activated MLP layers at appropriate depths, mimicking the sequential accumulation of higher frequencies on top of lower frequencies in wavelet filter banks.
This wavelet-inspired decomposition into spatial and frequency components, coupled with the association of specific resolutions with corresponding frequencies, allows for efficient learning of detailed signals while maintaining model compactness and fast convergence.


§ METHOD




We propose a multi-resolution framework that facilitates representing neural fields on meshes across both spatial and frequency domains. As illustrated in Figure. <ref>, our pipeline comprises three key stages: (1) Diffusing features across mesh vertices via multiple DiffusionNet components (Section <ref>) to capture spatial variations, (2) Transforming these diffused features through Fourier feature mapping (Section <ref>) to associate different frequency bands with the respective resolution levels, and (3) Composing the multi-resolution, multi-frequency signal representation using a sine-activated MLP in a wavelet-inspired manner (Section <ref>).
We delve into the details of each stage in the following sections.

r[-5pt]0.13


    
    < g r a p h i c s >

    *Illustrative example. Partitioning of mesh vertices.



To clarify our architecture's pipeline, we use a synthetic example featuring the Chinese dragon mesh with 125K vertices and 250K faces. The mesh is divided into three groups - Red, Yellowish, and Blue - each linked to a distinct function depicted in the inset figure. These groups represent increasing frequencies: Red corresponds to ϕ_1, Yellowish to ϕ_125, and Blue to ϕ_500, where ϕ_j ∈ℝ^n is the j-th eigenfunction of the Laplace-Beltrami operator on the mesh, and n is the number of vertices. We generate the target neural field by mapping the patchwork function to an RGB using the HSV colormap.
More details can be found in Subsection <ref>.



 §.§ DiffusionNet Layers




  
Motivation


Aligned with the strategy in <cit.>, our pipeline's first stage inputs features into a multi-component "layer," each component associated with a different resolution band, representing varying spatial resolutions of mesh features. Unlike NFFB <cit.>, our architecture replaces each hash grid with a DiffusionNet component. As discussed in Section <ref>, DiffusionNet utilizes diffusion layers to facilitate spatial communication and optimizes diffusion support for each feature channel.




The choice of adopting the DiffusionNet architecture is based on two main reasons.
The first stems from its inherent compatibility with irregular data structures, specifically triangle meshes, as opposed to hash grid structures suited for regular formats like images. Despite previous adaptations of triangle meshes to regular-structured architectures yielding favorable results, a geometry-aware approach like DiffusionNet has proven more accurate and efficient in various applications. Due to that, substituting the grid representation with Diffusionnet is particularly effective for mesh data. Additionally,  DiffusionNet facilitates discretization-agnostic learning, enhancing generalization capabilities of the overall architecture.





Second, DiffusionNet intrinsically facilitates a methodology akin to the multi-resolution hash-grid paradigm of NFFB. The diffusion time parameter can be utilized to adjust spatial resolutions via the initial values assigned to each component. Furthermore, employing the "spectral method" for the diffusion process enables each component to be associated with a distinct set of eigenvectors, enhancing their spatial resolutions as well.

Formally, the DiffusionNet component at the i-th level, δ_i, maps the input 3D coordinate x∈ℝ^3 to an F-dimensional feature space:
δ_i: ℝ^3 →ℝ^F. 
Let N denote the number of DiffusionNet components. 


  
Splitting the spectrum

Considering the total number of eigenvectors k_eig used for diffusion, we distribute the eigenvectors evenly across the levels, associating the eigenvectors corresponding to the lowest eigenvalues with level 1 and highest to level N.

For each level i ∈ [1, N], we define the range of eigenvector indices used for diffusion in the i-th DiffusionNet component as [r_m(i), r_M(i)] where

    r := linspace(0, k_eig, N + 1)   
     
    r_m(i) := r(i)   r_M(i) := r(i+1)

where linspace(start, end, steps) is a one-dimensional vector of size steps whose values are evenly spaced from start to end, inclusive.

The corresponding sets of eigenvectors Ψ_i and eigenvalues Λ_i used for diffusion at the i-th level are:

    Ψ_i := {ϕ_j}_j=r_m(i)^j=r_M(i)    Λ_i := {λ_j}_j=r_m(i)^j=r_M(i)




  
Splitting diffusion time
Recall the diffusion time parameter in DiffusionNet controls the spatial resolution of diffusion, theoretically ranging from local to global scales. However, in practice, such range isn't fully realized, as shown in Sec <ref>.
The diffusion process, as implemented by the "spectral method", acts as a low-pass filter due to the exponentiation e^-λ_j t, where t is the diffusion time and λ_j the j-th Laplacian eigenvalue. By creating multiple DiffusionNets, each with distinct eigenvalue ranges, we achieve a refined representation of high-frequency components.

Following the initialization scheme considered in Wu et al wu2023neural for the Gaussian distribution
variance (as in our Equation (<ref>)), we initialize the diffusion times of  the i-th DiffusionNet component by t(i) defined as

    t(i) := t_base· (t_exp)^i
 
Typically t_base is set to the squared mean edge length of the mesh.



Figure <ref> illustrates the output features of each level at the key pipeline stages, displaying the smoothest and least smooth feature channels per level and stage. See Supplemental Material Section 2 for measuring function smoothness.
The first row shows d_i := δ_i(x) for i∈[1,2,3]. Since we set F=1 for simplification, only one feature is output at this stage. We observe that the functions at the three levels in this stage exhibit smooth behavior. 





















 §.§ Fourier Feature mapping


As in NFFB, the Fourier feature mapping stage serves to associate each level of the multi-resolution representation with a distinct frequency band. Inspired by the Fourier feature mapping approach of <cit.>, we apply a sinusoidal transformation to the output features from the previous DiffusionNet stage. 

In more details, the Fourier feature at the i-th level is defined as a mapping from the DiffusionNet output features at the i-th level d_i ∈ℝ^F to an m-dimensional feature space:

    η_i(d_i) := [ sin(2π·d_i^T ·B_i,1), …, sin(2π·d_i^T ·B_i,m )]^T

where B_i,1, B_i,2, …, B_i,m are trainable parameters in ℝ^F forming the frequency transform coefficients on the i-th level, and m is a hyper-parameter.


The frequency ranges for each level are defined by the initialization of the B_i,j coefficients. Drawing on the Gaussian random Fourier feature mapping by Tancik et al. tancik2020fourier, we set these coefficients using a Gaussian distribution with a mean of 0 and a level-specific variance σ_i (Equation (<ref>)).
Finer resolutions levels, associated with higher frequencies, are initialized with greater variance, biasing them towards encoding higher frequency signal components. This adaptive initialization approach allows each resolution level to naturally associate with specific frequency bands without pre-setting fixed ranges.

Practically, let σ_base, σ_exp∈ℝ, we initialize the i-th level coefficients with variance σ_i ∈ℝ defined by

    σ_i := σ_base· (σ_exp)^i

where σ_base, σ_exp are hyper-parameters, and σ_exp≥ 1.

Referring again to Figure <ref>, the second row depicts the output features η_i(d_i) for i∈[1,2,3].
We observe that the frequency of the least smooth feature increases as the level increases.



 §.§ Composing the final output







The next stage composes the final output using the Fourier transformed features η_i(d_i) ∈ℝ^m across levels i ∈ [1, N].
Two critical observations from <cit.> inform this process: First, features across levels are not necessarily orthogonal, calling for learned layers for optimal combination and mitigation of non-orthogonality. Second, implementing residual connections helps aggregate and joint update of the multi-resolution features, maintaining consistent processing depth across all levels in the network.

We thus start by applying a sine-activated MLP <cit.> that takes in the Fourier features η_i(d_i) in a manner that sequentially accumulates higher-frequency components on top of lower-frequency components.

More formally, let us denote the i-th layer as L_i where i ∈ [1, N], using f_i to represent the output of L_i, and h_i to denote the combination of the i-th layer output with the next level's features:

    f_i := sin (α_i ·𝐖_i ·h_i-1 + 𝐛_i),   h_i := f_i + η_i(𝐝_i)

where for ease of notation we define h_0 := 𝐱.
Here, 𝐖_i ∈ℝ^m × m and 𝐛_i ∈ℝ^m are the trainable weight and bias parameters in layer L_i, and α_i is analogous to the w_0 hyperparameter in SIREN <cit.>, acting as a frequency scaling factor that allows controlling the frequency band that this level focuses on representing.

Next, as illustrated in Figure <ref>, we establish residual connections by concatenating the outputs h_i ∈ℝ^m from each level and passing them through an additional MLP with ReLU activations, while also transferring them to the subsequent layer L_i+1 as described earlier.
Alternatively, as suggested in <cit.>, instead of concatenating {h_i}_i=1^N, one could pass each feature through a per-level linear layer O_i and sum the outputs to obtain the final feature representation. We refer to <cit.> for further details.

The third row in Figure <ref>, representing the output features h_i for i ∈ [1,2,3], exhibits features that are significantly less smooth than those in previous stages.
Further, in this stage we can see that for both the smoothest and least smooth features, higher levels correspond to increasingly noisier features.








To gain insight into the resolutions levels learned by our trained network, Figure <ref> depicts the output neural field representing the RGB function during evaluation, with either the first or last level disabled. Disabling level N generally results in a blurry output, reflecting lower frequency components, whereas disabling level 1 produces a high-contrast function tied to higher frequencies. We analyze this by zooming in on three areas across the ground truth (GT), our model’s predicted function, and outputs with level N or 1 disabled. The output from disabling level N is notably blurrier, capturing only basic texture outlines compared to the GT. Conversely, disabling level 1 enhances contrast, often exaggerating transitions. For instance, areas that are light blue in the GT and surrounded by similar light green regions appear as a more distinct blue, despite their subtle difference in hue in the GT.




 §.§ Implementation Details

The implementation details such as the loss function, number of training epochs, and network size are adjusted for each experiment.
Outside of these customized components, the overall training setup remains consistent across all experiments.
We implement our method in PyTorch <cit.>, and utilize the Adam optimizer <cit.> with the default settings of β_1 = 0.9 and β_2 = 0.99.
The learning rate is set to 10^-4, and it is reduced by a factor of 0.7 every 700 iterations. We set the output dimension of DiffusionNet components as F=2, and the maximal index of the Laplacian eigenpair considered in the diffusion process, k_eig, is set to 500.
We run all experiments on a single NVIDIA A40 GPU.
For brevity, only the essential details are presented here; for a detailed description of the hyperparameters, see the Supplemental Material.



§ EXPERIMENTAL RESULTS




We evaluate our method on three neural fields: synthetic RGB function (Section <ref>), UV texture coordinates (Section <ref>), and vertex normals (Section <ref>).
Focusing on demonstrating our architecture, all models were trained using a supervised approach. 


We compare against two baselines: a single DiffusionNet component, and our method with N=1, denoted as the One-Level model.
We refer to our method as the N-Level model where N>1, with N determined empirically for each experiment.
For all models, we report the results of the best-performing configuration.



 §.§ Synthetic Example


To illustrate the effectiveness of our method, we start with a synthetic example, resembling the one in the Section <ref>.
In this experiment we define the target field to be a 3-channel RGB function y_rgb∈ℝ^n × 3 defined on a mesh, where n denotes the number of mesh vertices.
We train the model by minimizing the mean square error (MSE), hence our loss function for this task is

    ℒ_rgb(y) := 1/n‖y - y_rgb‖_2^2


Data.
We demonstrate this example using the Chinese lion mesh, composed of 50K vertices.



  
Neural Field Generation








To generate the function y_rgb, we first partition the mesh vertices into three groups based on their x coordinates, as visualized in Figure <ref>. We denote the Red, Yellowish, and Blue, groups by group_1, group_2, and group_3, respectively.
We assign to each group a scalar function: g_1 := ϕ_1 ∈ℝ^n (constant), g_2 := ϕ_125∈ℝ^n, and g_3 = g_p ∈ℝ^n generated as Perlin noise on the mesh <cit.>.
Note that the frequency of the functions g_i increases with i.
We define a patchwork function q∈ℝ^n on the mesh such that q[group_i] = g_i[group_i]. y_rgb is then derived by mapping q, normalized to [0,1], to a HSV colormap by defining the Hue parameter. See y_rgb in the GT figures in  Figure <ref>. 



  
Results


Figure <ref> shows the error distributions for the fields learned by the three models, clipping errors above 5 × 10^-4 for clearer visualization.
Our N-Level model outperforms the others, exhibiting fewer artifacts. To provide quantitative results as well, Figure <ref> presents the Cumulative Distribution Functions (CDFs) of vertex errors across four groups: all vertices, group_1 (Red), group_2 (Yellowish), and group_3 (Blue), quantifying the percentage of vertices at each error level.
For each vertex v, error is measured as MSE: 1/3‖y(v) - y_rgb(v)‖_2^2. The x-axis represents relative error, calculated by dividing vertex errors by the maximal error across all models for each subset. We note that the N-Level model shows superior performance for all groups except for the Red group, which corresponds to a constant function and is considered a trivial region.


 §.§ Discontinuities and Exponential Scale Variations




We further evaluate our method on neural fields representing UV coordinates of textured meshes, which are typically non-continuous and exhibit exponential scale variations when generated by conformal parameterization. Demonstrating our method's robustness, Figure <ref> shows the learned UV coordinates for a multi-component mesh with highly non-continuous UV coordinates. Figure <ref> presents UV coordinates from a conformal map, highlighting our method's ability to handle exponential scale variations.
As in Section <ref>, we train the model by minimizing the mean square error (MSE), hence our loss function for this task is

    ℒ_uv(y) := 1/n‖y - y_uv‖_2^2

where y_uv∈ℝ^n× 2 defines the UV texture coordinates of vertices.


  §.§.§ Discontinuity of Mesh and UV Coordinates



  
Data In this example, we use a Kangaroo mesh with texture, with a total number of 10K vertices. The geometry of this mesh is composed of multiple connected components.


  
Results
Figure <ref> displays the ground truth texture mesh (b) and error distributions for the three models (c, d, e), with errors above 1 × 10^-5 clipped for visualization.
The bottom row shows the 2D UV coordinates for each model. On the left (a), the CDFs of vertex errors are shown. Notably, DiffusionNet exhibits high errors at distinct features like eyes, nails, and tail tips. In its UV coordinates (g), DiffusionNet tends to squash and distort smaller regions.
The One-Level model, while showing significant errors at the head area, has less squashing than DiffusionNet but has distortions that cause overlaps with other texture components in UV (h).
Conversely, the N-Level model outperforms the others both qualitatively and quantitatively, with minimal distortions in its 2D UV coordinates, closely resembling the ground truth and with the best CDF results.







Figure <ref> compares three UV distortion metrics for each model: area distortion, angle distortion, and the percentage of flipped faces.
Area distortion is measured by the absolute difference from 1 of the ratio between ground truth and predicted triangle areas, with values over 0.2 clipped. Angle distortion involves the absolute difference from 1 of the mean ratio between ground truth and predicted triangle angles, clipping values exceeding 0.01.
The bottom row notes in text the flipped faces percentages: 12% for DiffusionNet, 16% for One-Level, and 1% for the N-Level model.
Overall, the N-Level model outperforms the others across all metrics.


  §.§.§ Exponential Scale Variations





  
Data
In this example, we utilize a truncated lion head mesh with 8K vertices and UV coordinates computed using conformal mapping <cit.>, which leads to exponential scale variations notably between the head and neck areas, see Figure <ref>.


  
Results



Figure <ref> displays the GT texture alongside the results of the three models.
The top row features the textured mesh, while the middle and bottom rows show zoomed-in regions of the texture and UV coordinates, respectively.
The DiffusionNet texture reveals high distortion areas, but both the One-Level and N-Level models accurately capture the UV field.






Figure <ref> evaluates three UV distortion metrics for each model. Area distortion values above 5×10^-3 and angle distortion values above 9×10^-4 are clipped. The bottom row notes flipped face percentages: 2% for DiffusionNet and 0% for both the One-Level and N-Level models.
Although the One-Level and N-Level models show similar performance, the N-Level model still demonstrates the highest accuracy in area and angle distortions.



 §.§ Mesh Generalization



In this experiment, we demonstrate our architecture's ability to generalize across multiple versions of a single mesh. Starting with a base mesh, we generate several subdivided versions via a variant of Loop subdivision, as implemented by MeshLab <cit.>, which avoids adding new vertices if triangle edges are below a specified threshold. We apply subdivision iterations until additional triangles are negligible. Since our base mesh is not overly coarse, not all triangles are subdivided in each iteration. However, before being fed into the network, each mesh is centered and normalized, making triangle additions significantly change the mesh embedding.

We aim to learn the neural field defined by mesh vertex normals, y_n ∈ℝ^n× 3. Focusing on the direction of these normals, we train the model by minimizing the mean cosine distance error <cit.>.
Thus, our loss function is given by:

    ℒ_n(y) := 1/n∑_v( 1 -  ⟨y(v), y_n(v)  ⟩/‖y(v) ‖_2^2·‖y_n(v) ‖_2^2)

where y_n(v) ∈ℝ^3 is the normal vector at vertex v.


  
Data
The base mesh used for the subdivision iterations is the smiling ogre mesh, which comprises of 20K vertices. We then generated five additional subdivided versions, with the largest containing 33K triangles and 65K faces. The dataset was split into training and testing sets; the training set comprises meshes subdivided through [0,1,2,3,4] iterations, and the test set contains only the mesh generated by the 5-th subdivision iteration.




  
Results
Figure <ref> displays the CDFs of vertex errors on the left, and the error distributions of the three models on the right. Both quantitatively and qualitatively, our N-Level model significantly outperforms the other two models.



§ CONCLUSIONS

Our multi-resolution framework shows strong capability in representing neural fields on triangle meshes, achieving high precision across various domains and functions.
Its detailed capture of fine features makes it ideal for high precision tasks in computer graphics, such as UV learning, where a generally low error that suffices for applications such as segmentation is not enough, and one needs to be able to achieve close to machine precision.
This framework can be integrated into architectures addressing applications such as texture reconstruction from images and mesh stylization.
Additionally, it has the potential to serve as an effective feature extractor for various other high-precision tasks in geometry processing.




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