# Structurally guided task decomposition in spatial navigation tasks

Ruiqi He1, Carlos G. Correa 2, Thomas L. Griffiths 3,4, Mark K. Ho5

# Structurally guided task decomposition in spatial navigation tasks

Ruiqi He1, Carlos G. Correa 2, Thomas L. Griffiths 3,4, Mark K. Ho5

###### Abstract

How are people able to plan so efficiently despite limited cognitive
resources? We aimed to answer this question by extending an existing model of
human task decomposition that can explain a wide range of simple planning
problems by adding structure information to the task to facilitate planning in
more complex tasks. The extended model was then applied to a more complex
planning domain of spatial navigation. Our results suggest that our framework
can correctly predict the navigation strategies of the majority of the
participants in an online experiment.

## Introduction

People plan hierarchically—from planning the next holiday to their long-term
career. Previous studies have investigated the principles that guide how
people structure their behavioral hierarchies based on how they can simplify
representational and computational costs (Solway et al. 2014; Correa et al.
2023) in simple and unstructured planning domains. We build on a task
decomposition framework introduced by Correa et al. (2023), which was able to
predict hierarchical behavior in people in graph-structured planning tasks.
While the framework accounts for the complexity of planning, it does not
explain how people can plan so efficiently despite limited cognitive
resources. One explanation could be the incorporation of structural
information about the task to facilitate planning. Binder et al. (2021)
suggest that people exploit visual structure to inform their planning. We thus
extended the framework of task decomposition to include structured information
about the planning task through a heuristic search based on spatial distance
and applied it to a more complex spatial navigation task. We tested our novel
framework on a navigation planning experiment that consists of a large family
of tasks, where participants must choose between two paths to take in order to
solve a maze (see Figure 1). Our results suggest that our framework can
predict the navigation choice of the majority of the participants in an online
experiment.

## Methodology

Our modeling approach builds on the framework of task decomposition by
minimizing the computational cost of planning while maximizing task utility
(Correa et al. 2023). The framework consists of three levels:

### Task decomposition

decomposes a task into subtasks such that overall computational costs are
minimized. We formalize the task as $(\mathcal{S},T,s_{0},g)$, which is
defined by the set of possible states, $\mathcal{S}$; an initial state,
$s_{0}$; a goal state, $g$; and the set of possible state transitions
$T\subseteq\mathcal{S}\times\mathcal{S}$, so that $s$ can transition to
$s^{\prime}$ when $(s,s^{\prime})\in T$. Each subtask $z$ is simply defined as
reaching a subgoal state $z_{sg}$111Note that the subtask is essentially an
option (Sutton, Precup, and Singh 1999) that terminates at the subgoal, making
subtask-level planning a semi-Markov decision process.. In our setting, we
assume the task has been decomposed into a fixed set of subtasks $\mathcal{Z}$
for simplicity, consisting of the two states adjacent to $s_{0}$, a simple
approach for formalizing the choice between the two halves of the maze.

### Subtask-level planning

decides which subtask to choose based on the expected reward and computational
cost of visiting it en route to the goal. The subtask that maximizes this
overall reward is

$z^{*}=\arg\max_{z\in\mathcal{Z}}R_{\texttt{Alg}}(s_{0},z)+R_{\texttt{Alg}}(z,g).$
(1)

We assume the overall task of reaching $g$ can also be a subtask for the
purpose of action-level planning. Note that while we fixed subgoals above, the
choice between subtasks here still balances task reward and computational
cost.

### Action-level planning

which finds a sequence of actions to accomplish a subtask. Action-level
planning finds a sequence of states from a start state $s$ until the subtask
$z$ is completed by reaching $z_{sg}$, $\pi=\langle
s_{0},s_{1},...,z_{sg}\rangle$. Shorter plans are preferred, formalized by the
reward function $R(\pi)=-|{\pi}|$. A planning algorithm Alg non-
deterministically returns a plan and algorithm run-time,
$P_{\texttt{Alg}}(\pi,t\mid s,z)$. So the expected reward for reaching the
subtask $z$, while accounting for algorithm run-time, is
$R_{\texttt{Alg}}(s,z)=\sum_{\pi,t}P_{\texttt{Alg}}(\pi,t\mid
s,z)\left[R(\pi)-t\right]$.

We incorporate structural information by using A* (Hart, Nilsson, and Raphael
1968) with a spatial heuristic cost based on the Manhattan distance defined as
$h(s;g)=|s_{x}-g_{x}|+|s_{y}-g_{y}|$ given states with coordinates
$s=(s_{x},s_{y})$. The algorithm run-time is computed by counting the number
of visited states during search.

## Experimental design and results

To test our hypothesis, we designed 6 base mazes and created 8 transformations
through rotations and flipping along diagonal, horizontal, and vertical axes
resulting in a total of 6 sets, each containing 8 mazes. Each of the resulting
48 mazes can be decomposed into two subtasks, where one subtask was designed
to have a higher planning cost than the other subtask. The optimal route for
both subtasks is the same but mirrored along the diagonal (see Figure 1). Our
model hypothesizes that people would choose the subtask with the lower overall
cost, which is the sum of the optimal path length and planning cost.

To test this hypothesis, we recruited 41 participants (one participant was
excluded due to not completing the task) on Prolific to each navigate through
12 different mazes (two mazes were randomly sampled from each set to ensure
that participants saw two of each base maze). Before playing the navigation
game, participants received instructions and played two practice trials to
familiarize themselves with the task. Participants were given up to one minute
in the beginning for planning, after which they were motivated to reach the
goal as quickly as possible through a performance-dependent bonus that
rewarded shorter paths.

Figure 1: One example of the maze, where the participants were asked to
navigate from the blue dot to the goal (yellow tile in the lower right
corner). The red and blue subgoals were not visible to the participants but
marked the two subtasks. The subtask associated with the red path has a lower
planning cost (planning cost 53, step cost 26, total cost: 79) than the other
subtask with the blue path (planning cost 63, step cost 26, total cost 89). In
addition to both subtasks having the same optimal path mirrored along the
diagonal, the number of walls in each triangle is also the same. Figure 2:
Proportion of the favorable subtask chosen and error bars showing the standard
error at each trial

We observed that after the third trial, the majority of participants chose the
subtask which, according to our model, is more favorable due to a lower
planning cost (see Figure 2). Modeling the path choice in the last trial using
logistic regression ($path\\_chosen\sim intercept$), where the more favorable
path was encoded as 1, resulted in an intercept of $0.731$
($sd=0.338,p=0.030$). In addition, we observed a learning process among the
participants. The proportion of the computationally simpler subtask increased
from 42.5% in the first trial to 67.5% in the last trial. This increasing
trend was confirmed by again fitting a logistic regression model
($path\\_chosen\sim intercept+trial\\_number$) resulting in a significant
positive slope with a coefficient of $0.063$ ($sd=0.027,p=0.021$) and $0.106$
intercept ($sd=0.174,p=0.542$).

## Discussion and future work

In this work, we presented an extension to the task decomposition framework of
Correa et al. (2023) and tested it on a more complex planning domain of
spatial navigation tasks. Our model can capture the navigation strategies of
the majority of the participants. However, we see that after 12 trials, 32.5%
of the participants did not choose the subtask with lower planning costs. This
suggests room for improvement in the choice of planner and/or heuristics. In
addition, we observe a learning process. Future work can therefore concentrate
on how people learn to decompose and select tasks.

This line of research elegantly bridges our understanding of the human mind
and comprehending human planning processes with the development of efficient
algorithms for advanced machine learning systems.

## References

  * Binder et al. (2021) Binder, F. J.; Mattar, M. G.; Kirsh, D.; and Fan, J. E. 2021.  Visual scoping operations for physical assembly.  In _Proceedings of the 43rd Annual Conference of the Cognitive Science Society_.
  * Correa et al. (2023) Correa, C. G.; Ho, M. K.; Callaway, F.; Daw, N. D.; and Griffiths, T. L. 2023.  Humans decompose tasks by trading off utility and computational cost.  _PLOS Computational Biology_ , 19(6): 1–31.
  * Hart, Nilsson, and Raphael (1968) Hart, P. E.; Nilsson, N. J.; and Raphael, B. 1968.  A Formal Basis for the Heuristic Determination of Minimum Cost Paths.  _IEEE Transactions on Systems Science and Cybernetics_ , 4(2): 100–107.
  * Solway et al. (2014) Solway, A.; Diuk, C.; Córdova, N.; Yee, D.; Barto, A. G.; Niv, Y.; and Botvinick, M. M. 2014.  Optimal Behavioral Hierarchy.  _PLOS Computational Biology_ , 10(8): 1–10.
  * Sutton, Precup, and Singh (1999) Sutton, R. S.; Precup, D.; and Singh, S. 1999.  Between MDPs and semi-MDPs: A framework for temporal abstraction in reinforcement learning.  _Artificial intelligence_ , 112(1-2): 181–211.


# Omni-Line-of-Sight Imaging
for Holistic Shape Reconstruction

Binbin Huang†, Xingyue Peng†, Siyuan Shen, Suan Xia, Ruiqian Li, Yanhua Yu,
Yuehan Wang,
Shenghua Gao, Wenzheng Chen, Shiying Li, and Jingyi Yu  Binbin Huang and
Xingyue Peng contribute equally to this work; Binbin Huang, Xingyue Peng,
Siyuan Shen, Suan Xia, Ruiqian Li, Yanhua Yu, Yuehan Wang, Shenghua Gao,
Shiying Li and Jingyi Yu are with School of Information Science and
Technology, ShanghaiTech University; Wenzheng Chen is with University of
Toronto;

###### Abstract

We introduce Omni-LOS, a neural computational imaging method for conducting
holistic shape reconstruction (HSR) of complex objects utilizing a Single-
Photon Avalanche Diode (SPAD)-based time-of-flight sensor. As illustrated in
Fig. 1, our method enables new capabilities to reconstruct near-$360^{\circ}$
surrounding geometry of an object from a single scan spot. In such a scenario,
traditional line-of-sight (LOS) imaging methods only see the front part of the
object and typically fail to recover the occluded back regions. Inspired by
recent advances of non-line-of-sight (NLOS) imaging techniques which have
demonstrated great power to reconstruct occluded objects, Omni-LOS marries LOS
and NLOS together, leveraging their complementary advantages to jointly
recover the holistic shape of the object from a single scan position. The core
of our method is to put the object nearby diffuse walls and augment the LOS
scan in the front view with the NLOS scans from the surrounding walls, which
serve as virtual “mirrors” to trap lights toward the object. Instead of
separately recovering the LOS and NLOS signals, we adopt an implicit neural
network to represent the object, analogous to NeRF [1] and NeTF [2]. While
transients are measured along straight rays in LOS but over the spherical
wavefronts in NLOS, we derive differentiable ray propagation models to
simultaneously model both types of transient measurements so that the NLOS
reconstruction also takes into account the direct LOS measurements and vice
versa. We further develop a proof-of-concept Omni-LOS hardware prototype for
real-world validation. Comprehensive experiments on various wall settings
demonstrate that Omni-LOS successfully resolves shape ambiguities caused by
occlusions, achieves high-fidelity 3D scan quality, and manages to recover
objects of various scales and complexity.

###### Index Terms:

NLOS Imaging, Neural Rendering, Surface Reconstruction

##  1 Introduction

Three dimensional (3D) shape reconstruction in computer vision has largely
followed “to reconstruct is to see” strategy, _i.e_., only parts that are
observable from the camera can be recovered. _Holistic shape reconstruction_
(HSR) aims to recover the $360^{\circ}$ surrounding geometry of an object. As
a long-standing problem, it plays a key role in various applications, ranging
from shape scanning in virtual and augmented reality to archaeological
preservation and even rescue missions [3, 4, 5].

Traditionally, HSR adopts the implicit assumption that the object can be
observed from front to back and up to bottom, _i.e_., the object is scanned in
an omni-directional way. The capture can be conducted by either using a hand-
held camera [6] or constructing specialized imaging apparatus such as
$360^{\circ}$ camera arrays [7, 8] or a mechanically rotating board [9], [10].
However, obtaining omni-directional imagery data towards the target object in
reality can be challenging and sometimes prohibitive. For example, a hand-held
camera requires a user to hold a camera and circle around an object, which
takes tremendous user effort. Alternatively, specialized imaging apparatuses
are expensive and have sophisticated synchronization or calibration
procedures.

There have been a number of attempts to overcome the omni-directional scanning
requirement and explore ways to conduct HSR from a single scan spot. In such a
scenario, a sensor watches an object at a fixed position only (Fig. 1). While
simplifying the capture condition, due to the limited field of view, only the
front part of the object appears in the direct line-of-sight (LOS) view of the
sensor. As such, HSR becomes challenging since it lacks the necessary
information to reconstruct the occluded back geometry. Existing methods
propose to use auxiliary mirrors to reflect lights toward the object, so that
the back part beyond LOS regions can be virtually observed [11, 12, 13]. Such
approaches are applicable to both regular RGB cameras [11, 12] as well as the
time-of-flight (ToF) cameras [13]. Albeit effective, these methods inherently
require mounting mirrors, and for large-scale objects, the approach can be too
expensive in both cost and labor.

Figure 1: Conceptual illustration of 4 different setups under single-scan HSR
task. When we observe an object from a single scan spot, (a) LOS imaging
techniques only see the front shape but fail to recover the back part. (b)
Mirror systems (_e.g_. [13]) help create virtual back views to reconstruct the
holistic shape but suffer from calibration and correspondence matching issues.
(c) NLOS imaging methods treat diffuse walls as “diffuse mirrors” to
reconstruct the occluded object. Still, they only recover the regions that are
visible to the wall. (d) Our Omni-LOS system enables new capabilities to
conduct HSR from a single scan spot, where we apply LOS signals to recover the
front, visible regions, as well as NLOS measurements to reconstruct the
occluded, back part.

We recognize the single-scan HSR problem partially resembles the non-line-of-
sight (NLOS) imaging setting, where both aim to reconstruct an object that is
occluded and not directly observable from the senor. In the single-scan HSR
case, the occlusion comes from the object itself, _e.g_., the back geometry is
occluded by the front part. Instead, in the NLOS imaging setting, it is
typically due to an occluder. Effective NLOS solutions are largely attributed
to the latest advances in imaging hardware and a range of inverse imaging
techniques. Particularly, by exploiting the laser and single-photon avalanche
diode (SPAD) to prob the scene, as well as developing powerful inversion
techniques such as LCT [14], $f$-$k$ [15] or Phasor Field [16, 17], NLOS
techniques have demonstrated excellent capabilities to reconstruct occluded
objects faithfully.

In theory, existing NLOS techniques can be deployed to holistic shape
recovery. By placing an object nearby diffuse walls (Fig. 1), one can employ
LOS methods to reconstruct the front, visible part while applying NLOS methods
to recover the back, occluded geometry. In reality, naively combining two
reconstruction results incurs multiple problems: 1) Due to the inconsistency
of separate reconstruction results, seams might happen across views,
especially in the boundaries between LOS and NLOS scans. 2) Part of the NLOS
signal comes from the front, visible regions of the object, which also appear
in the LOS measurements but are not effectively used in NLOS reconstruction
and vice versa; 3) LOS and NLOS measurements have distinct ray propagation
models but there is no unified framework to simultaneously handle both of
them.

In this paper, we present Omni-LOS, a holistic shape reconstruction technique
that combines LOS and NLOS imaging under a unified neural modeling framework.
As shown in Fig. 1, Omni-LOS employs a SPAD and a laser emitter to prob an
object. It fixes the sensor in front of the object and effectively employs the
surrounding walls as virtual mirrors, eliminating the need to move the sensors
or use tailored mirror systems. At the core of our method is a neural
representation [1, 2] that encodes the holistic shape of the object with level
set methods [18]. Following the “analysis-by-synthesis” principle, we derive
differentiable ray propagation models for both LOS and NLOS transient imaging
procedures and integrate them into the training pipeline, supervising the
network parameters by minimizing the differences between the rendered
transient images and the real measurements. By simultaneous training on both
LOS and NLOS signals, Omni-LOS enables accurate and reliable holistic geometry
and normal recovery, even when observing the object from a single scan spot.

We conduct comprehensive experiments to validate Omni-LOS on object
reconstruction under different wall settings, including a single wall, two
side walls, and with and without a ceiling. Compared with the brute-force
techniques that individually conduct reconstruction on LOS and NLOS scans and
then fuse the results, we show Omni-LOS significantly improves the quality of
the recovered shape, especially on boundary regions across different scans. We
further construct proof-of-concept hardware prototypes to demonstrate our
approach in real captures. Experiments on synthetic and real scenes
demonstrate Omni-LOS achieves significantly better 3D scan quality and may
provide a practical omni-directional reconstruction solution suitable for
recovering objects of various scales.

The main contributions of this work can be summarized as follows.

  * •

We introduce Omni-LOS, an end-to-end solution that enables HSR from a single
scan spot. It largely simplifies the capture condition, requiring neither
camera arrays nor surrounding mirror systems.

  * •

Omni-LOS derives differentiable ray propagation models for both LOS and NLOS
transient imaging procedures, as well as proposes an implicit level-set
representation to recover shape jointly from LOS and NLOS scans. Compared to
brute-force methods, our unified framework demonstrates significantly better
reconstruction quality.

  * •

We further build a real-world imaging system and validate Omni-LOS with real
captures, demonstrating its potential usage for real-world deployment. The
code and data will be publicly available to the community to stimulate future
developments in HSR reconstruction.

##  2 Related Work

#### Holistic Shape Reconstruction (HSR).

HSR relies on trapping light of the $360^{\circ}$ geometry of an object,
_i.e_., observing it in an omni-directional way. In traditional line-of-sight
(LOS) settings, due to the limited field of view, typically multi-view images
are captured to cover the full body of the object. The images can be either
taken from hand-held cameras [6, 19, 20], or special imaging apparatuses like
$360^{\circ}$ camera arrays [7, 8, 21, 22]. With the acquired multi-view
images, techniques such as multi-view stereo (MVS) methods [23, 20, 24] or
neural modeling approaches [1, 25, 19] can be applied to infer the underlying
geometry from appearance. However, obtaining omni-directional imagery data
sometimes is challenging in reality, potentially due to the tremendous user
efforts or the expensive hardware cost. To overcome the onmi-directional
scanning requirement, people propose to watch an object from a single spot but
utilizing auxiliary sources to serve as “virtual cameras”, which can also
provide the onmi-directional information of the object. Two typical examples
are mirrors and walls, which we discuss below.

#### Trapping Light with Mirrors.

Intuitively, due to the special reflective property, a mirror can be treated
as a virtual camera. The idea of using mirrors to synthesize more views and
applying them in 3D reconstruction has been proposed for a long time, ranging
from regular RGB cameras [26, 27, 28, 29] to time-of-fight (ToF) cameras [13].
Most existing works [26, 27, 28, 29] focus on extending more field of view
with mirrors. Similar to us, Xu  _et al_. [13] augment a ToF sensor with a
multi-mirror system to conduct HSR of an object from a single observation
spot, where in the mirror system, the light rays would bounce multiple times
within the trap volume to eventually reach nearly all regions on the object,
especially the side and the back part. However, mirror-based methods typically
have challenging calibration and correspondence matching problems and are not
suitable for performing reconstruction applications in the wild.

#### Trapping Light with Walls.

Instead of real mirrors, people also explore using the relay walls as “diffuse
mirrors”, which diffusely scatter light from the object. The scattered signals
can be used to recover the object nearby the wall. Such works are often
explored in non-line-of-sight (NLOS) imaging, where the goal differs slightly
from HSR. NLOS imaging aims to reconstruct an occluded object and not directly
observable by the sensor. Due to the weakness of the collected signals, time-
resolved imaging systems are often employed in NLOS tasks, such as streak
cameras [30, 31, 32, 33], photonic mixer devices [34, 35], and SPADs [36, 15,
14, 37, 38, 17, 2, 39, 40, 41]. Our method follows this fashion and employs a
laser emitter and a SPAD to capture transient information of an object.

Previous NLOS methods represent the scene by either volumetric albedo [14,
15], volumetric feature [42, 43] or parametric surface [38, 44]. Recently,
neural-based approaches have made remarkable progress [2, 45]. NeTF [2]
applies a density field to represent the scene, which fails to acquire a high-
quality surface. Concurrent to our work, NLOS-NeuS [45] represent NLOS scenes
with signed distance fields (SDF) [18, 46], which shows better surface
recovery. Different from them, our technique incorporates a more general level
set representation, on which we develop a unified transient rendering
formulation under both LOS and NLOS settings. Our method not only achieves
high-quality surface reconstruction but also eliminates the common Eikonal
regularization term and stabilizes the network training.

#### Joint Reconstruction with LOS and NLOS Scans.

There have been several pioneering works that try to combine LOS and NLOS
imaging together. Henley  _et al_. [47] use a SPAD and a laser to estimate the
shape of a 3D scene from the first- and second-bounce measurements, where the
hidden objects in the cast shadows can be recovered from the second-bounce
measurements. Rapp  _et al_. [48] propose an edge-resolved transient imaging
(ERTI) method to reconstruct the visible and hidden parts of the scene, where
they scan the scene along an arc around a vertical edge and analyze the
transient histogram differences to reconstruct the hidden objects. We differ
from them in both problem setup and methodology. Omni-LOS tackles the single-
scan HSR problem. Moreover, we develop a neural modeling technique to jointly
utilize both LOS and NLOS transients to reconstruct the holistic geometry of
the object.

##  3 Omni-LOS Framework

We consider the problem of single-scan HSR, where the goal is to reconstruct
the whole shape of an object observed from _a single scan spot_. As shown in
Fig. 1, it is extremely challenging for traditional LOS imaging methods (Fig.
1a). Due to the limited fields of view, LOS methods can only recover the
front, visible shape. Prior works propose to augment new views with mirrors
[13] (Fig. 1b). Such methods successfully estimate the complete geometry but
suffer from calibration and correspondence-matching issues. Inspired by NLOS
imaging methods (Fig. 1c) which estimate hidden objects with three-bounce
transient information from diffuse walls, we propose Omni-LOS (Fig. 1d), a
brandy new framework to conduct HSR by utilizing cues from both LOS and NLOS
measurements.

As illustrated in Fig. 1d, we place an object nearby diffuse walls and employ
a single-photon avalanche diode (SPAD) together with a laser emitter to shoot
rays and capture transient images of the object. Omni-LOS applies LOS methods
to estimate the visible, red regions, as well as taking the nearby walls as
“diffuse mirrors” and employing NLOS techniques to infer the occluded, blue
regions. We mainly consider a two-wall setting, which we find enough to cover
the near $360^{\circ}$ geometry of the object. However, it can be easily
applied to various wall settings like single-wall or with-ceiling cases. We
validate different wall settings in the experimental sections (Sec. 5 and Sec.
6).

To better utilize cues in both LOS and NLOS imaging measurements, we propose a
neural modeling framework to represent the object with level set based
implicit neural networks. We further derive differentiable ray propagation
models for both LOS and NLOS imaging conditions and jointly train the model
from all measurements. We find that joint training achieves a significant
performance boost compared to the separate reconstruction baseline.

In the below sections, we first detail our level set-based neural
representation and the differentiable ray propagation models in Sec. 4. Next
we analyze our method on a synthetic dataset in Sec. 5. Finally, we build a
proof-of-concept Omni-LOS hardware prototype and verify its effectiveness with
real-world experiments in Sec. 6.

Figure 2:  Omni-LOS Pipeline (a) We first sample 3D points along straight rays
in LOS measurements (red, visible regions) while over spherical wavefronts in
NLOS measurements (blue, occluded regions). (b) These 3D samples are then fed
into a hash encoding network [25] to predict the corresponding level set
values and albedo. To account for the light intensity difference between LOS
and NLOS, we predict a 2-channel albedo to represent them separately. (c) We
render the samples under both LOS and NLOS imaging settings and compare them
with the real measurements. (d) By representing the object with a unified
neural network and optimizing from both LOS and NLOS signals, Omni-LOS
successfully reconstructs the high-quality, holistic shape of the object.

##  4 Omni-LOS Shape Recovery

###  4.1 Neural level set Representation

Neural representations have emerged as a powerful tool to represent objects or
scenes with complex appearance, geometry and material [1, 18, 49]. Following
this fashion, we also employ neural representations as a flexible and compact
parameterization method to encode the holistic shape of an object.
Specifically, we choose a level set based representation for better geometry
modeling [50, 18]:

$\mathcal{F}_{\Theta}:(\mathbf{x},\textbf{d})\rightarrow(\varphi(\mathbf{x}),\rho(\mathbf{x},\textbf{d}))\
\ ,$ (1)

where we employ a neural network $\mathcal{F}_{\Theta}$ which takes as input
position $\mathbf{x}=(x,y,z)$ and ray direction
$\textbf{d}=(d_{x},d_{y},d_{z})$ and predicts both the level set value
$\varphi(\mathbf{x})$ and the view-dependent surface albedo
$\rho(\mathbf{x},\textbf{d})$. Here, the level set value $\varphi(\mathbf{x})$
implicitly defines the surface at zero, _i.e_., the surface
$\mathcal{S}=\\{\mathbf{x}|\varphi(\mathbf{x})=0\\}$. The level set values in
the outside regions are defined as positive while the inside values are
negative. Following IDR [50], the surface normal ${\mathbf{n}}$ can be
obtained by

$\mathbf{n}(\mathbf{x})=\frac{\nabla_{\mathbf{x}}\varphi(\mathbf{x})}{\|\nabla_{\mathbf{x}}\varphi(\mathbf{x})\|}\
\ .$ (2)

The level set representation also enables flexible conversion to an occupancy
indicator function [51], where the inside region is one while the free space
is zero. We adopt a sigmoid function to transform level set values to the
indication of occupancy:

$\mathbf{o}(\varphi)=\frac{1}{1+e^{s\varphi}}\ \ ,$ (3)

where $s$ is a scalar that controls the sharpness of the sigmoid function.

We highlight that both normal $\mathbf{n}$ and occupancy $\mathbf{o}$ are
crucial in our transient ray propagation models, which are described in Sec.
4.2 and Sec. 4.3, respectively. First, normal $\mathbf{n}$ participates in
calculating the radiometric quantities. Moreover, occupancy $\mathbf{o}$ plays
an important role in density modeling in the ray propagation equation(Eq.
(4)). Incorporating them in the neural level set representations allows for
reliable shape and accurate surface reconstruction compared to the density
volume representation [1].

In contrast to NeuS or NLOS-NeuS that directly use a signed distance field,
the level set is more flexible as it doesn’t require the magnitude of the
gradient to be constant, eliminating the requirement of tuning the common
Eikonal regularization term. We provide a detailed analysis in the
supplementary.

###  4.2 Neural Transient Rendering

We now describe how to render the level set representation into transient
measurements under both LOS and NLOS imaging conditions. Different from
regular RGB images, transient images can be described as
$\tau(\tilde{\mathbf{x}},t)$, where $\tilde{\mathbf{x}}$ is the light source
and camera positions (_i.e_., in LOS case, it denotes the SPAD position
$\tilde{\mathbf{x}}_{s}$ while in NLOS case it is typically referred to a wall
position $\tilde{\mathbf{x}}_{w}$ (See Fig. 2a), $t$ is the light traveling
time. Here, we consider a confocal setting [14] in both LOS and NLOS imaging,
where the SPAD sensor and the camera are located at the same position. Note
that our framework can be generalized to the non-confocal setting as well.

Rendering neural representations into transient images has been explored in
NeTF [2], where Shen  _et al_. choose a density-based neural representation
and focus on NLOS setting only. Instead, we derive both LOS and NLOS neural
transient rendering equations under level set representations. Our formula
demonstrates better geometry and normal reconstruction in NLOS imaging over
NeTF, as well as enables HSR from a single scan spot when LOS data is
applicable.

Without loss of generality, we define LOS and NLOS image formation models in a
unified integral equation:

$\displaystyle\tau(\tilde{\mathbf{x}},t)=\underset{\Omega}{\int}H(\tilde{\mathbf{x}},\mathbf{x})G(\tilde{\mathbf{x}},\mathbf{x})T^{2}(\tilde{\mathbf{x}},\mathbf{x})\sigma(\mathbf{x})\rho(\mathbf{x},\textbf{d})d\mathbf{x}\
\ .$ (4)

$\Omega$ is the 3D sampling space. In LOS setting it is a straight ray while
in NLOS setting it is a spherical wavefront.
$H(\tilde{\mathbf{x}},\mathbf{x})$ is Dirac term of the light transport,
_e.g_., the time that light travels from light source $\tilde{\mathbf{x}}$ to
the 3D point $\mathbf{x}$ and back to the sensor position $\tilde{\mathbf{x}}$
should be equal to $t$. $G(\tilde{\mathbf{x}},\mathbf{x})$ describes the
radiometric fall-off due to the light traveling and surface reflections.
$\sigma(\mathbf{x}),\rho(\mathbf{x},\textbf{d})$ is the volumetric density and
view-dependent albedo. $T(\tilde{\mathbf{x}},\mathbf{x})$ is a transmittance
term representing the mutual visibility from the sensor to the 3D surface
point:

$\displaystyle
T(\tilde{\mathbf{x}},\mathbf{x})=\mathrm{exp}{\left(-\int_{0}^{\frac{t}{2}}\sigma(\tilde{\mathbf{x}}_{s}+sc\textbf{d})ds\right)}\
\ .$ (5)

We apply a square $T^{2}(\tilde{\mathbf{x}},\mathbf{x})$ in the integral to
describe the active imaging process, where light travels from the SPAD to the
3D point and then reflects back to the sensor. $c$ is the constant light
speed. In the following, we separately define the rendering equations for both
LOS and NLOS settings. We include detailed derivation in the supplementary.

#### LOS Transient Rendering Equation.

The LOS rendering equation can be described as:

$\displaystyle\tau_{LOS}(\tilde{\mathbf{x}}_{s},t)=\int_{\Omega}\delta(\|\tilde{\mathbf{x}}_{s}-\mathbf{x}\|-\frac{1}{2}tc)$
(6)
$\displaystyle\frac{(\mathbf{n}\cdot\widehat{\mathbf{x}\tilde{\mathbf{x}}_{s}})}{\|\mathbf{x}-\tilde{\mathbf{x}}_{s}\|^{2}}T^{2}(\tilde{\mathbf{x}}_{s},\mathbf{x})\sigma(\mathbf{x})\rho(\mathbf{x},\textbf{d})d\mathbf{x}\
\ ,$

where we measure the transient intensity of a ray shooting from spad position
$\tilde{\mathbf{x}}_{s}$ along direction
$\textbf{d}=\widehat{\mathbf{x}\tilde{\mathbf{x}}_{s}}$ at time $t$, _i.e_.,
$\mathbf{x}=\tilde{\mathbf{x}}_{s}+\frac{1}{2}tc\textbf{d}$. Note that here we
fix the SPAD position $\tilde{\mathbf{x}}_{s}$ but varying view direction d to
capture different 3D points. The Dirac term
$H=\delta(\left|\tilde{\mathbf{x}}_{s}-\mathbf{x}\right|-\frac{tc}{2})$
constrains the light travel time from $\tilde{\mathbf{x}}_{s}$ to the 3D point
$\mathbf{x}$ should be $\frac{t}{2}$. The light fall-off
$G(\mathbf{x},\tilde{\mathbf{x}}_{s})={(\mathbf{n}\cdot\widehat{\mathbf{x}\tilde{\mathbf{x}}_{s}})}/{\|\mathbf{x}-\tilde{\mathbf{x}}_{s}\|^{2}}$.
This formulation resembles LOS rendering in TöRF [52]. The key difference is
that we further add normal in the light intensity calculation. Besides the
distance, we consider the cosine foreshortening term
$(\mathbf{n}\cdot\widehat{\mathbf{x}\tilde{\mathbf{x}}_{s}})$ and find it
brings more accurate surface reconstruction. Implement-wise, we ensure
$(\mathbf{n}\cdot\widehat{\mathbf{x}\tilde{\mathbf{x}}_{s}})$ to be larger
than zero by truncating the negative part.

#### NLOS Transient Rendering Equation.

On the other side, the NLOS rendering equation is defined as:

$\tau_{NLOS}(\tilde{\mathbf{x}}_{w},t)=\underset{\Omega}{\int}\delta(\|\tilde{\mathbf{x}}_{w}-\mathbf{x}\|-\frac{ct}{2})\cdot\\\
\frac{(\mathbf{n}\cdot\widehat{\mathbf{x}\tilde{\mathbf{x}}_{w}})^{2}(\mathbf{n}_{w}\cdot\widehat{\tilde{\mathbf{x}}_{w}\mathbf{x}})^{2}}{\|\mathbf{x}-\tilde{\mathbf{x}}_{w}\|^{4}}T^{2}(\tilde{\mathbf{x}}_{w},\mathbf{x})\sigma(\mathbf{x})\rho(\mathbf{x},\textbf{d})d\mathbf{x}\
\ .$ (7)

Here, we measure the transient intensity of a ray scatters from a relay wall
point $\tilde{\mathbf{x}}_{w}$ to 3D space and reflect back to
$\tilde{\mathbf{x}}_{w}$ again at a specific time step along $t$. The Dirac
term $H=\delta(\|\tilde{\mathbf{x}}_{w}-\mathbf{x}\|-\frac{ct}{2})$. After
scattering three times, the light fall-off is
${(\mathbf{n}\cdot\widehat{\mathbf{x}\tilde{\mathbf{x}}_{w}})^{2}(\mathbf{n}_{w}\cdot\widehat{\tilde{\mathbf{x}}_{w}\mathbf{x}})^{2}}/{\|\mathbf{x}-\tilde{\mathbf{x}}_{w}\|^{4}}$
where $\mathbf{n}_{w}$ denotes the surface normal at the relay wall spot
$\tilde{\mathbf{x}}_{w}$. As we consider at most three bounces, the light
travels as a spherical wavefront in NLOS. Following NeTF [2], we uniformly
sample in $(\theta,\phi)$ space for computing the spherical integral. Compared
to [2], we incorporate surface normal in transient rendering, which enables
more accurate surface reconstruction. Note that the view-dependent model
allows Omni-LOS to represent various materials. We demonstrate it in Sec. 6

###  4.3 Volumetric Rendering of Neural Surface

We now discuss how to model density $\sigma$ in the rendering equations. In
contrast to [52, 2] that aims to reconstruct volumetric fields, we aim to
reconstruct an implicit surface parameterized by neural networks. To do so, we
should model the volumetric density $\sigma(\mathbf{x})$ as a function of an
implicit surface $\varphi(\mathbf{x})$. In the following, we re-parameterize
all the 3D points $\mathbf{x}$ along a ray as
$\mathbf{x}(\eta)=\tilde{\mathbf{x}}+\textbf{d}\eta$. We slightly abuse the
notations $T(\eta)=T(\tilde{\mathbf{x}},\mathbf{x}(\eta))$ and density
$\sigma(\eta)=\sigma(\mathbf{x}(\eta))$ for clarity. We define the weight term
$w(\eta)$ of $\mathbf{x}(\eta)$ as

$w(\eta)=T^{2}(\eta)\sigma(\mathbf{x}(\eta))\ \ ,$ (8)

where $w(\eta)$ weights radiometric intensity to the transient measurements.
Since we aim to reconstruct the surface, we should ensure the radiometric
contribution mostly comes from the surface $\mathcal{S}$, _i.e_., the zero
level set. Following [18], we first assume there is only a single visible
plane when ray marches from outside into inside. With this assumption, one can
define the $w(\eta)=\delta(\varphi(\mathbf{x}))$ so that, ideally, only points
at the zero-level set contributes to transient measurement. To apply
differentiable rendering, we seek a smooth Dirac approximation $\delta_{s}$
instead of $\delta$. We notice that the predefined occupancy indicator
$\mathbf{o}$ (Eq. (3)) meets our purpose as it is a smooth Heaviside step
function and its derivative serves as a smooth Dirac function $\delta_{s}$.
Therefore, we model $w(\eta)$ as a function of level set $\varphi$ by

$w(\eta)=\delta_{s}(\varphi(\mathbf{x}(\eta)))=-\frac{d\mathbf{o}(\varphi(\mathbf{x}(\eta)))}{d\eta}$
(9)

By substituting Eq. (3) and Eq. (8) into Eq. (9) and solving the PDE, we can
derive the density $\sigma$ as

$\displaystyle\sigma(\mathbf{x}(\eta))$
$\displaystyle=\frac{-s}{2(1+e^{-s\varphi(\mathbf{x})})}\nabla_{\mathbf{x}}\varphi(\mathbf{x})\cdot\textbf{d}\
\ .$ (10)

Notice that the density is only valid for a single plane intersection where
the ray travels from outside to inside. To account for multi-plane
intersection, we adopt $\hat{\sigma}=\max(\sigma,0)$ [18] and composite
transmittance with Eq. (5) with $\hat{\sigma}$.

Remarkably, our method differs from existing neural surface rendering [18,
53]. While they employ signed distance fields that assume
$\|\nabla\varphi(\mathbf{x})\|=1$ everywhere, our formulation is based on a
more general level set function and has incorporated
$\nabla\varphi(\mathbf{x})$ in our forward model, so we do not need explicitly
impose eikonal regularization [54]. See the supplementary ablation study of
level set representation.

#### Weight Warm-up

. We notice that the weight of a sample in Eq. 9 is essentially view-
dependent, _i.e_., only locations where $(-\mathbf{n}\cdot\textbf{d})>0$ have
positive weights. If the network is not well initialized, there may be many
points not being optimized at all. To ensure every point stands a chance for
optimization, we apply a warm-up trick. We define

$w_{\text{train}}(\eta)=\alpha w(\eta)+(1-\alpha)\|\nabla\mathbf{o}(\eta)\|\ \
,$ (11)

where $\|\nabla\mathbf{o}(\eta)\|$ is a view-independent density. During
training, we first ignore the influence of the normal to get a good level set
field initialization. We increase $\alpha$ from $0$ to $1$ to switch from the
view-independent model to the view-dependent model.

Figure 3: Joint Training. Omni-LOS supports joint training with both LOS and
NLOS measurements and demonstrates significant improvements over the naive
baseline that separately reconstructs LOS and NLOS partial shapes and fuses
them together. (a,b) Training with only LOS or NLOS data results in good front
or back shape reconstruction. However, the other side is missing. (c) Fusing
the LOS and NLOS shape together has severe merging artifacts, especially for
the edge regions between two scans. (d) Our joint training framework
demonstrates much more accurate shape reconstruction.

###  4.4 Learning Objectives

Our method allows joint training of both LOS and NLOS transient images.
However, the measurement intensity in NLOS and LOS images are quite different,
_i.e_., NLOS intensity is much weaker. Therefore, in each 3D point, we predict
a 2-channel surface albedo $\rho$ to account for LOS and NLOS intensity
separately. Note that the geometry representation $\varphi(\mathbf{x})$ is
shared to ensure holistic surface reconstruction from multiple sources.

Besides, we find LOS measurements provide strong geometric priors. Encouraged
by [55], we detect the first returning photon from the LOS transient and
create a 3D binary mask $M$ to indicate occupancy. That’s to say, for the
regions where LOS data didn’t detect any object, those regions should be empty
($\mathbf{o}=0$). We supervise the network by minimizing the binary cross
entropy (BCE) loss between $M$ and the predicted occupancy value. Adding mask
priors brings faster convergence speed and better shape recovery. Taking all
these into consideration, we set our learning objectives as:

$\displaystyle\mathcal{L}_{b}$
$\displaystyle=\sum_{\tilde{\mathbf{x}}_{w}}\|\tau_{NLOS}(\tilde{\mathbf{x}}_{w},t)-\bar{\tau}_{NLOS}(\tilde{\mathbf{x}}_{w},t)\|^{2}$
(12)
$\displaystyle+\gamma\sum_{\tilde{\mathbf{x}}_{s}}\|\tau_{LOS}(\tilde{\mathbf{x}}_{s},t)-\bar{\tau}_{LOS}(\tilde{\mathbf{x}}_{s},t)\|^{2}$
$\displaystyle+\beta\sum_{\tilde{\mathbf{x}}_{s},t}\text{BCE}(\mathbf{o}(\varphi(\mathbf{x}_{\eta})),M(\tilde{\mathbf{x}}_{s},\eta))\
\ ,$

where $\gamma$ and $\beta$ are to balance the objectives and
$\mathbf{x}_{\eta}$ a sample point along the ray and $M$ is the corresponding
binary mask. We empirically set $\gamma=1$ and $\beta=0.01$. Fig. 3
demonstrates the superiority of our joint training framework.

###  4.5 Network Implementation

We design our network based on the fast neural training framework i-NGP [25].
It consists of a learnable hash table and two shallow MLPs representing
geometry $\mathcal{F}_{l}$ and directional albedo $\mathcal{F}_{a}$. We set
the hash table with $L=32$ and $D=8$ to output a $256$-dim hash embedding.
$\mathcal{F}_{l}$ is a three-layer MLP that takes as input the hash embedding
and outputs a 1-dim level set value and a 15-dim geometry feature.
$\mathcal{F}_{a}$ is a two-layer MLP that takes the geometry feature and the
encoded view direction and outputs a 2-channel surface albedo (representing
both LOS and NLOS) with a Softplus activation. A detailed network architecture
is included in the supplementary materials.

For each iteration, we sample a location from SPAD positions
$\tilde{\mathbf{x}}_{s}$ or wall positions $\tilde{\mathbf{x}}_{w}$ to conduct
LOS or NLOS training. Specifically, for LOS training, we sample $256$ points
along a ray to render $\tau_{LOS}(\tilde{\mathbf{x}}_{s},t)$. For NLOS
training, we sample $64\times 64\times 128$ points, where $64\times 64$ is the
$\theta,\phi$ angles of the wavefront and $128$ is different radius samples
(for transmittance computation). We adopt Adam [56] as our optimizer with a
learning rate beginning at $1\times 10^{-3}$ and decaying exponentially to
$5\times 10^{-5}$. By default, we conduct the training with 100k iterations
and find that 10k iterations can produce a reliable result (roughly 10 mins)
for most experiments. After optimizing the networks, we infer a $256^{3}$
level set grid and extract a mesh using marching cubes. All experiments are
implemented based on Pytorch and run on a server with an NVIDIA A40 GPU.

Figure 4:  Synthetic Evaluation. We show 4 reconstruction examples. For each
example, left is its capture setting. On the right we show the reconstructed
shape in both front and back views. Our method is able to reconstruct high-
quality, continuous and smooth surfaces, while the baselines produce poor
bumpy surfaces and have artifacts in the edges between LOS and NLOS scans. We
apply two diffuse walls for the rabbit, the terracotta warrior, and the
Pikachu examples. For the tripod, we additionally add a ceiling as another
NLOS relay wall to recover the concave shape.

##  5 Evaluation

###  5.1 Sythetic Datasets

We first evaluate our method under synthetic data to quantitatively analyze
the performance of our method. As we tackle a new single-scan HSR setting, we
need to synthesize LOS measurements in the front view and NLOS measurements in
multiple back views (assuming two walls). We adopt a publicly available NLOS
transient renderer from [43] and extend it to synthesize both multi-view NLOS
transients and LOS transients.

We first normalize the object and put it at the center of a $1m\times 1m$
ground. We then build two $1m\times 1m$ walls at the edges of the ground. From
each planar wall, we acquire transients at a resolution of $256\times
256\times 512$. To simulate LOS transients, we sample along a ray instead of a
wavefront and uniformly synthesize $256\times 256$ transients.

In the real world, the object might occlude the walls. However, many NLOS
techniques, _e.g_., LCT or $f$-$k$ [14, 57], require regular sampling on the
relay wall to reconstruct the hidden object. To compare with those existing
NLOS methods, we consider an ideal scenario. We assume that the walls are
fully visible and synthesize transients at all spots on the relay walls. We
also omit the inter-reflections between different walls in NLOS measurements.
Note our technique also supports training with irregular samples on the wall.
In Sec. 6, we demonstrate the capability of our method under a real setting.

TABLE I: Quantitative Comparisons. We measure the Chamfer Distance (cm) and
Normal Consistency (%) between the predicted shape and GT shape. Omni-LOS
outperforms other methods for most cases.

Methods | Rabbit | Pikachu | Terracotta Warrior | Tripod
---|---|---|---|---
CD$\downarrow$ | NC $\uparrow$ | CD$\downarrow$ | NC $\uparrow$ | CD$\downarrow$ | NC $\uparrow$ | CD$\downarrow$ | NC $\uparrow$
DLCT [57] | 2.28 | 77.59 | 1.73 | 85.38 | 1.46 | 77.45 | 1.63 | 79.61
$f$-$k$ [15] | 1.94 | 67.73 | 1.32 | 72.81 | 1.11 | 69.43 | 1.68 | 70.68
PF [16] | 2.07 | 63.83 | 1.62 | 67.92 | 1.43 | 61.81 | 2.76 | 59.59
BP [31] | 1.79 | 71.64 | 1.34 | 74.64 | 1.30 | 71.61 | 1.92 | 71.79
NeTF [2] | 2.10 | 69.01 | 1.86 | 71.07 | 1.88 | 71.67 | 2.56 | 72.65
Ours | 1.79 | 85.93 | 0.09 | 94.01 | 1.02 | 87.69 | 1.87 | 83.02

Figure 5: Ablation Study for BCE loss and Normal. (a) Our full model
effectively recovers the accurate surface of the object. (b) When the BCE loss
is removed, some floating regions appear. (c) Without the normal constraint in
the image formation models, the recovered surface becomes concave, such as the
nose and the neck of the rabbit.

###  5.2 Omni-LOS Surface Reconstruction

Our method applies level set implicit representations, which can be easily
converted into explicit meshes using marching cubes. We demonstrate our method
acquires more accurate surface reconstruction over prior arts. We mainly
consider NeTF [2] as the neural-based baseline as it also applies neural
representation. To make a fair comparison, we train NeTF with the same network
architecture as us but change the prediction to be density values and apply
its own image formation model. Since NeTF is designed for NLOS only, we apply
NeTF to reconstruct each wall separately, then extract multiple surfaces and
the LOS surface and fuse them together. We do the same things for DLCT [57]
since it also produces NLOS surfaces.

We further compare our technique to other volume-based NLOS methods, including
Back-projection (BP) [31], $f$-$k$ [15], Phasor Field (PF) [16], and LCT [14].
We apply them to reconstruct the NLOS volume from each wall separately. For
LOS transients, we warp it into a new volume with Back-projection. Given all
volumes, we fuse them into a single volume by summation. We then apply
marching cubes to the fused volume to acquire the holistic shape of the
object. Due to limited space, we put some qualitative results in the
supplementary. In the main paper, we only show the comparison with BP and
NeTF.

Figure 6: Capture System. Our prototype includes a pulsed laser, a SPAD with
TCSPC for photon counting, and a galvo for scanning on the relay wall.

We show qualitative results in Fig. 4. We also quantitatively evaluate the
shape reconstruction quality in Tab. I. For quantitative comparison, we check
the chamfer distance (CD) [58] and the normal consistency (NC) [59] between
the predicted shape and GT shape. We uniformly sample 20K points from the
reconstruction mesh and ground truth shape, which we denote as
$\mathbf{x}\in\mathcal{X}$ and $\mathbf{y}\in\mathcal{Y}$. Chamfer distance
(CD) and normal consistency (NC) are defined as

$\displaystyle\text{CD}(\mathcal{X},\mathcal{Y})$
$\displaystyle=\frac{1}{2}\text{D}(\mathcal{X},\mathcal{Y})+\frac{1}{2}\text{D}(\mathcal{Y},\mathcal{X})$
(13) $\displaystyle\text{D}(\mathcal{X},\mathcal{Y})$
$\displaystyle=\frac{1}{\|\mathcal{X}\|}\sum_{\mathbf{x}\in\mathcal{X}}\|\mathbf{x}-q_{\mathcal{Y}}(\mathbf{x})\|\
\ .$ $\displaystyle\text{NC}(\mathcal{X},\mathcal{Y})$
$\displaystyle=\frac{1}{2}\text{N}(\mathcal{X},\mathcal{Y})+\frac{1}{2}\text{N}(\mathcal{Y},\mathcal{X})$
(14) $\displaystyle\text{N}(\mathcal{X},\mathcal{Y})$
$\displaystyle=\frac{1}{\|\mathcal{X}\|}\sum_{\mathbf{x}\in\mathcal{X}}\|\mathbf{n}(\mathbf{x})\cdot\mathbf{n}(q_{\mathcal{Y}}(\mathbf{x}))\|~{}\
\ .$

Here, $q_{\mathcal{Y}}(\mathbf{x})$ means finding the closest point
$\mathbf{y}\in\mathcal{Y}$ from $\mathbf{x}$.

Several interesting observations can be found in the results. First, our
method achieves better reconstruction quality. As for the whole shape, Omni-
LOS achieves more accurate and complete shape recovery, even for the extremely
challenging concave tripod example. In contrast, baseline methods typically
have large holes or missing regions. We believe the success comes from our
level set representation and the unified neural modeling framework. On the
contrary, naively reconstructing the LOS and NLOS part separately and merging
them together produces strong artifacts, especially for the edges between LOS
and NLOS scans. When checking the LOS and NLOS surface reconstruction quality,
our neural modeling method produces a smooth surface while baselines get bumpy
or pixelate surface. In quantitative comparison (Tbl. I), we have better
scores in Rabbit, Pikachu and Terracotta Warrior examples, while it slightly
drops for the Tripod example. We hypothesize it is because all the methods
fail for the bottom part of the tripod since it is not observed at all. As a
result, this part strongly influences the score. Although $f$-$k$ and DLCT are
better than us, the three methods are pretty close and our normal is much
better.

Lastly, we find our results lose some fine details in Terracotta Warrior or
Tripod examples, potentially due to the limited measurement resolution.
Further improving our technique to recover fine details is an interesting
direction and we leave it for future work.

Figure 7: Noise Handling. We add a noise prediction branch to account for
noisy NLOS transients. We verify it on an NLOS scene [15]. Our noise modeling
successfully removes most artifacts degraded by NLOS noise.

###  5.3 Ablations

We apply normal in the image formation models (Eq. (6) and Eq. (7)) and BCE
mask in the objective function (Eq. 12). We ablate them two in Fig. 5. The
normal guides the surface directions (Eq. (10)) and helps recover a more
accurate surface. Ignoring normal in the rendering equation may generate a
surface with incorrect curvature, _e.g_., the concave shape shown in Fig. 5.

Conversely, BCE loss helps “carve” the space and gets clean reconstruction
results. Since the field of view of LOS scan is quite small, removing BCE loss
makes no constraints for the unobservable regions and generates floating
artifacts in those regions.

##  6 Validation with Prototype

Figure 8: Real World Experiments. We validate Omni-LOS on three objects with
different shapes and materials. The cat is made of retro-reflective material
and has a relatively simple geometry. Our approach precisely recovers the
cat’s head, feet, and tail. The ninja is made of cloth, while the turtle shell
is covered with retro-reflective material. Omni-LOS recover both the front of
the ninja and the back of the turtle shell. The horse is made of foam and
therefore presented as the most challenging case due to the low SNR. Our
approach recovers the head, back foot, and even the horse’s tail. However, it
fails to recover the horse’s front foot due to the limited number of returning
photons. On the other side, baseline methods generate poor geometry
reconstruction with missing regions, large holes, and pixelated artifacts in
all 3 cases.

###  6.1 Hardware

Fig. 6 shows our experimental setup. A supercontinuum laser (SuperK EXTREME
FIU-15) with a wavelength-tunable filter (SuperK VARIA) emits a collimated
light beam. We use the wavelength band between 665 nm and 675 nm with an
average power of 48 mW and a repetition frequency of 39 MHz. The beam passes
through a polarized beam-splitter (Thorlabs VA5-PBS251) and a 2D galvanometer
(Thorlabs GVS012) and scans the relay wall. Light bounces off diffusely three
times: on the relay wall, object surface, and back on the relay wall. A fast-
gated SPAD coupled with a PicoHarp 300 at 4 ps resolution then collects
photons that return through the galvanometer. We exploit a delayer (PicoQuant
PSD-065-A-MOD) to shield the LOS transients with a gate width of 12 ns and
record NLOS transients only. Our system is confocal, i.e., the laser spots
coincide with the detection points on the relay walls. The hardware is about
1.5 m from the scanning area of $1m\times 1m$ on each wall; the objects, e.g.,
cat, ninja, horse, are 0.5 m away from the relay walls.

Unlike traditional NLOS cases, our single-scan HSR setting involves measuring
an object both directly and from multiple relay walls. We calibrate our system
on each relay wall using the onsite method [60] and scan the object on a
uniform $128\times 128$. grid. We then split all the measured points into
irregular LOS and NLOS scans by determining whether the point hit the wall or
not. While most NLOS methods only support regular grid sampling measurements
[14, 15], our technique allows irregular sampling at arbitrary points. When we
capture under multiple walls, different walls will interfere with each other
in NLOS measurements. We apply LOS measurements to roughly divide the object
range and manually crop the interference NLOS signals from other walls. We do
this for all the methods. We compare our method with BP and NeTF, as they also
support irregular sampling measurements.

###  6.2 Handling Noise

NLOS transients typically contain strong noise due to the weak signals and the
imperfection of the hardware systems, _e.g_., timing jitter, pile-up or the
intractable multi-bounce scattering in the NLOS scene. To handle the noise,
following NeRF in the wild[61], we employ another network branch to predict
the residual error. This procedure resembles transient rendering but renders
noise instead of radiance fields:

$\small\tau_{\text{noise}}(\tilde{\mathbf{x}})=\underset{\Omega}{\int}\delta(\|\tilde{\mathbf{x}}_{w}-\mathbf{x}\|-\frac{ct}{2})\pi(\mathbf{x})d\mathbf{x}\
\ ,$ (15)

where we predict noise $\pi(x)$ for each 3D point $\mathbf{x}$. Then we add it
to the transient to get the final prediction:
$\hat{\tau_{NLOS}}(\tilde{\mathbf{x}},t)=\tau_{NLOS}(\tilde{\mathbf{x}},t)+\tau_{\text{noise}}(\tilde{\mathbf{x}})$.
We further regularize the noise intensity to be small:

$\small\mathcal{L}_{\text{NLOS}}=\sum_{\tilde{\mathbf{x}}_{w}}\|\tau_{\text{NLOS}}(\tilde{\mathbf{x}}_{w},t)-\hat{\tau_{\text{NLOS}}}({\tilde{\mathbf{x}}_{w}},t)\|+\lambda\|\tau_{\text{noise}}(\tilde{\mathbf{x}})\|^{2}\
\ ,$ (16)

where $\lambda$ is the noise coefficient related to the signal-to-noise ratio
(SNR). Transients with a higher SNR should impose a larger $\lambda$ to weaken
the effect of noise. The noise branch only exists in the training phase.
During inference, we ignore the noise branch and acquire the shape from the
regular geometry and albedo branches. Our technique enables reliable holistic
shape even from noisy transients. Note that we predict the noise only for NLOS
scans since the LOS transients are clean enough. We show an ablation study in
Fig. 7, where we apply our method on a purely NLOS statue scene [15].
Implementation details are included in Supplementary Material.

###  6.3 Validation

Omni-LOS models the view-dependent effects and can therefore support various
materials. We validate our method on three objects with different materials: a
retro-reflective cat, a ninja carrying a turtle shell, and a foam horse. The
first two examples are captured with two NLOS walls. As the horse has a
slender body, we tried a different wall setting and only applied the NLOS data
from the right wall. Fig. 8 shows the qualitative reconstruction results. In
the retro-reflective cat example, our method successfully reconstructs the
complete and clear shape, including the slender tail, limbs, and ears. This
demonstrates our method’s ability to handle objects of different materials. BP
and NeTF lose details of the object, such as the cat’s ears. In the case of
the ninja which contains complex geometry and material (the human is plastic
while the turtle is retro-reflective), our method also successfully
reconstructs the limbs and complete turtle shell, as well as the gaps between
the human body and the shell. BP fails to reconstruct the circular turtle
shell while NeTF reconstructs a messy, rough geometry. In the case of the
foaming horse, Omni-LOS produces a continuous geometry. In contrast, both
comparative methods show seams between the LOS and NLOS parts, as well as
pixelated LOS reconstruction results.

##  7 Conclusion

Figure 9: Our method has the potential for various future applications,
especially for archaeological tasks. (a) we show a flying drone probing a
dangerous historic site on the cliff. (b) is the enlarged photograph of the
orange region. (c) common LOS-based reconstruction methods only recover the
front face of the statue, while Omni-LOS can be applied to reconstruct the
whole shape of the statue, even its back part is hidden within cavernous
recesses.

3D shape reconstruction plays an important role in computer vision, with
various applications ranging from virtual and augmented reality to
archaeological preservation and rescue missions. In this paper, we proposed
Omni-LOS, a novel approach that enables holistic shape reconstruction from a
single scan spot. Our method opens a new door for in-the-wild reconstruction
tasks. For example, in some challenging and dangerous cases like
archaeological scenes or rescue missions, our system allows for flexible full-
shape reconstruction from the front view, eliminating the requirements of
archaeologists or rescue members going to dangerous, unknown back regions.
Future work can focus on developing a more portable and efficient system that
can be equipped in the real world, as well as further improving the
reconstruction quality to recover fine details of the object.

## Acknowledgments

The authors would like to thank Linlin Li for his help with the figure, Zi
Wang for helpful comments on the text, and Zhengqin Pan for his support in the
data capture and initial discussion.

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The above proof would be fairly straightforward, and included much sooner in
the paper, if we knew right away that very ampleness gave us a two-dimensional
very ample family of plane curves. The reason we had to wait until now to
present the result is that we needed Proposition 6.1 to get such a family.

Before moving on, we point out one more fact that is relevant to pseudoplanes.
As we have seen, very ampleness implies non-local modularity of strongly
minimal structures. In Corollary 6.12 below, we show, conversely, that non-
local modularity (in fact, non-1-basedness) implies the existence of a very
ample stationary type. This is true for stable theories, and the proof goes
through complete type-definable pseudoplanes. Recall the following (see [15,
§, 4 Definition 1.6]):

######  Definition 6.9.

A complete-type-definable pseudoplane consists of a complete type of a pair of
(potentially imaginary) tuples $p=\operatorname{tp}(b,c)$ such that:

  1. (1)

$b\notin\operatorname{acl}(c)$.

  2. (2)

$c\notin\operatorname{acl}(b)$.

  3. (3)

If $c^{\prime}\neq c$ and
$\operatorname{tp}(bc)=\operatorname{tp}(bc^{\prime})$ then
$b\in\operatorname{acl}(cc^{\prime})$.

  4. (4)

If $b^{\prime}\neq b$ and
$\operatorname{tp}(bc)=\operatorname{tp}(b^{\prime}c)$ then
$c\in\operatorname{acl}(bb^{\prime})$.

The following is proved in ([15, §4, Lemma 1.7]):

######  Fact 6.10.

A stable theory is 1-based if and only if it does not admit a complete-type-
definable pseudoplane.

######  Remark 6.11.

There is a subtlety in Definition 6.9. Namely, suppose $(P,L,I)$ is an
arbitrary type-definable pseudoplane. Then $(P,L,I)$ might not contain any
complete-type-definable pseudoplane: indeed, it is important that the partial
type defining the incidence relation be complete. In fact, Fact 6.10 fails if
one considers general type-definable pseudoplanes. Let us sketch why. We leave
it to the reader to verify that in a 1-based stable theory one can have a
definable set $X$ equipped with a definable function $f:X\rightarrow X$ with
all fibres infinite. One can then build a definable pseudoplane $(P,L,I)$ by
letting $P=L=X$ and letting $I$ be two copies of $f$ (one in each direction).

We now show:

######  Corollary 6.12.

A stable theory admits a very ample non-algebraic stationary type if and only
if it is not 1-based.

###### Proof.

Let $T$ be a stable theory. First suppose $T$ is 1-based. We show that no non-
algebraic stationary type is very ample. To see this, let
$p=\operatorname{tp}(a/c)$ be stationary, where $c=\operatorname{Cb}(p)$.
Since $p$ is not algebraic, there is $b\models p$ with $b\neq a$. Now by
1-basedness, $c\in\operatorname{acl}(a)$, so automatically
$c\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{a}b$, which
shows that $p$ is not very ample.

Now suppose $T$ is not 1-based. By Fact 6.10, there is a complete-type-
definable pseudoplane $\operatorname{tp}(bc)$. Let
$d=\operatorname{Cb}(\operatorname{stp}(b/c))$. Clearly,
$d\in\operatorname{acl}(c)$. On the other hand, by Definition 6.9, any
$c$-conjugate of $\operatorname{tp}(b/c)$ is not parallel to
$\operatorname{tp}(b/c)$. Thus $c\in\operatorname{dcl}(d)$, so in particular
$c$ is interalgebraic with $d$.

We now claim that $\operatorname{tp}(b/d)$ is very ample. To see this, let
$b^{\prime}$ be any realization which is distinct from $b^{\prime}$. We want
to show that $b^{\prime}$ forks with $d$ over $b$; by interalgebraicity, it is
enough to show that $b^{\prime}$ forks with $c$ over $b$. By symmetry, it is
equivalent to show that $r=\operatorname{tp}(c/bb^{\prime})$ is a forking
extension of $q=\operatorname{tp}(c/b)$. But this is clear, since $r$ is
algebraic and $q$ is not. ∎

##  7\. Very Amplness in Strongly Minimal Groups

In this final section we give some results related to very ampleness in
strongly minimal groups. We first prove that non-locally modular strongly
minimal groups admit non-affine plane curves (a result that has been long
assumed but does not exist in writing). We then use this result to show that
Question 3.30 has a positive answer for groups, and that divisible strongly
minimal groups are already very ample; finally, we apply the very ampleness of
divisible groups to to characterize fullness of strongly minimal ACF-relics
with definable divisible group structures, in particular officially answering
an old question on expansions of the multiplicative group.

###  7.1. Existence of Non-Affine Plane Curves

In [9] Hrushovski and Pillay show that a stable group ${\mathcal{G}}$ is
1-based if and only if every definable subset of $G^{n}$ (any $n$) is _affine_
, i.e., a boolean combination of cosets of definable subgroups of $G^{n}$.
Restricted to strongly minimal (expansions of) groups, it has been well known
among experts that non-1-basedness (equivalently, non-local modularity) even
implies the existence of a non-affine plane curve. While this result is often
used in practice, there does not seem to exist a full treatment anywhere in
writing333The result is claimed in [10, Proposition 4.2], but there is a
mistake in the proof, that we do not see how to bridge.. It can, however, be
deduced by a particular application of Proposition 5.1. Thus, we dedicate the
present section to proving this result as a service to the community.

Throughout this subsection, we fix ${\mathcal{G}}$, a strongly minimal
expansion of a group. We will say that a definable set $X\subseteq G^{n}$ is
_affine_ if it is a (finite) boolean combination of cosets of definable
subgroups of $G^{n}$; and almost affine if it is almost equal to an affine
set. Note that affine sets are almost affine, and the converse holds for one-
dimensional sets (since finite sets are affine); more generally, note that
arbitrary finite Boolean combinations of affine sets are affine.

######  Notation 7.1.

For any non-empty definable set $X\subset G^{n}$, we let
$\operatorname{Stab}(X)$ denote the set of $g\in G^{n}$ such that $g+X$ almost
coincides with $X$; so $\operatorname{Stab}(X)$ is a subgroup of $G^{n}$ which
is definable over any set of parameters defining $X$.

We will show:

######  Theorem 7.2.

Let $\mathcal{G}=(G,+,\dots)$ be a strongly minimal expansion of a group. Then
the following are equivalent:

  1. (1)

$\mathcal{G}$ is locally modular.

  2. (2)

Every definable plane curve in $\mathcal{G}$ is affine.

  3. (3)

Every definable one-dimensional set $X\subset G^{n}$, for any $n$, is affine.

  4. (4)

Every stationary definable set $X\subset G^{n}$, for any $n$, is almost
affine.

  5. (5)

Every definable subset of each $G^{n}$ is affine.

We will be brief, as many parts of the above proof are standard, and follow
from either well known arguments or the main result of [9]. Our contribution
will be the quick proof of the implication (3) $\Rightarrow$ (4) using
Proposition 5.1. In fact there is also a rather easy direct proof of the
result of [9] restricted to the strongly minimal case (using only the
equivalence of local modularity and 1-basedness). So, for completeness, we now
give this proof (so that our proof of Theorem 7.2 does not depend on [9]).

######  Lemma 7.3.

The following are equivalent:

  1. (1)

${\mathcal{G}}$ is locally modular.

  2. (2)

If $\\{X_{t}:t\in T\\}$ is a non-empty faithful family of definable subsets of
a definable set $Y$, each of dimension $k$, then $\dim(T)\leq\dim(Y)-k$.

  3. (3)

Every stationary definable set $X\subset G^{n}$ (for all $n$) is almost
affine.

  4. (4)

Every definable set $X\subset G^{n}$ (for all $n$) is affine.

Moreover, if (1)-(4) hold, then every definable subgroup $H\subset G^{n}$ (for
all $n$) is $\operatorname{acl}(\emptyset)$-definable.

###### Proof.

(1) $\Rightarrow$ (2): Assume ${\mathcal{G}}$ is locally modular, and let
$\\{X_{t}:t\in T\\}$, $Y$, and $k$ be as in (2). Without loss of generality
$\\{X_{t}\\}$ and $Y$ are $\emptyset$-definable. Let $t\in T$ be generic and
$x\in X_{t}$ generic over $t$. Then $\dim(xt)=\dim(T)+k$, and by 1-basedness
(equivalently local modularity) $t\in\operatorname{acl}(x)$, so
$\dim(x)=\dim(T)+k$, thus $\dim(Y)\geq\dim(T)+k$, which is equivalent to (2).

(2) $\Rightarrow$ (3): Let $X\subset G^{n}$ be stationary of dimension $k$,
without loss of generality $\emptyset$-definable. Applying (2) shows that the
family of translates of $X$ is at most $n-k$-dimensional (i.e. a generic
translate of $X$ has canonical base of dimension at most $n-k$). It follows
that $\dim(\operatorname{Stab}(X))\geq k$, which easily implies that $X$ is
almost equal to a coset of $\operatorname{Stab}(X)$, proving (3).

(3) $\Rightarrow$ (4): By induction on dimension. Namely, assume (3) holds,
let $X\subset G^{n}$ be definable, and assume that all definable sets of
smaller dimension than $X$ are affine. By (3) applied to each stationary
component of $X$, it follows that $X$ is almost affine. So there is an affine
set $Y$ almost coinciding with $X$. Then $X$ is a Boolean combination of $Y$,
$X\setminus Y$, and $Y\setminus X$, and the second two are of these are affine
by induction.

(4) $\Rightarrow$ (1): Let $S$ be any strongly minimal plane curve; we show
that $\dim(\operatorname{Cb}(S))\leq 1$, implying local modularity. Now we can
write $S=C_{t_{0}}$ for some generic $t_{0}\in T$, where
$\mathcal{C}=\\{C_{t}:t\in T\\}$ is a $\emptyset$-definable (not necessarily
faithful) family of plane curves, and $T\subset G^{n}$ for some $n$. Absorbing
more constants to the language, if needed, we may assume $T$ is stationary. It
follows that the total space $C=\\{(x,t):x\in C_{t}\\}$ of $\mathcal{C}$ is
stationary, so (by (4)) almost coincides with a coset $H$ of some connected
definable subgroup of $G^{n+2}$. Note that (since $H$ is a coset) there is
some (necessarily $\emptyset$-definable) subgroup $K\leq G^{2}$ such that each
non-empty fibre $H_{t}\subset G^{2}$ is a coset of $K$. In particular, since
$t_{0}\in T$ is generic, it follows that $S=C_{t_{0}}$ almost coincides with a
coset of $K$. Thus $\operatorname{Cb}(S)$ is definable over a single element
of the one-dimensional group $G^{2}/K$, which gives
$\dim(\operatorname{Cb}(S))\leq 1$, as desired.

So we have shown that (1)-(4) are equivalent. Now suppose ${\mathcal{G}}$ is
locally modular. If there is a connected definable subgroup $H\subset G^{n}$
for some $n$ (say of dimension $k$) which is not
$\operatorname{acl}(\emptyset)$-definable, then there is a
$\emptyset$-definable infinite faithful family $\\{H_{t}:t\in T\\}$ of
$k$-dimensional subgroups of $G^{n}$, whose generic members are connected.
Passing to a subfamily and adding parameters if necessary, we may assume that
$\dim(T)=1$. Then the family $\\{X_{s}:s\in S\\}$ of all cosets of all $H_{t}$
is a $\emptyset$-definable faithful $(n-k+1)$-dimensional family of
$k$-dimensional subsets of $G^{n}$, which contradicts (2). ∎

Before proceeding, let us also isolate the following two well known and easy
facts:

######  Lemma 7.4.

Let ${\mathcal{H}}$ be a group of Morley rank 1 definable in an
$\omega$-stable structure. Then any definable subset of $G$ is affine.

###### Proof.

Let $S\subseteq G$ be any definable set, and let $H_{1},\dots,H_{k}$ be the
cosets of $H^{0}$ (the definable connected component of $H$). Then each
$H_{i}$ is strongly minimal, so $S\cap H_{i}$ is either finite or cofinite for
all $i$. Thus $S$ has finite symmetric difference with
$\bigcup\\{H_{i}:\operatorname{RM}(S\cap H_{i})=1\\}$. Since the right-hand
side is affine, set and the symmetric difference is finite (therefore also
affine), the lemma follows. ∎

######  Lemma 7.5.

Let $S\subset G^{n}$ be stationary and $A$-definable. Then the following are
equivalent:

  1. (1)

$S$ is almost affine.

  2. (2)

For generic independent $x,y,z\in S$ over $A$, $x+y-z$ is also generic in $S$
over $A$.

###### Proof.

First assume (1) holds. Then by stationarity, $S$ almost coincides with a
coset $H$ of a subgroup of $G^{n}$. Then $H$ clearly satisfies (2), so also
$S$ does.

Now assume (2) holds. Fix $z\in S$ generic over the parameters defining $S$.
Then (2), equivalently stated, gives that $S-z$ is almost contained in
$\operatorname{Stab}(S)$ (since given $x,y,z$ as in (2) it is clear that
$(x-z)\in\operatorname{Stab}(S)$). Thus $S$ is almost contained in
$\operatorname{Stab}(S)+z$. On the other hand, since $z$ is generic in $S$ it
follows by definition that $\operatorname{Stab}(S)+z$ is almost contained in
$S$, so that in fact $S\sim\operatorname{Stab}(S)+z$. Thus, (1) holds. ∎

Let us now proceed with the proof of Theorem 7.2:

###### Proof.

The implications (1) $\Rightarrow$ (2), (4) $\Rightarrow$ (5), and (5)
$\Rightarrow$ (1) are contained in Lemma 7.3. We show (2) $\Rightarrow$ (3)
and (3) $\Rightarrow$ (4):

(2) $\Rightarrow$ (3): Let $C\subset G^{n}$ be definable of dimension 1. Since
a finite union of affine sets is affine, we may assume $C$ is strongly
minimal. Since $C$ is infinite, there is a projection $\pi:G^{n}\rightarrow G$
with cofinite image in $G$. Without loss of generality $\pi=\pi_{1}$ (the
leftmost projection). Now for $i=2,\dots,n$, let $C_{i}=\pi_{1i}(C)\subset
M^{2}$ (the image of $C$ in the first and $i$th coordinates). Then $C_{i}$ is
a strongly minimal plane curve, so by (2) almost coincides with a strongly
minimal coset $H_{i}$. Deleting finitely many points from $C$ if necessary, we
may assume $C_{i}\subset H_{i}$ for all $i$.

Using that $H_{i}$ is a coset and $\pi_{1}(C)$ is cofinite in $G$, it follows
that $\pi_{1}(H_{i})=G$. So for each $i$, there is an element $c_{i}\in G$
with $(0,c_{i})\in H_{i}$. Then replacing $C$ with $C-(0,c_{2},\dots,c_{n})$
if necessary, we may assume each $H_{i}$ is in fact a subgroup of $G^{2}$.

Now let $H=\\{(x_{1},\dots,x_{n}):(x_{1},x_{i})\in H_{i}\textrm{ for
}i=2,\dots,n\\}$. Then $C\subset H$, and it is easy to verify that $H$ is a
one-dimensional definable subgroup of $G^{n}$. So by Lemma 7.4, $C$ is affine.

(3)$\Rightarrow$(4): If ${\mathcal{G}}$ is locally modular this is immediate
by Lemma 7.3. So assume ${\mathcal{G}}$ is not locally modular. Let $X\subset
G^{n}$ be stationary of dimension $m$ and definable over $A$. Since finite
sets are affine, we may assume that $m\geq 1$. Then by Proposition 5.1, there
is $B\supset A$ and a strongly minimal definable set $Y\subset X$ over $B$,
such that any three independent generics in $Y$ over $B$ are also independent
generics in $X$ over $A$. Let $x,y,z$ be as such. By (3) $Y$ is affine, so by
Lemma 7.5 $x+y-z$ is generic in $Y$ over $B$, and thus (again by the choice of
$Y$) also generic in $X$ over $A$. By Lemma 7.5 again, this implies that $X$
is almost affine. ∎

###  7.2. Very Ampleness in Groups

We now apply Theorem 7.2 to show that strongly minimal groups have very ample
finite quotients, and conclude that divisible strongly minimal groups are
already very ample. We will need the following, which is an easy exercise (see
[4, Lemma 3.10] and Example 3.18):

######  Lemma 7.6.

Let $\mathcal{G}=(G,+,\dots)$ be a non-locally modular strongly minimal
expansion of a group, and let $C\subset G^{2}$ be a strongly minimal plane
curve.

  1. (1)

The following are equivalent:

    1. (a)

The family of translates $\\{C+t:t\in T^{2}\\}$ is faithful.

    2. (b)

The family of translates $\\{C+t:t\in T^{2}\\}$ is very ample.

    3. (c)

The group $\operatorname{Stab}{(C)}$ is trivial.

  2. (2)

The following are also equivalent:

    1. (a)

$C$ is affine.

    2. (b)

$\operatorname{Stab}(C)$ is infinite.

Now we show:

######  Theorem 7.7.

Let $\mathcal{G}=(G,+,\dots)$ be a non-locally modular strongly minimal
expansion of a group. Then:

  1. (1)

There is a finite subgroup $H\leq G$ such that $G/H$ is very ample. In
particular, $\mathcal{G}$ admits a very ample sort.

  2. (2)

If $\mathcal{G}$ is divisible, then $\mathcal{G}$ itself is very ample.

###### Proof.

First we prove (1). By Theorem 7.2, there is a strongly minimal plane curve
$C\subset G^{2}$ which is not affine. Then by Lemma 7.6, the group
$\operatorname{Stab}{(C)}\leq G^{2}$ is finite. Let $H\leq G$ be a finite
subgroup such that $\operatorname{Stab}(C)\leq H^{2}$ (for example, $H$ could
be the group generated by all coordinates of elements of
$\operatorname{Stab}(C)$). Let $C/H$ be the image of $C$ in $(G/H)^{2}$, by
applying the projection $G\rightarrow G/H$ to both coordinates. It follows
easily that, in the strongly minimal group $(G/H)$, the set $C/H$ is a
strongly minimal plane curve with trivial stabilizer. So by Lemma 7.6 again,
$G/H$ is very ample.

Now assume $G$ is further divisible; we prove (2). Let $H$ be as in (1), and
let $h=|H|$. Then the map $x\mapsto h\cdot x$ is well-defined on $G/H$, and so
gives a definable map $G/H\rightarrow G$. Moreover, by divisibility this map
is surjective with finite fibres. Thus, we are in the situation of Lemma 3.24
with $X=G/H$ and $Y=G$. So since $G/H$ is very ample, the lemma implies that
so is $G$. ∎

###  7.3. Expansions of Algebraic Groups

In the final subsection, we apply Theorem 7.7 to characterize ACF-definable
expansions of 1-dimensional divisible algebraic groups. Our motivation is as
follows. Suppose $(K,+,\times)$ is an algebraically closed field, and let
$K^{\textrm{lin}}$ denote the structure $(K,+,\\{\lambda_{a}:a\in K\\})$,
where $\lambda_{a}$ is the map $x\mapsto ax$. In the paper [14], Martin makes
the following two conjectures:

  1. (1)

There are no intermediate structures between $(K,\times)$ and $(K,+,\times)$.

  2. (2)

If $\operatorname{char}(K)=0$ then there are no intermediate structures
between $K^{\textrm{lin}}$ and $(K,+,\times)$.

(2) was proved by Marker and Pillay [12]; however, it seems that (1) has not
since been addressed. Our result in this subsection will in fact be a more
general statement implying both (1) and (2). Namely, we show:

######  Theorem 7.8.

Let $K$ be an algebraically closed field, and let $(G,\cdot)$ be the group of
$K$-points of a one-dimensional divisible algebraic group over $K$. Let
$G^{\mathrm{lin}}$ denote the structure endowing $G$ with the group operation
and all of its endomorphisms (as an algebraic group), and let
$G^{\mathrm{Zar}}$ denote the full $K$-induced structure on $G$. Then there
are no intermediate structures between $G^{\mathrm{lin}}$ and
$G^{\mathrm{Zar}}$.

######  Remark 7.9.

Since every endomorphism of the additive group is a scaling, and every
endomorphism of the multiplicative group is a power map (thus definable from
the group operation alone), we obtain (1) and (2) above. Moreover, note that
Theorem 7.8 also applies to elliptic curves over $K$.

###### Proof.

Let $\mathcal{G}=(G,\cdot,\dots)$ be a reduct of $G^{\mathrm{Zar}}$ which
properly expands $G^{\mathrm{lin}}$.

######  Claim 7.9.1.

There is a non-affine $\mathcal{G}$-definable set $X\subset G^{n}$ for some
$n$.

###### Proof.

It suffices to show that every connected definable subgroup of $G^{n}$ is
definable in $G^{\mathrm{lin}}$. So, let $H\leq G^{n}$ be a connected
definable subgroup, say of dimension $d$. So $H$ is stationary, which implies
there is an almost finite-to-one projection $H\rightarrow G^{d}$. Since $H$ is
a subgroup, this implies that $H\rightarrow G^{d}$ is everywhere finite-to-
one, and the fibers of $H\rightarrow G^{d}$ are cosets of a finite group, say
$K\leq H$. Let $k=|K|$, and let $kH$ be the image of $H$ under scaling by $k$.
Then the projection $kH\rightarrow G^{d}$ is the graph of a definable function
$f:G^{d}\rightarrow G^{n-d}$. Since $kH$ is a subgroup, $f$ is a homomorphism.
In particular, each coordinate component of $f$ (i.e. $G^{d}\rightarrow
G^{n-d}\rightarrow G$) is a definable homomorphism from $G^{d}$ to $G$, and
thus $f$ is the direct sum of $d$ definable endomorphisms of $G$. In
particular, this implies that $f$ is $G^{\mathrm{lin}}$-definable, and thus so
is $kH$.

Finally, let $H^{\prime}$ be the preimage of $kH$ under scaling by $K$; so
$H\leq H^{\prime}$. Note that by divisibility, scaling by $k$ is finite-to-one
on $G^{n}$; thus $\dim(H)=\dim(kH)=\dim(H^{\prime})$. Then, since $H$ is
connected, it must be the connected component $(H^{\prime})^{0}$. In
particular, $H$ is the image of $H^{\prime}$ under scaling by $l$ for some
$l$. Now to complete the proof, recall that $kH$ is
$G^{\mathrm{lin}}$-definable; thus so is $H^{\prime}$ by definition, and thus
(scaling by $l$) so is $H$. ∎

Now by the claim and the main result of [9] (or Lemma 7.3 if desired), it
follows that $\mathcal{G}$ is not locally modular. Then by divisibility and
Theorem 7.7, $\mathcal{G}$ is very ample. Finally, by [5] $\mathcal{G}$
satisfies the Zilber trichotomy; so by Theorem 4.12, $\mathcal{G}$ is full in
$K$. In other words, $\mathcal{G}$ is interdefinable with $G^{\mathrm{Zar}}$,
as desired. ∎

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