Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction
Authors: Thanh Tung Le
Khai Nguyen
shanlin sun
Kun Han
Nhat Ho
Xiaohui Xie
Venue: ICLR 2024 poster (2024)
Keywords: Mesh deformation
optimal transport
cortical surface reconstruction
computer vision
medical imaging.
Primary area: applications to neuroscience & cognitive science
TL;DR: Our new approach for 3D mesh deformation outperforms other methods in cortical surface reconstruction by encoding a mesh into a probability measure, using sliced Wasserstein distance for comparison, and employing a neural ODE for deformation.

Abstract: Mesh deformation plays a pivotal role in many 3D vision tasks including dynamic simulations, rendering, and reconstruction. However, defining an efficient discrepancy between predicted and target meshes remains an open problem. A prevalent approach in current deep learning is the set-based approach which measures the discrepancy between two surfaces by comparing two randomly sampled point-clouds from the two meshes with Chamfer pseudo-distance. Nevertheless, the set-based approach still has limitations such as lacking a theoretical guarantee for choosing the number of points in sampled point-clouds, and the pseudo-metricity and the quadratic complexity of the Chamfer divergence. To address these issues, we propose a novel metric for learning mesh deformation. The metric is defined by sliced Wasserstein distance on meshes represented as probability measures that generalize the set-based approach. By leveraging probability measure space, we gain flexibility in encoding meshes using diverse forms of probability measures, such as continuous, empirical, and discrete measures via \textit{varifold} representation. After having encoded probability measures, we can compare meshes by using the sliced Wasserstein distance which is an effective optimal transport distance with linear computational complexity and can provide a fast statistical rate for approximating the surface of meshes. To the end, we employ a neural ordinary differential equation (ODE) to deform the input surface into the target shape by modeling the trajectories of the points on the surface. Our experiments on cortical surface reconstruction demonstrate that our approach surpasses other competing methods in multiple datasets and metrics.

Published as a conference paper at ICLR 2024
DIFFEOMORPHIC MESH DEFORMATION VIA EFFICIENT OPTIMAL TRANSPORT FOR CORTICAL SURFACE RE- CONSTRUCTION
Tung Le', Khai Nguyen", Shanlin Sun', Kun Han', Nhat Ho & Xiaohui Xie
University of California, Irvine Thee University of Texas at Austin
ABSTRACT
Mesh deformation plays pivotal role in many 3D vision tasks including dynamic simulations, rendering, and reconstruction. However, defining an efficient discrep- ancy between predicted and target meshes remains open problem. A prevalent approach in current deep learning is the set-based approach which measures the discrepancy between two surfaces by comparing clouds from the two meshes with Chamfer pseudo-distance Nevertheless, the set-based approach still has limitations such lacking a theoretical guarantee for choosing the number of points in sampled point-clouds, and the pseudo-metricity and the quadratic complexity of the Chamfer divergence. To address these issues, we propose novel metric for learning mesh deformation. The metric is defined by sliced Wasserstein distance on meshes represented as probability measures that generalize the set- set-hased approach. By leveraging probability measure space, we gain flexibility in encoding meshes using diverse forms of probability measures, such as continuous, empirical, and discrete measures via varifold representation. After having encoded probability measures, we can compare meshes by using the sliced Wasserstein distance which is effective optimal transport distance with linear computational complexity and can provide a fast statistical rate for approximating the surface of meshes. To the end, we employ a neural ordinary differential equation (ODE) to deform the input surface into the target shape by modeling the trajectories of the points on the surface. Our experiments on cortical surface reconstruction demonstrate that our approach surpasses other competing methods in multiple datasets and metrics.
randomly sampled point-
INTRODUCTION
Mesh deformation is a fundamental task in 3D computer vision and computer graphics. A wide range of shape reconstruction tasks (Chen & Zhang, 2019. Jiang et al.] 2020 Mescheder et al.)
2019 Niemeyer et al.] 2020; Park et al. 2019 Sun et al. 2023 Yariv et al.
2020) and shape
registration (Ashburner) 2007 Dalca al. 2018 Balakrishnan et al. 2019 Le al. 2024: Krebs
et al. 2019 Dalca et al. 2019 Han et al. 2024; Sun et al. 2022 Han et al. 2023) leverages
state-of-the-ar mesh deformation methodology. One popular approach for mesh deformation is to estimate the vertex displacement vectors (3D offsets) while keeping their connectivity (Wang
al. 2019: Bongratz al. 2022). However, displacement-basee -based methods cannot guarantee the manifoldness of the resulting mesh and often produce self-intersecting faces. To address this issue, 1975; Arsigny 2004) is one effective way to deform a mesh while preserving its topology. As an instance of diffeomorphic surface deformation, Neural Mesh Flow (NMF) (Gupta, 2020 learns sequence of diffeomorphie flows between two meshes and models the trajectories of the mesh vertices as ordinary differential equations (ODEs). However, these methods have limited shape representation capacity and struggle to perform well on complex manifolds. To overcome this limitation, several diffeomorphic mesh deformation models have been proposed, such as CorteXODE (Ma et al. 2022) and CorticalFlow (Lebrat et al., 2022). which aim to handle hard manifolds, such the cortical surface. While CorticalFlow (Lebrat et al.) 2022) introduces diffeomorphic mesh deformation (DMD) modules to learn a series of stationary velocity fields, Cor (Ma et al. 2022) encodes spatial information of the MRI images along
diffcomorphic transformation (Ruelle & Sullivan

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3. T.Template
b. Target
Sample points
d. CD loss
SWD lass
Figure 1: 2D deformation toy example. We deform (a) template circle to a (b) target polygon via an optimization-basee setting. (c) Points are uniformly sampled from the two contours. CD loss is easily trapped at local minima, i.e. the green points are not uniformly distributed on the target contour as desired. In contrast, (e) SWD loss can find the optimal transport plan among discrete probability measures, i.e. the resulting points are distributed more uniformly along the contour. More details about this toy example can be found in the Appendix B
with vertices features using an MLP model and employs neural ODEs (Chen et al. 2018 to model the trajectories of the points. Nonetheless, these approaches often rely on Chamfer distance as the objective function, which might have disadvantages.
Selecting an appropriate metric to evaluate the dissimilarity between two meshes is a crucial step in learning deformation mesh models. Recent literature favors the approach of using set-based comparison due to its simplicity. In particular, the set- satd approach first samples two sets of points on the surface meshes, and then use Chamfer distance (CD) (Deng et al. 2018 Duan et al., 2019)
Groueix et al. 2018) to compare two meshes and optimize them. However, the CD loss tends to get trapped in local minima easily (Achlioptas et al. 2018 Nguyen et al., 202 lb; 2023 2024a). failing to distinguish bad samples from the true ones, as demonstrated by our toy example Fig. Although a weighted CD has been proposed to prioritize fitting local regions with high curvatures in Vox2Cortex (Bongratz al., 2022), the issue is only alleviated but not resolved completely, still resulting in suboptimal assignments between two sets of points. Therefore, we propose novel approaches to transform mesh into a probability measure that generalizes the set-based approach. Furthermore, by relying on the probability measure approach, we can employ geometric measure theory to represent mesh as an oriented varifold (Almgren 1966 Vaillant & Glaunes 2005 Glaunès et al. 2008) and get better approximation of the mesh compared to the random sampling approach. After that, we adopt efficient optimal transport compare these measures since optimal transport distances are naturally fitted to compare disjoint-support measures.
Wasserstein distance (Peyré & Cuturi 2019: Villani 2009) has been widely recognized as an effective optimal transport metric to compare two probability measures, especially when their supports are disjointed. Despite having a lot of appealing properties, the Wasserstein distance has high computational complexity. In particular, when dealing with discrete probability measures that have at most m supports, the time and memory complexities of the Wasserstein distance are O(m 3 log m) and O(m²), ), respectively. The issue becomes more problematic when the Wasserstein distance is computed on different pairs of measures as in mesh applications, namely, each mesh can be treated as probability measure.
To improve the computational complexities of the Wasserstein distance, by adding entropic regular- ization and using the Sinkhorn algorithm (Cuturi 2013), an e-approximation of Wasserstein distance can be obtained in (m²²/=) However, this approach cannot reduce the memory complexity of Omm²) due the storage of the cost matrix. Moreover, the entropic regularization approach cannot lead to valid metric between probability measures since the resulting discrepancy does not satisfy the triangle inequality. more efficient approach based on the closed-form solution of Wasserstein is sliced Wasserstein distance (SWD) (Bonneel et al. (015),, which is computed as the expectation of the Wasserstein distance between random onendimensional push-forward measures from two original measures. SWD can be solved in Om log m) time complexity while having linear memory complexity (m).
In this paper, we propose learning-based Diffeomorphic mesh Deformation framework via an efficient Optimal Transport metric, dubbed DDOT, that learns continuous dynamics to smoothly deform initial mesh towards intricate shape based volumetric input. Specifically, given 3D brain MRI volume, we aim to reconstruct the highly folded white matter surface region. We first extract the initial surface from the white matter segmentation mask of the brain MRI image, then
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deform the initial surface to the target surface by modeling its vertices trajectory via neural ODE (Chen et al.] 2018). Our deformation model is optimized via sliced Wasserstein distance loss by encoding the mesh as probability measure. We further represent mesh an oriented varifold and empirically show that our approach surpasses other related works on multiple datasets and metrics, namely, almost eeffiinerseccton-free while maintaining high geometric accuracy. It is worth noting that although our DDOT is developed within the scope of cortical surface reconstruction (CSR), the underlying concepts can be extended to other 3D deformation networks.
Contribution. In summary, our contributions are as follows:
1. We propose represent triangle meshes probability measures that generalize the common set-based approach in a learning-based deformation network. Specifically, we present three forms of mesh as probability measure: continuous, empirical, and discrete measure via an oriented varifold.
2. We propose a new metric for learning mesh deformation. Our metric utilizes the sliced Wasserstein distance (SWD), which operates meshes represented as probability measures. We demonstrate that sliced Wasserstein distance (SWD) is a valid computationally fast metric between probability measures and provide the approximation bound between the SWD between empirical probability measures and the SWD between continuous probability measures.
3. We conduct extensive experiments on white matter reconstruction by employing neural ODE (Chen et al. 2018) to deform the initial surface the target surface. Our experiments on multiple brain datasets demonstrate that our method outperforms existing state-of-the-art related works in terms of geometric accuracy, self relf-intersection ratio, and consistency.
Organization. The paper's structure is as follows. We first provide background about the set- based approach for comparing two meshes and the definition of Wasserstein distance as well as diffeomorphic flows for deforming meshes in Section 2 In Section 3 we propose probability measure encoding to represent mesh and further employ SWD as an objective function for diffeomorphic deformation framework as well as analyze their relevant theoretical properties. Section presents our experiments on reconstructing cortical surface and provides quantitative results compared to state-of- the-art methods. In Section[5] identify the limitations of our approach and outline potential avenues for future research. We then draw concluding remarks in Section [ Finally, we defer the proofs of key results, supplementary materials, and discussion on related works Appendices.
Notations. For any d 2 2, we denote gd-1 == {0 E Rdd ||011 = 1} and U($-=1) as the unit hyper- sphere and its corresponding uniform distribution. We denote Oin as the push-forward measures of ja through the function / Rd > IR that is f(x) 0 x. Furthermore, we denote 8, as Dirac distribution location x, and Pp (IR") is the set of probability measures over Rd that has finite p-moment, By abuse of notations, we use capitalized letters for both random variables and sets.
BACKGROUND
In this section, we first review the set-based approach for comparing two meshes. After that, we review the definition of Wasserstein distance and diffeomorphic flows for deforming meshes.
2.1 THE SET-BASED APPROACH: MESH TO POINT-CLOUD
Mesh to point-cloud. To sample a point p from mesh face f (I1, (21, 2z) ) is first sampled with the probability proportional to the area of the face. Then, the position p can be sampled by setting p == w + W222 + W3#3, where wi + W2 W3 are random barycentric coordinates which are uniformly distributed over a triangle (Ravi et al. 2020; Wang et al. 2019). The process is repeated until getting the desired number of points.
Comparing two point-clouds. After having representative point-clouds from meshes, a discrepancy in the space of point-clouds (sets) is used. The most widely used discrepancy for point-cloud is the set-based Chamfer pseudo distance (Barrow et al. 1977). For any two point-clouds X and Y, the Chamfer distance is:
CD(XYY)
1 X
Σ
min I wEY
1
Σ veY
min || BEX
v11..
(1)
where X denotes the number of points in X.
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2.2 WASSERSTEIN DISTANCE
We now review the definition of the Wasserstein distance for comparing two probability measures
J E (RR()) and
(IR)). The Wasserstein-p (Villani 2003) distance between f and as
follows:
v)( inf
|- yll.d((z, y),
(2)
=EIT(A.)) xR*R"
where (1(,,)) v) is the set of joint distributions that have marginals are μ and v respectively. A benefit of Wasserstein distance is that it can handle two measures that have disjointed supports.
Wasserstein distance between continuous measures. Computing the Wasserstein distance curately between continuous measures is still an open question (Korotin et al. (2021) due to the non-optimality and instability the minimax optimization for continuous functions which are the Kantorovich potentials (Arjovsky et al. 2017). Hence, discrete representations of the continuous measures are often used as proxies compare them. i.e., the plug-in estimator (Bernton et 2019).
In particular, let X1...., Xx,, iéd 1 and Y1, ..... Yn f.é.d V, the Wasserstein distance (,(,s, v) is ap- Σ== ōy, the corresponding
proximated by Wp(pm, Pm) with Pm empirical measures. However, the convergence rate of the Wasserstein distance is (m (Mena & Weed 2019). Namely, we have E (pm pm) Wp(,,))] I C' -1/d for an universal constant C. where Am. m Σn= 6x, is the corresponding empirical measure. Therefore, the Wasserstein distance suffers from the curse of dimensionality i.c., the Wasserstein distance needs more samples to represent the true measure well when dealing with high-dimensional measures. In the setting of comparing meshes, the Wasserstein distance will be worse if we use more features for meshes e.g.. normals, colors, and so on.
Ei=1 ōx, and 2m
Wasserstein distance between discrete measures. When μ and are two discrete probability measures that have most m supports, the time complexity and memory complexity to compute Wasserstein distance are Om (mmloggm) and O(m?) respectively. Therefore, using the plug-in estimator requires expensive computation since it requires a relative large value of m to the dimension.
2.3 DIFFEOMORPHIC FLOWS
Diffeomorphic flows be established by dense point correspondences between source and target surfaces. Given an input surface, the trajectories of the points can be modeled by an ODE, where the derivatives of the points are parameterized by deep neural network (DNN). Specifically, let (p, t) Q C: IR3 x [0, + f C I be a continuous hidden state of the neural network that defines a trajectory from source position p = (p, 0) to the target position p' = xpp 1). and Fş be a DNN with parameters ø. An ordinary differential equation (ODE) with the initial condition is defined as:
2 (p, t) (T(,)))) S.I. (pp 0) p.
Öt
If Fo is Lipschitz, solution Eq. 3 exists and is unique in the interval [0, 1]. which provides theo- retical guarantce that any two deformation trajectories do not intersect with each other (Coddington Levinson 1984).
3 DIFFEOMORPHIC MESH DEFORMATION VIA AN EFFICIENT OPTIMAL TRANSPORT METRIC
In this section, we generalize the set-based mesh representation by proposing three ways of trans- forming mesh into probability measure encoding. We further employ sliced Wasserstein distance as an objective function the diffcomorphic flow model for cortical surface reconstruction task.
3.1 THE MEASURE-APPROACEE MESI TO PROBABILITY MEASURE
We now discuss the approach that we rely on to compare meshes via probability metrics. In particular, consider transforming mesh into probability measure.
Mesh to a continuous and hierarchical probability measure. Let a mesh M have set of faces FM [fMM , JM (N 0) where face is represented by its vertices
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define probability measure over faces, namely, (f) Ei= Σソ= ) 8MM is the categorical distribution over the faces that has the weights proportional to the areas of the faces (volume of the convex hull of vertices). For example, in the case of triangle meshes, we have Ver(f) (I1, F2, 22) and Vol(f) 211(22 - 21) (Tz =1)ll- Given a face f. the conditional distribution for a point in the space is μ (x]f) = vol.7 Vr e (onvexHull(Ver(())) Therefore, the marginal distribution for point in the space induced by mesh M is (z (x) Ei= ( (MM
Ver(f) { ........ Fad (vf 2 3). We
Vol(s;
It is worth noting that given two meshes M, and M2, their corresponding probability measures yrM and yrMa are likely to have disjoint supports. Therefore, the optimal transport distances are natural metrics for comparing them.
Mesh to empirical probability measure. As discussed in the background, the most computa- tionally efficient and stable approach to approximate the Wasserstein distance is through the plugin estimator (Bernton et al. 2019). In particular, let T1,,. ................................ Im be a set of points that are independently identically sampled from fer and y.,..., Im be set of points that are independently identically sampled from M2 After that, we can define A represented empirical probability measure of the two meshes Mi and M2 respectively. Finally, the discrete Wasserstein distance is computed between i.M, and jim as the final discrepancy.
1 Σ 62, and j.m m Ei=10 6y, as the
Σn
m
Mesh to a discrete probability measure. To have richer representation of mesh, we consider the discrete probability measure representation through varifold. Let M be a smooth submanifold of embedded in the ambient space of R", e.g. = 3 for surface, with finite total volume Vol(M) < OC. For every point p E M, there exists a tangent space TμMM be a linear subspace of RR. To establish orientation of M, it is essential orient the tangent space Ty.M for every p E M. This ensures that each oriented tangent space can be represented as an element of an oriented Grassmannian. Inspired from previous works (Glaunes et al. 2004 Vaillant & Glaunes 2005 Charon & Trouvé 2013) Kaltenmark et al. (017). M can be associated as an oriented varifold i.e. a distribution on the position space and tangent space orientation Eº x g as follows: S.M (pp.t())) dVl((p)) where rp(p) is the unit oriented normal vector to the surface at p. Once established the oriented varifold for a smooth surface, an oriented varifold for triangular mesh M that approximates smooth shape M can be derived as follows:
dimension 2
T =
FI
LP
ES,
6 (on
(4)
where pt is the barycenter of the vertices of face ft and a; = Vol(f.) is the area of the triangle. To ensure that j.' possesses the characteristic of a discrete measure, we normalize a,s" such that they sum up to Note that provided the area of triangular sufficiently small, MiMM gives an acceptable approximation of the discrete mesh M in terms of oriented varifold (Kaltenmark et al. 2017)
3.2 EFFECTIVE AND EFFICIENT MESH COMPARISON WITH SLICED WASSERSTEIN
Sliced Wasserstein distance. The sliced Wasserstein distance (Bonneel et al. 2015) (SWD) between two probability measures μ PP (RR) and (Rd) ) is defined
SWE(e, v) Eg (Pt 04v)].
(5)
The benefit of SW is that (Oipu, 0tv) has a closed-form solution which is fo [4 -2)-= Fr [2)
with F-1 denotes the inverse CDF function. The expectation is often approximated by Monte Carlo sampling, namely, it is replaced by the average from o,.... Or. (I is the number of projections) that drawn i..d from Ld (Sd-1). In particular, we have:
.....,,
1 Σ wp(Ortu, Octr). L
(w,
=
(6)
The computational complexity and memory complexity of the Monte Carlo estimation of SW are O Lm log m) and (Lm) (Nguyen & Ho 2024) respectively when f and v are discrete measures with at most supports. Therefore, the SW is naturally suitable for large-scale mesh comparison.
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Convergence rate. We now discuss the non-asymptotic convergence of the Monte Carlo estimation of the sliced Wasserstein between two empirical probability measures to the sliced Wasserstein between the corresponding two continuous probability measures surface meshes.
idd Theorem For any two meshes Mi and M2, let XI...., Xm ~ M (x)) Yi,...,Y, Y Lid (x), (M (=)= = = Ei= - ōx, and p=Ma(x) E by, be the corresponding empirical distribution. Assume that yrM, and μ have compact supports with the diameters that are most R. we have the following approximation error:
(d + log
E
SW (PM) - L.- -W Maaay
S RCC,
m
Var [Wp 0ap=Ma)] [X1.m V1.m
1/2
EE
for universal constant Cp, n > O. The variance is with respect to 0 U($-11).
Theorem suggests that when using empirical probability measure to approximate meshes, the emror between the Monte Carlo estimation of sliced Wasserstein distance between empirical distributions over two sampled point-clouds and sliced Wasserstein distance between two continuous distributions on meshes surface is bounded by the rate of m m-21/2 and T.-1/2, It means that, when increasing the number of points and the number of projections, the error reduces by the square root of them. This rate very fast since it does not depend exponentially on the dimension, hence, it is scalable to meshes with high-dimensiooal features at vertices, such as normals, colors, and so on. Leveraging the scaling property and the approximation of varifold mesh M mentioned in See Sec we can represent meshes as discrete measures j and optimize SW p. (ū Mi; L) as the objective function. The proof of Theorem iis given in Appendix C It worth noting that a similar property not able be derived for Chamfer since it is a discrepancy on sets that cannot be generalized to compare meshes.
3.3 SLICED WASSERSTEIN DIFFROMORPHIC FLOW FOR CORTICAL SURFACE RECONSTRUCTION
Diffeomorphic deformation framework for reconstructing cortical surfaces. this section, we present DDOT that incorporates diffeomorphic deformation and sliced Wasserstein distance to reconstruct the cortical surface. Specifically, our goal is to derive a high-resolution, 2D manifold of the white matter that is topologically accurate from 3D brain MR image. Let I R Dxxxx be MRI volume and M = FV. be a 3D triangle mesh. The corresponding vertices of the mesh are represented by R3. Firstly, we train a U-Net model to automatically predict the white matter segmentation mask from I. Then, signed distance function (SDF) is extracted from the binary mask before employing Marching Cubes (Lorensen & Cline 1987) to get the initial surface. After getting the initial surface Mo. the trajectory of each coordinate vo is modeled via the ODE with initial condition from Eq.3 Inspired from (Ma et al. 202T; 2022), we concatenate the point features with the corresponding cube features sampling from I as feature vector. The new feature is passed through a multilayer perceptron F, to learn the deformation. More implementation settings included in the Appendix [
Sliced Wasserstein distance as a loss function. To train the DDOT model, we minimize the distance between predicted mesh M and the ground truth mesh M We adopt novel way of transforming mesh into probability measure and leveraging sliced Wasserstein distance (SWD) as loss function optimize two discrete meshes. As discussed, the SW is valid metric on the space of distribution and can guarantee the convergence of the probability measure. Moreover, as shown in Theorem II] the sample complexity of the SW is bounded with parametric rate, hence, is suitable to use the SW to compare empirical probability measures as the proxy for the continuous mesh probability measure. Therefore, we sample points on M and M as probability measures and compute SWD loss between these two measures without regularization terms. Additionally, we can represent mesh as discrete probability measures, i.e. oriented varifold, and utilize the same objective function. Based our observations, the varifold representation has shown better performance compared to encoding using empirical probability measures, which provide more detailed comparison in ablation study in Sec. 4.3) As a result, we assume that our experiments in the following sections will be conducted using the oriented varifold approach unless otherwise specified.

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Table 1: Quantitative results of white matter surface reconstruction in terms of earth mover's distance (EMD), sliced Wasserstein distance (SWD), average symmetric surface distance (ASSD), Chamfer normals (CN), and self-intersection face ratio (SI) on ADNI and OASIS datasets. Best values are highlighted. EMD, SWD, ASSD results are in mm. All results are listed in the format "mean value standard deviation". DDOT represents the reconstruction results from our proposed approach. While means smaller metric value is better, 1 indicates larger metric value is better.
ADNI daiact
Left WM
Right WM
Method
EMD (mm)
SWD (mmm
ASSD (mm)
CN
(16)
EMD (mm) 1
SWD (mm) 1
ASSD crum)
CN 1
(16)
1.368 +-721
+-010
1.350 +:390
1.357
+:172
936 -01
1-173
0.073
5:011
-219
1,048 #:11
811 2-21
927 -010
CFFP
205
.21
5:00
(158 +-082
821
.473
4:073
-000
ConCODDE
.36 4:33
234
5:030
1113 5:001
782
384
231 4:002
939 =00
013
9:233
.302 5.04
#:012
<
.702
.36
0:006
938 =0012
OASIS damset
Left WM
Right WM
Mothed
(mm)
SWD (nm)
ASSD (mmm)
CN
S1
FMD (mom)
SWD (mun)
ASSD (mmm)
CN
(1))
DespCSR
.587 2.787
1,020 s: 892
.312 2.124
941 1: 010
.900 2:740
1,072 2: 2.419
.344 1.158
941 a: 011
Vox2Coetex
.994 2.216
.876 1.053
302 2.037
925 1. 000
.994 2.193
.574 2.256
872 2.002
.305 1.042
929 4: 019
1,022 1516
CTPP
.511 2.212
844 1.020
225 2.038
937 1: 007
254 1.000
.475 2.2
840 1.077
.227 1.04
955 1: 00s
1076 008
CiDE
.25 1-8
765 9:047
185 4.036
943 + 007
003 +.525
.444 2:0
757 8:065
.182 +.052
043 + 008
012 =2000
Ours
.488 +- 192
.779 05055
.661 4:000
.999 + 005
< 10
.299+.20 +-20
.770 +.070
.160 +.04
.49 + 008
< 100
EXPERIMENTS
Within this section, we first provide detailed settings and conduct extensive experiments on multiple datasets. Furthermore, we also give comprehensive ablation study to validate our findings.
4.1 EXPERIMENTAL. SETTINGS
Datasets. We conduct experiments on three publicly available datasets: ADNI dataset (Jack Jr ct al. 2008), OASIS dataset (Marcus et al., 2007). and TRT dataset (Maclaren et al., 2014). The eseudo-ground truth surfaces are obtained from Freesurfer v5.3 (Fischl 2012). We want to emphasize that using pseudo pseudo-ground ground truth as a reference is a standard practice in brain image analysis, adopted by all of the methods have included in comparison (Cruz et al. |021 Bongratz al. 2022
Santa Cruz et al., 2022; Ma et al. 2022). Therefore, using seeudo- truth does not limit the significance of contributions within the context of this paper. Regarding data split, carefully stratified (at the patient-level) each dataset into train, valid, and test sets, We select the best checkpoint based on train and validation set, subsequently reporting the outcomes on the unseen test set. Additional dataset details are available in Appendix
Baselines. We reproduce all competing methods using their official implementations and recom- mended experimental settings based on our split for a fair comparison. Specifically, we reproduce DeepCSR (Cruz et al.) (02T) in both the occupancy field and signed distance function (SDF) and report SDF results due to better performance. For Vox2Cortex (Bongratz. et al. 2022), we use the authors' suggestion with a high-resolution template with as 168, 000 vertices for each structure to get the best performance. Regarding CorticalFlow (Lebrat et al.] 2021), we reproduce with their improved version settings CFPP (Santa Cruz et al. 2022). Finally, we retrain CorteXODE (Ma et al.) 2022 with default settings.
Metrics. We employ various metrics including earth mover's distance (EMD), sliced Wasserstein distance (SWD). average symmetric surface distance (ASSD), Chamfer normals (CN), and self- intersection faces ratio (SI). We sample 100K points over the predicted and target surface to compute EMD, SWD, ASSD, and CN. Due the large number of sampled points, we estimate EMD using entropic regularization and the Sinkhorn algorithm from (Feydy et al. 2019). Furthermore, we determine SI faces using PyMesh (Zhoul 2019) library.
4.2
RESULTS & DISCUSSION
4.2.1
RESULTS
Geometric accuracy. As shown in Tab. our DDOT provides more geometrically accurate surfaces than other competing methods in multiple metrics. Qualitative results from Fig. also indicate that our proposed method is closer to the ground truth than competing methods.
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DeepCSR
Vox2Cortex
CFPP
CortexODE
Ours
GT
Figure 2: Qualitative results of white matter surface reconstruction. The color represents the point-to-face distance, i... the darker color is, the further the predicted mesh to the pseudo-groumd ground truth. More visualization is given in Appendix F and the video supplementary.
Self-intersection. Compared to GNN deformation-bar based methods such as Vox2Cortex with no SI-free theoretical guarantee, diffeomorphic deformation-base) methods such as CFPP, CortexODE, and our proposed approach has much less SI. However, despite the nice property existence and uniqueness of the solution of ODE models, CFPP and CortexODE still introduce certain amount of SI faces since they both rely on the optimization of Chamfer divergence on discretized vertices of the mesh. Our approach, on the other hand, represents mesh probability measures and has strong theoretical optimization support by employing efficient optimal transport metric, thus can approximate the mesh much better with almost SI faces, i.c. less than 10 10-4%, It is worth noting that DDOT is 100x better in terms of SIF score compared to CortexODE, the current SOTA in CSR task. DeepCSR introduces no SI thanks to Marching Cubes (Lorensen & Cline 1987) from the implicit surface but often has other artifacts and requires extensive topology correction.
Consistency. We compare the consistency of our DDOT, Vox2Cortex, CortexODE (which are all trained on OA- SIS), and FreeSurfer on the TRT dataset. We reconstruct white matter cortical surfaces from MRI images of the same subject on the same day and evaluated the EMD, SWD, and ASSD of the resulting reconstructions. The expectation that the brain morphology two consec- utive scans taken the same day should be similar to each other, except for the variations caused by the imaging process. To align pairs of images, we utilized the iterative closest-point algorithm (ICP) following (Cruz et al. (021), As presented in Tab. 2 we outperform both EMD and ASSD, and only Freesurfer (Fischl 2012) result has the better performance in SWD score than us.
CD (OV)
CD+Reg
CD+Reg (OV)
SWD
SWD (OV)
Time (s)
100
300
500
700
Number of supports (x1000)
Figure 3: Running time comparison.
The diagram indicates the scalability of three presented losses. The lines imply the losses computed between two sets of points in 3D-coordinate, the dashed lines with dots represent the loss computed between two varifolds. OV denotes ori- ented varifold, Reg denotes regulariza- gratz et al.) 2022 Santa Cruz et al.) 2022). Firstly, tion.
4.2.2 RUNNING TIME ANALYSIS
We compare the running time of CD loss, CD loss with regularization, and SWD loss, as shown in Fig. 3 The regularization of CD loss includes mesh edge loss, normal consistency, and Laplacian smoothing, which are com- monly employed in mesh deformation frameworks (Bon-
the number of supports increases, SWD loss consistently
exhibits significantly faster performance compared to CD losses. This empirical finding further substantiates our assertion regarding the theoretical running time complexities of SWD (O(m log m)) and CD (O(m?) with denotes the number of supports. Secondly, regarding high-dimensional measures, e.g. varifolds, while the CD losses rigorously scale w.r.t. the dimension of the supports, SWD loss shows minimal variation the number of supports increases, further support Theorem
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In conclusion, our proposed metric is scalable w.r.t. both the number of supports and the dimension of those supports, thus demonstrating the efficiency of our methods in learning-baat based frameworks.
4.3 ABLATION STUDY
Setups. We evaluate the individual optimization design choices and report the WM surface reconstruction on ADNI dataset. For fair comparisons, we conduct ablation studies on the same initial surface and the same number of supports, i.e. the number of faces on mesh. We train all of them for 300 epochs and get the best checkpoints on the validation set. The result is reported in Tab. 3 on the holdout test set.
White matter surface reconstruction consistency com- parison in terms of EMD, SWD, ASSD on TRT dataset.
Table
Method
EMD
SWD
ASSD
Vox2Coutex
.86 130
.855 +-70
263 +:112
CortesODE
799 1.038
.44 1:20
241 1.040
Comparisons. To begin with, we conduct experiments where we independently identically sample points on the surface and used SWD loss to optimize two sets of points. The results showed that this approach did not perform as well as optimizing SWD oriented varifold representation. This indicates that the varifold method provides better mesh approximation compared to using random points on the surface. In the second part of our ablations, we use CD to optimize two oriented varifolds. Our results show that SWD loss outperforms CD on varifold, which further supports our Theorem I Finally, we employ the Sinkhorn divergence, implemented by (Séjourné et al. 2019), , as the loss function to optimize two oriented varifolds. is worth noting that Sinkhorn divergence is the approximation of Wasserstein distance, but cannot lead to a valid metric between probability measures since the resulting discrepancy does not satisfy the triangle inequality. Experiments on both left and right WM show that our SWD loss outperforms Sinkhorn in both metrics.
PreeSurfer
4:213
.38 2.275
.25
Ours
1.14
.229 1.010
LIMITATIONS AND FUTURE WORKS
Our work the first learning- based deformation approach that tackles the local optimality problem of Chamfer dis- tance on mesh deformation by employing efficient optimal transport theory on meshes as probability measures. Yet, is not without its limitations, which present intriguing avenues for future exploration. Unlike the set-based ap- proach that predefines the number of sampling supports, our optimization settings work best the deterministic supports correlated with the mesh resolution, thus intro- ducing stochastic memory during training. Our future work will focus on mitigating this issue by either employ- ing remeshing techniques or ensuring consistent cutoff number of supports for both the predicted mesh and the target mesh. Secondly, though the underlying proposed techniques have potential applications in other deformation tasks beyond CSR, within the context of this paper, we only focus on this task. It is intriguing to explore the potential applications of our approach in diverse domains.
Table 3: Ablation Study in terms of
EMD and SWD metric. P.S. and O.V. are short for point sampling and ori- ented varifold representation, respec- tively. Best values are highlighted.
WM
Right
EMD
SWD
EMD
SWD
CD
334 -11
494 34
888
510 4:300
Sirdhom nV
1023 109
528 1 107
1003 6 071
SWD O.V.
-18
9:3
700 5.00
6 CONCLUSION
In this paper, we introduce a learning based diffcomorphic deformation network that employs sliced Wasserstein distance (SWD) as the objective function to deform an initial mesh to an intricate mesh based on volumetric input. Different from previous approaches that use point-clouds for approximating mesh we represent mesh probability measure that generalizes the common set-based methods. By lying on probability measure space, we can further exploit statistical shape analysis theory to approximate mesh as an oriented varifold. Our theorem shows that leveraging sliced Wasserstein distance to optimize probability measures can have a fast statistical rate for approximating the surfaces of the meshes. Finally, we extensively verify our proposed approach in the challenging brain cortical surface reconstruction problem. Our experiment results demonstrate that our method surpasses existing state-of-the -f-the-artttttt- competing works in terms of geometric accuracy, self- eeff-intersection ratio, and consistency.

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7 ACKNOWLEDGEMENTS
We thank Xiangyi Yan and Pooya Khosravi for their feedbacks on the paper. NH acknowledges support from the NSF IFML 2019844 and the NSF AI Institute for Foundations of Machine Learning.
ETHICS STATEMENT
Potential impacts. Our work experiments the human brain MRI dataset. Our proposed model holds the potential to assist neuroradiologists in effectively visualizing brain surfaces. However, is crucial to emphasize that our predictions should not be utilized for making clinical decisions. This is because our model has solely undergone testing using the data discussed within this research, and we cannot ensure performance on unseen data in clinical practices.
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Supplement to 'Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction"
In this suppiementary, we first discuss some related works in Appendix AA] Then, provide additional materials in Appendix B including the detailed setup and more insight about our toy example mentioned in Fig. [ Secondly, we demonstrate complete proof for the Theorem [ in Appendix C Next, we delve into the implementation details in Appendix D and provide detailed information about the datasets in Appendix E Finally, present additional visualization of our experiment results in Appendix
A RELATED WORKS
Deformation network for surface reconstruction. 3D surface reconstruction can be obtained from various approaches such as volumetric, implicit surfaces, and geometric deep learning methods.
While volumetric- based (Choy et al., 2016; Häne et al. 2017: Tatarchenko et al. 2017 Wang et al.
2018b Cruz et al.) 2021) and implicit surface-based (Mescheder al., 2019 Park et al. [2019] Xu et al. 2019) methods can directly obtain surface by employing iso-surface extraction methods, such Marching Cubes (Lorensen & Cline, 1987), they often require extensive post-processing
to capture the high-quality resulting mesh. In contrast, geometric deep learning approaches use mesh deformation achieve the target mesh while maintaining vertex connectivity (Pan et al. 2019 Smith et al. 20199 Wang et al. 20188) Sa Wickramasinghe et al. 2020; Nguyen et al. [2024b] Gupta 2020: Bongratz et al. (022). Among deformation based approaches. diffeomorphic deformation demonstrates its capability to perform well on complex manifolds while keeping the "manifoldness' property (Gupta 2020; Ma al., 2022, Lebrat et al., 2021). However, those methods often use Chamfer divergence as their objective optimization, which is sub-optimal, especially on intricate manifolds such as cortical surfaces, i.e. as illustrated Fig. 4 Therefore, in this work, we address the problem by employing efficient optimal transport in optimizing mesh during training diffeomorphic deformation models.
Mesh varifold representation. Varifolds were initially introduced in the realm of geometric measure theory as a practical approach to tackle Plateau's 's problem (Almgren 1966). which involves determining surfaces with specified boundary that has the least area. Specifically, varifolds provide convenient representation of geometric shapes, including rectifiable curves and surfaces, and serve as an effective geometric measure for optimization-
shape matching problems (Charron & Trouvé 2013 Charon 2013; Kaltenmark et al. 2017 Hsich & Charon 2020, Rekik et 2016 et al. 2010 In this work, we focus on employing varifold as a discrete measure approximating the mesh. To the best of our knowledge, we are the first to exploit oriented varifolds as discrete probability measures in the sarming-basedd deformation framework.
B TOY EXAMPLES
Setups. In this toy example, we aim to deform the template circle the target polygon in an pptimization-based We uniformly sample the template circle and the target polygon into 2D points. The number of sampled points on both the template circle and target polygon are 678 points. To optimize the position of predicted points and target points, we employ Chamfer loss implemented by (Ravi al. 2020), and the sliced Wasserstein distance with p = 2 approximated by Monte Carlo estimation with 100 projections. We optimize the two sets of points with stochastic gradient descent (SGD) optimizer with a learning rate of 1.0 and momentum of 0.9 for 1000 iterations.
Discussion. As shown in Fig. 5, we can see that the set of points optimized by Chamfer distance often gets trapped in some specific region, e.g. the acute region of the polygon in this example. This confinement occurs due to the nature of Chamfer distance, which primarily focuses on optimizing nearest neighbors, inhibiting the points from escaping the local region during the optimization process. To alleviate this issue, practitioners often introduce multiple losses regularizers to aid Chamfer distance in escaping local minima. However, determining the appropriate weights for each auxiliary loss is a challenging task, as they tend to vary across different tasks, thus making the optimization process harder. SWD loss, on the other hand, can find the optimal transport plan for the whole set of points, thus resulting in better solution when compared Chamfer distance.
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Figure 4: Comparison between SWD loss (left) and CD loss (right). The mesh obtained through probability measure representation and SWD optimization exhibits a more uniformly surfaced appearance compared to the set-based approach that optimizes with CD loss.
CD loss
SWD loss
Iter
Iter 25
Iter 50
Iter 200
Figure 5: Visualization of the optimization process of 2D toy example. The set of green points, i.e. sampled points from the template circle, optimized by CD loss often concentrates around the acute region of the polygon and easily gets trapped some local regions. Nonetheless, the set of points optimized by SWD loss distributes more uniformly along the edge of the polygon, thus making the optimization process more robust.
C PROOF OF THEOREM 1
Using the triangle inequality of the L, norm, we obtain:
E swi (DiM MMM L) SWP(WM Ma
< ssw (iM pMMa L) SWP(TM Ma)
E [SWP(MM ( SWP MeM
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Now, we bound the first term E sw p (Ami, (m;;1) SWp pm pm) the SW and its Monte Carlo estimation, we have:
By the definitions of
E
SW Capplin
(im , iim L) Sip (Ma))
1
E
Ve 1 - B(W ,000 ( Y1ml
27
E
wg(Ortnm ,000 "0 | Yium]
27 ( Yisml)
1
WO ,0EPMM)) E[ (010mM))
where the inequality is due to the Holder's inequality. Using the fact that Σ (0 IX X1sm) = (0 ) Yiml since
01..... OL sid u($=11), we have:
s Var (
w W2(0:tpm ( X1mm Xamn]] )])
SW (Prm p= (W " AmM)
Σ
1 Var (Wz(0tpmm) @
X1:m Y1=m]]
VI
Now, we bound the second term E [[Swp " SW.(a inequality, obtain
Using the Jensen
E [|SWB(DM pM1)) SWS M2)
= [E[ - - B[WB(0EuM
S E [W -0000)) - Wg(00n 0th
BI (O M1) w (OtúMM Ma)]
E| (Oraa Ma)))
where the second inequality is due to Lemma 4 in (Goldfeld et al. 2022). We now show that:
(0 ( Mi) S CC. (2) (d + 1) log m
m
Following (Nguyen al. 2021a N guyynn & Ho 2023).
have:
(o5i Mi)
< E
max Wi Wi (0 Otp X1sm, Vi.m 8E84-1
E
max pERC1100|aSS
(110a Mi X Yim
/ [Fmb(2)) (2)ldz) Vi DERTITIO!
E
max
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where ( (2) is the inverse CDF of fiM F3 (z) is the inverse CDF of MM Using the assumption that the diameter of M at most R, we have:
[
[F (z)d Xim, Vimm
max OER
1101
E
max
[Fmo(y) - Po(w)) X1mm X1m
0ER
Σ RE
| Fm, - Fo(e))
sup
pm (C) - م Mi (C))
RE
sup
CEB
where B e 1 I y) is set of half spaces for and y. Since the
(VC) dimension of B is upper bounded by d d (Wainwright, 2019), applying the VC inequality results:
-() S scv (d + 1)log
sup CEB
Am Mi (C)
for a constant proof.
0. Absorbing constants, we obtain the final result. Therefore, we conclude the
D IMPLEMENTATION DETAILS
Network architecture. First of all, for the white matter segmentation model, we train a vanilla 3D U-Net model with sequential 3D convolutional layers with kernel size 3 R we linearly transform it to 128 128-dimensional feature
3. Secondly, for
the deformation network, for each vector. For each Vp, we find corresponding cube size 5 x on I. This process can be repeated across multiple scales of I, resulting in the extraction of multiple cubes. Then, apply 3D convolution layer followed by a linear layer to transform the spatial cubes to a 128-dimensional feature vector, i.e. same the feature of co- Once having the 0o 's features and its corresponding spatial features, we concatenate them as new feature before passing through an MLP, namely Fs, to learn the deformation. As discussed, represent the predicted mesh M and the target mesh M as probability measures y.Mh and yrM'. respectively. In practice, measures, e.g. oriented varifold, as j. and jiM respectively. The loss function is computed as follows:
substitute discrete probability
((M,M') = SW (T- aM))
T. Wg(0rtm 0i
where W. M²) ) is the Wasserstein-p Villann 2003) distance between ji and jiM We fix L 100, p- for all of our experiments. terms oriented varifold, for each support, we concatenate the barycenter of vertices of the face and the unit normal vector single vector in R The training procedure is described in Algo.
Training details. We optimize both segmentation and deformation networks with Adam opti- mizer KKngmma & Ba 2014) with a fixed learning rate 10-4 We train the segmentation and the deformation networks for 100 and 300 epochs, respectively, and get the best checkpoint the validation set. All experiments are implemented using Pytorch and executed on a system equipped with NVIDIA RTX A6000 GPU and Intel i7-7700K CPU.
E DATASET INFORMATION
Dataset split. As discussed in Sec. 4. employ three publicly available datasets: the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset (Jack Jr et al. (2008). the Open Access Series of Imaging Studies (OASIS) dataset (Mareus et al. 2007), and the test-retest (TRT) dataset (Maclaren
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Published as a conference paper at ICLR 2024
Algorithm 1 Training cortical surface reconstruction with SWD distance
Input: MRI volume I, initial mesh MMg (Vo.F), learning rate 7,, max iter T, number projections
L.
Initialization: Deformation network IF (o(
while é not converge or reach '" do
V + 0
> Zero gradient.
V < ODESolver(E Vo)
> Deform source vertices Vo to V.
M + (V.F)
Get predicted mesh. Transform to discrete measures.
iM + ToMeasure( HM + ToMeasure(M"))
V V El VoWp(OttiM Mi
Update gradient. > Update parameters.
0 " V
end while
Return: QM-M
al. 2014). A subset of the ADNI dataset (Jack et al. 2008) is employed, consisting of 419 T1-weighted (TIw) brain MRI from subjects aged from 55 to 90 years old. The dataset is stratified into 299 scans for training (Rs 70%), 40 seans for validation(<< 10%). and 80 scans for testing (ee 20%). Regarding the OASIS dataset (Marcus et al. 2007). all 416 TI-weighted (TIw) brain MRI images are included. We stratify the dataset into 292 scans for training (=s 70%), scans for validation (=s 10%), and 80 scans for testing (z= 20%). As for the TRT dataset (Maclaren et al. 22014) consists of 120 scans obtained from three distinct subjects, with each subject undergoing two scans within a span of 20 days.
Preprocess. We strictly follow the pre-processing pipeline from (Bongratz et al. 2022). Specifically, first register the MRIs the MNI152 scan. After padding the input images to have shape 192 x 208 x 192, we resize them to 128 x 144 x 128. The intensity values are min-max-noomaalne the range [0, 1].
F VISUALIZATIONS
We provide more visualization of work in Fig. 6 We randomly select the prediction meshes from the test set and compute the point-to- cant-to--urface distance. The color is encoded as how far the point is to the surface. The figures say that the darker color is, the further the predicted mesh the pseudo-groutd truth.
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Published as a conference paper at ICLR 2024
ADNI
OASIS
Ours
CorteXODE
CFPP
Vox2Cortex
DeepCSR
Figure More examples of predicted mesh color-coded with the distance to the ground-truth surfaces as shown in Fig. of the main paper.
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Reviews:
[{"review_id": "t4HXGAxZnA", "reviewer": "Reviewer_As4U", "summary": "This work addresses the challenge of efficiently measuring the discrepancy between predicted and target 3D meshes, a key component in various 3D vision tasks. The paper introduce a novel metric for learning mesh deformation, defined by the sliced Wasserstein distance. This metric operates on meshes represented as probability measures, which generalize the set-based approach. This approach offers computational efficiency, flexibility in encoding mesh representations, and outperforms other methods in cortical surface reconstruction experiments.", "questions": "1. Is the initial surface in your method the same as the initial surfaces used in competing methods, which are also extracted from the white matter segmentation mask of the brain MRI image?\n2. How about the topological information of the initial surface and the ground truth surfaces? Are they all genus-0 surfaces? Does the DDOT framework maintain the topological information during the deformation process?\n3. It is better to give a brief introduction of oriented varifold for paper's readability.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This work introduces a novel representation for triangle meshes as probability measures, which generalizes the common set-based approach within a learning-based deformation network. To be precise, the paper outlines three forms of mesh representation as probability measures: continuous, empirical, and discrete measure through the utilization of oriented varifolds. And the authors present a novel learning-based framework DDOT for Diffeomorphic mesh Deformation framework via an efficient Optimal Transport metric, which leverages an efficient Optimal Transport metric. DDOT enables the learning of continuous dynamics to smoothly deform an initial mesh into a complex shape. \n\nThe paper is presented in a clear and structured manner, making it easy for readers to understand the proposed methods and experimental results. The work demonstrats improved performance over existing state-of-the-art methods in experiments on multiple brain datasets, particularly in terms of EMD, SWD, ASSD, CN an SI.", "weaknesses": "The authors depict a mesh as an oriented varifold, a pivotal concept in this paper. However, for enhanced clarity and accessibility, it's better to provide a concise introduction to oriented varifolds, even though the foundational idea is rooted in earlier works. This brief overview will promote a fundamental understanding of oriented varifolds and their importance in our study, improving the paper's readability.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "dGmazLpJk7", "reviewer": "Reviewer_4B9Y", "summary": "This work introduces a learning-based diffeomorphic deformation network that employs sliced Wasserstein distance as the objective function to deform an initial mesh to a complicated mesh based on volumetric input. The work proposes to present triangle meshes as probability measures that generalize the common set-based approach in a learning-based deformation network; the work proposes the sliced Wasserstein distance as a metric  for learning mesh deformation, and proved the convergence rate is solely determined by the number of samples, independent of the dimensionality; the work conducts extensive experiments on white matter reconstruction by employing neural ODE, which show the proposed method outperforms the SOTA in terms of geometric accuracy, self-intersection and consistency.", "questions": "1. What is the cost function defined on the position-orientation space? \n2. Does the Wasserstien distance vary when one mesh is transformed under a rigid motion ?\n3. Why use the position-orientation space to represent the mesh measure? Why not just use position space?\n4. Can we say something about the regularity of the optimal transportation map? Is it diffeomorphic ?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This work has solid theoretic foundation, especially the theorem 1 is novel and promising, which shows sliced Wasserstein distance metric   has faster convergence rate than others, and suitable for practical applications.\n2. The experimental results are thorough and convincing, the reconstruction mesh quality is good for geometric analysis purposes.\n3. The work is well written, the representation is clear, the logic is clean, and the theoretic deduction is explained in details.", "weaknesses": "The proposed method to represent a mesh as a distribution in the position-orientation space, and the optimal transportation map is carried out in this space. It should be explained the cost function, and also this representation depends on the position and the orientation of the mesh, and the Wasserstein distance varies if one mesh is transformed by a rigid motion.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "dGuXIgDIWg", "reviewer": "Reviewer_9F6e", "summary": "This paper presents a learning-based diffeomorphic deformation network that employs sliced Wasserstein distance (SWD) as the objective function to deform an initial mesh to an intricate mesh based on volumetric input. Different from previous approaches that use point-clouds for approximating mesh, it represents a mesh as a probability measure that generalizes the common set-based methods. By lying on probability measure space, it can exploit statistical shape analysis theory to approximate mesh as an oriented varifold. It proves a theorem that shows that leveraging sliced Wasserstein distance to optimize probability measures can have a fast statistical rate for approximating\nthe surfaces of the meshes. The main application is on brain cortical surface reconstruction. Experiment results demonstrate that the proposed method surpasses existing state-of-the-art competing works in terms of geometric accuracy, self-intersection ratio, and consistency.", "questions": "The genus zero requirement might be another limitation? can we extend it to shapes of other topology?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper proposes a new metric for learning mesh deformation defined by sliced Wasserstein distance on meshes represented as probability measures that generalize the set-based approach. By leveraging probability measure space, it can gain flexibility in encoding meshes using diverse forms of probability measures, such as continuous, empirical, and discrete measures via varifold representation. The new metric seems novel and works well.", "weaknesses": "The paper is very math-heavy and is relatively hard to read for non-experts. The results seem limited to the brain surface. The authors do mention the limitation, which I assume would be a high requirement on the mesh quality. I would like to see more details on that, i.e. how applicable this method is on a common dataset/mesh. The genus zero requirement might be another limitation? can we extend it to shapes of other topology?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "N86v7tgtwZ", "reviewer": "Reviewer_LCQZ", "summary": "The paper proposes to learn a diffeomorphic flow using sliced Wasserstein distance (SWD), instead of classical chamfer distance (CD) and earth mover distance (EMD). Such a design choice is not tried before in the context of mesh optimization. The authors shows that SWD leads to better performance, compared to some diffeomorphic flow baselines using other losses.", "questions": "1. I am not sure if the computation of chamfer distance (CD) should be slower than that of SWD. Some paper argues that the complexity of SWD is similar to CD [2]. And by using KD-tree for nearest neighbor retrieval or some recent methods [3], one should be able to compute CD much faster. I would like the authors explain the speed difference between CD and SWD shown in Fig. 3.\n\n2. There are some improved variants of CD, such as [4]. Just out of curiosity: how do they perform compared to SWD?\n\n[3] Bakshi, Ainesh, et al. \"A Near-Linear Time Algorithm for the Chamfer Distance.\" arXiv preprint arXiv:2307.03043 (2023).\n\n[4] Wu, Tong, et al. \"Density-aware chamfer distance as a comprehensive metric for point cloud completion.\" arXiv preprint arXiv:2111.12702 (2021).", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The proposed solution is simple: just to replace CD with SWD.\n- The authors provide theoretical justification for the benefit of using SWD.\n- The authors show better quantitative and qualitative results on mesh optimization to reconstruct the cortical structure, compared to some baselines on the same task.", "weaknesses": "- The novelty of the idea is limited -- optimizing meshes with CD and/or earth mover distance (EMD) between sampled point clouds is not new (for instance [1]). Replacing EMD with another distributional distance seems more or less some pure trial-and-error endeavor. Indeed, the paper [2] has shown that optimization with SWD on point clouds is beneficial. It is therefore not surprising that SWD can be used for mesh optimization.\n\n- I feel that the math for the probabilistic interpretation, although probably not explicitly presented in the context of mesh optimization, is unnecessary at least from the practical point of view, especially given that EMD has already been used for mesh optimization.\n\n- The presentation of the paper needs to be improved. There are many notations not explained (e.g., the # operator in Eqn. 5).\n\n- While the proposed method seems general, the scope of this paper is quite limited to cortical structures. The authors may consider trying the proposed method on general shapes and compare the results on common benchmarks (if any).\n\n[1] Wang, Weiyue, et al. \"3dn: 3d deformation network.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2019.\n\n[2] Nguyen, Trung, et al. \"Point-set distances for learning representations of 3d point clouds.\" Proceedings of the IEEE/CVF International Conference on Computer Vision. 2021.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (poster)