Title: NeuroGF: A Neural Representation for Fast Geodesic Distance and Path Queries
Venue: NeurIPS 2023
Accepted: yes
Keywords: geodesic distance

implicit representation

3D geometry
Abstract: Geodesics play a critical role in many geometry processing applications. Traditional algorithms for computing geodesics on 3D mesh models are often inefficient and slow, which make them impractical for scenarios requiring extensive querying of arbitrary point-to-point geodesics. Recently, deep implicit functions have gained popularity for 3D geometry representation, yet there is still no research on neural implicit representation of geodesics. To bridge this gap, we make the first attempt to represent geodesics using implicit learning frameworks. Specifically, we propose neural geodesic field (NeuroGF), which can be learned to encode all-pairs geodesics of a given 3D mesh model, enabling to efficiently and accurately answer queries of arbitrary point-to-point geodesic distances and paths. Evaluations on common 3D object models and real-captured scene-level meshes demonstrate our exceptional performances in terms of representation accuracy and querying efficiency. Besides, NeuroGF also provides a convenient way of jointly encoding both 3D geometry and geodesics in a unified representation. Moreover, the working mode of per-model overfitting is further extended to generalizable learning frameworks that can work on various input formats such as unstructured point clouds, which also show satisfactory performances for unseen shapes and categories. Our code and data are available at https://github.com/keeganhk/NeuroGF.

NeuroGF: A Neural Representation for Fast Geodesic Distance and Path Queries

Abstract

Geodesics play a critical role in many geometry processing applications. Traditional algorithms for computing geodesics on 3D mesh models are often inefficient and slow, which make them impractical for scenarios requiring extensive querying of arbitrary point-to-point geodesics. Recently, deep implicit functions have gained popularity for 3D geometry representation, yet there is still no research on neural implicit representation of geodesics. To bridge this gap, we make the first attempt to represent geodesics using implicit learning frameworks. Specifically, we propose neural geodesic field (NeuroGF), which can be learned to encode all-pairs geodesics of a given 3D mesh model, enabling to efficiently and accurately answer queries of arbitrary point-to-point geodesic distances and paths. Evaluations on common 3D object models and real-captured scene-level meshes demonstrate our exceptional performances in terms of representation accuracy and querying efficiency. Besides, NeuroGF also provides a convenient way of jointly encoding both 3D geometry and geodesics in a unified representation. Moreover, the working mode of permodel overfitting is further extended to generalizable learning frameworks that can work on various input formats such as unstructured point clouds, which also show satisfactory performances for unseen shapes and categories. Our code and data are available at https://github.com/keeganhk/NeuroGF .

1 Introduction

The computation of geodesic distances and paths on 3D mesh models has been extensively studied over the past few decades and plays a critical role in many geometry processing tasks, including texture mapping, shape description, correspondence, and deformation. Traditional computational geometry methods, such as discrete wavefront propagation [25, 8, 39, 41, 32] and geodesic graphs [46, 2], excel at computing exact or high-quality approximate geodesic distances and paths on meshes with arbitrary triangulation. However, these methods often suffer from computational inefficiency or require a pre-computation step. On the other hand, partial differential equation (PDE) methods [17, 10, 36] are renowned for their efficiency, flexibility and ease of implementation, but the accuracy of their results is sensitive to the quality of mesh triangulation.

It is also worth mentioning that existing geodesic algorithms are primarily designed for computing single-source all-destination (SSAD) or multiple-sources all-destination (MSAD) geodesics on meshes. While there are some works focused on all-pairs geodesic computations, such as [40, 4, 12], these methods are built upon relatively complex computational and optimization processes.

Dealing with the trade-off among accuracy, speed, and complexity for geodesic computation remains technically non-trivial.

As 3D deep learning continues to advance, integrating geodesic information into deep learning frameworks becomes increasingly important for a wide range of applications, enabling more accurate and efficient analysis and manipulation of complex 3D shapes. However, until now, none of the existing geodesic algorithms have been seamlessly integrated into deep neural networks.

In this paper, we make the first attempt to utilize deep neural networks for accurately and efficiently answering geodesic distance and path queries. Towards this goal, we propose neural geodesic fields (NeuroGFs), an overfitting paradigm for representing geodesic distances and paths of a given 3D shape in a single neural model in an implicit fashion. Our approach is highly desirable to applications that require extensive and frequent online queries of point-to-point geodesic computations. Evaluations on common 3D models demonstrate that NeuroGFs excel in solving both SSAD and point-to-point geodesic problems. They achieve exceptional performance and strike a satisfactory balance between representation accuracy, model compactness, and query efficiency. Specifically, NeuroGFs are capable of achieving computation times of less than 1 ms for solving SSAD on 3D meshes consisting of 400K vertices, surpassing traditional methods, such as the heat method and discrete geodesic graphs, by two orders of magnitude. Besides, NeuroGFs offer the unique advantage of encoding both 3D geometry and geodesics in a unified representation. It is important to note that despite its compact size, with only 259K hyperparamters, the NeuroGF network consistently delivers high accuracy, with a relative mean error of less than 0.5%. Moreover, to facilitate more diverse applications, we further investigate generalizable learning frameworks of NeuroGFs by introducing different types of 3D shape feature encoders, which can well generalize to unseen shapes and novel categories. These results strongly demonstrate that NeuroGF is an effective and efficient tool for geodesic queries on 3D meshes, opening up exciting possibilities for a wide range of potential applications in the field of 3D deep learning. The main contributions of this work can be summarized as follows:

We make the first attempt to explore neural implicit representations for answering geodesic queries on 3D meshes.

In addition to geodesic distances, we model shortest paths as discrete sequences of ordered 3D points and customize a learning process of feature-guided curve deformation, which can flexibly approximate the shortest path between any pair of source and target query points.

We construct a unified representation structure for both geometric and geodesic information of given 3D shapes. Compared with traditional computational and optimization-based approaches, the proposed NeuroGF framework is easy to implement, highly efficient for online query, and naturally enjoys the powerful parallelism of modern GPUs.

We evaluated the performances of our NeuroGF representation paradigms in both overfitted and generalizable working modes, demonstrating the great potential and technical extensibility of our approach.

2 Related Work

Discrete Geodesics. The existing techniques for computing geodesics on 3D surfaces can be broadly classified into six categories: computational geometry methods, graph methods, PDE methods, surface partitioning methods, and iterative methods.

Traditional computational geometry methods [25, 8, 39, 47, 41, 32] are capable of computing exact geodesic distances and paths on arbitrary meshes. However, these methods are computationally expensive, especially when dealing with large meshes. Additionally, while these methods are well-suited to solving the single-source/multiple-sources and all-destinations problem, they are not practical for computing all-pairs geodesic distances due to their high computational costs.

Graph-based methods [46, 2] exploit the local nature of the discrete geodesic problem by constructing a sparse graph, where each edge represents a short geodesic path. These paths are computed locally by using exact geodesic algorithms. By transforming the mesh-based geodesic problem into a graphbased shortest-path problem, graph-based methods can use efficient algorithms such as Dijkstra's algorithm to compute geodesic distances and paths. Furthermore, their results are guaranteed to be a true distance metric and have controllable accuracy. However, achieving high-accuracy geodesic

computation often requires significant memory for storing the graphs, as the complexity of the graph increases with the desired accuracy.

PDE methods are highly flexible and can work for a wide range of discrete domains, including point clouds, implicit functions, polygonal meshes, regular grids and even broken meshes. They solve the Eikonal equation either directly [17, 44, 5, 37] or indirectly [10, 3, 36]. PDE methods are efficient and easy to implement. However, since they compute only a first-order approximation of geodesic distances, they typically require meshes with fairly good tessellation. To obtain geodesic paths using PDE methods, the negative gradient of the computed geodesic distances must be back-traced. However, the accuracy of the resulting paths also depends on the mesh quality, as the gradient may not be continuous or may have sharp discontinuities at the mesh edges. Nevertheless, PDE methods remain popular for their efficiency and versatility in computing geodesics on various discrete domains, including those beyond polygonal meshes.

Surface partition methods aim to break down the discrete geodesic problem into small sub-problems. The geodesic tracks (GT) method [23] places evenly-spaced, source-independent Steiner points on edges. Given a source vertex, it constructs a Steiner-point graph that partitions the surface into mutually exclusive tracks. Within each triangle, the tracks form sub-regions with approximately linear change in the distance field, enabling efficient approximation of geodesic distances. While the GT method supports high-quality geodesic path and isoline tracing, it remains inefficient for point-to-point distance and path queries due to the need for constructing the Steiner point graph using a Dijkstra-like sweep. The separator-based method (SEP) [12] performs a nested bisection of the input mesh using separator curves. It approximates the distances between each mesh vertex and a small relevant subset of these curves using polynomials. Consequently, the geodesic distance between any two mesh vertices can be approximated by solving a small number of simple univariate minimization problems. The SEP method efficiently handles point-to-point distance queries but does not provide geodesic paths. Furthermore, it is limited to query points located on mesh vertices, and it may not perform well on anisotropic meshes.

Additionally, there are iterative approaches like the edge flipping method [33] and the curve shortening method [38] that specially target the computation of geodesic paths and loops. These methods typically rely on an initial path and proceed in an iterative manner. Their time complexities depend on the area of the region the path sweeps. In contrast, our method does not require an initial path and can efficiently compute geodesics with a single forward pass through the network. It provides flexibility in solving both SSAD/MSAD and point-to-point geodesics. Furthermore, our method takes advantage of the high parallelism offered by modern graphics hardware, resulting in impressive runtime performance.

3D Deep Learning with Geodesics. Geodesics provide valuable structural information that can be seamlessly integrated into deep learning frameworks for diverse tasks in shape analysis. Several existing works have leveraged geodesic information within deep learning models. For example, GCNN [22] introduces manifold-adapted convolutional operators, constructing local geodesic coordinate systems for patch extraction. MeshCNN [14] combines specialized convolution and pooling layers operating on mesh edges by leveraging intrinsic geodesic connections for performing feature extraction on triangular meshes. Geodesic-Former [28] proposes geodesic-guided transformer model for few-shot instance segmentation of 3D point clouds, where geodesic distance information is leveraged to tackle the issue of imbalanced point density. GSA [18] designs geodesic self-attention to facilitate capturing long-range geometrical dependencies of point cloud objects. In the above studies, geodesics are exploited as additional input information to boost the subsequent learning models. Differently, GeoNet [15] explicitly learns geodesics-informed representations from unstructured point clouds in a supervised manner, which can be considered as a pre-training process of the target backbone networks. More recently, GraphWalks [31] proposes a differentiable geodesic path estimator learned in a supervised manner, but its prediction accuracy is still unsatisfactory. In terms of working modes, [15, 31] targeted at learning generalizable deep models for geodesics prediction, while in this work we tend to explore both overfitted and generalizable learning frameworks. Concurrently, [49] proposes a graph-based geodesic learning network that works on mesh models, while our implicit field-based modeling paradigm shows better flexibility for various input formats such as unstructured point cloud data.

Neural Implicit Representation for 3D Shape Geometry . The working mechanisms of existing neural implicit representations can be categorized as generalizable [29, 24, 9, 30, 6, 16] and overfitted

FC Layers

embedding

shortest path

guidance vector

feature-space

difference

append

Point-Wise MLPs

FC Layers

regress

signed distance

concatenation

FC Layers

deform

line primitive

FC Layers

regress

geometric query

target geodesic query

source geodesic query

geometric query

target geodesic query

source geodesic query

geodesic distance

FC Layers

embedding

shortest path

guidance vector

feature-space

difference

append

Point-Wise MLPs

FC Layers

regress

signed distance

concatenation

FC Layers

deform

line primitive

FC Layers

regress

geometric query

target geodesic query

source geodesic query

geodesic distance

paradigms [34, 35, 27, 21, 26, 45, 19]. Essentially, the former category of approaches tends to learn generalizable shape priors from a large collection of shapes at the cost of a lower-fidelity representation, while the latter category of approaches aims at reproducing a single shape with high-fidelity by accordingly overfitting a single neural network model. In this paper, we start by following the representation paradigm of the latter overfitted neural implicit models, i.e., separately overfitting a single neural network on a single target mesh model, to validate the representation power of our NeuroGFs. Then, we further make efforts to extend generalizable schemes to learn geodesics for unseen shapes and novel categories. Generally, despite the proliferation of research on deep implicit geometric information representation, to the best of our knowledge, there is still no such effort on implicit geodesics representation.

3 Proposed Method

3.1 Problem Formulation

Denote by S a given geometric shape represented in the form of a 3D mesh model M S = ( V , E , F ) , where V , E , F are the sets of vertices, edges, and faces, respectively. In an overfitted representation paradigm, our ultimate goal is to construct a neural model N Θ parameterized by network weights Θ , which are particularly optimized on the given shape S for fitting geodesic information. Specifically, given an arbitrary pair of geodesic queries, we deduce their corresponding point-to-point geodesic distance and shortest path through N Θ , which can be formulated as:

where q s and q t denote the source and target 3D points located on the underlying surface of shape S , d ∈ R + is a scalar value representing the geodesic distance between the two geodesic query points, and c s → t ∈ R M × 3 represents the shortest path from source q s to target q t modeled as a discrete sequence of M ordered 3D points, i.e., c s → t = { p m } M -1 m =0 , where the subscript m indicates the order index of each point.

In the offline training phase, we employ a supervised learning approach to overfit the neural network N Θ on a specific shape S . The training is performed using a collection of ground-truth geodesics pre-computed from its corresponding 3D mesh M . Once the overfitting process is completed, we can efficiently compute point-to-point geodesics for arbitrary online queries via a single forward pass through the trained network N Θ .

3.2 Architectural Design

In principle, we stick to the most concise technical implementation of the overall learning pipeline, without highly introducing sophisticated modules or mechanisms, for exploring the viability and potential of our proposed NeuroGF representation paradigm. As illustrated in Figure 1, all required learnable components of N Θ are simply built upon the stack of either fully-connected (FC) layers or point-wisely shared multi-layer perceptrons (MLPs). The input query points are individually lifted to high-dimensional feature vectors using FC layers, enabling subsequent manipulation of the data. For ease of representation, the feature embeddings of a pair of source and target queries, q s , q t ∈ R 3 , are denoted as e s , e t ∈ R D e . These embeddings are obtained by passing the inputs through three consecutive FC layers, progressively increasing the output channels. After obtaining the source and

target feature embeddings e s and e t , we proceed by deploying two separate learning branches to predict the geodesic distance d and the shortest path c s → t .

In addition to the geodesic distance and shortest path learning branches, we also incorporate an auxiliary learning branch for surface geometry representation by predicting the corresponding signed distance field (SDF). Given a query point q ′ ∈ R 3 and its lifted D e -dimensional feature embedding e ′ , we predict its signed distance s ′ ∈ R . This allows us to obtain a unified representation that encodes both geodesics and geometry. It is worth emphasizing that, during training, q ′ is sampled from the entire bounding sphere of the given shape S , while q s and q t are selected from the mesh vertex set V . During testing, q s and q t can be arbitrary points on the underlying shape surface, not limited to the vertices of the given mesh.

Overall, after the initial query point embedding, our neural network architecture consists of three parallel learning branches for predicting geodesic distances, shortest paths, and signed distance fields. While the three types of outputs are closely correlated, we deliberately design the intermediate learning processes in an inter-independent manner. This means that there is no explicit interaction (such as feature fusion or propagation) among the branches. We make this design choice to avoid redundant computations in scenarios where only one specific property is required for online query.

Neural Fitting of Geodesic Distances . The geodesic distance regression is modeled as a learnable function of the 'feature-space difference' between source and target queries. We compute an absolute difference vector between query embeddings e s and e t , which is further mapped to a scalar value of the geodesic distance through a stack of FC layers, which can be formulated as:

where | ∗ | means element-wisely taking the absolute value of each vector entry, and the geodesic distance learning branch B gdist : R D e → R + is implemented as three consecutive FC layers, where the output dimension of the last layer is set to 1 .

Neural Fitting of Shortest Paths . The shortest path generation is achieved as a feature-guided curve deformation process by adapting previous 'folding-style' 3D shape reconstruction decoders [43, 13], where pre-defined 2D regular grid primitives conditioned on unique shape signatures are smoothly deformed to the target 3D shape surface. Inheriting the same working mechanism, we tend to deform the straight line segment between a pair of source and target geodesic query points to the 3D curve of their shortest path, which is conditioned on a learnable deformation guidance vector exported from source and target point embeddings.

Formally, we start by sequentially sampling M discrete 3D points with uniform intervals along the directed straight line segment from source q s to target q t , as given by:

where l s → t ∈ R M × 3 serves as the initial line primitive to be deformed. In the online query phase, one can flexibly adjust the density of curve points according to the actual requirements by simply sampling the initial straight line segment with different uniform intervals. Next, the required curve deformation guidance vector can be deduced by:

where g s → t ∈ R D e uniquely determines the deformation result of the corresponding l s → t , [ ∗ ; ∗ ] denotes channel concatenation, ℏ : R D e + D e → R D e represents a learnable mapping function that can be simply implemented as a single FC layer.

To perform feature-guided curve deformation, we append g s → t to each point in the initial line primitive l s → t . The resulting M -by-( D e +3) feature matrix is row-wisely embedded for shortest path generation, which can be formulated as:

where B spath comprises four point-wise MLPs with the output dimension of the last layer set to 3 .

(Auxiliary) Neural Fitting of Signed Distances . As discussed before, there is an auxiliary learning branch that consumes the feature embedding e ′ of the given geometric query point q ′ as input and regresses a scalar value of its signed distance, which can be formulated as:

where B ′ sdist : R D e → R is implemented as three consecutive FC layers with the output dimension of the last layer set to 1 .

3.3 Learning Objective

Corresponding to the three types of outputs, the overall learning objective of NeuroGF representations comprises three loss function terms for the supervision of predicted geodesic distances d , curve points of shortest paths c s → t , and signed distances s ′ . Additionally, we also impose two auxiliary constraint terms on the generated shortest paths to enhance the consistency of the three output shape properties. Specific mathematical formulations are given as follows.

Supervision of Geodesic Distances . We compute L 1 loss between the predicted and ground-truth geodesic distances d and ˜ d as:

Supervision of Shortest Paths . We point-wisely compute L 1 losses between the generated and ground-truth curve points of shortest paths c s → t and ˜ c s → t as:

Note that since we uniformly use M curve points for approximating shortest paths during training, the raw ground-truth geodesic paths typically contain different numbers of points. For the convenience of supervision, during data pre-processing, points in the raw ground-truth geodesic paths are densely interpolated and then resampled to the desired number of M points.

Supervision of Signed Distances . We compute L 1 loss between the predicted and ground-truth signed distances s ′ and ˜ s ′ as:

Consistency Constraint of Curve Lengths . For the same input pair of geodesic query points, we can simultaneously obtain their geodesic distance d and shortest path c s → t = { p m } M -1 m =0 . To promote the consistency between d and c s → t , we explicitly minimize the difference of curve lengths between the predicted c s → t and its ground-truth ˜ c s → t = { ˜ p m } M -1 m =0 as:

where we roughly approximate the curve length as the summation of pair-wise Euclidean distances between adjacent points.

Distribution Constraint of Curve Points . Theoretically, each curve point p m of the shortest path is supposed to be located on the underlying shape surface, with zero signed distance value. However, under the generative modeling framework, there is no strict guarantee to eliminate deviation between the generated geodesic curves and the corresponding shape surface. To this end, we further constrain the spatial distribution of the generated curve points by minimizing the absolute values of their signed distances, as given by:

where N ϕ : R 3 → R represents an independent neural model overfitted on the given shape for the fitting of signed distance fields in advance, whose network parameters are fixed. Given an arbitrary spatial query, N ϕ outputs a scalar of the corresponding signed distance value, offering a natural way of constraining the generated curve points in a differentiable manner.

4 Experiments

4.1 Implementation Details

Testing Shapes . We experimented with a variety of 3D shape models that are commonly adopted in the geometry processing community, covering different complexity in terms of both geometry and topology, as displayed in Figure 2. Detailed mesh statistics can be found in Table 1. The anisotropy τ quantifies the level of anisotropy in the meshes. Meshes with τ > 3 are classified as anisotropic models. In our experiments, all the testing models have already been uniformly scaled into a unit sphere.

armadillo

bimba

bunny

cow

dragon

fandisk

heptoroid

maxplanck

nail

bucket

Training Data Preparation . We employed DGG-VTP [1] and fast discrete geodesic graphs (fDGG) [2] to generate the ground-truth geodesic distances and shortest paths, respectively. Specifically, we set the accuracy control parameter ε = 10 -7 for both DGG-VTP and fDGG. This parameter ensures that the computed geodesic distances and paths have accuracy comparable to the results obtained from the exact VTP method [32] using single-precision floating points. It is worth emphasizing that training NeuroGF on a given mesh does not require the utilization of geodesics between every pair of mesh vertices, which would be computationally expensive. Instead, we selected a subset of vertices and then uses the distances and paths between them. In our implementation, we sampled 20K vertices using mesh simplification tools, such as QSlim [11], for each input mesh. Our experiments show that the down-sampling strategy is sufficient for achieving satisfactory learning results. We also observed that exhaustive preparation of all-pairs geodesics for the input mesh did not further bring any obvious gain of accuracy.

Network Configuration . The overall learning framework is composed of four network components. The initial query point feature embedding is achieved through a stack of three FC layers with output dimensions of { D e / 4 , D e / 2 , D e } , where D e is an adjustable hyperparameter that controls the whole network size. Then comes the subsequent three learning branches for the fitting of geodesic distances, shortest paths, and signed distances, respectively. Specifically, B gdist comprises three FC layers with output dimensions of { D e / 4 , 64 , 1 } , B spath is a stack of point-wise MLPs with output dimensions of { 128 , 64 , 32 , 3 } , B ′ sdist comprises three FC layers with output dimensions of { D e / 4 , 64 , 1 } .

We adopted D e = 256 for constructing our baseline representation model, which totally contains 259 K network parameters. Note that the same number of parameters applies to all different testing shapes without changing the network structure.

Optimization Strategy . We adopted the popular AdamW [20] optimizer for parameter updating with 500 training epochs, with the learning rate gradually decaying from 0 . 01 to 0 . 0001 scheduled by cosine annealing. During each training epoch, we randomly sampled around 30 K spatial points as signed distance queries, 90 K paired mesh vertices as geodesic distance queries, and 20 K paired mesh vertices as shortest path queries, which are repeatedly consumed as inputs for 200 iterations. In the whole training phase, we specified the number of geodesic curve points as M = 128 and M = 32 for long and short shortest paths, respectively, to facilitate batch-wise processing.

Evaluation Protocol . We evaluated the geodesic representation accuracy and online SSAD query efficiency of the proposed NeuroGF learning framework and made necessary quantitative comparisons with two representative computational approaches HM [10] and fDGG [2]. The quality of geodesic

Ground-Truth Curve Points

Our Predicted Curve Points

distance representation is measured by the mean relative error (MRE), which can be formulated as | d -˜ d | / ˜ d × 100% . For shortest path evaluation, we densely interpolate the predicted and ground-truth curves to around 1 K points and compute their similarity as ChamferL 1 distance.

4.2 Quantitative Evaluations and Visualization Examples

We performed quantitative evaluations on geodesic distances produced from different approaches, as compared in Table 1, where our NeuroGF achieves the lowest mean relative errors for most testing shapes. Particularly, the performance of the heat method (HM) [10] suffers from obvious degradation when dealing with anisotropic meshes (especially for dragon and bucket with anisotropy degree τ > 10 ), while our approach shows satisfactory robustness. We also visualized geodesic distance fields using isolines in Figure 4, which shows the smoothness of our results. We further compared the time efficiency of different approaches (including exact VTP [32] for ground truth geodesic distances and paths), where [32, 10, 2] run on the CPU (Intel i5 7500), while NeuroGF runs on the GPU (NVIDIA GeForce RTX 3090). fDGG allows the user to specify the desired accuracy of geodesic distances and paths. We set its accuracy parameter ε = 2 . 5% so that its results have an accuracy similar to ours, which enables us to make a fair comparison in terms of speed. Computational results show that our approach is much faster than the others for answering online SSAD queries.

In addition, the quality of the shortest paths exported from NeuroGFs is measured in Table 2 and visualized in Figure 3. We can observe that our predicted geodesic curves are highly close to ground truths even for heptoroid , which is characterized by highly complicated topological structures.

4.3 Extension to Generalizable Learning Frameworks

We made further efforts to extend the overfitting working mode of NeuroGFs introduced previously to generalizable learning frameworks. More specifically, as illustrated in Figure 5, we designed three versions of generalizable NeuroGFs using (a) autodecoder-based (i.e., DeepSDF-like [29]), (b) point transformer-based , and (c) graph convolution-based feature extraction strategies. Notably, NeuroGF equipped with (a) or (b) for shape encoding can directly work on point clouds during testing. We used the popular ShapeNet [7] mesh dataset pre-processed by [42], covering 13 different shape categories. We collected 3000 models from 8 categories as our training set. For each model, we only sparsely generated 2K ground-truth training pairs. For evaluation, we constructed different testing sets: 1)

FC Layers

embedding

shortest path

guidance vector

feature-space

difference

append

Point-Wise MLPs

FC Layers

regress

signed distance

concatenation

FC Layers

deform

line primitive

FC Layers

regress

geodesic distance

'

No need to change

'

'

Replaced by

'

concatenation

'

extracted from input shape

'

global descriptor

FC Layers

query point

feature

embedding

(a) Autodecoder-based (AutoDec)

shape_1

shape_2

shape_N

assigned with

assigned with

assigned with

code_1

code_2

code_N

(c) Graph Convolution-based (GraphConv)

'

Can be achieved in

multiple different ways

'

Set Abstraction

Set Abstraction

Self-Attention

Self-Attention

point cloud

Pooling

Pooling

EdgeConv

EdgeConv

b

e

b

e

8

ER

gs-t

110

Cs→t

=

{Pm}m=0

A-1

et

d ∈ R+

(V,E,F)

3.5

0.85

1

armadillo

3

bimba

.bucket

0.8

(%)

MeanRelativeError

2.5

Kuunq.→

cow

uobep.

(sw)

Cost

0.75

0.7

fandisk

2

-.heptoroid

maxplanck

Time

0.65

1.5

nail

0.6

SSAD

0.55

0.5

0.5

0.45

0

0.4

0

100

200300400500600700800

9001000

0

100

200300400500600700800

9001000

NumberofNetworkParameters(K)

SN-Airplane, SN-Chair, and SN-Car are collected within the same categories of airplane , chair , and car , each of which containing 500 models; 2) SN-8x50 is collected from the same 8 categories as in the training set, but each shape is unseen during training. 3) SN-5x50 is collected from the other 5 different categories.

Quantitative results are reported in Table 3. The testing results on (a) validate the category-specific representation capability (with about 3% MRE). The testing results on (b) show that our extended approach equipped with a powerful deep point encoder works well on point clouds for both seen and unseen categories. (c) further incorporates mesh connectivity cues, thus achieving better performance.

4.4 Ablation Studies

In the preceding experiments, we uniformly configured D e = 256 , corresponding to 259 K network parameters. Here, we further investigated the effects of scaling the network size by changing D e to 64 , 128 , and 512 , leading to three different variants with 35 K and 84 K, and 916 K parameters, respectively. As illustrated in Figure 6, as the network complexity increases, the geodesic representation accuracy gets stably enhanced. Furthermore, to reveal the specific influences of the different learning components and supervision objectives involved in our approach, we performed necessary ablative analyses as presented in Table 4. We can observe that removing any of them leads to some degree of performance degradation, which demonstrates their necessity and effectiveness.

5 Conclusion and Discussion

This paper made the first effort to investigate learning-based neural implicit representations for 3D surface geodesics in both overfitted and generalizable working modes. The proposed NeuroGF achieves accurate and highly efficient point-to-point queries of both geodesic distances and shortest paths, which can be naturally combined with the learning of signed distances to further produce a

unified geodesic and geometric representation structure. Experiments demonstrated that our approach achieves comparable, if not better, geodesic representation accuracy against previous representative computational and optimization-based approaches. We believe that NeuroGF opens up new and promising paradigms for the complicated problem of geodesic representation and computation.

0%

5%

0%

5%

While we have demonstrated superior runtime performance of NeuroGF in answering point-to-point distance queries and solving the SSAD problem, our approach is still subject to a few limitations. There is no guarantee that the generated geodesic path strictly lies on the underlying surface, as the path points may exhibit varying degrees of deviation from the surface. To enhance the accuracy of geodesic paths, post-projection/refinement techniques can be employed, utilizing direction cues from signed distance fields. As shown in Table 5, the shortest path representation accuracy will further improve after pre-processing for making curve points locate exactly on the underlying surface. Although our approach achieves highly stable and satisfactory mean relative errors, it is worth noting that the metric of maximal errors can sometimes reach higher values, as depicted in Figure 7, where the relative error can rise up to approximately 5 %. To address this, incorporating more advanced optimization strategies, such as online hard example mining, may be necessary.

References

Y. Y. Adikusuma, J. Du, Z. Fang, and Y. He. An accuracy controllable and memory efficient method for computing high-quality geodesic distances on triangle meshes. CAD , 150:103333, 2022.

Y. Y. Adikusuma, Z. Fang, and Y. He. Fast construction of discrete geodesic graphs. TOG , 39(2):14:1-14:14, 2020.

A. G. Belyaev and P.-A. Fayolle. On variational and pde-based distance function approximations. CGF , 34(8):104-118, 2015.

O. Burghard and R. Klein. Simple, robust, constant-time bounds on surface geodesic distances using point landmarks. In VMV , pages 17-24, 2015.

M. Campen and L. Kobbelt. Walking on broken mesh: Defect-tolerant geodesic distances and parameterizations. CGF , 30(2):623-632, 2011.

R. Chabra, J. E. Lenssen, E. Ilg, T. Schmidt, J. Straub, S. Lovegrove, and R. Newcombe. Deep local shapes: Learning local sdf priors for detailed 3d reconstruction. In Proc. ECCV , pages 608-625, 2020.

A. X. Chang, T. Funkhouser, L. Guibas, P. Hanrahan, Q. Huang, Z. Li, S. Savarese, M. Savva, S. Song, H. Su, et al. Shapenet: An information-rich 3d model repository. arXiv preprint arXiv:1512.03012 , 2015.

J. Chen and Y. Han. Shortest paths on a polyhedron. In SoCG , pages 360-369, 1990.

Z. Chen and H. Zhang. Learning implicit fields for generative shape modeling. In Proc. CVPR , pages 5939-5948, 2019.

K. Crane, C. Weischedel, and M. Wardetzky. Geodesics in heat: A new approach to computing distance based on heat flow. TOG , 32(5):152, 2013.

M. Garland and P. S. Heckbert. Surface simplification using quadric error metrics. In SIGGRAPH , page 209-216, 1997.

C. Gotsman and K. Hormann. Compressing geodesic information for fast point-to-point geodesic distance queries. In SIGGRAPH Asia , pages 1-9, 2022.

T. Groueix, M. Fisher, V. G. Kim, B. C. Russell, and M. Aubry. A papier-mâché approach to learning 3d surface generation. In Proc. CVPR , pages 216-224, 2018.

R. Hanocka, A. Hertz, N. Fish, R. Giryes, S. Fleishman, and D. Cohen-Or. Meshcnn: a network with an edge. TOG , 38(4):1-12, 2019.

T. He, H. Huang, L. Yi, Y. Zhou, C. Wu, J. Wang, and S. Soatto. Geonet: Deep geodesic networks for point cloud analysis. In Proc. CVPR , pages 6888-6897, 2019.

C. Jiang, A. Sud, A. Makadia, J. Huang, M. Nießner, T. Funkhouser, et al. Local implicit grid representations for 3d scenes. In Proc. CVPR , pages 6001-6010, 2020.

R. Kimmel and J. A. Sethian. Computing geodesic paths on manifolds. PNAS , 95:8431-8435, 1998.

Z. Li, X. TANG, Z. Xu, X. Wang, H. Yu, M. Chen, and X. Wei. Geodesic self-attention for 3d point clouds. In Proc. NeurIPS , 2022.

C. Lin, N. Mitra, G. Wetzstein, L. J. Guibas, and P. Guerrero. Neuform: Adaptive overfitting for neural shape editing. Proc. NeurIPS , 35:15217-15229, 2022.

I. Loshchilov and F. Hutter. Decoupled weight decay regularization. In Proc. ICLR , 2019.

J. N. P. Martel, D. B. Lindell, C. Z. Lin, E. R. Chan, M. Monteiro, and G. Wetzstein. Acorn: Adaptive coordinate networks for neural scene representation. SIGGRAPH , 40(4), 2021.

J. Masci, D. Boscaini, M. Bronstein, and P. Vandergheynst. Geodesic convolutional neural networks on riemannian manifolds. In Proc. ICCVW , pages 37-45, 2015.

W. Meng, S. Xin, C. Tu, S. Chen, Y. He, and W. Wang. Geodesic tracks: Computing discrete geodesics with track-based steiner point propagation. TVCG , 28(12):4887-4901, 2022.

L. Mescheder, M. Oechsle, M. Niemeyer, S. Nowozin, and A. Geiger. Occupancy networks: Learning 3d reconstruction in function space. In Proc. CVPR , pages 4460-4470, 2019.

J. S. Mitchell, D. M. Mount, and C. H. Papadimitriou. The discrete geodesic problem. SICOMP , 16(4):647-668, 1987.

L. Morreale, N. Aigerman, P. Guerrero, V. G. Kim, and N. J. Mitra. Neural convolutional surfaces. In Proc. CVPR , pages 19333-19342, 2022.

L. Morreale, N. Aigerman, V. G. Kim, and N. J. Mitra. Neural surface maps. In Proc. CVPR , pages 4639-4648, 2021.

T. Ngo and K. Nguyen. Geodesic-former: A geodesic-guided few-shot 3d point cloud instance segmenter. In Proc. ECCV , pages 561-578, 2022.

J. J. Park, P. Florence, J. Straub, R. Newcombe, and S. Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In Proc. CVPR , pages 165-174, 2019.

S. Peng, M. Niemeyer, L. Mescheder, M. Pollefeys, and A. Geiger. Convolutional occupancy networks. In Proc. ECCV , pages 523-540, 2020.

R. A. Potamias, A. Neofytou, K. M. Bintsi, and S. Zafeiriou. Graphwalks: Efficient shape agnostic geodesic shortest path estimation. In Proc. CVPRW , pages 2968-2977, 2022.

Y. Qin, X. Han, H. Yu, Y. Yu, and J. Zhang. Fast and exact discrete geodesic computation based on triangle-oriented wavefront propagation. TOG , 35(4), 2016.

N. Sharp and K. Crane. You can find geodesic paths in triangle meshes by just flipping edges. TOG , 39(6), 2020.

V. Sitzmann, J. Martel, A. Bergman, D. Lindell, and G. Wetzstein. Implicit neural representations with periodic activation functions. Proc. NeurIPS , 33:7462-7473, 2020.

T. Takikawa, J. Litalien, K. Yin, K. Kreis, C. Loop, D. Nowrouzezahrai, A. Jacobson, M. McGuire, and S. Fidler. Neural geometric level of detail: Real-time rendering with implicit 3d shapes. In Proc. CVPR , pages 11358-11367, 2021.

J. Tao, J. Zhang, B. Deng, Z. Fang, Y. Peng, and Y. He. Parallel and scalable heat methods for geodesic distance computation. TPAMI , 43(2):579-594, 2021.

O. Weber, Y. S. Devir, A. M. Bronstein, M. M. Bronstein, and R. Kimmel. Parallel algorithms for approximation of distance maps on parametric surfaces. TOG , 27(4):104:1-104:16, 2008.

S.-Q. Xin, Y. He, and C.-W. Fu. Efficiently computing exact geodesic loops within finite steps. TVCG , 18(6):879-889, 2012.

S.-Q. Xin and G.-J. Wang. Improving Chen and Han's algorithm on the discrete geodesic problem. TOG , 28(4):104, 2009.

S.-Q. Xin, X. Ying, and Y. He. Constant-time all-pairs geodesic distance query on triangle meshes. In I3D , pages 31-38, 2012.

C.-X. Xu, T. Y. Wang, Y.-J. Liu, L. Liu, and Y. He. Fast wavefront propagation (FWP) for computing exact geodesic distances on meshes. TVCG , 21(7):822-834, 2015.

Q. Xu, W. Wang, D. Ceylan, R. Mech, and U. Neumann. Disn: Deep implicit surface network for high-quality single-view 3d reconstruction. Proc. NeurIPS , 32, 2019.

Y. Yang, C. Feng, Y. Shen, and D. Tian. Foldingnet: Point cloud auto-encoder via deep grid deformation. In Proc. CVPR , pages 206-215, 2018.

L. Yatziv, A. Bartesaghi, and G. Sapiro. O ( N ) implementation of the fast marching algorithm. JCP , 212(2):393-399, 2006.

W. Yifan, L. Rahmann, and O. Sorkine-Hornung. Geometry-consistent neural shape representation with implicit displacement fields. arXiv preprint arXiv:2106.05187 , 2021.

X. Ying, X. Wang, and Y. He. Saddle vertex graph (SVG): A novel solution to the discrete geodesic problem. TOG , 32(6):170:1-12, 2013.

X. Ying, S.-Q. Xin, and Y. He. Parallel Chen-Han (PCH) algorithm for discrete geodesics. TOG , 33(1):9:1-9:11, 2014.

W. E. Lorensen and H. E. Cline. Marching cubes: A high resolution 3d surface construction algorithm. ACM SIGGRAPH Computer Graphics , 21(4):163-169, 1987.

B. Pang, Z. Zheng, G. Wang, and P.-S. Wang. Learning the geodesic embedding with graph neural networks. arXiv preprint arXiv:2309.05613 , 2023.

A Appendix

1) Visualization of MSAD Geodesic Distance Fields . We randomly selected 5 source points on the shape surface to produce the resulting MSAD geodesic distance field, as illustrated in Figure A1. Besides, we also visualized the triangulation of the testing meshes in Figure A2.

20S

armadillo

bimba

bucket

bunny

cow

nail

dragon

fandisk

heptoroid

maxplanck

2) Visualization of SSAD Geodesic Distance Fields . We provided more visual comparisons of the SSAD geodesic distance fields exported from different approaches, as presented in Figure A3. To facilitate comparisons, we also marked the quantitative metrics of geodesic representation accuracy below each example computed from different approaches. In particular, Figure A4 presents close-up views for regions on the anisotropic nail model.

VTP

HM

fDGG

NeuroGF

0.51%

0.59%

1.03%

0.68%

0.46%

10.6%

0.46%

0.57%

0.67%

0.50%

0.42%

2.71%

0.18%

0.96%

3.35%

0.35%

0.66%

0.88%

0.87%

0.48%

1.75%

9

VTP

HM

fDGG

NeuroGF

3) Offline Training with Limited Ground-Truth Geodesics . For the preprocessing stage of preparing the required ground-truth geodesics for offline training of NeuroGFs, during which 4 data generation scripts run in parallel, we explored the effects of using different numbers of source points, i.e., 10000 , 1024 , 512 , 256 , 128 , and 64 . For each source point, we only preserved its geodesic distances between 4096 target points and its shortest paths between 2048 target points. As compared in Table A1, our approach can still achieve relatively satisfactory geodesic representation performances when only a limited amount of source points are exploited to produce ground-truth training data. Besides, we can also observe that using a much larger amount of training samples (i.e., with 10000 source points) only leads to insignificant performance gains.

Furthermore, note that here our preprocessing stage is not well-optimized, since we need to do lots of disk writing operations. We can directly integrate the approaches used for data preparations (fDGG [2] and DGG-VTP [1]) to Python to avoid costly disk writing time in the future. Comparatively, the recent work SEP [12] requires around 30 minutes for preprocessing a mesh with 20 K vertices. Its time complexity of preprocessing is O ( f ( n )+ m 3 n ) √ n ) , where n is the number of mesh vertices and f ( n ) is the time complexity of the used SSAD algorithm, which is at least O ( n ) time for fDGG [2]. We can see that this time complexity increases more than linearly in the number of mesh vertices. Thus, we can estimate that for the dragon mesh model with more than 400 K vertices, SEP would require more than 10 hours to finish preprocessing. This is much longer than our preprocessing stage using 1024 source points, which achieves geodesic representation accuracy comparable to SEP.

In addition, we also evaluated training with highly sparse ground-truth pairs. As shown in Table A2, when training NeuroGFs with only 8K and 2K ground-truth pairs, the errors do not rise up drastically, still maintaining relatively satisfactory.

4) Memory Footprint of Online Point-to-Point Geodesic Query . We compared the memory footprints of different approaches for online answering the geodesic distance between an input pair of source and target points. The quantitative results are reported in Table A3, where we can observe that our approach consistently maintains satisfactory memory efficiency. Thanks to our implicit querying paradigm, the resulting memory footprint is independent of the complexity of the given shape. Instead, for both the two competing approaches of HM [10] and fDGG [2], their memory footprints increase when dealing with larger meshes.

5) Evaluation on Large-Scale and Real-World Meshes . We experimented with a much denser version of the dragon model with 1.5M vertices and another classic graphics model lucy with 6.9M vertices. As reported in Table A4, our approach works well on these two highly dense mesh models. As illustrated in Figure A5, in addition to the object models used in the paper for testing, here we further selected several representative large-scale real-world mesh models, covering (a) indoor room, (b) urban scene, and (c) street view, which are from diverse dataset sources. It can be observed that our approach still achieves satisfactory performances on the three scene-level meshes with MRE rates of 0.37%, 1.14%, and 0.23%, respectively.

Dataset: ScanNet

#V: 1.96M

(a)

Dataset: SUM-Helsinki

#V: 0.83M

(b)

(c)

#V: 1.09M

Dataset: ETHZ-CVL-RueMonge

MRE: 1.14%

MRE: 0.23%

MRE: 0.37%

6) Geometry Representation Recovery . As pointed out in the paper, NeuroGFs jointly encode both 3D geometry and geodesics in a unified neural representation structure, although geodesic information is our major focus and geometric information just serves as a byproduct in this work.

Here, we provided necessary qualitative and quantitative evaluations for signed distance field information encoded in NeuroGFs. We deduced the predictions of signed distance values on a uniformlydistributed 3D grid with the resolution of 512 3 , and then measured the L 1 differences between the predicted and ground-truth signed distances, as reported in Table A5. For visualization, we applied the Marching Cubes [48] algorithm for isosurface extraction. The resulting mesh reconstructions are displayed in Figure A6.

7) Time Cost of Per-Model Overfitting . Practically, since NeuroGF is an offline per-input overfitting process, users can flexibly control the trade-off between training time and fitting accuracy. Typically, the training time cost for achieving comparable performances as reported in the paper is "minute-level". Table A6 provides statistics averaged on all the 10 testing models, showing that the convergence speed is fast. For example, it only takes less than half a minute to reach the MRE lower than 3%. From our experience, in most cases, the whole training effects will saturate within 10 minutes. Further extending the training process can only bring marginal performance gains at a slow pace.

ns

8) More Ablation Studies . We conducted more ablation studies in Table A7. We applied position encoding to input query point coordinates before high-dimensional feature embedding. We can observe that the resulting performances on the two complex meshes dragon and heptoroid are improved in different degrees, demonstrating the potential of exploring more advanced position encoding strategies. We also explored different ways of formulating the learning objectives in terms of the choice of distance metric. For example, we can choose to use L 2 loss instead of L 1 to calculate all learning objectives. Besides, for the formulation of shortest path supervision (i.e., formulated as Eq. (8) in the paper), we can also choose the popular Chamfer distance to supervise the curve deformation process. We can observe that in most cases using L 2 loss causes different degrees of performance degradation, and Chamfer distance leads to further performance improvement. Still, in our implementation, we did not adopt Chamfer distance as our final choice because of its additional computational burden and memory cost.

Peer reviews:
comments: SUMMARY:
The paper proposes a neural implicit representation of a 3d surface that enables querying for geodesic lengths and paths between points on the surface. A neural network is overfitted to one given surface. The input query points are embedded into high dimensional features whose euclidean distance represents the desired geodesic distance. Another branch learns to receive the high dimensional features to deform a straight line to the geodesic between the two points. A final branch learns to represent the SDF of the surface. The network is trained via strong supervision w.r.t given geodesics computed on the mesh. The results are shown to produce high accuracy when overfitted to well-known graphics models.

STRENGTHS:
The idea of overfitting to represent a geodesic paths can be quite useful, especially considering the ability to parallelize many queries through the network as one batch.

WEAKNESSES:
I find the work to provide a good technical solution to the given problem, however I cannot champion the paper: The technical contribution is quite limited. Beyond the core idea of overfitting to geodesics of a given shape, the rest of the approach is quite straightforward. As such this feels more as an application than a NeurIPS paper.

QUESTIONS:
1. Why use the SDF at all? why not provide the first 2 branches + the original mesh as the full representation? if the SDF is completely decoupled, what good does it bring.
2. is there any guaranteed that the geodesic paths actually lie on the isosurface of the SDF? if they are not, how is the difference reconciled?

LIMITATIONS:
Yes
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a solution to the estimation of the geodesic distance between a pair of points (source, target) on a mesh. The proposed approach relies on an implicit field. In particular, the implicit field learns/memorizes the distance between each pair of points. The authors sample a subset of mesh vertices and pre-compute the geodesic distance between them. These points are later used to optimize the network weights (implicit field). Each point is converted into a feature vector by a shared MLP. The difference between source and target features is used to predict the geodesic distance between the two points. On top of this, a second branch predicts the implicit field representing the shape itself (using the shared MLP features as input). Finally, a third branch using the same features, as concatenation, predicts the geodesic path connecting the source to the target as a set of points.\n\nThe authors demonstrate the effectiveness of the paper on a handful of meshes, showing numerical and qualitative results, while comparing with other approaches.\n\nSTRENGTHS:\nThe paper addresses an important problem. The manuscript describes the idea intuitively, clearly, and in a well-structured manner. The solution is interesting, original, and well-crafted. It leverages past approaches to achieve a different goal. Furthermore, the method is incredibly faster than chosen baselines while being accurate - both in terms of geodesic distance, geodesic path, and reconstruction error. This is possible with the use of NN, as it requires a single forward pass, compared to classic methods which require extensive computations.\n\nWEAKNESSES:\nThe proposed method uses a shared MLP to predict a set of features from 3D points. However, the input is not limited to points belonging to the surface. The same is the output of the geodesic path (one of the branches).\nThis seems to be quite an important limitation which may severely prevent the application of this method. Indeed, the authors introduce an extra loss term 5) to \"force\" the points on the curve to belong to the surface. Yet there is no guarantee this happens.\nFurthermore, to apply this technique we have to optimize a neural network, which takes time while the classic methods do not require such expensive pre-processing.\n\nQUESTIONS:\nThe solution is interesting and leaves me with some questions, it would be great if the authors can address some of these:\n- how much time does the pre-processing time require? ie optimizing the network + sampling points + estimating distances for training\n- the current formulation for the path relies on AtlasNet/Foldingnet, have the authors thought about using a different approach that cannot predict points off the surface? For example, the use of a transformer that predicts the index of the next point on the path.\n- how well does the method scale with very large meshes? (#V  > 1M)\n- how robust is the method on sparse meshes where very few vertices can be used to define geodesic distances? For example, on a cube where the only vertices are at the corners, and the query points are pairs on the faces (assuming this is possible).\n\nI thank the authors for the time spent addressing any of these concerns.\n\nLIMITATIONS:\nThere are no negative social impacts of this work.\nThe authors discussed the limitation of this work, although they did not detail properly the time requirement of the method (such as pre-processing time). I would recommend the authors add these details either in the main paper or in the supplementary. Furthermore, it would be great if the authors could evaluate the approach on a very diverse set of meshes, such as a cube and a super-dense version of \"dragon\".", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a framework to effeciently compute pairwise geodestic distances and shortest geodesic paths on the surface of a given mesh. This is accomplished by overfititng a neural learning model to a given mesh to regress the distances, paths and a SDF to reconstruct the surface. The geodesic distances/paths are learned in an implicit manner given pairs of points on the surface - specifically, points are sampled along the line between the points and the corresponding points on the geodesic path are predicted by a neural network.\n\nSTRENGTHS:\nThe problem being tackled, namely of effecient geodesic computations is important for the graphics community. The method is simple and easy to understand and reproduce.\n\nWEAKNESSES:\n1. In my opinion, the overfitting setting weakens the contribution significantly, especially considering that GraphWalks [Potamias et al] performs this task with a dataset and can predict even if with lower accuracy, on unseen test shapes. This work can be extended to learn on datasets, for eg simply by learning a global shape descriptor and appending it to the existing learned point features. \n\n2. Since its in the overfitting sitting, the results are less impressive. Neural networks are known for the ability to interpolate between seen data - i.e, if the model is trained to predict geodesics between enough number of pairs, its not too difficult to then get a good performance on the remaining pairs at inference. \n\n3. Additionally, since it is in the overfitting setting and thus needs to be trained for every new mesh, the training time itself should be considered as part of the time taken to compute the geodesics. The training time for each mesh is not mentioned in the paper.\n\n4. The network takes in point coordinates (via FC layers) and thus the neighborhood of a point is not encoded in any way - a better representation would be to encode as well the immediate neighborhood of the given pair of points. However, this has not been tried in the paper.\n\n5. The role of the SDF is simply as a byproduct, as noted by the authors in the supplemental. Table 3 shows that adding the SDF only provides a small improvement.\n\nQUESTIONS:\n1. The computation of the Distrubution constraint (eq 11) is not clear - the text says that a separate network is trained to predict the SDF of the generated curve poitns P_m, and that it is trained in advance and kept fixed. However, P_m itself is computed during training (from the line sampels), so how is the network trained in advance?\n\nLIMITATIONS:\nAs discussed in the weakness section, the primary limitation is that the method has to be trained for every given mesh, thus I'd consider the training time as part of the geodesic computation time. I'd suggest the authors to extend the method to learn on datasets with a global shape descriptor to make it more useful for the community. In my opinion, the overfitting setting weakens the result and is not as useful. The work would be very useful if it can be used to perform online computations on unseen shapes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper develops a neural network architecture to estimate the geodesic distances and shortest geodesics between query points on a given 2D surface. It also provides a signed-distance function field evaluated at the given query points. The architecture consists of a set of FC layers followed by Pointwise MLPs. The learning process is supervised. That is, the output of the network is compared with the ground truth distances and geodesic paths. The paper demonstrates these ideas for a number of 3D surfaces and compares the results with two previously published approaches. The results are found to be more accurate and take less computing time.\n\nSTRENGTHS:\nThere are many techniques in shape analysis of 3D objects that require computing geodesics or geodesic distances between arbitrary points. This paper develops a learning based solution that provides a slightly more accurate estimates and a somewhat faster pace.\n\nWEAKNESSES:\nIn my opinion this problem is not that challenging. The paper uses about 20K points on a surface and generates ground truth quantities (geodesic distances, geodesics, etc) from them. This seems to me like a very dense sampling. If on a 2D surface we have pairwise distances between 20K (or some such number) points, then the task of finding distances between remaining arbitrary pairs does not seem that challenging. Indeed, the errors across different methods are not that different – mostly within a one percentage point or so. \n\nIn terms of regression, one is learning a function f: S x S --> R_+ using millions of data points for the geodesic distance estimation. How about training the algorithm it only hundreds or even thousands of paired distances.\n\nQUESTIONS:\nThe conclusion states that the paper learn “neural implicit representations for 3D surface geodesics”. I do not quite understand what this means. Is there a mathematically precise statement that can replace this ambiguous claim.\n\nLIMITATIONS:\nAs the authors state, there is no guarantee that the shortest path lies on the actual surface. This seems like a big limitation. How much error does the post-processing introduces in this?", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-06"}

comments: SUMMARY:
This paper proposed to employ neural geodesic fields to implicitly represent (1) geodesic distance, (2) signed distance filed, (3) shortest geodesic path query using the overfit-paradigm. Experiments are conducted to demonstrate the effectiveness of the proposal.

STRENGTHS:
• The writing of this paper is clear and easy to follow. 
• The extension to use  implicit neural representation to represent geodesic distance and efficiency query is novel. It is a simple and yet seems effective approach under the overfitting paradigms for geodesic implicit representation.

WEAKNESSES:
1. The diversity of the mesh shape used in the experiments is limited. No realistic real-world mesh is experimented. The original scale of mesh shape in the experiments is not diverse enough: e.g. from small toy scale to large skyscraper scale mesh to validate the proposal in a more realistic setting.

QUESTIONS:
1. Page 4, section 3.2, any ablation study on input points embedding? Any comparison regarding FC layers vs position embeddings for point coordinates? Usually position embedding might work better for avoiding over-smoothing of the input signals for complicated shapes. It might worth exploring.
2.  Section 3.3, any ablation studies on the learning objectives? Why L1? How about others? 
3. Table 1, please annotate the best and worst performing entries in comparison. 
4. Section 4.3, more ablation analysis is needed. For example, more conclusive analysis is needed for Table 3.

LIMITATIONS:
NA
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-10