NeuroGF: A Neural Representation for Fast Geodesic Distance and Path Queries
Authors: Qijian Zhang
Junhui Hou
Yohanes Yudhi Adikusuma
Wenping Wang
Ying He
Venue: NeurIPS 2023 poster (2023)
Keywords: geodesic distance
implicit representation
3D geometry
TL;DR: the first attempt to represent geodesics on 3D mesh models using neural implicit functions and exceptional performance in solving the single-source all-destination and point-to-point geodesics

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
Qijian Zhang', Junhui Hou' Yohanes Yudhi Adikusuma', Wenping Wang', Ying He² Department of Computer Science, City University of Hong Kong, Hong Kong SAR, China Schooo of Computer Science and Engineering, Nanyang Technological University, Singapore DDpartmeet of Computer Science and Engineering, Texas A&M University, Texas, USA gijizhang3- cityu. edu. .kk, jh. hou@cityu. edu. hk Yohanes. adikusuma@ntuu edu. sg, wenpingStamu, edu, yheêntu.edu..
Abstract
Geodesics play a critical role in many geometry processing applications. Traditional algorithms for computing geodesics on 31) 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, 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 -ll-s geodesics of given 3D mesh model, enabling to efficiently and accurately answer queries of arbitrary point-to-poing 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 convenient way of jointly encoding both 3D) geometry and geodesics in a unified representation. Moreover, the working mode of per- model overfitting is further exlended 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: //////iitttt github. com Som/koeganbk//eeuroe.
Introduction
The computation of geodesic distances and paths on 3D mesh models has been extensively studied over the past few decades and plays critical role in many geometry processing tasks, including texture mapping, shape description, correspondence, and deformation. Traditional computational geometry methods, such discrete wavefrrnt 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-soune sources all-destination (MSAD) geodesics on meshes. While there are some works focused all-pairs geodesic computations, such as [40, 4, 12], these methods are built upon relatively complex computational and optimization processes.
"Corresponding author
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Dealing with the trade-off 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 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 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 NeuroGFss 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 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, NeuroGF's offer the unique advantage of encoding both 3D geometry and geodesics a unified representation. is important to note that despite its compact size, with only 259K hyperparamters, the NeuroGF network consistently delivers high accuracy, with relative mean error of less than 0.5%. Moreover. to facilitate more diverse applications, we further investigate generalizable learning frameworks of NeuroGFFs 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 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 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 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 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, 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 sorglll-oourremuullllee-ooo practical for computing all all-pairs geodesic distances due to their high computational costs.
and all-destinations problem, they are not
Graph-basee based methods [46, 2] exploit the local nature of the discrete geodesic problem by constructing 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 graph- based shortest-path problem, graph-based methods can use efficient algorithms such as Dijkstra's 's algorithm to compute geodesic distances and paths. Furthermore, their results are guaranteed to be true distance metric and have controllable accuracy. However, achieving high-accuracy geodesic

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computation often requires significant memory for storing the graphs, as the complexity of the graph increases with the desired aceuracy.
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 mack-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-pon distance and path queries due to the need for constructing the Steiner point graph using a Dijkstra- like sweep. The separator- baseed 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 small number of simple univariate minimization problems. The SEP method efficiently handles point-to-poing 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 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 coor- dinate 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. Geodessic-Fomas [28] proposes geodesic-gus 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-altention 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 gcodesics-ifformed representations from unstructured point clouds in supervised manner, which can be considered as pre- pre-training process of the target back- bone networks. More recently, Graph Walks [31] proposes a differentiable geodesic path estimator learned in 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 unstructured point cloud data.
In
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
3

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append
Point Winee MlP,
FC Layers
FC FC Layyer
deform
embedding
concatenation
guidence vector
line primitive
shortest path
Layers
FC Layers
geometric query
feature space difference
geodesic query
target geodesic quary
signed distance
geodesic distance
Overall workflow of querying point-to-point geodesic distances and shortest paths well as extracting signed distance fields for surface geometry representation from NeuroGFs.
Figure I:
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 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 deep implicit geometric information representation, to the best of our knowledge, there still no such effort on implicit geodesics representation.
paradigms [34, 35, 27, 21, 26, 45, 19].
3 Proposed Method
3.1 Problem Formulation
Denote by S a given geometric shape represented in the form of 3D mesh model F). where V, &, F are the sets of vertices, edges, and faces, respectively. In an overfitted representation paradigm, our ultimate goal is to construct neural model We parameterized by network weights O, 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-poin geodesic distance and shortest path through No. which can be formulated as:
{d, c3-+1
where q, and qe denote the source and target 3D points located on the underlying surface of shape S, d E IR+ is scalar value representing the geodesic distance between the geodesic query points, and C. E R x33 represents the shortest path from source qs to target qı modeled as a discrete sequence of M ordered 3D points, i.c., Cs-t = 1 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 Ne 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 single forward pass through the trained network Ng.
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 Ne are simply built upon the stack of either fully-connected (FC) layers or poiety 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 pair of source and target queries, q., E R3, are denoted as es, e; E R.D. These embeddings are obtained by passing the inputs through three consecutive FC layers, progressively increasing the output channels. After obtaining the source and

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learnable deformation guidance vector exported from target feature embeddings e, and e,, we proceed by deploying two separate learning branches to predict the geodesic distance and the shortest path Cz-st-
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' E R and its lifted D,-dimensional feature embedding e'. we predict its signed distance s' R. This allows us 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 qs and qu are selected from the mesh vertex set V. During testing, 9s and qi 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 required for online query.
Neural Fitting of Geodesic Distances. The geodesic distance regression is modeled as a learnable function of the "feature-spate space difference" between source and target queries. We compute absolute difference vector between query embeddings e, and e₁, which is further mapped to a scalar value of the geodesic distance through stack of FC layers, which can be formulated as:
d d= Bydiss (le, - e.)),
(2)
where means element-wisely taking the absolute value of each vector entry, and the geodesic distance learning branch Bydist R. - IR+ 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 achieved as 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 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, to target q.. as given by:
1,-,t = (q. + (qi q.)/(M - 1) m) m
(3)
where 1 E RMx3 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:
ga+e e()e,;e:)),
where ga-r E De uniquely determines the deformation result of the corresponding 1,-0, [+ denotes channel concatenation, fi DD + D. represents a learnable mapping function that can be simply implemented as a single FC layer.
To perform feature- guided curve deformation, append g--s to each point in the initial line primitive 1y-.t. The resulting M- 33 feature matrix is row- low-wisely embedded for shortest path generation, which can be formulated
C2-+ = Bepatti([gg-a l--e)),
(5)
where Bepath comprises four woint-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:
s' = B'diac''ee)
(6)
where B B'diis P. + IR is implemented as three consecutive FC layers with the output dimension of the last layer set to 1.
5

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3.3 Learning Objective
Corresponding to the three types oulputs, 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,-tt 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.
1) Supervision of Geodesic Distances. We compute L loss between the predicted and ground-truth geodesic distances d and d as:
Eydist = | dd
2) Supervision of Shortest Paths. We point-wisely compute I, losses between the generated and ground-truie curve points of shortest paths Ca-st and č,-st as:
Espath ||c,-+1- -
(8)
Note that since 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- pre-processing, points in the raw ground-truth geodesic paths are densely interpolated and then resampled to the desired number of M points.
3) Supervision of Signed Distances. We compute L1 loss between the predicted and ground-truth signed distances and 5' as:
Extiist ||s - 3'11
4) Consistency Constraint of Curve Lengths. For the same input pair of geodesic query points, we can simultaneously obtain their geodesic distance and shortest path C--xt { To promote the consistency between d and C.-st, we explicitly minimize the difference of curve lengths M-1
M
between the predicted Ca-st and its ground-truth čr- = {pm } as:
M-1 Σ
M-1
Lee
(llmm 111-
Σ (l Pm-
(10)
m=1
where we roughly approximate the curve length as the summation of pair-wise Euclidean distances between adjacent points.
5) Distribution Constraint of Curve Points. Theoretically, each curve point Pm of the shortest path 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:
1 M-1
Ldep
Σ (o((m))
(11)
M
m==
where Ng. : 1 + IR 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, Na outputs scalar of the corresponding signed distance value, offering a natural way of constraining the generated curve points differentiable manner.
4 Experiments
4.1
Implementation Details
Testing Shapes. We experimented with 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? Detailed mesh statistics can be found in Table 1. The anisotropy quantifies the level of anisotropy in the meshes. Meshes with > 3 are classified as anisotropic models. In our experiments, all the testing models have already been uniformly scaled into unit sphere.

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armadillo
bimba
bucket
bunny
COW
dragon
fandisk
heptoroid
maxplanck
nail
Figure 2: Visualization of selected testing shapes, among which the dragon, bucket, and nail meshes are highly anisotropic, and the heptoroid mesh is with high genus.
Table Comparison of geodesic representation accuracy and time efficiency for SSAD querying.
Running Time (ms) of SSAD Query
Mean Relative Error (%)
Mesh
#V (K)
VTP [32]
HM [10] IDGG [2] NeuroGF
HM [10]
IDGG [2] NeuroGF
armadillo
173
1.3
1778
194
59
0.5
1.03
0.59
0.51
bimba
75
1.1
985
82
20
0.5
0.67
0.57
0.46
bucket
35
14.1
500
18
16
0.5
3.35
0.96
0.18
bunny
35
1.4
374
29
10
0.5
0.87
0.58
0.44
cow
46
1.6
593
28
11
0.5
2.19
0.57
0.51
dragon
436
12.7
6209
246
145
0.7
10.6
0.46
0.68
fandisk
20
1.4
359
14
0.5
0.88
0.66
0.35
heptoroid
287
2.6
5789
212
86
0.6
1.75
0.48
0.87
maxplanck
49
1.2
797
33
11
0.5
0.79
0.57
0.39
nail
2.4
4.6
16
1.4
0.6
0.4
2.71
0.42
0.50
Training Data Preparation. We employed DGG-VTP [1] and fast discrete geodesic graphs (IDGG) [2] to generate the ground-truth geodesic distances and shortest paths, respectively. Specifi- cally, we set the accuracy control parameter è = 10-7 for both DGG-VTP and fDGG. This parameter ensures that the computed geodesic distances and paths have accuracy comparable
the results ob- tained 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 /4, De/2, De}. where D, 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, Bgdist comprises three FC layers with output dimensions of (D./4, 64, 1), Bepath is stack of point-wise MLPs with output dimensions of {128, 64, 32, 3), B'dii comprises three FC layers with output dimensions of [D./4, 64,
We adopted D, 256 for constructing 259K network parameters. Note that the same number of parameters applies to all different testing shapes without changing the network structure.
baseline representation model, which totally contains
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 30K spatial points as signed distance queries, 90K paired mesh vertices geodesic distance queries, and 20K paired mesh vertices as shortest path queries, which are repeatedly consumed as inputs for 200 iterations. In the whole training phase, specified the number of geodesic curve points M 128 and M 32 for long and short shortest paths, respectively, to facilitate batch-wise 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
7

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Mesh
Chamfer-LL (x10-
armadillo
1.366
bimba
1.301
Ground-Trune Truth Curve Points
Our Predicted Curve Points
nail
0.354
bunny
1.559
cow
0.941
dragon
1.319
fandisk
0,822
heptoroid
2.244
maxplanck
1.434
bucket
1.183
Table 2: Chamfer-L, errors between ground-truth and our predicted shortest path points.
Figure 3: Visual comparison of shortest paths.
Figure 4: Visualization of our predicted geodesic distance fields. Q Zoom in to see details.
distance representation is measured by the mean relative error (MRE), which can be formulated as |d d d x 100%. For shortest path evaluation, we densely interpolate the predicted and ground truth curves to around 1K points and compute their similarity Chamfer-L, L distance.
4.2 Quantitative Evaluations and Visualization Examples
We performed quantitative evaluations 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 10), while 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). IDGG allows the distances and paths. We set its accuracy parameter similar ours, which enables to make fair comparison in terms of speed. Computational results show that our approach is much faster than the others for answering online SSAD queries.
specify the desired accuracy of geodesic 2.5% so that its results have an accuracy
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 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 -ased (i.e., DeepSDF-like [29]). (b) point transformer-base and (c) graph convolution-baasee 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 categories as our training set. For each model, we only sparsely generated 2K ground-truthe training pairs. For evaluation, constructed different testing sets: 1)

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"No need to change"
QaN pain
uebedding
conolleatton
aronddig
"Con ochieved In multiple different ways""
"Replaced by"
A.C.
ailged
eniipred with
(a) Autodecoderr-aaeeeeeeeooee [AutoDec]
Point Transformer-basee based (PointTr)
[c] Graph Convolution (GraphConv)
Figure 5: Illustration of extending our NeuroGFs to generalizable learning frameworks. From the architectural point of view, only need replace the single-object query point embedding module (red box) by multi-object embedding module (orange box).
0.8
Moan Relative Error (50
SSAD (mo)
0.75
0.7
landisk
heptorold
0.66
magglance
1.5
nall
0.8
0.5
0.45
0.4
500
Number of Network Parameters (K)
Number of Network Parameters (K)
Figure 6: Statistics of SSAD geodesic distance querying with different network complexity.
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, each shape unseen during training. 3) SN-5x50 is collected from the other 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 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, uniformly configured D. 256, corresponding to 259K network parameters. Here, we further investigated the effects of scaling the network size by changing D, to 64, 128, and 512, leading three different variants with 35K and 84K, and 916K parameters, respec- tively. 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 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 surface geodesics in both overfitted and generalizable working modes. The proposed NeuroGF achieves accurate and highly efficient point-to-pon queries of both geodesic distances and shortest paths, which can be naturally combined with the learning of signed distances to further produce

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Table 3: MRE (%) performances of generalizable NeuroGF learning frameworks equipped with
different global shape feature extractors, including: (a) AutoDec, (b) PointTr, and (c) GraphConv.
SN-Airplane SN-Chair SN-Car
(b)
SN-8x50 SN-5x50
(c)
SN-8x50
SN-5x50
AutoDec 3.03
3.91
2.78
PointTr 3.28
4.16
GraphConv
2.94
3.55
Table 4: Influences of different learning components and supervision objectives, where the results are averaged on all testing shapes. The right two columns show the representation accuracy of our predicted geodesic distances and shortest paths (the lower, the better). The averaged statisties of our full implementation in termss of the two metrics are 0.49% and 1.25 x 10 -2 In particular, we mark the relative change within each bracket facilitate comparison.
B Bedist
Bapath
B'diit
leel
Edep
Mean Relative Error (%)
Chamfer-Ly (x10 3)
1.46 (1 0.21)
0.61 (1 0.12)
X
0.57 (1 0.08)
1.31 (1 0.06) 1.35 (1 0.10) 1.38 (1 0.13)
X
0.52 (1 0.03) 0.50 (1 0.01)
X
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-bant -based approaches. We believe that NeuroGF opens up new and promising paradigms for the complicated problem of geodesic representation and computation.
Table 5: Chamfer-Ly L1 (x10 3) errors between ground-truie and our predicted shortest path points after post-pun pest-processing.
armadillo
bimba
nail
bunny
dragon
fandisk
heptoroid
maxplanck
bucket
average
1.131
1.126
0.347
1.325
0.804
1.070
0.766
1.994
1.248
1.079
1.09
Figure 7: Error distribution of our predicted geodesic distance fields. Q Zoom in to see details.
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 guarantee that the generated geodesic path strictly lies on the underlying surface, the path points may exhibit varying degrees of deviation from the surface. To enhance the accuracy of geodesic paths, post-projcctioo/refieeeet 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-pro 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 where the relative eтror can rise up to approximately 5%. To address this, incorporating more advanced optimization strategies, such as online hard example mining, may be necessary.
Acknowledgement
This work was supported by the Hong Kong Research Grants Council under Grants CityU 1 11202320 and CityU 11219422.
10

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12

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A Appendix
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.
Figure A1: Visualization of MSAD geodesic distance fields computed by our approach. Q Zoom in
to see details.
armadillo
bimba
bucket
bunny
dragon
fandisk
heptoroid
maxplanck
nail
Figure A2: Visualization of mesh triangulations.
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.
13

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VTP
NeuroGF
fDGG
HM
0.51%
0.59%
1.03%
0.46%
0.57%
0.67%
0.18%
0.96%
3.35%
0.68%
0.46%
10.6%
0.50%
0.42%
2.71%
0.35%
0.66%
0.88%
0.87%
0.48%
1.75%
Figure A3: Visual comparison of SSAD geodesic distances. Below each example, also marked the corresponding mean relative error (%). Q Zoom to see details.
14

---

VTP
NeuroGF
fDGG
HM
Figure A4: Close-up views for geodesic distance fields the anisotropic nail model.
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, only preserved its geodesic distances between 4096 target points and its shortest paths between 2048 target points. As compared in Table A 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 need to do lots of
disk writing operations. We can directly integrate the approaches used for data preparations (fDGG [2]
and DGG-VTP [1]) Python avoid costly disk writing time in the future. Comparatively, the
recent work SEP [12] requires around 30 minutes for preprocessing a mesh with 20K vertices. Its
time complexity of preprocessing is (f +n)) where 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 400K vertices, SEP would require more than 10 hours to finish preprocessing. This much longer than our preprocessing stage using 1024 source points, which achieves geodesic representation accuracy comparable to SEP.
In addition, 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.
Table A1: Effects of offline training with different amounts of paired ground-truth geodesics on the dragon model. In the right two columns, respectively record the preprocessing (Prep.) time cost for generating ground-truth geodesic distance (G.D.) and shortest path (S.P.) data.
#Sources
MRE (%)
Chamfer- LA (x10-2)
Prep. G.D. (minutes)
Prep. S.P. (hours)
10000
0.66
1.297
119.1
21.6
1024
0.68
1.319
12.2
2.2
512
0.75
1.397
6.1
1.1
256
0.83
1.580
3.0
0.6
128
1.09
2.044
1.5
0.3
64
1.63
2.926
0.8
0.14
32
2.21
3.725
0.4
0.07
Table A2: Effects of training NeuroGFs with highly sparse ground-trui truth pairs.
#GT-Pairr
MRE (%)
#GT-Pairs
MRE (%)
#GT-Pairs
MRE (%)
Original 0.53
8K
4.05
2K
5.62
4) Memory Footprint of Online Point-to-Poine 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.
15

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Table A3: Comparison of memory efficiency for online point-to-point geodesic query.
Memory Footprint (MB) of Point-to-Poiis Geodesic Query
Mesh
#V (K)
HM [10] (on CPUs)
fDGG [2] (on CPUs)
NeuroGF (on GPUs)
armadillo
173
272
60
10
bimba
75
138
26
10
bucket
35
45
15
10
bunny
35
56
13
10
cow
46
70
17
10
dragon
436
597
152
10
fandisk
20
29
10
heptoroid
287
690
108
10
maxplanck
49
85
17
10
nail
2.4
2.9
1.3
10
5) Evaluation on Large-Scale and Real- World Meshes. We experimented with much denser version of the dragon model with 1 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-worke 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. 1,4,, and 0.23%, respectively.
Table A4: MRE (%) performances for much denser version of the dragon model and another popular lucy model.
Mesh
#V (M)
MRE (%)
Chamfer- L1 (x10-2)
dragon
1.5
0.70
1.335
lucy
6.9
0.58
1.404
(a)
[b)
0.37%
MRE: 1.14%
#V: 1.96M
#V: 0.83M
#V: 1.09M
MRE: 0.23% Dataset: ETHZ-CVL- CVL-Re
Dataset: ScanNet
Dataset: SUM--Hlsinkl
Figure A5: Experimental evaluations large-scale real-world meshes (with up 2 million vertices).
6) Geometry Representation Recovery. As pointed out in the paper, NeuroGFs jointly encode both 3D geometry and geodesics in unified neural representation structure, although geodesic information 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 infor- mation encoded in NeuroGFs.. We deduced the predictions signed distance values on uniformly- distributed 3D grid with the resolution of 5123, and then measured the J, 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 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 statisties averaged on all the testing models, showing that the convergence speed is fast. For example, it only takes less than half 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.
16

---

Table A5: Mean L1 errors between our predicted and ground-trutie signed distances.
Mesh
armadillo
bimba
bucket
bunny
cow dragon fandisk heptoroid maxplanck nail
L1 (x10
1.22
0.97
0.68
0.97
0.86
1.28
0.67
1.01
1.03
0.45
Figure A6: Visualization of mesh reconstruction deduced from NeuroGFs.
Table A6: Time cost for overfitting a single input 3D model.
MRE
<3%
<2%
<1 %
Training Time Cost
0.4min
1.7min
7.9min
Table A7: Ablation studies different variants of our technical implementations, where "variant (1)" means adding position encoding before fed into FC layers for query point feature embedding, "variant (2)" means replacing L1 loss with L2 loss for supervisions, and "variant (3)" means computing Chamfer distance between generated and ground truth curve points shortest paths C.- and Č.- for the formulation of spath (Eq. (8) in the paper).
dragon
heptoroid
Implementation Variants
MRE
Chamfer-Li
MRE
Chamfer-Li Li
(0) Original Implementation
0.68
1.319
0.87
2.244
(1) Adding PosEnc
0.62
1.253
0.79
1.921
(2) L2 Supervisions
0.71
1.345
0.88
2.227
(3) Chamfer Loss for Espath
0.66
1.286
0.82
2.149
8) More Ablation Studies. We conducted more ablation studies in Table A7. We applied position encoding to input query point coordinates before sigh-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 choose all learning objectives. Besides, for the formulation of shortest path supervision (i.e., formulated Eq. (8) in the paper), deformation process. We can observe that in most cases using L2 loss causes different degrees of performance degradation, and Chamfer distance leads 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.
L2 loss instead of L, to calculate
also choose the popular Chamfer distance to supervise the curve
17


Reviews:
[{"review_id": "Y5htGZpMzE", "reviewer": "Reviewer_Racd", "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. ", "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.\n2.  Section 3.3, any ablation studies on the learning objectives? Why L1? How about others? \n3. Table 1, please annotate the best and worst performing entries in comparison. \n4. Section 4.3, more ablation analysis is needed. For example, more conclusive analysis is needed for Table 3.\n\n", "limitations": "NA", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "• The writing of this paper is clear and easy to follow. \n• 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. \n\n", "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.\n\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer Racd]**\n\n### **[W1]** *Not enough shape diversity and scale of the used testing meshes.*\n\n\n**Response:** We would like to remind that some of the testing models used in our experiments (such as *armadillo*, *bunny*, *dragon*) are real-world meshes, which are created from real-scans. Please have a check from the website of *The Stanford 3D Scanning Repository*. \n\nStill, we do agree with the reviewer that it is valuable to perform evaluations on more diverse mesh models. To this end, here we considered both *1) data volume scale* and *2) scene representation scale*. \n\nFor the former, we experimented with a much denser version of the *dragon* model with 1.5M vertices and another classic graphics model *lucy* (which is also created from real scans) with 6.9M vertices. As reported in the Table below, our approach works well on these two highly dense mesh models.\n\n\n| Mesh | #V (M) | MRE (%) | Chamfer-$L_1$ ($\\times 10^{-2}$) |\n| :----: | :----: | :----: | :----: |\n| dragon | 1.5 | 0.70 | 1.335 |\n| lucy | 6.9 | 0.58 | 1.404 |\n\nFor the latter, we further performed evaluations on three large-scale real-scanned meshes covering (a) indoor room, (b) urban scene, and (c) street view, which are from diverse dataset sources. Since these realistic scene scans are highly messed up, we turn to use the classic Dijkstra algorithm to compute geodesics as training and testing data. As shown in Figure R2 of the uploaded PDF file, our approach still achieves satisfactory performances (with MRE rates of 0.37\\%, 1.14\\%, and 0.23\\%, respectively) on the three scene-level meshes.\n\n### **[Q1]** *Ablation study on the position encoding of query point coordinates.*\n\n**Response:** Thanks very much for your valuable suggestions. Accordingly, as reported in Table R1 of the uploaded PDF file, we conducted the corresponding ablation study by applying a classic position encoding operation (as used in NeRF, ECCV-20') to input query point coordinates before the subsequent FC layers for high-dimensional feature embedding. Indeed, 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.\n\n### **[Q2]** *Ablation study on the distance metric used to formulate the learning objectives.*\n\n**Response:** Thanks very much for your detailed evaluation of our specific choices of technical implementation. In fact, we did experiment with some 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., $\\ell_\\mathrm{spath}$ formulated as Eq. (8) in the paper), we can also choose the popular Chamfer distance to supervise the curve deformation process. The resulting performances obtained from such two different implementation variants are reported in Table R1 of the uploaded PDF file. 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.\n\n### **[Q3]** *Annotating the best and worst performing entries in Table 1 of the paper.*\n\n**Response:** Thanks very much for your useful suggestion to help improve the formatting quality of our manuscript. We will accordingly revise Table 1 in the final version.\n\n### **[Q4]** *Enriching conclusive analyses in Section 4.3 of the paper.*\n\n**Response:** Thanks very much for your valuable comments, and we are sorry for not being able to place detailed ablative analyses in the paper due to page limits. According to your suggestion, we will supplement more adequate and in-depth analyses to facilitate readers' understanding of our approach in the supplementary material."}, {"review_id": "390QGtrwSt", "reviewer": "Reviewer_oZNk", "summary": "This 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. ", "questions": "The 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. ", "limitations": "As 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?", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "\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. ", "weaknesses": "\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. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer oZNk]**\n\n\n### **[W1]** *Very dense sampling of ground-truth training data.*\n\n**Response:** There seems to be a misunderstanding about preparing ground-truth pairs for training. In our experiments, we will create a simplified version of mesh model with around 20K vertices if the original mesh is of larger scale (note that the testing data are still obtained from the original input mesh). Then as presented in the supplementary material (page 3, lines 17-19) we compute and preserve paired ground-truth geodesics between 1024 source points and 4096/2048 target points (4096 for geodesic distances, 2048 for shortest paths), rather than all pairs between the 20K vertices. The proportions of \"our used training pairs\" to \"all pairs between vertices of the original mesh\" for geodesic distance ground-truth preparation are given below. We can see that the ratio of our used pairs is very small, except for the sparse *nail* model.\n\n\n| Mesh &#124; | armadillo | bimba | bucket | bunny | cow | dragon | fandisk | heptoroid | maxplanck | nail |\n| :----: | :----: | :----: | :----: | :----: | :----: | :----: | :----: | :----: | :----: | :----: |\n| **Ratio** &#124; | 0.03% | 0.15% | 0.69% | 0.69% | 0.40% | 0.004% | 2.10% | 0.01% | 0.35% | 71.2% |\n\n\nMoreover, we did experiment with different numbers of ground-truth training data, as comprehensively evaluated in Table S1 of our supplementary material, where the second row (\\#Sources=1024) corresponds to the setup we used in the paper. We can observe that the resulting performance (MRE of 2.21\\% on the challenging dragon model) can still be competitive even when we only use 32 source points (the last row).\n\n### **[W2]** *Training with highly sparse ground-truth pairs.*\n\n**Response:** To better address your concern, in addition to the experiments conducted in Table S1 of our supplementary material, we further explored much sparer settings of paired ground-truth preparation. Due to limited time, here we only used five testing meshes *armadillo*, *bunny*, *cow*, *dragon*, and *nail*, and listed the average performances in Table R3 of the uploaded PDF file. When training NeuroGFs with only 8K and 2K ground-truth pairs, the errors still maintain relatively satisfactory.\n\nBesides, please also refer to our newly-added experiments (Table R4 of the uploaded PDF file) of extending NeuroGFs to be generalizable learning models (as asked by Reviewer yBGS), where we only prepared about 2K ground-truth pairs for each training mesh.\n\n### **[Q1]** *Ambiguous meaning of neural implicit representations for 3D surface geodesics.*\n\n**Response:** The concept of \"neural implicit representation\" has been popular in recent years, especially for representing 3D signals, such as DeepSDF (CVPR-19'), NeRF (ECCV-20'), and their numerous follow-up works. Generally, corresponding to \"implicit representation\", traditional 3D data structures like meshes, voxels, and point clouds are known as \"explicit representation\". That is, they have a finite number of explicitly stored data elements, like vertex/triangle, voxel occupancy status, and spatial point. Differently, neural implicit models tend to represent the target signal with infinite resolutions in a \"query-and-answer\" fashion. For example, given a certain 3D query point position, we feed it into a neural network, which outputs the scalar value of its signed distance. Thus, by densely sampling query points and collecting their outputs, we can flexibly reconstruct the surface geometry from the signed distance field with arbitrary resolution (depending on the number of queries).\n\nIn our case, the formal mathematical description of the proposed \"neural implicit representations for 3D surface geodesics\" has already been given by Eq. (1) in the paper. Specifically:\n- The input query is a pair of source and target points ($\\mathbf{q}_s$;$\\mathbf{q}_t$) located on the underlying surface. \n- The neural model outputs a scalar value of geodesic distance $d$ and a sequence of shortest path points $\\mathbf{c}_{s \\rightarrow t}$.\n\n### **[L1]** *1) No guarantee that the shortest paths lie on the surface; 2) Effects of post-processing.*\n\n**Response:** As both quantitatively and qualitatively illustrated in the paper, the generated shortest paths are close enough to the underlying iso-surface, meaning that we can conveniently deduce post-processed shortest paths whose curve points are exactly located on the surface by straightforward local projection. Here, we simply implement this process by locally sampling surface points and then performing nearest-neighbor matching for the raw outputs of curve points. As shown in Table R2 of the uploaded PDF file, such a post-processing procedure consistently brings further accuracy improvement for the prediction of shortest paths on ALL the testing meshes. On average, the error decreases from the original $1.25 \\times 10^{-2}$ to $1.09 \\times 10^{-2}$. Hence, we beg to differ with the judgment that \"This seems like a big limitation\"."}, {"review_id": "kxRpxZeHZt", "reviewer": "Reviewer_yBGS", "summary": "This 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.", "questions": "1. 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?", "limitations": "As 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.", "rating": 3, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "The problem being tackled, namely of effecient geodesic computations is important for the graphics community. The method is simple and easy to understand and reproduce.", "weaknesses": "1. 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer yBGS]**\n\n\n### **[W1]** *Weakened contribution due to the overfitting setting; Extension to learn on datasets.*\n\n**Response:** Thanks for your insightful advice, yet we beg to differ with the judgment that the overfitting setting \"significantly\" weakens our contribution. In terms of the overfitting paradigm, NeuroGF inherits from previous works like SEP [13] (C. Gotsman, et al., SIGGRAPH Asia-22'), which can also be viewed as the process of \"compressing geodesics\". As summarized in [13], this setting well suits many interactive applications where rapid computation of arbitrary point-to-point geodesics on very large meshes is required.\n\nStill, we do agree with the reviewer's advice that it is very promising to extend NeuroGFs to be generalizable models. We made efforts to such extensions and performed comprehensive experimental evaluations. As shown in Figure R1 of the uploaded PDF file, we designed three versions of generalizable NeuroGFs using (a) *autodecoder-based*, (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. These modifications are technically straightforward since we only need to replace the query point embedding stage with a feature-conditioned process.\n\nWe used the popular ShapeNet mesh dataset pre-processed by DISN (Q. Xu, et al., NeurIPS-19'), which covers 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) 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 (each category with 50 models), but each shape is unseen during training. 3) SN-5x50 is collected from the other 5 different categories (each category with 50 models).\n\nResults are provided in Table R4 of the uploaded PDF file. 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. And (c) further incorporates mesh connectivity cues, thus achieving better performance.\n\n\n### **[W2]** *Not difficult to obtain good performance with enough number of training pairs.*\n\n**Response:** Please refer to our detailed responses to Reviewer oZNk [W1] and [W2]. Briefly speaking, the actual ratio of \"our used training pairs\" to \"all pairs between vertices of the original mesh\" for geodesics ground-truth preparation is typically small, and our approach can still maintain relatively satisfactory performances even with much sparser (e.g., thousands of) ground-truth training pairs.\n\n### **[W3]** *Time cost of training NeuroGF for overfitting each new mesh.*\n\n**Response:** Practically, since NeuroGF is an offline 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\". Below we provide 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\\%.\n\n| MRE | <3\\% | <2\\% | <1\\% |\n| :----: | :----: | :----: | :----: |\n| **Training Time** | 0.4min | 1.7min | 7.9min |\n\nFrom 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.\n\n### **[W4]** *Encoding neighborhood of given pair of query points.*\n\n**Response:** Query point encoding is indeed an important issue, as presented in our response to Reviewer Racd [Q1] and our newly-added ablation study (variant (1) in Table R1 of the uploaded PDF file) of adding NeRF-style position encoding to point coordinates. However, in our processing pipeline, the technical soundness of encoding neighborhood of query points migh be questionable. This is because, in real application scenarios, users may randomly specify query points with varying density and distribution. In an extreme case, if a user only specify a single pair of query points, the neighbor aggregation process would fail. Hence, we still adopt our original design.\n\n\n### **[W5]** *Role of the SDF learning branch.*\n\n**Response:** Essentially, we include the SDF branch with the goal of forming a unified representation for encoding both 3D geometry and geodesics information, and it also brings actual benefits. In addition to the boosting effect (though not very significant) as verified in our ablation studies, the learning of SDF also speeds up the overall convergence speed of geodesics branches. The average training time needed for reaching MRE lower than 3\\% and 2\\% will respectively increase to 0.9min and 2.4min if we remove the SDF branch.\n\n### **[Q1]** *How the separate SDF network is trained for the distribution constraint.*\n\n**Response:** It seems that there exists a misunderstanding of this learning procedure. Specifically:\n- We remind that the notation $\\mathcal{N}_\\phi$ used in Eq. (11) is a separate network for SDF fitting.\n- This separate SDF network is different from the notation $\\mathcal{N}_\\Theta$ as given in Eq. (1). It will cause confusion if mixing them up. \n\nThe separate SDF network $\\mathcal{N}_\\phi$ is not optimized on the generated curve points $\\mathbf{p}_m$. Instead, it is optimized on randomly sampled spatial queries in advance. After finishing training, we freeze its parameters and apply it to infer SDF values of the generated curve points $\\mathbf{p}_m$."}, {"review_id": "UfaEERCBDn", "reviewer": "Reviewer_nwTQ", "summary": "The 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.", "questions": " The 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.", "limitations": "There 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\".", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The 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.", "weaknesses": "The 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", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer nwTQ]**\n\n### **[W1]** *1) The input is not limited to points belonging to the surface; 2) The output geodesic path is not guaranteed to lie on the surface; 3) Have to optimize a neural network for each new input.*\n\n**Response:** For the first issue, we argue that it is not an influencing factor in practice, because users will naturally input paired query points that are either vertices of the target mesh model or sampled from the mesh surface. Thus, it would be unnecessary to impose restrictions on the input geodesic query points.\n\nFor the second issue, as both quantitatively and qualitatively illustrated in the paper, the generated shortest paths are close enough to the underlying iso-surface, meaning that we can conveniently deduce post-processed shortest paths whose curve points are exactly located on the surface by straightforward local projection. Here, we simply implement this process by locally sampling surface points and then performing nearest-neighbor matching for the raw outputs of curve points. As shown in Table R2 of the uploaded PDF file, such a post-processing procedure consistently brings further accuracy improvement for the prediction of shortest paths on ALL the testing meshes. On average, the error decreases from the original $1.25 \\times 10^{-2}$ to $1.09 \\times 10^{-2}$.\n\nFor the third issue, as can be found in our below response to your question [Q1], although our approach is an overfitting process, its convergence speed is fast. Moreover, in terms of the overfitting paradigm, NeuroGF inherits from previous works like SEP [13] (C. Gotsman, et al., SIGGRAPH Asia-22'), which can be viewed as the process of \"compressing geodesics\". As summarized in [13], this setting well suits many interactive applications where rapid computation of arbitrary point-to-point geodesics on very large meshes is required.\n\n### **[Q1]** *Time cost of the whole pre-processing process (including network training).*\n\n**Response:** The time cost statistics of the data pre-processing procedures (not including network training) have already been provided in Table S1 of the supplementary material, and detailedly discussed on page 3, lines 15-33 of the supplementary material.\n\nAs for the network training, practically, since NeuroGF is an offline 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\". Below we provide 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\\%.\n\n| MRE | <3\\% | <2\\% | <1\\% |\n| :----: | :----: | :----: | :----: |\n| **Training Time** | 0.4min | 1.7min | 7.9min |\n\nFrom 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.\n\n\n### **[Q2]** *Think about different approaches that cannot predict path points off the surface.*\n\n**Response:** Thanks for your valuable comments and insightful consideration. For the current learning framework implemented in a generative fashion, it is basically impossible to ensure that all generated points exactly lie on the surface. We agree that predicting the next point along the path might be a viable scheme worthy of further exploring. Yet, considering the limited time for rebuttal and the complexity of designing a completely new framework, we cannot conduct systematic explorations for this issue. We may leave it as a promising direction for future study.\n\n### **[Q3]** *How well does the method scale with very large meshes (\\#V > 1M).*\n\n**Response:** Following your instructions, we added more diverse experiments on very large-scale meshes. Here, 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 the Table below, our approach works well on these two highly dense mesh models.\n\n| Mesh | \\#V (M) | MRE (\\%) | Chamfer-$L_1$ ($\\times 10^{-2}$) |\n| :----: | :----: | :----: | :----: |\n| dragon | 1.5 | 0.70 | 1.335 |\n| lucy | 6.9 | 0.58 | 1.404 |\n\nMoreover, we further performed evaluations on three large-scale real-scanned meshes covering (a) indoor room, (b) urban scene, and (c) street view, which are from diverse dataset sources. As shown in Figure R2 of the uploaded PDF file, our approach still achieves satisfactory performances (with MRE rates of 0.37\\%, 1.14\\%, and 0.23\\%, respectively) on the three scene-level meshes with million-scale amounts of vertices.\n\n\n### **[Q4]** *Robustness to extremely sparse meshes with very few vertices (e.g., a cube model with only vertices at corners).*\n\n**Response:** As you suggested, here we add an experiment with a cube model, with only 8 vertices at corners and 12 triangles. The total number of available ground-truth pairs is 28. The testing data are prepared by performing decimation on the cube to deduce a much denser mesh, on which we calculated enough pairs of geodesic data for performance evaluation. It turns out that our approach fails under this extreme case, achieving about 70\\% MRE rate. \n\nYet, we argue that in practice we can conveniently perform necessary mesh decimation to enable preparing more ground-truth pairs. For example, when we decimate the cube model to about only 100 vertices for ground-truth preparation, the resulting performance drastically improves to 5.24\\% MRE.\n\nIn addition to this extreme case, we would also refer the reviewer to our response to Reviewer oZNk [W2] for our results when the ground-truth data are sparse."}, {"review_id": "SPVUVxtti3", "reviewer": "Reviewer_Ujtd", "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. ", "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.\n2. 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", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 2, "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. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer Ujtd]**\n\n### **[W1]** *Limited technical contribution due to the quite straightforward approach.*\n\n**Response:** Thanks for your recognition of the usefulness of our core idea and technical solution. **Still, we beg to differ with the judgment that our work shows limited contribution only because our technical implementations are quite straightforward.** \n\nOn one hand, as the very first attempt to adapt the neural implicit modeling paradigm for geodesics representation, we resort to a concise yet highly effective learning framework to verify the potential of such a completely new geodesics representation paradigm. \n\nOn the other hand, the formulation of the shortest path as a discrete sequence of ordered curve points requires an appropriately designed learning structure, and our corresponding adaptation of previous \"folding-style\" 3D shape reconstruction decoders [14, 43] to a 1D version for curve deformation also has its technical value and thus is not quite straightforward as the reviewer commented.\n\nBesides, we also refer the reviewer to our extensions of generalizable NeuroGFs (asked by Reviewer yBGS), as given in Figure R1 and Table R4 of the uploaded PDF file as well as **our responses to Reviewer yBGS [W1]**. These explorations are also of significance.\n\nIn general, we believe that this work does bring novel insights and meaningful technical contributions for the problem of geodesics representation and computation and thus qualified for the NeurIPS research community.\n\n### **[Q1]** *Necessity of the SDF learning branch.*\n\n**Response:** There seems to be a **misunderstanding** of our approach. Maybe our descriptions in lines 180-185 of the paper is not clear enough. In fact, during the testing phase, we can say that the SDF learning branch is completely decoupled. However, during the training phase, since the query points for both SDF and geodesics share the same FC layers for feature embedding, the learning process of the SDF branch will have an impact on the other branches. And in our ablation studies (Table 3 of the paper), the effectiveness of the SDF learning branch has been validated (i.e., removing the SDF learning branch causes performance degradation of geodesics). Moreover, as discussed in **our response to Reviewer yBGS [W5]**, the learning of SDF also benefits the convergence speed of the geodesics learning process.\n\n### **[Q2]** *Difference between the geodesic paths and the iso-surface of the SDF.*\n\n**Response:** As both quantitatively and qualitatively illustrated in the paper, the generated shortest paths are close enough to the underlying iso-surface, meaning that we can conveniently deduce post-processed shortest paths whose curve points are exactly located on the surface by straightforward local projection. Here, we simply implement this process by locally sampling surface points and then performing nearest-neighbor matching for the raw outputs of curve points. As shown in Table R2 of the uploaded PDF file, **such a post-processing procedure consistently brings further accuracy improvement for the prediction of shortest paths on ALL the testing meshes**. On average, the error decreases from the original $1.25 \\times 10^{-2}$ to $1.09 \\times 10^{-2}$."}]

Author rebuttal:
### **[Global Response]**

We sincerely thank all five reviewers for their time and efforts in reviewing this paper and providing different aspects of valuable suggestions and helpful comments. In summary, during the rebuttal period, we made the following major efforts to address reviewers' concerns. The critical contents and results will be included in our paper or supplementary material to further improve the quality of our paper.

- We experimented with additional testing meshes with more diverse types and larger scales.

- We conducted more ablation studies to explore different choices of our specific technical implementations.

- We demonstrated the effectiveness of our NeuroGF when trained with a limited amount of ground-truth pairs.

- We introduced the post-processing procedures for making the generated shortest path points lie exactly on the underlying surface, which can further boost the representation accuracy.

- We conducted extensions of generalizable NeuroGF learning frameworks and evaluated their effectiveness on the popular and widely-used ShapeNet dataset.

- We clarified/presented the statistics of network training (as well as other pre-processing procedures) time cost.

- We explained the goal, necessity, and effects of adding the auxiliary SDF learning branch.

- We carefully answered reviewers' questions about some confusions/misunderstandings of our approach and some setups/procedures to facilitate better assessments of this work.

For convenience, below we will briefly summarize each raised Weakness (W), Question (Q), and Limitation (L), and provide our response item by item. Note that there is also a one-page PDF file uploaded for presenting more figures and tables.

Meta-review: The paper introduces an efficient method for estimating geodesic paths on 3D meshes using neural implicit functions. All the reviewers appreciated the straightforward method and acknowledged the relevance of the problem, together with a notable efficiency and the good experimental results that are reportedly competitive with respect to prior (and extensive) art. Major weaknesses were identified in the limited diversity of shapes and on the projection step, required for the estimated paths to lie on the surface. The authors provided a rather extensive rebuttal, including several additional shapes, more ablation studies, more datasets, additional evaluation and discussion addressing all the above points. The AC also read through the paper, acknowledging the previously raised points, as well as these being satisfactorily addressed by the rebuttal. We therefore recommend acceptance, while strongly recommending the authors to incorporate all the new material in the final revision of the paper.
Decision: Accept (poster)