Improving Neural Network Surface Processing with Principal Curvatures
Authors: Josquin Harrison
James Benn
Maxime Sermesant
Venue: NeurIPS 2024 poster (2024)
Keywords: geometric deep learning
geometry processing
shape analysis
discrete differential geometry
Primary area: machine_vision
TL;DR: Neural networks architectures designed to process surfaces can be improved by representing input surfaces via principal curvature

Abstract: The modern study and use of surfaces is a research topic grounded in centuries of mathematical and empirical inquiry. From a mathematical point of view, curvature is an invariant that characterises the intrinsic geometry and the extrinsic shape of a surface. Yet, in modern applications the focus has shifted away from finding expressive representations of surfaces, and towards the design of efficient neural network architectures to process them. The literature suggests a tendency to either overlook the representation of the processed surface, or use overcomplicated representations whose ability to capture the essential features of a surface is opaque. We propose using curvature as the input of neural network architectures for surface processing, and explore this proposition through experiments making use of the shape operator. Our results show that using curvature as input leads to significant a increase in performance on segmentation and classification tasks, while allowing far less computational overhead than current methods.

Improving Neural Network Surface Processing with Principal Curvatures
Josquin Harrison
James Benn Inria Sophia Antipolis ames. benn@inria fr
Inria
Sophia Antipolis
josquin haw..hhniiiioiiii........
Maxime Sermesant Inria
Sophia Antipolis
maxime ser.esant
Abstract
The modern study and use of surfaces research topic grounded in centuries of mathematical and empirical inquiry. From a mathematical point of view, curvature is an invariant that characterises the intrinsic geometry and the extrinsic shape of a surface. Yet, in modern applications the focus has shifted away from finding expressive representations of surfaces, and towards the design of efficient neural network architectures to process them. The literature suggests tendency to either overlook the representation of the processed surface, or use overcomplicated representations whose ability to capture the essential features of surface is opaque. We propose using curvature as the input neural network architectures for surface of the
processing, and explore this proposition through experiments making shape operator. Our results show that using curvature as input leads to significant increase in performance segmentation and classification tasks, while allowing far less computational overhead than current methods.
Introduction
Surfaces are natural representation for many real world objects ranging from organs and organisms to archaeological artefacts. They also central tool in virtual environments such computer games, or computer-aidee design. This ubiquity has resulted in a large body of work dedicated to mathematical methods developed for the efficient use of surfaces, well their analysis.
The goal of traditional computational surface analysis is to find a representation of surface that is expressive enough to capture details relevant for the problem or task at hand, while being com- putationally weigh.. However, the effectiveness Convolutional Neural Network (CNN) in image processing opened new doors to surface processing. The design of efficient convolution-lie operations adapt neural networks (NN) to surfaces alleviated need for complex and detailed rep- resentations, the point where most state of the art architectures use extrinsic vertex coordinates as input, letting the NN models learn the surface structure at multiple scales. While some attempts were made well known representations inputs, yielding some increase in performance the general consensus is that the model should be able to learn it by itself [25]. While it is true that neural networks efficient capturing surface features at multiple scales, the of local surface representation that is more expressive and more natural to interpret than extrinsic coordinates should naturally improve the performance of the network. The optimal choice of representation should be somewhere between coarse extrinsic vertex coordinates, and more complex representations.
[38] (3),,
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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In Riemannian geometry, the shape operator is the main tool linking the intrinsic geometry of surface with its bending and curving in ambient space, Its eigenvalues are the principal curvatures. Their product and evenly divided sum give precisely the Gauss and mean curvature of the surface, respectively. We hypothesize that the optimal choice for local surface representation that meets the requirements of surface processing is the set of principal curvatures: they characterize the surface up to isometry (location and orientation in space); they are purely local, which allows the neural network to decide on more general surface features; and they are lightweight, leaving very little computational overhead in any scenario. This work tests our hypothesis against two widely used representations of surfaces in three state of the art NN architectures.
In the next section we give an overview of surface processing, and introduce shape representations and learning methods. In section 3 we give an introduction to the Shape Operator, although we should remark now that this not the first appearance of the shape operator in the surface processing literature. The shape operator has already become an efficient tool in surface processing and is, among other things, used to define local tangent frames and compute surface features like creases 33.. Following these introductions, we then conduct extensive experiments in section 4 comparing principal curvature with three other representations in conjunction with three different NN architectures, on two segmentation datasets and one classification dataset, that shows how principal curvature enhances any state of the art model in different tasks. In addition to outperforming other methods, we show that this more concise representation is faster to compute, leaving minimal computational overhead when added to pipeline.
2 Related work
The first step to surface processing is usually its discretisation mesh point cloud, which is particularly useful for visualisation or rendering. From this starting point, novel representations have been derived in an effort provide tools for different surface related tasks. These tasks include surface matching, semantic segmentation, classification, or even shape retrieval.
Historically, the general trend has been to find compact descriptors of a shape which could be then compared within a dataset. A long list of such descriptors exist, among them signature- based descriptors, such as Heat Kernel Signature (HKS) [42] wave kernel signature 44. proved to be particularly efficient. Closely related are histogram methods, which are often combined with signatures to provide expressive representations such the SHOT 37 or the Echo [26] deseriptors. Geometric measure theory has also been a source of inspiration for developing efficient finite elements, or kernel-based currents 43) and varifolds [10] tailored for shape deformation. Such representations can be used in conjunction with classical statistical analysis tools, e.g (26 37 5 although they are often building blocks for
representations, such as geometric currents [5] leveraging
specialised methods on surfaces such as LDDMM 49 functional maps 29 or spectral-based analysis 445)
With the advent of deep learning, many methods previously stated were re-written with the help of neural networks resulting in Deep functional maps [20), and ResNet-LDDMM (2), to name but few. Representations of surfaces themselves were proposed as neural networks, such as DeepSDF images, 32 or DeepCurrents [30]_ As convolutions proved particularly effective when learning
ie structured grids, some work proposed voxel-based solutions to the study of surfaces [23]. Others suggested representing surfaces as geometric images [40], on which convolutions can be applied. A second generation of geometric deep learning has focused on building network architecture specifically tailored work directly on surfaces, i.e meshes or point clouds. From point cloud perspective, Point Net (34] and its extension Point Net ++ (35) consider the surface as a set of points by applying set operations on them. Among others, DGCNN (46] applies a convolution- operation on dynamic graphs constructed layer-wise. MeshCNN [66] on the other hand fully leverages the mesh structure develop operation unique to triangulations. Transformer based architectures have also appeared for the specific purpose of surface processing [18]. Among them, and in similar vein as before, GaTr [8] proposes to represent geometric data in an algebra of choice and designs an architecture with operations belonging to this algebra. An effort to have efficient generalisation of convolutions on surfaces was proposed by [22], although the lack of global coordinates creates ambiguity in local operations. To alleviate this problem, a large body of work has proposed rotation equivariant operations, namely GemCNN [22,, augmenting graph NNs, or field convolutions [25]. Finally, recent models propose to bypass the problem of generalising convolutions by focusing
2

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on well defined operations on surfaces, such as discrete exterior calculus in Hodge Net [41] heat diffusion in Diffusion Net 38 or a suit of known operators in Delta Net [48].
As model architectures include more and more knowledge of shapes, the need for a better repre- sentation of the input to these models has decreased. Outside of models that contain operations proper to the structure of choice (c.g [6 ( most models naturally accept as input the coordinates at every point. Some papers propose to augment the model by inputing higher dimensional descriptors initially designed for more direct analysis, such as the ones previously mentioned (e.g HKS, WKS, SHOT). Such proposals can be seen in 38 where HKS interacts well with the diffusion part of the architecture, or [35] were they combine HKS. WKS, Gaussian curvature through concatenation and PCA. However, recent work has dismissed this idea [25]. citing the results of Diffusion Net (38 which show no great improvement when moving from coordinates to HKS.
As methods for learning on surfaces have evolved, suggest that better input representation is a simpler one, yet is more expressive than coordinates. We suggest that we can scale back to the simplest differentiable invariant of surface: its curvature.
3 The Shape Operator
Here we give a conceptual introduction to the Shape Operator and describe how its eigenvalues completely characterize the surface to which it belongs (section 3.1). In section 3.2 we describe an explicit calculation of the Shape Operator which igl's 's implementation of the principal curvatures is based on this is the implementation familiar with the differential geometry of curves and surfaces may skip ahead to section 4; for others this section serves as concise introduction although, we do rely on basic understanding of functions of several variables and their derivatives, and surfaces and their tangent spaces.
use in experiments (see section 4.1). Those already
Surfaces in 3 will be denoted by S and S. points in surfaces by p's and q's, and the tangent space to 5 at a point p E S by S. Maps from TR33 to itself will be denoted by F : R.3 - and their derivatives at a point p by DF the Jacobian matrix. parameterisation X of a smooth surface is diffeomorphism between open set U C IR2 and an open set V S, and provides mathematical description of S as it lies in The standard Euclidean inner product on Rd will be signified by ), and it's restriction to surface S and its tangent bundle TS = Lp Sp by gs (:-)), which we call the induced metric. A normal vector to S at p is which is orthogonal to every vector in T,S (measured in (:, -)) and will be denoted by Np: if we have a field of normal vectors in an open set around p then this field will be denoted simply by N.
The Shape Operator of a surface at a point p E S measures the rate which surface normal vectors N separate around p. which is precisely the bending of the surface in space:
Definition 1 Given point p on surface S C CR and the unit normal vector N defined on neighbourhood U of p. the shape operator is the linear map
Sp T S
v
V
where S denotes the tangent space of S at point p
In other words, the shape operator S, tells us how the normal vector changes as we move S, the direction of from p. One possible way to visualise the shape operator is through the Gauss map, which identifies each point p E S C IR3 with its unit normal vector Np. now thought of as a point in S?. The shape operator is then the differential of the Gauss map at p and a tangent vector to S at the image Np of p. as illustrated in figure]] [.
The operator Sp is linear for each p E S. and self-adjoint in the Euclidean inner product (1 +): (V), w) = (v, (w))) It can therefore be represented by a symmetric 2 x 2 matrix [Sp] T,S T each point p E S. It is well-known that symmetric matrices admit a complete system of orthonormal eigenvectors e2), spanning the space on which they act. The matrix representation [S,] with respect to the basis (ei, e2) has the simple form: [Sp] =
K1 0
(2)
K2
3

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nip)
n
npp)
6
n(4)
another via [DF,]. As per Theorem 9.2 Figure 1: The shape operator may be visualised via the Gauss map.
Where 8i and K2 the eigenvalues of Sp-
Definition 2 Let S be surface in R2, a point in S, Sp the shape operator at p and [Sp] its matrix representation.
1. The eigenvalues (p) and k2(p) of (Sp) al the principal curvatures of S al p, and their corresponding eigenvectors Er and eg are the principal directions:
2. The Gauss curvature K. of S at p is the product 81 (1(p) (2(()) of the principal curvatures;
3. The mean curvature Hp is the average (1(()) of the principal curvatures
The Gauss and mean curvatures can be equivalently interpreted as the determinant and half the trace of (Sp), respectively.
The importance of these quantities two-fold: (1) two surfaces differ only in location and orientation in space if and only if they have the same principal curvatures (Theorems 9.1 and 9.2 in 28 that is, the shape operator completely characterizes the shape surface; and (2) the Gauss and mean curvature generate all possible differential invariants of surface (see Guggenheim [13]. Olver (27) in particular, Gauss and mean curvature are fundamental characteristics of the shape of a surface, and the inclusion of the higher order invariants they generate into representation could improve the results shown here.
3.1 Congruence
To explain how Gauss and mean curvature completely describe the shape of surface we need few more definitions.
An isometry of R33 is a map F R + R whose differential preserves the angles between tangent vectors every point of I:
(w,w)p (DFp v, DF, w)ps Ve, E T,R..
gs is the Riemannian inner product induced on TS by the Euclidean inner product (.))) then an isometry between two surfaces is map 1) - whose differential preserves the angles between tangent vectors to S:
(s w) (D U. Dn w).
Every isometry F R restricts an isometry of surfaces Fls = 7 need not be true, unless an additional hypothesis on the shape operators is satisfied.
F(S), but the converse
Two surfaces and are congruent there exists an isometry F R + g such that F(S) = 3:
that is, congruent surfaces surfaces which differ only in their location and orientation in space. It is clear that the shape operators S and S of two congruent surfaces are related by
DF, (p) Sp(p (DF, u), Ve E TyS;;
(5)
in particular, the matrices [Sp] and [Sp] conjugate
of 28 if there exists isometry n 5 S such that
Dnp Sp(v) Sy(P) (Dnp v), Yu T,S,
(6)

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Le. such that the matrices p and [Sp] are conjugate to one another via the matrix representation [Dpp), then there exists an isometry F IR3 + R such that Fls(5) = 77S) = S, and the two surfaces are congruent. The conclusion of this brief mathematical digression is that two congruent surfaces have the same intrinsic geometry and shape in space, and two surfaces with the same intrinsic geometry and shape in space are congruent. This what sought after when representing shapes with intrinsic quantities.
3.2 Discrete curvature
As well as being an important theoretical tool, curvature is a central notion in mesh processing. large body of work has been dedicated to estimating its discrete counterpart. Among them, many methods propose to infer Gaussian curvature directly, such as in [24], or involve the use of geometric measure theory [33 in [ Interestingly, many efficient methods propose to first discretize the shape operator in order to compute the Gaussian curvature from This is done either directly on mesh triangles, such as in [36], or by first locally fitting a function to the surface, and then computing explicitly the shape operator. To get a better feel for why this is natural construction of the shape operator, consider a surface S, given point E S. Then the surface around as X(u, 2) with (u, v) E RR2. The inner product at S, also called the first fundamental form, is then given for any two tangent vectors v; by:
be parameterized
E F F G
(v, D
W,
where E = be defined as
== (0,X, X) and G (0,X,, 0,X). And the surface normal p can
n 0,, v) X v)
(8)
(u, v) x 0, (u, 2i
We can now define the second partial derivatives of in the normal direction n, quantity called the second fundamental form, noted II:
L
M
II where L (0mnX, n). = (0ve n),, and = (0n.., n). The partial derivatives of the surface normal can then be expressed via the Weingarten equations, in terms of the components of the first and second fundamental form:
M
N
FM- GL
FL EM
on n
FG F2 0,X +
EG 0,X
0,n PN- -GM
FM EN
EG - F2 x,X +
EG F O,
This enables us to write the matrix form of the shape operator at as:
T.G-MP ME LF NE MF)
S = -EG- F
ME-LF LF
(10)
From these derivations, it becomes interesting to find good local parametrisation of surfaces, that is, bi bi-variate scalar function such that:
X X(u, v) (u, fu, v))
(11)
The shape operator can then be easily derived from the first and second derivatives of f. An efficient way find such functions is via osculating jets, proposed in (.. For the following experiments, use a multi-scale version of this, proposed in (31], in which the shape operator is computed by using neighbourhoods of varying size around point, yielding robust method for estimating curvature on a mesh.
4 Experiments
We test the representation surfaces by the principal curvatures K1, K2 and Gaussian curvature against the three most commonly used representations: the HKS [42,, the SHOT descriptor (37), and
5

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(a)
(b) n2
Figure 2: Principal curvature visualisation of a Louis XIV statue.
the extrinsic coordinates. The HKS is purely intrinsic representation derived from the Laplace operator, and constitutes the most widely used signature-based method to represent shapes. The SHOT representation is descriptor mixing signature and histogram-based methods to describe shapes, and is therefore an extrinsic representation. As they belong to two different classes of surface representations believe they are the most adequate for benchmarking our proposed curvature representation.
All representations tested with three different architectures. We regard Diffusion Net 38 the state of the art in NN architectures, as it shows the most promising results on general benchmark tasks. In addition, it shows very little difference in performance when changing the input from coordinates to HKS, making it the hardest test for our representation. Point Net ++ 35 has been designed as a general method process shapes arising in many situations, including controlled environments as in our case - but also from segmented images encountered in the autonomous driving field [35). As such Point Net ++ uses the least geometric structure describe surface: all one needs point set. We believe that in this case, using better surface information for the input will greatly enhance the performance of the model. The authors of PointNet++ have already touched on this subject, recommending a linear combination of HKS, WKS and Gaussian curvature, followed by PCA projection, leading to a 64 dimensional feature per point. We aim to show that 2d (or even Id) input of curvature information is more relevant for smaller computational cost. Delta Net [48 proposes an architecture intrinsic to surfaces by design, by combining four operators defined on the surface: Laplacian, divergence, curl and norm. Most papers that propose other surface descriptors rather than coordinates as input, do it solely to have an intrinsic representation of the surface. Curvature gives isometry invariance (section 3, and we further believe it is also more robust, numerically. Better performance from a curvature based representation in this architecture would support this belief.
Finally, we pick three tasks of varying complexity to measure the impact of each method: human segmentation (21). molecular segmentation (6), and shape classification [19]. Examples from each dataset are shown in figure 3
4.1 Implementation
The performance of each representation is strongly dependent on the chosen implementation. We have tried to be as fair as possible by not developing our own implementations of existing work and instead using implementations which have already been tried, tested, and validated the literature. For calculating the discrete principal curvatures via quadratic surface fitting, we have used igl 's implementation with a fixed neighbourhood radius of 5; the Gaussian curvature k then computed directly as the product of k1, K2- HKS depends on the Laplacian, and we have used the method implemented in robust- -aplaacin based on [39 this also consistent with what used in Diffusion Net. The eigendeccomposition of the Laplacian is then performed with scipy. For the

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(a) Human poses 21
(b) Molecules 16.
(c) Shree' 11 19
Figure 3: Samples of the segmentation and classification datasets used for experiments.
SHOT representation, we use the implementation in the pcl library, which computes 352 features per vertex, in this case normalise all shapes and use ball of radius .1; all other parameters left untouched.
Regarding the neural networks, use the implementations made publicly available by the authors, modifying only when needed to accommodate more than just coordinates as input. We also use the same parameters proposed in each paper when they known, which we detail for every task below. We make all our code and experiments available at battps://gg htt.bbbbtt..../// thub cwmmIooiiiiaeeeeeeeeeeeeeeeeeeee Asclepios/ shape-nets
4.2
Time Complexity
As a first experiment, we compute" for each representation method, the computation time as function of the number of points in surface. The performances are reported in figure 4 HKS and curvature are both efficient for meshes with up to 100k points. However, curvature is consistently faster, even for larger meshes (up to 500k points), displaying the very small overhead incurred by the use of curvature in surface models.
shot
20
Time
15
10
10
10
Number of points
Figure 4: Time of computation for each representation with respect to the number of points in mesh.
'Computed on Apple M2 chip

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4.3 Human Anatomy segmentation
We first segment the human parts from the composite dataset proposed in 21 containing samples from other human dataset, namely FAUST 77I. SCAPE [3), Adobe [.. MIT 44 As in the original paper, we use the SHREC07 dataset as test set. Similar to 47, we differ from 21 by evaluating on vertices rather than faces. For Point Net ++ and Delta Net we resample each shape to 1024 points, and we leave the meshes untouched for Diffusion Net, as per the experiments conducted in each paper. We optimise the negative log-likelihood for 100 epochs, with the ADAM optimiser and scheduler step every 20 epochs. We ran the experiment 5 times and have reported the mean test accuracy in I
and SHRECO7 [14).
ryz
shol16
shot64
hks
R1, Rig
Point Net ++
69.6
71.4
72.4
78.1
80.6
74.5
Delta Net
72.4
58.1
66.2
68.9
86.8
60.0
Diffusion Net
94.7
95.0
95.0
95.1
97.4
95.4
Table 1: Test accuracies (%) on the Human part segmentation task.
The results highlight the assumption that better representations lead to better performance. PointNet++ showed the greatest improvement when moving away from coordinates: this is due to its architecture having the least amount of geometrie information at baseline. The better results come from the principal curvatures, and show how expressive this representation is. The effects the principal curvatures
more pronounced in the Delta Net experiment, where K1, K2 greatly outperform all other methods. It's s interesting to note that in this experiment the coordinate representation performs better than the other more complex representations. Diffusion Net may show that it is more robust to the type of input, as long as it loosely describes the shape, however the improvement brought by the principal curvature is still significant. To further demonstrate the impact of good representation, even in the case of Diffusion Net, we show in the worst cases for xyz and Ki, K2 inputs. The clear improvement in this case may be even more important than general accuracy in some cases, e.g with human-in Human-in-the-lle in-the-loop type corrections.
(a) K1, R2. 80.9% accuracy.
xyz. xyz. 29.2% accuracy.
(c) HKS. 12.3% accuracy.
Figure 5: Human part segmentation with Diffusion Net. Worst cases for different representations, blue shows the correct prediction, red the error.
4.4 Molecular segmentation
The molecular dataset made available by 66 and first proposed [33. can be considered harder segmentation task then the Human part dataset: it proposes wider range of shapes in the form of RNA molecules, and a 260-way part segmentation task. We resample all meshes to 2048 points, except in the case of Diffusion Net where we kept the original discretisation. We evaluate all our baselines on 5 random splits with a train-test ratio of 80-20. We run the models for 200 epochs, and report the mean test accuracies in table 2
8

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zyz
shot16
shot64
hks
R1. R2
Point Net ++
35,4
70,9
71.9
70.2
72.4
69.0
Delta Net
29.2
45.6
56.5
49.6
55.5
29.2
Diffusion Net
82.6
88.4
89.1
85.6
89.4
84.0
Table Test accuracies (%) on the Molecular segmentation task.
Again, we see significant improvement when using better representation of the surface in the case of PointNet++, going from failing in the case of coordinates to outperforming Delta Net with principal curvatures giving the best performances. Diffusion Net shows non-negligable jump in performance as well. Although the SHOT descriptor outperforms other representations in the case of Delta Net, the general performance of this architecture is underwhelming. We believe this is due to the accumulation of errors in the discretisation surface operators used. Indeed, one layer computes chain of operators the surface: since the RNA shapes very irregular, the error for each operator could be significant.
4.5 Classification
In addition to segmentation tasks, we propose to compare representations in the context of classifica- tion. This experiment should show whether or not geometrically informative inputs interact well with pooling-type operations. We choose the widely adopted baseline Shrec11, proposed in [19). It is 30-way classification dataset with shapes per category. We choose the simplified mesh dataset and the harder version of training, using only 10 samples per class and evaluating the test. We perform our experiments on 5 random splits. We train our baselines for 100 epochs with a scheduler step at epoch 50 and optimise the cross-entropy loss with a label smoothing factor of 0.2. Resulting mean test accuracies are shown in table 3
ryz
shot16
shotgs
hks
K1 R2
Point Net +
71.5
69.8
60.7
60.8
85.7
96.2
Delta Net
75.7
54.9
60.4
98.6
90.1
98.8
Diffusion Net
80.3
52,6
67.4
98.9
94.2
99.1
Table 3: Test accuracies (%) the Shree classification task.
Yet again we observe a significant improvement when turning to better representations, even more so when using Gaussian curvature Additionally, figureloshows that all geometric representations yield less variability across each folds. In addition, HKS, Gaussian curvature k, and principal curvature 61, K2 converge much faster than all others.
1.0
Test accuracy
0.6
k1k2
hks
chot18
shot
0.0
20
40
80
80
100
Epooh
Figure 6: Evolution of the test accuracy with 95% confidence interval by epochs per representations across folds, for the Shrec07 dataset using Diffusion Net.
The fact that gaussian curvature, closely followed by HKS outperform principal curvature in this classification task seems to indicate that Gaussian curvature interacts better with pooling operations present in classification architectures. Interestingly, all three architectures tested here have different

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ways of performing the pooling operation. Although it is hard to give any analytical reasoning to this behavior, we believe simply the fact that gaussian curvature already an aggregation of the principal curvatures, that it shows better performance in classification tasks. For each experiment, additional metrics be found in Appendix A.
4.6
Noisy data
We propose one final experiment to highlight the robustness of input features to noisy data. We focus on three representations: HKS, known to robust to noise as it computes representation at multiple scale; extrinsic coordinates that are directly impacted by the noise; and principal curvatures, known to be less robust to noise as purely local descriptor. To compare these representations we pick the diffusion net trained on the human pose dataset, and we add noise to the dataset at inference time. Specifically, we add gaussian noise with a standard deviation of 1%, 3%, 5%, 7% and 10% of the diagonal length of the bounding box of each shape. Examples of the noisy data can be seen in Appendix A.2 Results show that the accuracy for all three features worsen at the same rate, as shown in figure showing that principal curvature can be viable choice even in the presence of noisy data.
1.0
Input feature
k1k2
09
xyz
.....n Noss
0.8
Test accuracy
0.7
0.5
044
0.3
Noise (%)
Figure 7: Evolution of the test accuracy on the human pose segmentation task for inputs kk), HKS and the extrinsic coordinates when noise is added to the shapes.
Conclusion
In this work we have shown that curvature should be the representation of choice when it comes processing surfaces with neural networks. In almost all experiments the principal and Gaussian curvatures performed better than any other choice of input, both qualitatively and quantitatively. In particular, this representation can be obtained with minimal computational overhead. Its combination with PointNet+++, the architecture that has the least prior information about the surface, showed that it can help the network better understand the surface structure. When combined with Delta Net, which contains only intrinsic operations, the improvement indicates that curvature gives more than just rigid transformation invariance. Even in the case of Diffusion Net, where the diffusion operation seems to interact nicely with any representation, curvature as input showed significant amelioration. For these reasons, we believe curvature should become the standard practice when using models to learn on surfaces. Finally, although experiment have shown that gaussian curvature outperforms principal curvatures on classification tasks, we would like to further define those guidelines in future work, as well as compare representations in a wider range of tasks and architectures.
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6
Acknowledgements
This Work has been funded by PARIS ERACoSysMedd grant number 15087, by G-Statistics ERC grant number 786854 and has been supported by the French government through the National Research Agency (ANR) Investments in the 31A Côte d' Azur (ANR-19-P3)) 19-P31A--00).. The authors are grateful to the OPAL infrastructure from Université Côte d' Azur for providing computing resources and support.
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A Appendix
A.1 Segmentation and classification detailed results
We present below the complete results for each experiment. For each dataset, and each neural network architecture, we show the Accuracy, Balanced accuracy, Fl score, and Specificity. They were all measured using 5-fold cross validation, and we give the results on the test sets in the form of mean + standard deviation.
Accuracy
Balanced accuracy
FI score
Specificity
zyz
.696:..306
.004..393
.662+.2933
.9574.49
shot16
.714::061
.815=.15
.6894.061
.59-1.231
shote4
.74
827=.13
.693+.085
.61
hks
.781=.07
.655..159
.754+.08
969+22003
Ki, R2
086++.079
.871.114
.799+.089
192+.196
k
.74
826+.227
.714
.644.332
Table 4: Human pose segmentation
Point Net ++ results.
Accuracy
Balanced accuracy
Fl score
Specificity
zyz
.724+.265
.23340076
.23+.0076
.7244.265
shot1e
.951
.215=.05
.215..05
.581=.095
shote4
.662+t.07 075
.302:t.84
.302+.084
.6622.075
hks
.303=0043
.303+.043
.689-.14
61, Kig
.8..1.128
.2991.114
.2991.114
.8281.128
.611.087
.1631.054
.163.1.04
.6110087
Table 5: Human pose segmentation Delta Net results.
Accuracy
Balanced accuracy
F1 score Specificity
zyz
9471.013
.9331.015
.9431.015
.471.003
shot16
.951.017
.9461.018
.961.0088
.951.017
shots1
.951.018
.944..021
.9441.021
.951.018
hks
.951+.04
.969+.055
.947+.089
.993+-07
61,
.955+.014
.91
.971+.015
975+-014
.54+t.014
.911
.9514.016
.954..014
Table 6: Human pose segmentation
Diffusion Net results.
Accuracy
Balanced accuracy
Fl score Specificity
ryz
354.L.008
.202:1.001
.202-10011
.354..008
shot
.70911.04
.59211016
.5921.016
.7091.004
shotg4
.719+.009
.602=.01
.602+.01
.719+.009
hks
.703+.017
.574+.003
.574+.003
.703+.017
Ki, K2
.724+-013
.597+.013
-5974.013
.724+-013
.694.008
.572+.008
.5724.008
.694.008
Table RNA molecules segmentation PointNet++ results.
Accuracy
Balanced accuracy
FI score
Specificity
zyZ
.292+.011 018
.182:.012
.182+0012
.2924.018
shot
.45++.008
.352+.016
3522.016
.456+.008
shotg4
.565+.010
.2994.008
.299+.008
.565+.010
hks
.96+t.017
.392+.008
.392+.008
.496+.017
61, R2
.5551.023
.4961.011
.4961.013
.5551.023
.2221.008
.142.1008
.1421.008
.292.1.08
Table 8: RNA molecules segmentation Delta Net results.
14

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Accuracy
Balanced accuracy
FT score
Specificity
zyz
826+.0001
186
.704+.003
.999f.129
s
8844.002
.765=.008
.765+.008
.874.0002
shots1
.8911.001
.7811.008
.7812.008
.8791.001
hks
.8661.596
.823.1.62
.7581.536
.9991.697
R1,
8941.008
.8731.04
.7831.025
.9991.022
844-0088
.862+.113
.718+.011
.999+.12
Table 9: RNA molecules segmentation Diffusson Net results.
Accuracy
Balanced accuracy
FI score
Specificity
zyz
.715
.8521.019
.709:.034
.991.001
shot16
.698+.035
.8441.018
.6941.036
.994.001
shotg4
.607+.031
.7971.011
.5944.038
986:1.001
hks
.6081.181
.7971.003
.6021.184
.86110006
Ki, K2
.877+.007
.9262.004
.852+.007
.995..0
.622t.011
.98+006
.611.011
.999..0
Table 10: Shree classification Point Net ++ results.
Accuracy
Balanced accuracy
FT score Specificity
ayz
.777+.027
.753+.031
.753+.031
.757+.028
shot16
5549+.029
.533+.026
.532+.026
.549+.029
sholg4
6004+.026
.597+.019
.597+.019
.604+-026
hks
.95-+.005
.985+0006
.986+.006
.986+-005
K1, R2
887:.000
.819
.881#0019
887=.020
k
.988+. 004
.988=.004
988+.004
.888.0004
Table I1: Shree classification Delta Net results.
Accuracy
Balanced accuracy
FI score
Specificity
zyz
8034.155
8984.082
.791#.169
.9934.005
shot
.526:.054
.755=.028
.513+.053
.84++0002
shotg1
.674.t003
.831..017
.668+.086
.989t.001
hks
.9891.008
.9941.003
.8911.006
1.04.0
922+.0099
.961.004
.9191.008
.9971.0
.911:.003
.9951.001
.9112.003
1.04.00
Table 12: Shree classification Diffusson Net results.
A.2 Noisy data exemples
15

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(a) 1%.
(b) 3%.
(c) 5%.
(d) 7%
(e) 10%.
Figure 8: Different quantity of noise added to shape from the human pose dataset, from 1% 10% of the diagonal of the bounding box of the shape.
(a) Initial shape.
(h) 1%
(c) 3%.
(d) 5%
(e) 7%.
(I) 10%.
Figure 9: Different quantity of noise added shape, from 1% 10% of the diagonal of the bounding box of the shape.
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22


Reviews:
[{"review_id": "pw5r9ft4p2", "reviewer": "Reviewer_qKsN", "summary": "The paper proposes to use curvature instead of previous surface descriptors in neural networks that process shapes. They show that the principal curvatures and/or mean curvature are better surface descriptors for three very different neural network pipelines, and is much faster to compute. They point out previously known properties of curvature for why curvature makes for such good shape representation.", "questions": "-", "limitations": "None given, and in the checklist have said it is not necessary given it is a comparative study. However given they also try to motivate why curvature is approriate theoretically and it depends whether to use principle curvature or Gaussian curvature, it would be nice for them to either try to identify when to use principal or Gaussian curvature(s) (theoretically or hypothesis-wise)", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- While there is clearly performance improvement, it is impressive that the benefit is clear on three very different neural network pipelines\n- Furthermore the simplicity and low dimension show of the proposed descriptors really demonstrate their effectiveness\n- Reduced computation time scaling means it can be scaled to very large sampling\n- \"The performance of each representation is strongly dependent on the chosen implementation. We have tried to be as fair as possible by not developing our own implementations of existing work and instead using implementations which have already been tried, tested, and validated in the literature\" - Very happy with with mentality, this is very important and a lot of papers in the literature do not do this.", "weaknesses": "- The reduction in performance for the principle curvatures for classification is a bit puzzling, especially since Gaussian curvature can trivially be computed from the principal curvatures. Surely the complexity of Delta Net and Diffision Net can cope with that, but there is a large performance reduction compared to HKS. The paper mentions that it indicates that Gaussian curvature interacts better with pooling, but that is not really justified. In the task HKS performs a lot better than shot16/64 indicating that there is some difference in the task. Given that this paper is more of a discovery of the benefit of a well known quantity, I would expect more analysis into this. Especially since for the other two tasks, Gaussian curvature performs worse than the principle curvatures, and intuitively this made sense as it decreased the amount of information, but then discarding this argument for the classification task is not appropriate.\n- It would be nice to also benchmark against the pipeline PointNet++ recommended, the \"linear combination of HKS, WKS and Gaussian curvature, followed by a PCA projection, leading to a 64 dimensional feature per point\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive comments, and for their very valuable remarks. \n\nWe acknowledge that our results raise some interesting and important questions that are not answered in the work, as pointed out in the 'weaknesses' section. However, we do not see these as weaknesses of the work: discovery has always preceded explanation in science and it's important that discoveries are shared with the community and circulated. The fact that there are not easily answered questions arising from the discovery is an invitation to investigate our constructions further, which we believe to be a scientific strength. \n\nNonetheless, we are grateful to the reviewer for their suggestions, that represent excellent directions for future work. We will make sure to include these discussion points in revisions of our work.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nHOmA685QU", "reviewer": "Reviewer_LGgy", "summary": "This paper proposes to use surface curvatures as input to neural networks to improve performance on 3D tasks. The main hypothesis is that as the curvatures are intrinsic properties of the surface, they will enable more effective learning for the relevant tasks. The paper carries out some evaluation comparing performance of network with curvatures as input against existing approach that uses point coordinates or other intrinsic properties as input. The proposed approach achieves better results on a few category of tasks.", "questions": "- How well does the method perform when dealing with noisy data?\n- How well does the method perform when compared against transformer-based architectures?", "limitations": "There is no discussion on limitations. In particular, the dependencies on orientation, the difficulty of orienting the surface globally, and the sensitivity to noises are not discussed sufficiently.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper carries out evaluation with some existing methods to demonstrate the benefit of using surface curvature as input.\n- The paper is overall well-written and easy to understand.", "weaknesses": "- The use of surface curvature in neural networks is not really new, so the novelty is limited.\n\n- The paper fails to discuss some inherent limitations when using curvatures as input. e.g.\n   *) Surface curvature requires 2D manifold structure. In a lot of applications, we are dealing with point clouds rather than a well-defined surface structure such as meshes. On point clouds with thin structures, some points not in the neighborhood of a point on the manifold may be closed actually close to the point in 3D space, causing difficulty in correcting identifying the neighbors and affecting the accuracy of curvature computation.\n   *) As a high-order differential property, curvature is sensitive to noises.\n   *) The signs of curvatures are dependent on the orientation of the surface (i.e., among the two possible normal directions, which one points to the outside and which one points to the inside). Given a point cloud, it is non-trivial to properly orient the normals to enforce a globally consistent orientation. If some points on the point cloud have incorrect orientation, their curvatures will have negated signs and may mislead the neural network.\n   These real-world scenarios can cause difficulties when using curvature rather than point coordinates as input to neural networks. There is insufficient evaluation on such real-world cases.\n\n- In recent years, transformers have shown better performance than convolution networks on many tasks. There is a growing body of transformer-based networks for 3D tasks (see https://github.com/lahoud/3d-vision-transformers), but there is no comparison with such methods in this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their work and time. \n\nWe clarify misunderstandings and respond to the questions raised below:\n\n> \"The main hypothesis is that as the curvatures are intrinsic properties of the surface, they will enable more effective learning for the relevant tasks.\"\n\nThe idea that curvature should be the representation of choice is not, as stated by the reviewer, solely for its intrinsic properties. Many other methods are intrinsic, such as HKS, which we show is outperformed by curvature. \n\n>\"The proposed approach achieves better results on a few category of tasks\"\n \nThe inflection of this statement is misleading, especially since we achieve better results on *all* categories of tasks competing against the state-of-the-art neural network architectures with state-of-the-art surface representations.\n\n> \"The use of surface curvature in neural networks is not really new, so the novelty is limited.\"\n\nOur methodology and results *are* new and were missed by the rest of the community. What is  important is novelty of results, not novelty of subject matter, since otherwise this point of view shuts down almost all avenues of scientific inquiry. We also make sure to reference papers that use curvature in our work, and explain clearly how we differ from them.\n\n>\"In recent years, transformers have shown better performance than convolution networks on many tasks...\"\n\nAlthough they have shown tremendous improvements in the NLP domain, one could argue the verdict on transformers outperforming CNN in image processing is not out yet. More so, regarding shape data, which is the context of our work and results, we have selected three architectures that are the most used in the domain and can undeniably be considered the state-of-the-art in neural networks for shapes.  Nonetheless, we are open to including a comparison with one of the emerging 3D transformer architectures in revisions of our work; as we show that curvature representation outperforms other methods on a variety of architectures, we believe a transformer model would also benefit from such representation. \n\n>\"The paper fails to discuss some inherent limitations when using curvature as input... In a lot of applications we are dealing with point clouds rather than a well-defined surface structure such as meshes.\"\n\nIt is true that point clouds are different from meshes. But there are several things to consider here: (1) In our paper, and within the domain of surface processing, we are focusing on shapes for which standard representation is a mesh, as mentioned explicitly in our paper. Application domains making use of mesh shape data are numerous, and large, including medical imaging, biology, architectural artifacts, so the context in which our results are valid is not at all limited. (2) We believe curvature as shape representation could be extended to noisy point cloud data. Perhaps more so than other representations, since discrete curvature knows a long history of research specialized in noisy data. In the paper we reference methods (that have been implemented and tested) from geometric measure theory, that can be used to define curvature for point clouds without an obvious orientation of normals everywhere.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OsnNIv2J4N", "reviewer": "Reviewer_ifJ1", "summary": "The paper promote to use the principal curvature as the surface presentation that can be better used in the modern neural network architectures. To support their hypothesis, the paper compares three different representations: HKS, SHOT descriptor and extrinsic coordinates in the use by PointNet++, Delta Net and Diffusion Net. Among all the providing experiment figures, principal curvatures provides the best performance.", "questions": "Refer to the weaknesses.", "limitations": "Refer to the weaknesses.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- The paper provides an overview of the classic surface representations, it also provides an overview of the popular achitectures for geometric deep learning in the exsiting body of work. \n- The paper work on a relatively meaningful topics in geometric deep learning, as it analysis different surface representations to be used by some popular surface-processing architectures.", "weaknesses": "- Quality: \n  - Although section 3 provides the way to calculate the principal curvature, it takes up too much space but does not adding equivalently significant value to the paper, and thus appear to be a bit redundant.\n  - The mathematical presentation in the paper is quit disorganized. Most of the formulation are not indexed. The definitions are given without a term [line 104, line 118]. \n\n- Incompliteness:\n  -  Section 6 Acknowledgement is empty, which becomes another major drawback regarding to the presentation quality.\n\n- Limited Contributions:\n  -  The surface representations being analyzed in the paper is relatively limited. The task and the methods are not new, not does the paper provide an intelligence-stimulated combination of the existing method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the time they have spent on reviewing our work. \n\nHowever, we are puzzled by several statements they have made, making it difficult to give a meaningful rebuttal. We will respond, point by point, as best as we can: \n\n> \"... [section 3] takes up too much space but does not adding equivalently significant value to the paper, and thus appears to be a bit redundant.\"\n\nThis is a value judgement and not a scientific judgement. Since curvature and the shape operator are the main tools used in the work, it is expected to state and define what these tools are, and give references. We believe this is normal scientific practice. If the reviewer believes otherwise, we ask they be more specific about what they expect or like to see instead.\n\n>\"The mathematical presentation is quit disorganised. Most of the formulation are not indexed. The definitions are given without a term\"\n\nThe mathematical presentation is confined to two and a half pages, and is a little more than a sequence of definitions introducing the main mathematical objects we use. The definitions are clearly labelled, and since we make no explicit use of any particular formulas in later sections there is no real need to number them. We feel the reviewer is voicing, once again, a value judgement, which can be interpreted as deliberately obstructive and combative.\n\n> \"Section 6 Acknowledgements is empty, which becomes another major drawback regarding to the presentation quality.\"\n\nThis is perhaps the most puzzling statement made by the reviewer. This being a blind peer review process, we believe it is of common knowledge that acknowledgements should be left empty at this stage, to avoid compromising our anonymity. Using this argument for saying our paper is incomplete seems obstructive and combative.  \n\n> \"The surface representation being analyzed in the paper is relatively limited. The task and the methods are not new, not does the paper provide an intelligence stimulated combination of the existing methods\"\n\nThe belief that the analyzed representations (i.e. data) is limited is a value judgement, not a scientific judgement. The work addresses shape data which is used in numerous broad application domains, including  medical  imaging,  biology, architectural artifacts, etc. We have already addressed the point of novelty in our general answer, however we do not understand the problem raised by the reviewer with the tasks themselves not being new. They are commonly used as state-of-the-art benchmarking tasks, appropriate for comparing methods, with completely open sourced data, and varying levels of difficulty. We are unsure what you mean by the final clause in the second statement.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We would like to first thank the reviewers for their time, and thoughtful comments and questions that will certainly help improve the revised paper. In particular we thank the reviewers for acknowledging the qualities of our work:
\begin{equation*}
    \text{"... it is impressive that the benefit is clear on three very different neural network pipelines."}
\end{equation*}
\begin{equation*}
    \text{"... a lot of papers in the literature do not do this."}
\end{equation*}
\begin{equation*}
    \text{"The paper is overall well written and easy to understand"}
\end{equation*}
\begin{equation*}
    \text{"The paper works on a relatively meaningful topic in geometric deep learning..."}
\end{equation*}

We are particularly happy with the comments of reviewer qKsN who perfectly grasped the main message of our work, and highlighted some interesting questions.

Although reviewers LGgy and ifJ1 have pointed out that the use of curvature in surface processing is not new, this is specifically what makes our contribution all the more valuable and all the more publishable. Despite the existence of related work using curvature, our results were missed by the rest of the community, and the questions our work raise are novel and non-trivial.

Finally, we believe some comments made by reviewer ifJ1 are, unfortunately, value judgements and not scientific judgements, and we are concerned about the extent to which they actually engaged with the work. 

We respond point by point to each reviewers' remarks in individual comments below.

Meta-review: To characterize shape properties the authors proposed to use the principal curvatures and/or mean curvature. They demonstrated that these descriptors can be efficiently computed and leads to a significant increase in performance on segmentation and classification tasks for three very different neural network pipelines when used as input. The proposed recipe is very easy to implement.

The main achievement of the article is a very simple practical recipe. Since the approach is empirical, its validity should be assessed under different conditions as much as possible. Despite the positive reviews, I would insist on adding additional important experiments, see comments below.

In fact, 
- the authors performed testing using relatively old and not very representative datasets such as shrek
- they did no study how robust the results are with respect to changing the hyperparameters of the considered neural network architectures
- the authors did not study to what extent the results change in the presence of noise in the data, how this effect depends on the noise level. It is important to discuss how to deal with noisy data and to which extent the noise corrupts the observed effects
- is this approach suitable only for 3d shapes? will it work efficiently for 3D models of mechanical parts?

I personally would recommend to consider more datasets to verify the proposed approach, e.g. using so-called ABC dataset (collection of one million Computer-Aided Design (CAD) models) to predict sharp features (see the follow up paper DEF of the same authors).
Decision: Accept (poster)