Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data
Authors: Xinran Liu
Yikun Bai
Rocio Diaz Martin
Kaiwen Shi
Ashkan Shahbazi
Bennett Allan Landman
Catie Chang
Soheil Kolouri
Venue: ICLR 2025 Spotlight (2025)
Keywords: Optimal Transport
Spherical Data Analysis
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We introduce a fast Linear Spherical Sliced Optimal Transport metric for comparing spherical probability measures in various applications.

Abstract: Efficient comparison of spherical probability distributions becomes important in fields such as computer vision, geosciences, and medicine. Sliced optimal transport distances, such as spherical and stereographic spherical sliced Wasserstein distances, have recently been developed to address this need. These methods reduce the computational burden of optimal transport by slicing hyperspheres into one-dimensional projections, i.e., lines or circles. Concurrently, linear optimal transport has been proposed to embed distributions into $L^2$ spaces, where the $L^2$ distance approximates the optimal transport distance, thereby simplifying comparisons across multiple distributions. In this work, we introduce the Linear Spherical Sliced Optimal Transport (LSSOT) framework, which utilizes slicing to embed spherical distributions into $L^2$ spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures. We establish the metricity of LSSOT and demonstrate its superior computational efficiency in applications such as cortical surface registration, 3D point cloud interpolation via gradient flow, and shape embedding. Our results demonstrate the significant computational benefits and high accuracy of LSSOT in these applications.

Published as a conference paper at ICLR 2025
LINEAR SPHERICAL SLICED OPTIMAL TRANSPORT: A FAST METRIC FOR COMPARING SPHERICAL DATA
Xinran Liu', Yikun Bai', Rocío Díaz Martín', Kaiwen Shi', Ashkan Shahbazi', Bennett A. Landman Catie Changl 1.3 SShhil Soheil Kolouri for the Alzheimer's Disease Neuroimaging Initiative' *Department of Computer Science, Vanderbilt University, Nashville, TN, 37240 *Department of Mathematics, Tufts University, Medford, MA 02155 3 DDepartment of Electrical and Computer Engineering, Vanderbilt University, Nashville, TN, 37240
ABSTRACT
Efficient comparison of spherical probability distributions becomes important in fields such as computer vision, geosciences, and medicine. Sliced optimal transport distances, such as spherical and stereographic spherical sliced Wasserstein distances, have recently been developed to address this need. These methods reduce the computational burden of optimal transport by slicing hyperspheres into one- dimensional projections, i.e., lines or circles. Concurrently, linear optimal transport has been proposed to embed distributions into T.2 spaces, where the 1.2 distance approximates the optimal transport distance, thereby simplifying comparisons across multiple distributions. In this work, we introduce the Linear Spherical Sliced Optimal Transport (LSSOT) framework, which utilizes slicing to embed spherical distributions into L spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures. We establish the metricity of LSSOT and demonstrate its superior computational efficiency in applications such as cortical surface registration, 3D point cloud interpolation via gradient flow, and shape embedding. Our results demonstrate the significant computational benefits and high accuracy of LSSOT in these applications.
INTRODUCTION
Spherical signal analysis has received increasing attention in many domains such computer vision.
astrophysics (Jarosik et al., 2011). climatology (Garrett et al. 2024) and neuroscience (Elad &
Kimmel 2002 Zhao et al. [019 2023).. In practice, certain types of data intrinsically defined on
a sphere, namely omnidirectional images, ambisonics, and earth data. In addition, some forms of
data are studied canonically in hyperspheres by spherical mappings (Gotsman et al., 2003 Cu et al.)
& Makedon 20066; Chung & Chen (2022) from the original data spaces. Among them, notable category data the genus-zero surfaces, which can be conformally mapped to unit spheres according to the Poincare uniformization theorem (Abikoff) 1981). Cortical surface-based based analysis, for example, can be facilitated by the inherent spherical topology (0- (0-genus) of the cerebral cortex. Compared to conventional 3D volumetric analysis, applying spherical representations of cortical surfaces offers a more streamlined and accurate geometric framework to examine the brain. This approach proves particularly advantageous in both cross-scctional and longitudinal studies of brain structure and function (hhao et al.) 2021: 2023).
Analyzing spherical data through machine learning techniques necessitates metrics that can effectively compare signals while maintaining data structure and geometry. In this context, optimal transport (OT) has emerged provides a framework for comparing probability distributions in various domains (Kolouri et al.
versatile and widely adopted tool in the machine learning community. OT
2017
Peyré al. 2019: Kolkin et al., 2019 Arjovsky et al.! 2017 Seguy al. [017]] Makkuva 2020 Cang et al. 2023). OT has demonstrated remarkable effectiveness not only in Huclidean
et al.
*Data used in preparation this article were partially obtained from the Alzheimer's 's Disease Neuroimaging Initiative (ADNI) database (aani, Ioni, i usc. edu).. As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this paper. A complete listing of ADNI investigators be found at: hhtpp: / i. oon usc. edu/ wp-conte wwwtennttnttseee
ttp__popppppp__p__ ADD _.ckkokkowl edgement_iiiieeeeeeeeeeeeee st..tpp

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spaces, but its theoretical foundations have also been finmly established for various other manifolds
(Fathi & 2ig1 McCann & Topping 2010), especially on spheres (Cui et al. 2019 Hamfeldt
& Turqquss 2022). However, the calculation of OT distances is prohibitively expensive with
cubic cost ( (N+). This computational burden poses a substantial challenge to applications where the input size N is large. In light of this, researchers have enthusiastically pursued and developed
faster alternatives, including entropic optimal transport (Cuturi, 2013) and sliced optimal transport
(Bonnotte 2013 Bonneel et al. 2013 Kolouri al. 2016 2018 Deshpande et al. 2018 Kolouri
et al. 2019).
Sliced Optimal Transport (SOT) accelerates computation by projecting high-dimensional data onto one-dimensional subspaces and applying efficient OT solvers for the resulting one-dimensional probability distributions. In this paper, we are interested in sliced optimal transport methods on hyperspheres. Related work includes the Spherical Sliced Wasserstein (SSW) discrepancy (Bonet al. 2023), which proposes slicing the sphere by the spherical Radon transform and leveraging circular optimal transport; and the Stereographic Spherical Sliced Wasserstein (S3W) distance (Tran) ct al. 2024). which utilizes stereographic projection to solve the SOT problem in Euclidean space with minimal distortion.
Many machine learning applications, from manifold learning to k-nearest neighbor classification and regression, rely on pairwise distance calculations among members of a set, often requiring distance evaluations on the order of the square of the number of objects. To accelerate pairwise calculation of OT distances between probability measures, Linear Optimal Transport (LOT) (Wang et al.) 2013) was developed as a framework that embeds distributions into L spaces where pairwise distance computations can be performed with significantly higher efficiency. For example, Wasserstein Embedding and Sliced Wasserstein Embedding have been applied in Euclidean spaces to enable efficient image analysis Basu et (2014).) faster graph learning (Kolouri et al., (020), supervised task transferability prediction (Liu et al.) (022), point cloud retrieval (Lu et al. (2024), etc. A more recent work, Martin et al. (2023) focused on ş and established a computationally efficient Linear Circular Optimal Transport (L.COT) framework to embed circular distributions into a 1,2 space.
In this paper, we build on the work of Martin et al. (2023) and extend the linearized methodology from circles to hyperspheres. We develop novel embedding technique for spherical probability distributions, which significantly enhances the efficiency of group analysis spherical domains.
Contributions. In summary, our contributions are as follows:
1. We propose Linear Spherical Sliced Optimal Transport (LSSOT) to embed spherical distri- butions into L space while preserving their intrinsic spherical geometry.
2. We prove that LSSOT defines a metric, and demonstrate the superior computation efficiency over other baseline metrics.
3. We conduct a comprehensive set of experiments to show the effectiveness and efficiency of LSSOT in diverse applications, from point cloud analysis to cortical surface registration.
2 BACKGROUND
2.1 CIRCULAR OPTIMAL TRANSPORT AND LINEAR CIRCULAR OPTIMAL TRANSPORT
Consider two circular probability measures μ, v E P(S!), where S² denotes the unit cirele in R². Let us parametrize S with the angles in between 0 and and consider the cost function c(z, y) == h(x yls!). where h : I + R+ is a convex increasing function and Jæ yls == minflæ - yl. 1 xx - y/} for x, E [0, 1). The Circular Optimal Transport (COT) problem between Ia and defined by the following two equivalent minimization problems:
CO inf sef((uM) xs
( c(x, (1r,))d((z,)) v) = inf c
- -x (x-)d)) (1)
where in the first expression I'(p, 1/) the set of all couplings between μ and v, and in the second expression F2 (respectively, F.) is the cumulative distribution function of μ on S (i.c., (y) == ([oo, y)) = -h" du, Vy E 1)) extended on R by Fy + n) = Fy(y) + Vy E [0. 1), n E Z, and its inverse (or quantile function) is defined by F-1(y) (v) = inf (:: (x) > v}. When h(x) = | P for 1 15 p < oc, we denote COT,(-.)) as COT,(- -). and COT,(-.)) (-.-->>> defines a metric on P(S)). Moreover, if μ U and h(x)) I the minimizer Cx,, of ①1 is the antipodal of (E()) i.e.,
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a. B(v) - 1/2. We refer the reader to Delon al. 20100) Rabin al. 201 Bonet al. (2023)
for more details.
Recently, Martin al. (2023) expanded the linear optimal transport framework (Wang et al. 20 12) to encompass circular probability measures, introducing the Linear Circular Optimal Transport (LCOT) embedding and metric. For the reader's convenience, summarize the key definitions from their work below.
The LCOT embedding corresponds to the optimal circular displacement that comes from the problem of transporting a reference measure to a largel measure with respect to COTx(:;-) Precisely, given a reference measure μ P(tt)) that is absolutely continuous with respect to the Lebesgue measure S', the LCOT embedding of target measure E P(S") is defined by
Duuh (z) := FF an.v) I, Vr [0, 1),
where ap is minimizer of ① When h(x) |0/|" for 1 S p < co, this embedding gives rise to the LCOT distance between circular measures:
LCOT
1 1s u.h (d),
Vn E P(S
(3)
where
LP(S' du) = (f s' + [1 = / (f(t)s, du(t) o0).
(4)
If H = Unif(S'), we use the notation L'((') = L dp). If also h(x) = |x| the LCOT embedding becomes
(2- E-- ( x.
= F x
Ve e e0,10,1)
(5)
and we denote the LCOT distance simply as LCOT2(;)).
2.2 SPHERICAL SLICING
Bonet al. (2023) recently presented the Spherical Sliced Wasserstein (SSW) discrepancy for probability measures supported the sphere gd- c: RR that projects Vzr gd-1 great circles C to:
arg min (arcooscccss( v)), v
(6)
where arccos((:,)) ():)) is the geodesic distance on gd-1 Since each great circle is obtained by inter- secting the sphere with 2-dimensional plane in Fd the authors proposed average over all great circles by integrating over all planes in Ra (i.e., the Grassmann manifold (Bendokat et al. (2020)) and
then project the distributions onto the intersection with gd-1 However, due to the impracticality of the Grassmann manifold, the set of rank 2 projectors is used instead:
(2( {U Rdx2 U 12).
where 1/2 ((R)) the Stiefel manifold (Bendokat et al. 2020 Hence, the integration process can be performed over V2 (IR") according to the uniform distribution ((V)(R))). The projection onto E (R()) can be computed as in (Bonet et al. 2023 Lemma 1) by
UTT
Vz E gd-1
\ fz 10:U0x=-
||Tx/|12
Finally, given u,v E P(gd-1), the Spherical Sliced Wasserstein discrepancy between them, for 1 < p < co, is defined in (Bonet et al. 2023) by
(SSW,
(COT, #1, ( dU)..
(8)
(o(R
METHOD
In this section, first introduce the Linear Spherical Sliced Optimal Transport (LSSOT) embedding and discrepancy, and then state the main theorem LSSOT indeed defines a metric in P P(Sd-1))
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3.1 LINEAR SPHERICAL SLICED OPTIMAL TRANSPORT (LSSOT)
Definition 3.1. Consider spherical measures V1 E P((S-1). We define the Linear Spherical
Sliced Optimal Transport (LSSOT) embedding 14, i = 1,2, with respect to slice U E V2 (2((R)) by
D. == (=) (P"(0)- -PP)) v) + 1 +22
x xE ses UE VV2(
(9)
where PV Us denotes the pushforward measure on S of vs by the map pV. We also define their LSSOT discrepaney by
I (LCOT2 11 P, v2)) do(U)
(LSSOT((V1:2))) =
p
(10)
12(R4)
- Vosce (,U) (,)))) do(U);
Remark 3.2 (LSSOT extends LCOT), When d = pU 12 for every U E V2(R2) since (IR coincides with the set of orthonormal basis on R.2. Thus, for any v E P(S!) we have that 8 (rr T (ar, V) for all U. V V2R(R),, so the LSSOT embedding and the LSSOT discrepancy coincide with the L.OT embedding and with the LCOT distance respectively, as the integration is normalized to I on V2 (TR2 (which isomorphic to the orthogonal group (2) of dimension 1).
Theorem 3.3 (Metric property of LSSOT). LSSOOT(<;)) defines pseudo-metric in P((t-)), and a metric when restricting to probability measures with continuous density functions. We will refer to it as LSSOT distance.
We refer the reader to the Appendix for detailed proof. The symmetry of LSSOT2(;.) is casy to deduce from its definition. The triangle inequality holds as an application of Minkowski inequality by using the fact that LCOT2(,;) -) is a metric (see Proposition D.I in Appendix D). For proving the identity of indiscernibles, the idea is to give an alternative expression of the LSSOT, involving Spherical Radon Transform R: see (45) in Appendix C. which is inspired by Bonet et al. (0023).
Such formula reads for probability density functions fI,J2 E (gd-111))) as
[ va(Ke)
(L))
(LCOI,
do(U)
(1I)
where R: LS + (V2(())) x 8 is the integral operator
Rf(U,2) [ 11EE80-1.-=P000) f(z)dz,
where the surface/line integral is taken over (d (d-22) 2)-dimmnnion surface, which coincides with a (1- dimensional) great semicircle when d = 3. We will call this transformation as Semicircle Transform when d = 3. and (d - 2)-HHemispherrcal Transform
gd-1 in general. We describe R. acting on ('(S2) by using different and detailed parameterizations in Appendix A and generalize for any dimension d in Appendix B In Appendix C R is extended to the set of measures on the sphere, and the disintegration theorem from measure theory is used to obtain the general version of (IT). The novelty is that will relate R with an integral transformation presented in Groemer (1nemerr19993 (1998), instead of relating with the Hemispherical Transform in Rubin (1999) as done in BBontt et al. (2023) (see Remark BB. for discussion in this regard). By doing so, we can prove the injectivity of R on continuous functions
( Corollary B.7)) which is the key property to show that if LSSOT2 p2) = 0 then V V22
As a byproduct, we obtain that SSW,(,)) is not only pseudo-metric as proven in Bonet al. (203). but also a metric when restricted to probability measures with continuous density functions (Theorem D.3 in Appendix D)
3.2 LSSOT IMPLEMENTATION
Following the Monte-Carlo approach for com- puting the classical Sliced Wasserstein distance, we consider samples from the uniform distribu- tion E ((V((())) for generating the slices.
Figure 1: Semicircle Transform projects non-polar points to the closest location on the circle, and the north/south poles to everywhere on the circle.
(which is equivalent considering two absolutely continuous probability measures 1d1
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Algorithm 1 Linear Spherical Sliced Optimal Transport (LSSSTT)
Require: Spherical distributions μ - > N2 reference size M, threshold
i=1 agdm; and Σラ= Uzōnzi number of slices L;
1 to I. do
Construct a matrix Z4 E TRdx2 with entries randomly drawn from (V 1) Apply QR decomposition on Zs to get an orthogonal Q matrix Ui
Project { J N, and {ys و on I2 to get a U. x, and 0/ U//y, yo. Vi.j
Find 17- = }i = { such that || || S E or 1 S
Apply c-cap
around poles
oi = Exer ai*, bj = === by:: N, = N₂ - |1 No = No - |J
i 1i=
Di= (a: Ny. 1544 in شز=
N,
- (bg + N. )6
Project all at and 8j on s P( | P())) T65
Calculate p.2 | by 3 where iitros
Σxn (a; + Nr )r(ety upros
10 + 3 N. p(of)) ji is the discrete uniform reference measure of size M
end for
return ISSOTT(,,U) R (+ ICOTS m
To do so, a standard procedure from the standard normal distribution (0, 1), then use the Seaam-chhmidt algorithm to orthonor- malize the column vectors of the matrix, or apply the deR-dccomposiioonn and finally normalize the orthogonal matrix Q (dividing each column vector of Q by its Euclidean norm). In contrast to the Spherical Sliced Wasserstein (SSW) numerical implementation in which projections of poles are not well defined (when U x 0 in Z LSSOT maps the polar points everywhere the equator by definition. To achieve this mapping numerically, we put an c-cap around the poles, and redistribute all the mass inside the c-cap evenly onto the discrete points outside the cap. See Figure [ for visual depiction of the projection and I detailed implementation steps.
first construct matrices Rdxx whose components are drawn
3.3 TIME COMPLEXITY AND RUNTIME ANALYSIS
Let L denote the number of slices and d denote the dimension. For a discrete spherical distribution with N samples, the slicing and projection require OLdNN) For single slice, the time complexity of calculating the LCOT embedding (Equation [5) of the corresponding circular distributions with respect to the uniform distribution is (NN after sorting. Therefore the time complexity of calculating LSSOT embedding of is O(LN + log N 1)), where LN log N comes from the sorting $1.
For a set of K distributions { each with N samples, the pairwise LSSOT distances can be boiled down K LSSOT embeddings and LK(K-1) distances in (S thus cost (d 1+ (-1)). However, other methods such as spherical Optimal Transport (OT), Sinkhorn divergence, Sliced Wasserstein Distance (SWD), SSW, and Stereographic Spherical Sliced Wasserstein distance (S3W) all require the time complexity O(K2D), where D is the complexity of each distance, e.g. O(K2N 2 N) for spherical OT and O(K* log N) for SSW, compared to O LN) for our method. Figure 2 shows the pairwise distances computation time of LSSOT along with the methods mentioned above. All methods distributions are randomly generated with sample sizes N = 1000, 5000, 10000, 12500, 15000. For each fixed sample size, we observe that LSSOT stands
2
the CPU for fair comparison. The
the fastest in all methods under the increase in the number of distributions. In the Appendix we also provide the runtime comparisons for slice-based methods with varying numbers of slices.
1000
5000
Y- 10000
N-12500
1500000
SUN
SUN
seconds
107
Shefic
10
10
32
37
10
127
distiibutions
number
disributions
distributions
Figure 2: Pairwise distances runtime (log scale) comparison w.r.t the number of distributions. N denotes sample sizes in each distribution. The number of slices is 500 for all alice-based methods.
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4 EXPERIMENTS
To demonstrate that the proposed LSSOT is an effective and efficient metric of spherical distributions while preserving the geometry, we design and implement the following experiments on toy examples, cortical surfaces, and point clouds.
4.1 ROTATION AND SCALING IN SPHERICAL GEOMETRY
We test the LSSOT metric spherical geometric transformations such as rotation and scaling.
For rotations, we generate set of 20 Von Mises-Fisher (VMF) distributions {p' (2;; jas, k) = 2m(en-e exp(xp; = with the same concentration parameter K, but the mean directions fus are 20 distinct points on the Fibonacci Sphere, i.e. spread evenly on S?. Since the 20 VMFs are of the same shape with varying mean directions, each can be interpreted as a rotated version of another. We calculate the pairwise LSSOT distances among the 20 distributions and visualize the embeddings in 3-dimensional space using multi- mulli-dimeenional scaling (MDS). We use the Scikit Learn (Pedregosa et al. (201I) sklearn. manifold. function for MDS implementation. As shown in sklearn.manifold..o
Figure 3 (lett), the embeddings spread almost evenly on the sphere, implying the LSSOT captures the rotation geometry.
To study the scaling geometry, we generate 6 VMF distributions with fixed μ but varying k. and then apply the same LSSOT pairwise distances calculation followed by the MDS embedding procedure. Figure 3 (right) illustrates that the LSSOT metric preserves the linear transitions in scaling geometry, with embeddings falling approximately onto a straight line.
LSSOT for Rotating VMFs
LSSOT for Scaling VMFs
10
25
25
25
Figure 3: Generated VMF distributions and manifold learning results by LSSOT. Left: group of 20 VMFs generated by rotating source VMF, and the corresponding 3-dimensional visualizations by LSSOT and MDS. Right: VMFs generated by scaling a source VMF with 30, and the same 3-dimensional visualizations.
4.2 CORTICAL SURFACE REGISTRATION
We further verify the validity and efficiency of the proposed LSSOT method in comparing spherical cerebral cortex data for registration tasks. Cortical surface registration seeks to establish meaningful anatomical correspondences across subjects or time points. This process is essential for surface-based analysis in both sectional and longitudinal neuroimaging studies. Cortical surface scans are often acquired from diverse spatial frameworks, necessitating a preliminary alignment process to situate them within a common space. This alignment is crucial for enabling meaningful group analyses and comparisons. The unit sphere $ is usually chosen to be this common space, owing to the inherent genus-O spherical topology of the cortical surfaces (Wang et al. 2004 Gu et al. 2004 Lui et al. 2007). Thus, such aligning involves mapping each hemisphere of the convoluted cerebral cortical surface onto the 8 space with minimal distortions. Then the goal is to register spherical cortical surfaces to template, i.e. computing smooth velocity field on the sphere which is used to deform the moving vertices into the fixed template. Through this deformation, correspondences are established across a group of cortical surfaces.

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4.2.1 EXPERIMENTAL SETUP
We leverage the Superfast Spherical Surface Registration (S3Reg) (haao et al., 2021) neural network to perform
registration, which focuses on registering all surfaces to atlas surface. S3Reg is an end-to-end unsupervised diffeomorphic registration network based on the Voxelmorph framework (Balakrishnan et al. 2018) and Spherical Unet (Zhao et al. 2019) backbone, offering flexibilities in choosing the similarity measure for spherical surfaces in the training objectives. Following the S3Reg experimental setup with resolution level at the 6-th icosahedron subdivisions (i.e. 40962 vertices on each surface), we replace the original mean squared error (MSE) similarity loss with our LSSOT distance and other baselines, and evaluate the registration performance achieved by each similarity measure. We use the Freesurfer fsaverage (Fischl 2012) surface as the fixed (atlas) surface. All models are implemented in PyTorch (Paszke al., 2l.]2001) and run on Linux server with NVIDIA RTX A6000 GPU. More experimental details are documented in Appendix F)
Dataset and Preprocessing. We perform registration on the NKI dataset (Nooner et al. 2012) and ADNI dataset (Jack Jr et al. 2008). NKI contains W1-weie brain MRI images from 51, subjects, which are processed to obtain reconstructed cortical surfaces, spherical mappings, parcellations, and anatomical features using the FreeSurfer pipeline (Fischl 2012). A subset of the ADNI dataset that contains 412 subjects is used and processed in the same FreeSurfer pipeline. We choose the sulcal depth as the single-channel feature for registration and normalize features into probability vectors. For parcellations, we choose the DKT Atlas (Klein & Tourville 2012 Klein et al.) 2017), which parcellates each hemisphere into 31 cortical regions of interest (ROIs). We split both datasets into 70% scans for training, 10% scans for validation, and 20% scans for testing.
Training Losses. The training loss has three components: the similarity loss for aligning the moving surfaces to the fixed atlas surface, the Dice loss to impose biological validity, and the regularization loss to enforce the smoothness of the deformation field. Specifically, we employ the parcellation maps generated by FreeSurfer and the torchmetrics. Dinee function from the torchmetrics library (Nicki Skafte Detlefsen al., 2022) for Dice loss. The training loss is a linear combination of the three components. In our experimental design, we keep the regularization and the Dice terms in all setups. Our primary focus on investigating the impact of various similarity losses on the registration performance, including LSSOT distance and the following baselines.
Baselines. Since the original S3Reg framework used MSE as part of the similarity loss, we include MSE as one of the baselines. Besides, Spherical Sliced-Wasserse Wasserstein (SSW) (Bonet et al. (2023) and Stereographic Spherical Sliced Wasserstein Distances (S3W) (Tran et al. 2024) are powerful spherical distances to test LSSOT against. We use the SSW implementation from the official repo and the Amortized Rotationally Invariant Extension of S3W (AI-S3W) from the official implementation with 10 rotations. We also add the Slice Wasserstein distance (SWD), which is defined in Euclidean space 3 instead of $2 The implementation of the SWD algorithm is from the Python Optimal Transport (POT) library (Flamary al.. 202T). We also included the classical freesurfer registration results for general comparison. However, note that peep-learningg-asee frameworks can efficiently register unseen test data in a single feed-forward pass, whereas FreeSurfer iteratively solves for each pair of scans.
Metrics We quantitatively evaluate the similarities between the registered moving surfaces and the fixed atlas surface using mean absolute eтror (MAE) and Pearson correlation coefficient (CC) on features, as well as LSSOT, SWD on the spherical distributions (represented by vertex locations and feature probability vectors). We calculate the Dice score of the registered parcellations and the fixed parcellation. Moreover, we measure the area (resp. edge) distortions between the original moving surface meshes and the deformed surface meshes as relative changes in the areas of all faces (resp. the lengths of all edges) of meshes.
4.2.2 RESULTS
Table summarizes the performance of registration evaluated by the metrics for four scenarios: left/right hemispheres of the NKI and ADNI datasets. As the two best methods overall, our proposed LSSOT metric and SSW are on par with each other, yet LSSOT costs less than 10% of the training
hhng:://gthhb.ccm///boo///hhhhep hwrpr//neee//phhcaaaaaasceeseeen Wasserstein
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Preesurfer
MSE
SWD
LSSOT (Ours)
NKI Hentspitere)
LSSOR()
0.2877+:0.092
0.2754=000011
0.229=00.054
0.2411: 0.0366
0.2079:000066
0.1891
SWD())
0.006010.00000
0.00111000017
0.0053-00.010
0.00270000011
0.00591100001
0.005210.0011
MAR())
0.1053+0.00214
0.1129-0.0471
0.2278-0.0572
0.2688-0.041
0:2145+0.0266
0.2516+0.8997
OC(()
0.8190+0.0224
0.8299000425
1.9216-0.0174
0.8722-0.0312
0.8671+0.0011
0.8649+0.0225
Diee (1)
0.7541=0.0651
0.774620.0861
0.8498:100067"
0.798410.0539
0.8:29:10.0992"
0.8462:0.0548
Edge (ist(()
0.3207=0.0436"
0.3060=0.0404"
0.392240.0647
0.3442:0.0366'
0.2476:0.0038"
0.2365:0.0344
Arca Disi(()
0.4652=0.0698"
0.4405=
0.4048:200072"
0.4073+00.0368
0.2733=2004022
0.2897
Time(sccondSX.))
73.07
118.96
1350.96
101.01
NKI (Right Hemtsspttue)
LSSOT(())
0.251311..0557
0.2847.0.0502
0.2062=000510
0.245420.0396
0.1349110.020
0.11211:000253
SWD())
0.0104+000018
0.0048+0.0015
0.0046-0.0099
0.0019-000003
10.0017
0.0033+0.0005
MAR(D
0.2353+0.0304
0.2224 0.0324
0.2379+0.0548
0.2268+00.0551
0.2489+0.0568
OC(T)
0.899820.0217
0,9334:00011
0.9145=0.0130
0.9191=000263
0:8979:000088
0.9284:0.0599
Dice (1)
0.7502=00.039
0.0318"000943""
00.0777
0.6972:40.0200*
0.7996:0.00766 0.7952::00051
Edge Dist(()
0.252100.0911
0,2812=000421
0.1395:0000709
0.2160=0.0406"
0.1460=0.0411"
0.1428:0.0162
Area Dist(()
0.4785
0.3452.0.0422
0.3125.10.0676
0.2924.1.0412
1..0091.
0.191010.0119
Time(seconds)()
72.14
116.91
1362.16
100.68
Homisspheee)
L.SOO())
0.1325+0.0436
0.1476+0.0396
0.1340-0.000
0:1412+0.0499
0.1041
SWD()
0.004920.0058
0.0016:0.0011
0.0041-0.0011
1.002700.0009
0.0052:+0.0014
0,0063#0.0016
MAE())
0.2671=:00059
0.2784:000367
0.3295=0.0127
0.3627=0.0477
0,3153:200485
0.3332:0.0873
CC(1)
0.9085
0.024
0.9077=00.007
0.9139-00.000
0:9032:10.0139
0.9036:0.1125
Dioc (t)
0.7721=0.0736
0.7899=0.0949"
0,7733:20.0816
0.7457=00.068""
0.7967::0.0775
0.8119:
Edge Dist(U)
0.3564_0.0603"
0.2836.10.0750 0.0241
0.2379.10.4459
0.1527
0.0979
Area Dist(()
0.008-
0.2990_0.0000'
0.1937.1.0884
0.1930.1.004500
0.100810.0003
0.1303
Time(sconndS())
100.78
96.69
ADNI (Ragit Henisshare)
LSSOT())
0.152010.0416
0.1807=0.0347
0.1367=000375
0.1497:0.0509
0.1349:00.051
0.1295:0.0281
SWD())
0.0078#0.0217
0.0014
0.0046:0.0011
0.0020:0.0005
0.0041
0.003710.0020
MAE(()
0.2633:1.0024
0.2433:10.0273
0.3229.0.0386
0.2618:000450
0.3236:10.0519
0.3493.0.0732
OC(T)
0.8257+000655
0.0330=0.013
0.9139=0.0204
0.8985=0.0391
0,9375::0.4377
0.8013+0.0851
Dice (f)
0.7611
0.8073=0.09833
0.7617.10.0806
0.8122:1.00645*
0.851810.0859 0.8622.10.0915
Edge Dist(()
0.380--.0799**
0.3671-0.0627*
0.2868+-.08835*
0.3690000513*
0.1226+0.0213
Area Dist.()
0.5919-0.0448
0..05-0.0568**
0.3109+0.09877
0.4123+0.0458--
0.1939 00,099
0.1948-0.0325
Timofsconds)L))
59.69
101.35
97.52
1114,83
85.37
Table 1: Evaluation metrics on the test datasets (mean + standard deviation) for sulcal depth registration of both hemispheres from NKI dataset and ADNI dataset. The best two methods are bolded. The average training time for each epoch is also included in the bottom row of each scenario. We would like to note that the number of samples is 40,962, using spherical optimal transport or even Sinkhorn divergence is computationally intractable both time and space. We perform significance testing for the difference on Dice, Edge Dist. and Arca Dist. against LSSOT." denotes p < 0.001 and denotes < 0.01.
time. We emphasize that LSSOT is the leading method in the alignment of the parcellations (i.e. Dice Score), indicating that LSSOT yields more biologically meaningful registrations. Besides, with the least area/edge distortions. LSSOT finds the simplest solution of the deformation field to align the moving surface to the fixed surface. Figure[4ssows the registered sulcal depth map and parcellation map of a random subject. More visualizations can be found in the Appendix G It's worth noting that the LSSOT embedding iix of the fixed surface calculated once and for all before training, and in each training iteration, only the L.SOT embedding of the deformed moving surface & is calculated and then compared to ag Other competing methods, however, need to process both surfaces and calculate the distance during each iteration, which accounts for LSSOT taking less time than SWD. This makes LSSOT a perfect fit for atlas-based registrations, as the template is fixed.
4.3 POINT CLOUD INTERPOLATION
LSSOT can also be utilized in point cloud analysis, once each point cloud is endowed with spherical representation. In this experiment, we explore the interpolations between point cloud pairs from the ModelNet dataset (Wu et al. 2015). Specifically, we train autoencoder to project the original point
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Moving
MSE
S3W
SWD
SSW
LSSOT
Fixed
Sulcal Depth
Parcellation
Figure 4: Qualitative registration results (middle columns) from moving surface (left column) the fixed surface (right column). Both sulcal depth and parcellations are visualized with global close-up views. This moving surface is from Subject A00065749 in the NKI dataset. Visualizations for more subjects can be found in the Appendix G
clouds spherical latent space, to represent each of them as a spherical distribution. Then we apply gradient flow between the pairs of spherical distributions using the LSSOT metric along with SSW and Spherical OT. Finally, this transformation reconstructed in the original space by the trained decoder, resulting in interpolation between the original pairs. See Figure (top) for illustration of this process.
4.3.1 MAPPING POINT CLOUDS TO SPHERICAL LATENT SPACE VIA LSSOT AUTOENCODER
As point clouds are set-structured data, we leverage the Set Transformer (Lee et al. 2019 architecture to build the autoencode El The encoder &enc is composed of three consecutive Set Attention Blocks, followed by a linear layer and normalization layer to output set points on $ The decoder Vaec consists of four Set Attention Blocks with a linear layer at the end. For set of point clouds {X,) (each point cloud X; is a set points), the autoencoder is trained to reconstruct each X, under a soft regularity constraint in the latent space. That is, the objective function is defined as
L Σ dec
Vi ( (X:)) X A ISSSOT Xunit)
(12)
where Xunit is a set of samples from the uniform distribution on $2,
After training, the encoder &enc can serve spherical mapping from R3 to S², and the decoder dee acts as an inverse mapping. Enforcing the latent representation to closely approximate the uniform distribution offers significant benefits. It minimizes the occurrence of singularities in both the spherical mapping and its inverse function, resulting in fewer distortions or discontinuities, hence the structural and geometrical characteristics of the original point clouds better preserved. In implementation. we select À = le 4 through hyperparameter search. We set the number of slices to be 1000, and the reference size as M 1024 for the LSSOT, function.

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4.3.2 GRADIENT FLOW AND INTERPOLATIONS OF SPHERICAL REPRESENTATIONS
Given two spherical distributions as the source and the target, we can attach gradients on the source distribution and drive it toward the target by minimizing the distance between them. For the distance, we use LSSOT, SSW, and Spherical OT for comparisons. Here the Riemannian Gradient flow algorithm (see Appendix I.3 is implemented on S?. With slight abuse of notation to mimic the constant speed geodesic, we sample time points denoted as 7 = 0, 1/4, 2/4, 3/4, 1, from the beginning (7 0) to the loss converging (7 = 1), such that the loss goes down by approximately the same amount each interval. In Figure (bottom), we visualize the interpolations for two example pairs of point clouds.
Interpolation using Gradient Flow
T-214
enc
Váees
Vdeec
Vaze
@ pnc
T=Q/4
T-1/4
T-2/4
T-414
T-2/4
SSW
Schenica
Figure Top panel: the interpolation process between pair point clouds using gradient flow the latent space S?, Bottom panel: gradient flow interpolations from a range hood to bottle (left), and from a stool to chair (right) using three metrics LSSOT, SSW and spherical OT. See Appendix I.4 for more interpolated pairs.
LIMITATIONS AND FUTURE WORK
To the best of our knowledge, this work is the first to adapt the Linear Sliced Optimal Transport methodology onto hyperspheres, enabling rapid comparisons among group of spherical distributions. However, spherical data/signals may not come in the form of distributions in practice, and normalizing data into probability distributions may result in the loss information, thus one limitation of our work is that LSSOT is defined only for probability measures. Immediate future work includes partial extension of LSSOT for positive signals with unequal mass, and transport LP [hhope et al. (2017): Liu et al. Crook al. (2024) extension for non-positive spherical signals, both of which will be studied on S as the first step, and extended to @ by slicing.
6 CONCLUSION
In this work, we propose Linear Spherical Sliced Optimal Transport (LSSOT) as novel and fast metric for defining the similarities between spherical data. We provide rigorous mathematical proofs establishing LSSOT as a true metric. also demonstrate the computation efficiency of the LSSOT, superior to other baselines in the setting of pairwise comparisons of spherical data. We verify that LSSOT captures the rotation and scaling geometry on spheres through experiments on the Von Mises-Fisher distributions. In the task of atlas-based cortical surface registration various datasets, LSSOT delivers top registration performance across multiple evaluation metrics while reducing the training time significantly. Furthermore, LSSOT is shown yield meaningful interpolations between point cloud pairs. The metric's s effectiveness in both point cloud analysis and practical neuroimaging applications demonstrates its broad potential in 3D data analysis.
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REFERENCES
William Abikoff. The uniformization theorem. American Mathematicall Monthly, 88:574-592, 1981.
URL httts://: wewant wtt.://o.......oooooooou.... 1t:CeppuSD::241122222 124716245
P-A Absil, Robert Mahony, and Rodolphe Sepulchre. Optimization algorithms on matrix manifolds. Princcton University Press, 2008.
Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In International conference on machine learning, pp. 214-223. PMLR, 2017.
Guha Balakrishnan, Amy Zhao, Mert Rory Sabuncu, John V. Guttag, and Adrian V. Dalca. Vox- elmorph: A learning framework for deformable medical image registration. IEEE Transactions Medical Imaging, 38:1778-1800 2018. URL https:://stt semanticscco ar.chhlaaroorr...
[orpus ID: 52281312
Saurav Basu, Soheil Kolouri, and Gustavo K Rohde. Detecting and visualizing cell phenotype differences from microscopy images using transport-based morphometry. Proceedings of the National Academy of Sciences, 111(9)3448-344) 2014.
Thomas Bendokat, Ralf Zimmermann, and P-A Absil. A grassmann manifold handbook: Basic geometry and computational aspects. arXiv preprint arXiv:2011 133699, 2020.
Clément Bonet, Paul Berg, Nicolas Courty, François Septier, Lucas Drumetz, and Minh-Tan Pham. Spherical Sliced- Wasserstein. ICLR, 2023.
Nicolas Bonneel, Julien Rabin, Gabriel Peyré, and Hanspeter Pfister. Sliced and Radon Wasserstein barycenters of measures. Journal of Mathematical Imaging and Vision, 51:22-45, 2015.
Nicolas Bonnotte. Unidimensional and evolution methods for optimal transportation. PhD thesis, Université Paris Sud-Paris XI; Scuola normale superiore (Pise, Italic), 2013.
Nicolas Boumal, Pierre-Antoine Absil, and Coralia Cartis. Global rates of convergence for nonconvex optimization on manifolds. IMA Journal of Numerical Analysis, 39(1):1-33, 2019.
Zixuan Cang, Yanxiang Zhao, Axel A Almet, Adam Stabell, Raul Ramos, Maksim V Plikus, Scott X Atwood, and Qing Nie. Screening cell-cell communication in spatial transcriptomics via collective optimal tramsport. Nature methods, 20(2):218-25 2023.
Moo Chung and Zijian Chen. Embedding of functional human brain networks sphere. arXiv preprint N.XIv:2204.03655 2022.
Oliver M Crook, Mihai Cucuringu, Tim Hurst, Carola-Bibiane Schönlieb, Matthew Thorpe, and Konstantinos C Zygalakis. linear transportation Ip distance for pattern recognition. Pattern Recognition, 147:110080) 2024.
Li Cui, Xin Qi, Chengfeng Wen, Na Lei, Xinyuan Li, Min Zhang, and Xianfeng Gu. Spherical optimal transportation. Computer-Aided Design. 115:181-193, 2019.
Marco Cuturi. Sinkhor distances: Lightspeed computation of optimal transport. Advances in neural information processing systems, 26, 2013.
Julie Delon, Julien Salomon, and Andrei Sobolevski. Fast transport optimization for Monge costs on the circle. SIAM Journal on Applied Mathematics, 70(7):2239 2222222292222,, 2258, 2010.
Ishan Deshpande, Ziyu Zhang, and Alexander Schwing. Generative modeling using the sliced wasserstein distance. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 3483-3491, 2018.
A. Flad and R. Kimmel. Spherical Flattening of the Cortex Surface, pp. 77-89. Springer Berlin Heidelherg, Berlin, Heidelberg, 2002.
Albert Fathi and Alessio Figalli. Optimal transportation on non-compact manifolds. Israel Journal of Mathematics, 175:1-59, 2010.
11

---

Published as a conference paper at ICLR 2025
Bruce Fischl. Freesurfer. Neuroimage, 62(2):77--711
Rémi Flamary, Nicolas Courty, Alexandre Gramfort, Mokhtar Z. Alaya, Aurélie Boisbunon, Stanislas Chambon, Lactitia Chapel, Adrien Corenflos, Kilian Fatras, Nemo Fournier, Léo Gautheron, Nathalic T.H. Gayraud. Hicham Janati, Alain Rakotomamoniy. levgen Redko, Antoine Rolet, Antony Schutz, Vivien Seguy, Danica J. Sutherland, Romain Tavenard, Alexander Tong, and Titouan Vayer. Pot: Python optimal transport. Journal of Machine Learning Research, 22(78):1-8 2021. URL (tR..http://///
org wwtp:o/pppppp////aaaa.p 20015102455111 ht.ml
Robert C Garrett, Trevor Harris, Bo Li, and Zhuo Wang. Validating climate models with spherical convolutional wasserstein distance. arXiv preprint arXiv:2401.14657 2024.
Daniela Giorgi, Silvia Biasotti, and aaura Paraboschi, Shape retrieval contest 2007: Watertight models iesmanticschie ar httph//l......gooooooooooaaseee/
track. 2007. htttsss::///ap
645500)
Craig Gotsman, Xianfeng Gu, and Alla Sheffer. Fundamentals spherical parameterization for 3d meshes. In ACM SIGGRAPH 2003 Papers, pp. 358-363. 2003.
Manuel Gräf and Daniel Potts. Sampling sets and quadrature formulae on the rotation group. Numerical Functional Analysis and Optimization, 30(7-8):665-688 2009.
Groemer. On a spherical integral transformation and sections of star bodies. Monatshefte für Mathematik, 126(2):117-124, 1998.
Xianfeng Gu, Yalin Wang, Tony F Chan, Paul M Thompson, and Jhing-Tung, Yau. Genus zero surface conformal mapping and its application brain surface mapping. IEEE transactions on medical imaging, 2399)9949-558, 2004.
Brittany Froese Hamfeldt and Axel GR Turnquist. convergence framework for optimal transport on the sphere. Numerische Mathematik, 151(3):627-677, 2022.
Ralf Hielscher, Daniel Potts, and Michael Quellmalz. An svd in spherical surface wave tomography. New Trends in Parameter Identification for Mathematical Models, pp. 121-144, 2018
Clifford Jack Jr, Matt A Bernstein, Nick C Fox, Paul Thompson. Gene Alexander, Danielle Harvey. Bret Borowski, Paula Britson, Jennifer L. Whitwell, Chadwick Ward, et al. The alzheimer's disease neuroimaging initiative (adni): Mri methods. Journal of Magnetic Resonance Imaging: An Official Journal of the International Society for Magnetic Resonance in Medicine, 27(4):685-691, 2008.
N Jarosik, CL Bennett, J Dunkley, B Gold, MR Greason, M Halpern, RS Hill, G Hinshaw, A Kogut, Eicchiro Komatsu, et al. Seven-year wilkinson microwave anisotropy probe (wmap*) observations: Sky maps, systematic errors, and basic results. The Astrophysical Journal Supplement Series, 192 (2):14, 2011.
Arno Klein and Jason Tourville. 101 labeled brain images and consistent human cortical labeling protocol. Frontiers in neuroxcience, 6:171, 2012.
Arno Klein, Satrajit S Ghosh, Forrest S Bao, Joachim Giard, Yrjö Häme, Eliezer Stavsky, Noah Lee, Brian Rossa, Martin Reuter, Elias Chaibub Neto, et al. Mindboggling morphometry of human brains. PLoS computational biology, 13(2):e1005350, 2017.
Max Kochurov, Rasul Karimov, and Serge Kozlukov. Geoopt: Riemannian optimization in pytorch, 2020.
Nicholas Kolkin, Jason Salavon, and Gregory Shakhnarovich. Style transfer by relaxed optimal transport and self-similarity In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pp. 10051-10060, 2019.
Soheil Kolouri, Yang Zou, and Gustavo K Rohde. Sliced wasserstein kernels for probability distribu- tions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5258-5267, 2016.
12

---

Published as a conference paper at ICLR 2025
Soheil Kolouri, Se Rim Park, Matthew Thorpe, Dejan Slepcev, and Gustavo K Rohde. Optimal mass transport: Signal processing and machine-learning applications. IEEE signal processing magazine, 34(4):43-59, 2017.
Soheil Kolouri, Gustavo K Rohde, and Heiko Hoffmann. Sliced wasserstein distance for learning gaussian mixture models. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3427-3436, 2018.
Soheil Kolouri, Kimia Nadjahi, Umut Simsekli, Roland Badeau, and Gustavo Rohde. Generalized sliced Wasserstein distances. Advances in neural information processing systems, 32, 2019.
Soheil Kolouri, Navid Naderializadeh, Gustavo K Rohde, and Heiko Hoffmann. Wasserstein embed- ding for graph learning. In International Conference on Learning Representations, 2020.
Juho Lee, Yoonho Lee, Jungtack Kim, Adam Kosiorek, Seungjin Choi, and Yee Whye Tch. Set trans- former: A framework for attention- based cermutation-invariie neural networks. In International conference on machine learning, pp. 3744-377 3753. PMLR, 2019.
Xinran Liu, Yikun Bai, Huy Tran, Zhanqi Zhu, Matthew Thorpe, and Soheil Kolouri, Ptlp: Partial transport f distances. In NeurlPS 2023 Workshop Optimal Transport and Machine Learning-
Xinran Liu, Yikun Bai, Yuzhe Lu, Andrea Soltoggio, and Soheil Kolouri. Wasserstein task embedding for measuring task similarities. arXiv preprini arXiv:2208. 1726,, 2022.
Yuzhe Lu, Xinran Liu, Andrea Soltoggio. and Soheil Kolouri. Slosh: Set locality sensitive hashing via innserseesn embeddings. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 2566-2576 2024.
Lok Ming Lui, Yalin Wang, Tony F Chan, and Paul Thompson. Landmark constrained genus zero surface conformal mapping and its application brain mapping research. Applied Numerical Mathematics, 57(5-7):847-858, 2007.
Ashok Makkuva, Amirhossein Taghvaei, Sewoong Oh, and Jason Lee. Optimal transport mapping via input convex neural networks. International Conference on Machine Learning, pp. PMLR, 2020.
Daniel S Marcus, Michael P Harms, Abraham Z Snyder, Mark Jenkinson, J Anthony Wilson, Matthew Glasser, Deanna M Barch, Kevin A Archie, Gregory C Burgess, Mohana Ramaratnam, et al. Human connectome project informatics: quality control, database services, and data visualization. Neuroimage, 80:202 219, 2013.
Rocio Diaz Martin, Ivan Medri, Yikun Bai, Xinran Liu, Kangbai Yan, Gustavo K Rohde, and Soheil Kolouri. Leot: Linear circular optimal transport. arXiv preprint (Xiv:23300000000 2023.
Robert McCann and Peter M Topping. Ricei flow, entropy and optimal transportation. American Journal of Mathematics, 132(3):711-730, 2010.
Nicki Skafte Detlefsen, Jiri Borovec, Justus Schock, Ananya Harsh, Teddy Koker, Luca Di Liello,
Daniel Stanel, Changsheng Quan, Maxim Grechkin, and William Falcon. TorchMetrics Measuring Reproducibility in PyTorhh, February 2022. URL https: /ttp:////ttttttttt com/ Lightning-At/
torchmmt cs
Kate Brody Nooner, Stanley J Colcombe, Russell Tobe, Maarten Mennes, Melissa M Benedict, Alexis L Moreno, Laura Panek, Shaquanna Brown, Stephen T Zavitz, Qingyang Li, et al. The nki-rockland sample: a model for accelerating the pace of discovery science in psychiatry. Frontiers in neuroscience, 6: 152, 2012.
Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeViot,, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017.
Pedregosa. G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Pretten- hofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12:2825-288 28300, 2011.
13

---

Published as a conference paper at ICLR 2025
Gabriel Peyré, Marco Cuturi, et al. Computational optimal transport: With applications data science. Foundations and Trends@ in Machine Learning, 11(5-6):355-6077 2019.
Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas Guibas. Pointnet++: Deep hierarchical feature learning on point sets in metric space. Advances in neural information processing systems, 30, 2017.
Michael Quellmalz, Robert Beinert, and Gabriele Steidl. Sliced optimal transport on the sphere. Inverse Problems, 39(10):105005, 2023.
Julien Rabin, Julie Delon, and Yann Gousseau. Transportation distances on the circle. Journal of Mathematical Imaging and Vision 41:147 2011.
Boris Rubin. Inversion and characterization of the hemispherical transform. Journal d' d'Analyse Mathématique, 17:105-128, 1999.
Vivien Seguy, Bharath Bhushan Damodaran, Rémi Flamary, Nicolas Courty, Antoine Rolet, and Mathieu Blondel. Large-scale optimal transport and mapping estimation. arXiv preprint arXiv: 1.000223, 2017.
Li Shen and Fillia Makedon. Spherical mapping for processing of 3d closed surfaces. Image and vision computing, 24(7):743-761 2006.
Matthew Thorpe, Serim Park, Soheil Kolouri, Gustavo K Rohde, and Dejan Slepčev. A transportation 1" pplp distance for signal analysis. Journal of mathematical imaging and vision, 59:187-210, 2017.
Huy Tran, Yikun Bai, Abihith Kothapalli, Ashkan Shahbazi, Xinran Liu, Rocio Diaz Martin, and Soheil Kolouri. Stereographic spherical sliced wasserstein distances. arXiv preprint arXiv:2402.02345, 2024.
Wei Wang. Dejan Slepčev, Saurav Basu, John A. Ozolek, and Gustavo Kunde Rohde. A linear optimal transportation framework for quantifying and visualizing variations in sets of images. International Journal of Computer Vision, 101:254 269, 2012. htl.htttp.//ppp...enttiiccchhorr
org/Corpus 59:207772
Wei Wang, Dejan Slepčev, Saurav Basu, John A Ozolek, and Gustavo K Rohde. A linear optimal transportation framework for quantifying and visualizing variations in sets of images. International journal of computer vision, 101:254 2013.
Yalin Wang. Xianfeng Gu, Tony F Chan, Paul M Thompson, and Shing-Tung Yau. Intrinsic brain surface conformal mapping using a variational method. In Medical Imaging 2004: Image Processing, volume 5370, pp. 241-252 SPIE, 2004.
Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015.
Fenqiang Zhao, Shun ren Xia, Zhengwang Wu, Dingna Duan, Li Wang, Weili Lin, John H. Gilmore, Dinggang Shen, and Gang Li. Spherical u-net on cortical surfaces: Methods and applications. Information processing in medical imaging proceedings of the conference, 11492:855-866 2019. URLEttpp.
api semant icscholar wwtg/orppssiiiiccooooooooo 9D0TD00022277777
Fenqiang Zhao, Zhengwang Wu, Fan Wang, Weili Lin, Shunren Xia, Dinggang Shen, Li Wang, and Gang Li. S3reg: Superfast spherical surface registration based on deep learning. IEEE Transactions on Medical Imaging,
2021. doi: 10. 10.
Fenqiang Zhao, Zhengwang Wu, Li Wang, Weili Lin, and Gang Li. Fast spherical mapping of cortical surface meshes using deep unsupervised learning. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 163-173. Springer, 2022.
Fenqiang Zhao, Zhengwang Wu, and Gang Li. Deep learning in cortical surface- based neuroimage analysis: systematic review. Intelligent Medicine, 3(1):46-55 48, 2023. ISSN 2667-1026.
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A SPHERICAL RADON TRANSFORM IN S²
In Bonet et al. (2023). Spherical Radon transform is introduced and is used to give an equivalent formula of the SSW discrepancy (8). In this section, will discuss such Spherical Radon transform and its relation with the Semicirele Transform.
A.1 NOTATIONS ON SPHERE $
Let E (RR),, where
V2(R9)= = (U E R3x2 U = 12)
Each U determines 2-dimensional plane spanned by the columns U:
H(U) span [U[: 1], [1. C R
The set [U]: 1], U[: 2]} is an orthonormal basis for HUU). The subspace (U)) can be equivalently defined by normal unit vector rig, which is the uniquely determined by U (up to scalar =1):
ny E S2,
U nu = 02-
(13)
Thus, the plane (U) can be defined
= = : fo R x² ny = 0}.
Note that S(U) HU) n defines great circle.
A.2 PROJECTION ONTO THE GREAT CIRCLE S(U)
Given U V2(RS), the operator UUT R3 R3 defines projection. Indeed, for each point
8 E @2 {+my). J. the mapping
U
U |UT s|l
U[;,1] |UT
U[:22] 2) Ul 1]+
~ (s) (+) = |UUTA||
Ul S(U) c R
1 $||
(14)
defines a projection onto the great circle S(U), where in 14p we used that
| a|| s JUTTUUT SUU ==|
(see Bonet et al. 2023 Lemma 1)). Note that in Bonet et al. (023). the above projection is written
as
U |UT'
s - p 3:=
(15)
which is map from 82 (+my) } to RR2 that saves the coordinates U'':,11''s 2|:,) This function
maps 8 into a 1-dimensional circle R $1, which can be regarded as aeppesemaation/paaaaetizaaaa of the great circle S(U.).
An equivalent method that allows us to define such projection is introduced by the following parameterization system Quellmalz et (2023):
Each E S2 can be written in spherical coordinates:
ccos a sin 8, sin a sin B, cos 3
where a E [0, 2T) called the azimuth angle of 8, and / E [0, m)] the zenith angle of s. We will use the notation azz(s), 8 zen(s).
Remark A.1. When [ 0. 1]. the above parametrization is unique. As in Quellmalz et al. 2023), we set a 0, = for [0, 0, 1] and a = 0, 3 = π for = [0, 0, - 1]. Thus,
$ + (a, ())
becomes bijection (see Queelmmaz el al.
2023
page 6)).
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Let us define the following mappings:
[0, 2x) x [0, n|
(a, [) + $ B) == [cos sin 5, sin sin 3, cos 3)² = 8 S
(16)
8² 3 + azi(s) =: a E [0, 2T)
(17)
s² 3 + zen(s) =: 3 E [0, x
(18)
In addition, we define the following matrix:
Ccos a
sin a
cos /
sin 8 0
cos y sin 7 0
sin ?
L (a, (,,,
sin a
COS
0
0
cos
0
0 1
sin
cos
1
By Quellmalz et al.
2023 page 7) and Graf & Potts (2009),
8²
[0, 2x) a (s, 7) + (azi(s)) zen(s), E SO(3)
(19)
is a bijection, where SO(3) is the special orthogonal group in R.::
SO(3) {G g GG = G Is, det(G) 1}.
In the special case ? 0, we adopt the following notation:
Coos cos
sin cos sin 0 sin a sin 8 cos B
(s) = 4 (,0
=
sin a cos 3
cos a
sin 3
where, given 8 E S2, a and B are as in by (16)).
and 18 (or equivalently, given (a, 8 S is defined
is straightforward to verify
(s) 0,0,
That is, we can regard (s)) as rotation matrix that can rotate s back to the North Pole [0, 0, 1TT Equivalently, (s) a rotation matrix, which can rotate [0, 0, 1 to
Define the Equator by
E = ( sin 0] E 00, 2m)}.
Given U V2(R)), consider the following set
E ( E}
obtain the following characterization of great circles for $2 C 123
Lemma A.2. Given U E V2(T))), it defines a great circle S(U) ng? which is characterized by the following identity:
S(U) = (s S (nny) E).
(20)
Proof. First, since ((ny is a rotation matrix, (ing) E s² for all E S² as (mn) )
||||||
cos Now, write ny = |cos a sin 8r, sin a sin cos 8T1T where a = azi(ny) and 3 = zen(nu).
For any s S(U), we have (ny) ) 0.
We have to prove that the third coordinate of (mn))²s E S² is 0:
(nv) ) [|]]] [cos a sin 3*, sin a² sin 8*, cos B*1s (nv) )" s= 0.
That is, for every & E SUU), we have obtained that s E {s E (Tny) E E
Similarly, if we consider E {s $ (nn)) E}, we have (w)r) $ Ts.
Thus, s E S(U).
Therefore, have the equality in 20
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Lemma A.3. The mapping
[0, 2m 3 00, = -cos y; sin s 0, E
and the mapping
3 3 (nn)ss S(U)
are bijections. Thus, the mapping
[0, (x, s (7, S(U)
is also bijection.
Proof. It is trivial to verify that the first mapping, Y + |cos sin y, 0]", is bijection.
By the Lemma A.2 the mapping rotation matrix, is bijection g + g whose inverse is (mv), and so the mapping s + (nu)s ) well-defined bijection from & to S(U): that is, if we restrict y ((ne) to E, by Lemma A.2 the S(U).
(ny)) is well-defined from S(U') E. Since (nn )" is
matrix V (nv) defines a bijection from & to (nu
Now, we define the following composition map P (s):
82 {=nr} 3 7 azi((( + P (s)) == v (Mn)) x/2) E S(U)
(21)
Proposition A.4. The mappings PS(()) (defined by 14] and Pis(y) (defined by 27) coincide on
8² {
Proof. Case 1: Suppose ny = [0, 0, 1"" Consider s = [cos a sin 6, sin a sin 3, cos 87 E S (+ng) i.e., where az)(s) and / zen(s) with E (0, x). In this case, from 13 U33 02. Thus, [U[1: 2, [1 2, 2] forms an orthornomal basis for TR2.. From [4]. we have
U = T2 07
s [cos a sin B, sin a sin 3, 0,
0
Thus,
T 5
Pl S(U)) (8) [UT 5||
[cos at sin a, 0j".
In addition, in this case, (n() Iz. Thus, 7 = azi( azi(V(no)) = azi(s) a. Therefore, Pr (s) T (nu ( n 7/2) ( 0) [cos a, sin a, x/2
Thus, 14 and coincide in this case.
Case 2: Suppose
* 0.. Each point E g? can be written as v (nu)s* where s' (ng) s. In addition, when * iny. we can parametrize s' via s' (cos sin 3, sin ain 3, cos 3.", where 3 E (0, r). We also have that (nn))T = [0, 0, 1j=.
Let U (no)) TU. We have (U')T 0, = UJ
UTTT ny 02-
Thus ne 00, 0, 1J. Besides, we have that
UTTs x = V(
(Wo)) = (U(
Thus,
Pst (s) (nu)) (5'), Vs Vs tnu. In addition, by using the parametrization of s' in spherical coordinates, we have == azv()(nv)) s) azi(s') = a.
Thus,
Ps (s) = v (nv) (cos sin 0 = (n ) (u)))
From Case 1, we have and from the the fact that s' + s v (nv)) ss is a bijection on $2, we get
P1 P
P ('wrr(''),
P P (8) (s)
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Lemma A.5. Given 2 E S', V2(R)), there exists a² (0,2n), [0, [0, such that
ny "0a,,"" = acos sin 8", sin a* 3 cos 3d17
(22)
Uz (nn)) sin
(23)
Vice-versa: Given a" E (0,2m) 5" E 1), E [0, 2T), there exist & E ss, U E V2(RR) for which formulas 22 and 23 hold.
Proof. Given U E V2(R)), can find ny S via (13). (Note that if ny satisfies
[3). then -RG
also satisfies (33 and so ny is unique up to sign.) Then, once ny is chosen, we can uniquely find (a", as (88).
In addition, for E C CR? since
UUTUZ Uz
P P((D)(U)) (U2) |UTUZ]]
= U2,
|||||
we obtain that Uz E S(U). By Lemma A.3 ((n))=U E Then, we can uniquely determine ?
(see Lemma A.3 Thus, 22 and 23 are satisfied.
For the other direction, consider angles a", , y and let us separate the analysis into the following cases:
Case 1: a====== = 0,
{0, æ). this case, we have ny = [0, 0, +1] and thus we can set
U = 12
In addition, set = [cos y, sin 7|a E St and so the identities 22 and 23 are satisfied.
Case 2: a² * 0, then set ny by 22 Choose orthornormal basis {wr, u2) for nt {ar
R² xx ny 0}, and set U jul U2) Then (U,ng) satisfies 13
By Lemma A.3 we have V/ (nz) [cos 7, sin 7, 07² E S(U). Then,
U v [cos sin 0J S(U),
and by setting
2 =: U==UT((m))) sin sin 7,0",
have
[cos sin 7 0j
Uz = UUT (nrr)coo [T /cos y, sin 0]
PP (W 0 (T (cos 0j"
Thus
23
satisfied.
Now, given E 2T), and ny = (a" (8*) E S we can define the following semicircle:
SCT := - {s 82 (z 7].
(24)
Remark A.6. When # O, we can replace the above definition by
SCT = SCa-.g- = E (+nu): azi(V((v)) Ts)) = =
(25)
Proposition A.7. Let E SS.U E (TR3). Consider a² E 20, 8r E [0, E 10, 2m) in
Lemma A.5 then, find U according to Lemma A.5) We have:
(or (or equivalently, first, fix (a ,3*) E ((0,2m) [0, U (0) x {0, E [0, 2mr). and
SCh, [+no) -SC% \ [(nu)= (8 E )
(+no): p8 s= == 2).
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Proof. We have that
{s E S? {+mt}: : p = 2) { | ( {=mo):
UU 1 bl = Uz
{ s 8²) { Ps Ps(()(s)= = Uz}
S [tnu}: Pisc (s) V(nn))cos sin } (by Proposition A.4 and (23
{s E \ [tne): (((o)) (azi ) {), x22) = (m ) |cos 7: s; 0 (by definition 21
{s E S² \ :
: (ne) ))) /22) cos sin 7, 0" (applying T (ny) on both sides)
{ S" : ( = =
(by Lemma A.3
5chy \ (+nu)= = SCa g. }
Remark A.8. With the notation of the above proposition. For each fixed pair (z,U) E V22 (R*). we choose distinct (a 3,377) and (ā", (3,5) such that the conditions in Lemma A.5 are satisfied,
we obtain
SCT-
SC .3-
A.3 SEMICIRCLE TRANSFORM AND SPHERICAL RADON TRANSFORM
Let E (S²). The Normalized Semicircle Transform of f is function S - IR defined as:
1 { f(s) sin(zen(s))dSe)
WS(a",
(26)
4m Jscyy
where dSC), mg is the curve differential on the semicircle SCY, Its un-normalized version is defined
uWf(a ,);) = 4x Jscu
s(8)dSC))
(27)
The Spherical Radon Transform of f E (S( for d 2, defined Bonet et (023 the mapping Rf (RR) S! + R given by:
Rf(U,2)= = | f(s)0(== pD ()))((=>(ss)))
(28)
When considering 3, we obtain the following relation between the above-detined transforms. Proposition A.9. The Unbounded Semicircle Transform and the Spherical Radon Transform 28 are equivalent, In particular, given z E S E VV2(R²),, consider a [0, 2 E [0, (T, 7 E [0, 2T defined as Lemma A.3 then
1 Rf(,U,
4T
Proof. From Proposition A.7
we have SCT .. = fs pli = z}. thus
Rf,
f(s)dSCT, = (A ,8,,))
Remark A.10. We notice that there are other versions of the Spherical Radon Transform. For example, in Groemer 1998) the spherical Radon transformation defined by integrating over the full great circles S(U)
-S0 fdS(U).
Rf(no) JS(U)
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In this paper are using the variant given in Bonet al. (2023), which we have proven coincides with the integral transformation over semicircles. That is, this version coincides with the integral transformation denoted by B in Groemer (7998. It holds that
R(ne) 2) Rf(U,-2)
We will review this in Appendix B when generalizing from d = 3 to higher dimensions.
We end this section with discussion about some alternative formulations for Wf and U. For each 8 E SC a* .-- A (0, }. we have
a2i( ,0, 0))) =
Thus, we can write as
- 30,00,0000,7333 = (00,,, x) (recall that (00, [sin(E), 0, (E) and see Groemer (1998) for more
for some & E x0, details).
Similar to Remark A.6
redefine the semicircle SC = (V B, (0, <) : (0,T)),
and "a"",""00 is called primal median of SC) -.
Combining this with the fact that ,"",,"," (a" 7)) is bijection, rewrite (26). and
(27 as
Wf(Q) 1
4 / (() sin(E)dg
1 n
uws(Q) 4m Jo
[(Q E))
where Q E SO(3) (by using the identity & = (0, {). for an appropriate Q E 5O(3)).
In Hielscher al. (2018), the Un-normalized Semicirele Transform is defined
uW(((Q) = f f(Q" 7/2)
Q E SO(3).
(29)
The equivalence relation between (F(() and (f( reads as follows: Lemma A.11. For any Q E SO(3), we have
uws(Q) uws
-1
Q-)
0
[ 0
07
Proof. Given Q E SO(3), E 10t
0
-1
Q ,5' "''="' = - - Then,
[0 1 0
Q'[L, =1[1:]=Q[:.11 Q]:11]
Q'22. [3 - -[3,3]
Q'33. = Q" [2.] = 2[:
and
(0(0,))
Q(sin E. 0,,...os50 E
Ql: 1] sj Q[: 3] cos
Ql: 1] cos(E') ) Q[: 3 sin(E')
Q' [11 cos E + Q' [2, sin
(Q') (Q')T(cos sin ,', , rr
= (Q))TT ( 7/2)
Thus,uws(Q) uws(Q).
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Another formulation has been introduced in Groemer 1998). Since it allows us define f in higher dimensions (d 2 3), we will discuss it in the next section.
B SPHERICAL RADON TRANSFORM IN S
B.1 GREAT CIRCLE AND SEMICIRCLE/ / (d 2)- 2J-DIMNNSIONAL HEMISPHERE IN gá-1
We set d 2 and consider
V2(RR) = IU RRdx2 U = 12).
Similar to the previous section, we can define 2D hyperplane
H(U) span U 11, 2):
and U- == H(U)+
In addition, we define the corresponding great circle: S(U) = HU) ns Note that the Equator a specific great circle:
E 1 s 0):
where = [0, 0, O, 1 E gd-1 c Rd. Similar to the previous section, the corresponding projection onto the great eircle is
UU
(s) (8) |
8 SES UT
(30)
Similar to Proposition A.7 and 24 we define the (d- 2)-dimensional hemisphere:
HS(U,,) 2) = {s p = 2} U (U- nst-1
(31)
Remark B.1. When d 3, U- = {-ny} and if we ignore these two points, 31 is consistent to 224)
By introducing the hemisphere, the projection 30 can be regarded as the following: Let s E gd XU, have: E HS(U, z) where 11"
Then the function Piscyy maps to the intersection point |
HS 2) n SC(U).
Proposition B.2. The following identity holds for great circles:
S(U) = P (8a - = (U2:2
(32)
Proof. Since (/) is the projection, we have Ps (Sa \U)) ) S(U).
For the other direction, choose E S(U) and we decompose s as
su + 8t1,
where st H(U), sp E
By definition of S(U), we have sy = 0. Thus Su. From the fact ((U))u- = {0a). we have & & U-, Therefore, & E gd-1 U-. Thus s = P (s). Therefore, we have
S(U) ( (Sd \Ut).
It remains verify the second equality. Pick St. we have
P1 (Uz) = UU
= Uz
|UTU>|
Thus Uz E S(U). For the other direction, pick & E S(U), thus
= SU T
In addition, = ||||| = | s|| = ||TS|| s||. Thus, E {U> : 2 E S') and the second equality
holds.
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B.2 GREAT CIRCLES AND HEMISPHERES IN GROEMER 1998)
In this subsection, we re-parametrize the great circles and semicircles by using the notation in Groemer (1998).
Let Ba {(u, v) u. E su-1, = 0}. We define, as in Groemer 1998). the hyperplane H [u, v) suan(u, of and the corresponding "great circle"
S(u, v) s n ((,
The (d - 2) dimeessiia hemisphere is defined by HS,u v) = {s gd-1 s² = 0, S. 1222 0} Lemma B.3. Pick U E V2((R)), St, there exists (u, v) E Bd such that span (U) = span v))
(33)
(34)
U²u=
(35)
and vice-versa. In addition, if 2, u, satisfy (34) and (35). we have:
S(U) S(u, v)
HS(U, z) HSSuu v)
Proof. First, we pick U. We set v Uz. There exists a uniquely determined (up to scalar =1) E H(U) no+ with ||u|| = 1. Since v = U2 = we have that (34), and 33 hold.
Now consider (u, v) Bd We U = [u, o]. (Note that the choice of is not unique, for example. we can also set U = [- 2) or U uu].) Let 2 U.D v = [0, 117. We have that (34 and 335 hold.
Suppose U, 2, v satisfy 33 and
35
We have 2U) = (u, v), and thus S(U) S(u, v).
Consider E HSSU, 2), we decompose as 8 = su + sy,, then (by Proposition B.2 Ps (s) U2 = - == v. The last equation holds from the fact E HU). Since u, v HUU). we have
ST sTu (UUT s)T o= -ITT s110 0 TTu sTu (UUT S) ||UT slluT
So, E HS,u v) and thus For the other direction, we pick E HSSu,, v). We have I O.
HS,U,,) 2) C (SS(,,)) v).
Since E HU),, we obtain:
0 - S u s + sT = stu
By combining this with the fact that 8y E H(U) = span( (u. v)), we have su = av for some a E TR.
In addition, from sTa 2 O, we have 2 0.
Case 1: If a we have U thus (U,
Case 2: If a > 0, we have:
UTs
UT av = -TT
p s P su
||TT|| a
Thus, s E HS(U,2). 2). Therefore HS(u, u) HS.U,2).
Remark B.4. When d 3, then = U|: 1). U[; and Tuxxell u x (cross product, normalized).
Moreover. these (d - 2) -dimensional hemispheres H. generalize the semicircles SC (see (24 given for d = 3.
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B.3 SPHERICAL RADON TRANSFORM AND HEMISPHERICAL TRANSFORM
We call the integral transformation defined in Grremer 1998 Section 2) as the "(d 2)- Hemispherical Transform gd-1n, it is the d- dimensional extension of the Un-normalized Semicircle Transform given by (7 for = 3. For E 1), we define
Hf(u,o) = J HSS(U.)) [(6)dHS(u,))
where d'HS(u, v) is the surface area differential on H., v). (We note that in the article Groemer (1998), the notation for the above transformation is B.)
Remark B.5. In the literature, the so-called Hemispherical Transform integrates over a full (d - 1)-
dimensional hemisphere of gd-1, that is,
f(2)
f(x)dx
(a E
(36)
1=E8--1:77:20
See, e.g. Rubin 1999)
Groemer 1998).
The Spherical Radon Transform in gd-1 is defined Bonet et al. (2023) as the linear and bounded
integral operator R: I. (8--1)- I (V/(RR x 81) with the formula:
R.)(U22) = [
Jgh-1 8(p4s (s)
(37)
Thus, we can re-write it as
{ JSL 8(P5s== 8= =))(s) ==)
INSIG..) (())HHS(U,z)
RJ((U=)) =
(38)
We define the following variant of such Spherical Radon Transform:
H()),,2) = { - J
8(P%s s =s)f(s)ds= JASCIEANA F(s)dHS(U z).
(39)
Proposition B.6. For U E V2((R)) and St, consider (u, v) as in Lemma
B.3 Then
Rf(U,2) = H.fuu, v) = H.)(U,z), V
Proof. By Lemma we have HSS(U,) 2) HS,u v). Then, by Rf((U,)) (H,
Thus, it remains to show Kf = RRf. For each (U, ! is a (d 2)- -dimensional subspace. Thus, sd-1 n is (d dimeennsia sub-sphere.
Since (s E sd-1 U-J, - gd disjoint, have:
HS( [{s E gd-1 \ U+ == (U=) nu = -U- gd
(40)
In addition, since U- n is (d - 3) dimdeenniaa sub-sphere, while HS(U, HS(u, 2) is (d - -)-dimmnsionnal hemisphere, we have
Rf(U,2)
HT((Uz)) -(,(,))
= Jurrgan [(x)HHS(UU=)
and we have completed the proof.
Corollary B.7. Given continuous function f.g C(Sd-1), ifR Rgg, then f g.
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Proof. For every 2), there exists u, E gd-1 such that, by Lemma B.6
R.( H u): Rg(U, =) = Hg((u, v)-
Thus, Hffu, v) Hg(u, v). From (Groemer 1998 Theorem 1). we have f =g.
9.
Remark B.8. In the article Bonet et al. (2023) the authors relate the Spherical Radon Transform R with the Hemispherical Transform F given in (36). Indeed,
Rf(U,2) F()
where fiswn denotes the restriction of (S to the ( (dd=2)-dinnnsional "great circle"
S(U) g and the above operator the Hemispherical Transform in L (8 By doing so, the authors can use the characterization of K (er(F) provided in Rubbn (999)) to prove their characterization of Ker(R) (see (Bones al. 2023 Proposition 4)). Also, Bonet al. (2023) the authors claim that they leave for future works checking whether the set Ker(R) is null not. In this paper we show that it is more natural to relate R with the integral transformation given in Groemer 1998 Section 2) that we have named as the "(d 2)-Hemissherical Transform S g and denoted by H. Thus, (Groemer 1998 Theorem 1) we can show the injectivity of R for absolutely continuous measures the sphere with continuous density functions. (Since Sd is compact, the space of continuous functions C(gá-1) is dense in (Sd()) under closure with respect the -norm. However, still can guarantee that the injectivity property of extends from C(R to L (gd-1), which represents the densities of absolutely continuous measures).
C SPHERICAL RADON TRANSFORM FOR MEASURES ON S
Similarly to the classical Radon Transform Euclidean spaces, the dual-operator of R, called the back-projection operator and denoted as R (V2((BR)) 8 C(gd-1), given in Bonet et al. (2023) by the following formula:
J
Rd (+)((x)
VIRRR) (UU Pd (z)))o(U),
ZES E x
(41)
That is, R" is such that the following identity holds:
[ Sevnare (U
f(s) R (s)dds
(42)
for f E L'(8""),, E C)(((RR)) x S (see (Bonet et al.
2023 Proposition 2)). Then, the the space of measures supported on
Spherical Radon transform R can be extended from L' gd-1 by defining Ru as a new measure on 1/2 (Rd) x St by the "duality expression":
Saxiscee (U, R(p(UU)) J R'v(s)dp(s),
V CO(((R))))
(43)
Remark C.l. For each E gd-1, ifs E U+, then U|: U1; st. Thus
a(UU (2( :EE
=o((U E V2(RR) U|: 1|, E E =0
In Bonet et al. (2023), the authors mention the above that definition is considered a.e.. Via the above property, have that 411 is well-defined for all E Sd-1
Now, consider probability measures μ E P( In analogy to the classical Radon Transform in Euclidean spaces, by the disintegration theorem in classic measure theory, there exists a o-almost everywhere uniquely determined family of 1-dimensional probability measures ( (ee,)(R)) on the unit circle S1 such that for any (IZ( x S²) we have
J v(U,z) dR
[ Jrcce) /
(11.0) d(RR)) (=) (o))
(44)
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In (Bonet et al.] 2023 Proposition 3) the authors prove that if My is absolutely continuous probability measure the sphere Sd-1, that is, it has density L then
(Ruy) P# (1y)
The same computation can be extended to any measure supported on gd to get (Ru)" = P (1)).
Remark C.2. the particular case of fus, that is, for functions (,1 we have that the measure Ruy has density R.f. This implies that the function (R((U,)) (i.., the function - Rf)(U2 defined for z E S' is the density of (R and therefore (U, is also the density of P (py)-
C.I LINEAR SPIERICAL SLICED OPTIMAL TRANSPORT IN TERMS OF THE SPHERICAL RADON TRANSFORM
By using the previous expressions, given D², ,v² E P(S we can re-write LSSOT2(s.) follows:
| JVS(RA) (LCOT,(P p4 do(U)
(LSSOTZ(VI, 22)))
- Va
(LCOI2((RM)) (()))) du(U).
(45)
D METRIC PROPERTY OF LINEAR SPHERICAL SLICED OPTIMAL TRANSPORT
Proposition D.1. LSSOT2(:)) is a well-defined pseudo-metric in P((S-1), that is, LSSOT2(V,,)) 0 for all E ), it is non-negative, symmetric, and satisfies the triangle inequality.
Proof. Let,,, E P(Sd-1).
It is straightforward to verify positivity (i.e., LSSOT3(4,,)) 2 0), and symmetry (i.e., LSSOT,(V), 22) LSSOTT((), v1))
If v1 for each U E VR( we have P 11 = Pf thus i (-. U) = 1 (.. U). It implies that O, and LSSOTh (V1,
[))()))) D (. (.
1|2 (.
Finally, we verify the triangle inequality. Let U E (2 By Martin et al. 2023 Theorem 3.6), we have LCOT2(--) defines a metric. Thus, the following triangle inequality holds:
LCOT, vn, PH122= & LCOT, P PH + LCOT, P Ps) vs).
Thus,
LSSOTg(M,.25)
1/2
(fr (LCOT,(P))) Ph pH22)))o(UU)
1/2
( VSCRR))
(ICOT2 v3) + I.COT2 (D4 v2, Pyl us))
(LCOT, v1 PH uu)o)ddoUU) + *( (LCOT Ptd Pol
1/2
1/2
Va(Rd)
(46)
T.SSOT((M, v3) + T.SSOT,
where (46) holds by Minkowski inequality in space (V(()(R))), do)..
Theorem D.2 (Theorem 3.3 LSSOT2(:.) is well-defined metric in the set of absolutely continu-
ous measures P( with continuous density function, that is, it pseudo-metrie satisfying also the identity of indiscernibles.
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Proof. From Proposition D.1 only have prove that if LS 0, then 11 1/- To do so, we will the expression the LSSOT2(":) in terms of a Spherical Radon Transform given in L/, 1/ P(Sd-1) such that
given in 45
(LS5 = Jrsce,, (LCOT)(RR))) (Rv)) do(().
Thus,
ICCT)((RV)), ) =0 o(U)
Since LCOT((;.) defines metric in P(St) (see again (Martin et al. 2023 that
Theorem 3.6)), we have
(Ru) (Ruz)
(U)
As the disintegration is family a-almost everywhere uniquely determined, we have that
Rv = Rv2.
Finally, if v1, have continuous density functions, by injectivity of the operator R (Corollary B.7 we have that V1 V/-
As byproduct of our deductions, we obtain that the SSW in Bonet et al. (2023) and defined in 8 is also a metric in P()d-1):
Theorem D.3. The Spherical Sliced Wasserstein discrepancy SS,,(, for in the set of absolutely continuous measures in P(G0-1) with continuous density function.
p 2 1, defines a metric
Proof. First, notice that since the sphere is compact, there is need to ask that P(S has have finite p-moment, as it automatically has such property. In (Bonet et al. 2023 Proposition 5), the authors show that (..,(..)) is pseudo-metric. Finally, by using the expression
(SSW (H, = / Juscee (COT,(PH pkD) do(U)
(47)
= - JystRo)
((M)) (RU) do(())
(48)
and repeating the arguments in the proof of Theorem 3.3 with the Wasserstein distance COTp(:: in the unit circle in place of the LCOT distance, we get that SSW,(..) satisfies the identity of indiscernibles when restricted to probability measures with continuous densities.
RUNTIME ANALYSIS WITH RESPECT TO NUMBER OF SLICES
Here provide further running time comparisons of the slice- based methods for pairwise distance calculations under different numbers of slices L = 500, 1000, 5000. All methods are run on NVIDIA RTX A6000 GPU.
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Xm-000
N=-000
N=10000
Ne12500
N4m 15000
L= 500
34000dd
00.
1000
N-5000
1=00000
No 2500
N- 15000
L= 1000
number dlstr buttons
number of distributions
number cistribucions
number distributions
number of distributions
15007)
L= 5000
seronds
177
number distriuutions
number dRST butions
number dicrnbuttoos
number diertbuttoons
number of dittibuttons.
Figure 6: Running time comparison under varying number of slices. denotes the number of slices used in all the bise- methods in each N is the number of samples in each distribution for each column.
F IMPLEMENTATION DETAILS OF CORTICAL SURFACE REGISTRATION
F.1 ICOSPHERES AND RESAMPLING
S3Reg relies on Spherical- Unet (Zhao et al. 2019) as the backbone, which capitalizes on the regular structure icospheres to perform convolution and pooling on spheres. Spherical surfaces are resampled on icospheres in order to be input to the S3Reg network.
lcospheres are generated from a regular convex icosahedron with 12 vertices, then by recursively adding new vertex to the center of each edge in each triangle. The number of vertices the icospheres is increased from to 42 (level 1), 162 (level 2), 642 (level 3), 2562 (level 4), 10,242 (level 5), 40,962 (level 6), 163,842 (level 7), etc. In this work, we focus on the level-6 icosphere with 40,962 vertices on the cortical surfaces.
For resampling in the preprocessing step, employ the simple but fast Nearest Neighbor method and implement it using sklearn.nn sklearn.neighbors.t .Drrre KDTree function from Scikit Learn (Pedregosa et al.) 201T)) The feature value each output point is represented weighted sum of the feature values on the nearest three vertices.
F.2 S3REG FRAMEWORK AND LOSS COMPONENTS
In nutshell, the S3Reg framework receives a concatenation of fixed and moving spherical surface maps defined on icospheres input, processes them through encoder-decoder architecture, and finally outputs the velocity field represented by tangent vectors on g In particular, as the convolution kernel in Spherical U-Net could cause distortions around the poles (Zhao et al.) 2019), complementary orthogonal Spherical U-Nets are combined to predict the deformation field so that the polar areas also regularly processed at least times. As the 3 predicted velocity fields should
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be orthogonal accordingly, loss term of inconsistency is added to the total loss. We refer the readers to Zhao et al. (022)) (Section IlI-A) for details of the consistency loss.
As the similarity losses thoroughly discussed in the main text, will now describe other loss components in more detail. Let denote the fixed image and M the moving image. Suppose pr. PM the corresponding parcellation maps of F and M, respectively. Let denote the velocity field predicted by the network.
Dice Score and Dice Loss Parcellation maps be integrated into the training process enforce biological validity. The Dice Score is a measure of similarity/overlap between two parcellations. Assume both pv and pm have K regions of interest (ROIs), then the Dice score is defined
1 K
K
p n (pār o Φ)
Dice(pF, PM 6)
(49)
where phr o is the warped moving image by Ø, the superscript & denotes the k-th ROL. Since the Dice Score is between 0 and 1, the Dice Loss can be defined as Liice (PF, PM
1 Dice(pp, PM o o).
Smoothness Loss The local spatial non-smoothness
the output velocity field Φ is penalized by Lsmooth E. VVgC((((v)) where the sum is over all the moving vertices C is the 1-ring convolution operator (Zhao et al. 2019). and Vga denotes the spherical gradients.
G MORE VISUALIZATIONSS OF CORTICAL SURFACE REGISTRATION
Moving
MSE
S3W
SWD
SSW
LSSOT
Fixed
Sulcal Depth
Parcellation
Figure Qualitative registration results from Subject A00060516 the NKI dataset.
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Moving
MSE
SWD
SSW
LSSOT
Fixed
Sulcal Depth
Parcellation
Figure 8: Qualitative registration results from Subject A00076798 in the NKI dataset.
Moving
MSE
S3W
SWD
SSW
LSSOT
Fixed
Sulcal Depth
Parcellation
Figure Qualitative registration results from Subject 0356 the ADNI datasct.
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Moving
MSE
S3W
SSW
LSSOT
Fixed
Sulcal Depth
Parcellation
Figure 10: Qualitative registration from Subject 6650 in the ADNI dataset.
Moving
MSE
S3W
SWD
SSW
LSSOT
Fixed
Sulcal Depth
Parcellation
Figure Qualitative registration results from Subject 7060 the ADNI dataset.
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H 3D SHAPE CLUSTERING AND RETRIEVAL
To further validate the effectiveness of LSSOT in evaluating the similarity of spherical distributions, we conduct additional experiments using a variety genus-0 3D shapes. Our methodology involves a two-step process: 1) mapping the 3D shapes onto spherical surfaces; and 2) performing clustering and nearest neighbor retrieval tasks on these spherical representations. For the spherical mapping process, we employ two distinct techniques, Inflation and Projection (IAP) and Deep Unsupervised Learning using PointNet++ Qi al. (2017).
H.1 DATASET
We conduct experiments mesh models from the Watertight Track of SHREC 2007 dataset (Giorgi et al. 2007). This dataset contains surfaces belonging to 19 categories, among them, we select genus-0 closed surface models, human, octopus, teddy bear, and Fish. Moreover, explored in the previous experiment, LSSOT is sensitive to rotations, therefore we remove the meshes with different orientations than the rest.
H.2 SPHERICAL MAPPING
Inflation and Projection. A direct projection from genus-O closed surface to the sphere by fixing an inside point source and normalizing the vertices may cause undesirable topologically incorrect foldings. One workaround is iteratively smoothing and inflating the surface before the final projection. This inflation and projection is a common spherical mapping strategy used neuroimaging data processing tools such as FreeSurfer (Fischl 2012), Connectome Workbench (Mareus et al. 2013).
Deep Unsupervised Learning using PointNet++. To accelerate the forward pass of spherical mapping, capitalize deep unsupervised learning and train PointNet++ network to map cortical surface on a sphere with minimal distortion. The trained network then serves as deep learning-based spherical mapping tool.
Loss function. Zhao et al. 2022) proposed an unsupervised approach of mapping cortical surfaces to a sphere using Spherical U-Net Zhao al. (2019) to learn the spherical diffeomorphic deformation field for minimizing the metric (distance), area, angle distortions. We follow their definitions of metric (Ja) and area (Ja) distortions as local relative differences, and conformal/angle distortion (Je) after market share normalization, but adapt them to general meshes instead of restricting to rrraheeddon-repaaameetiiizz surfaces. For pair of original input surface R and the predicted spherical mesh S, the loss function has three components:
1
N
S
d
Ja(R,5)
min FER
(50)
Ne
dR
1=1
N2
1
Ira.
a
Ja(R
min rERt
(51)
N₂
ap
N
S) 3N,
R
18
3 + 10 0
(52)
π
where N₂ is the total number of edges, N2 is the total number of faces. di and de are the length of i-th edge on original surface and predicted sphere. Similarly, at and af are the area of j-th face on original surface and predicted sphere; (a of ( and (a S6. 08) are the inner angles of the k-th face original surface and predicted sphere.
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Inflation Projection
Huraan
PointNet*+ Spherical Mapping
Human
Todidy
Teady
Fish
Finh
-20
Query
Retrieval Results
Query
Retrieval Results
Manifold learning of clustering results (top) and nearest neighbor retrieval results (bottom) using Inflation and Projection (left) and PointNet++ (right) spherical mappings.
Figure 12:
H.3 RESULTS
Figure 12 illusttras the effectiveness of the LSSOT metric under both spherical mapping techniques.
The results demonstrate that LSSOT successfully identifies inherent similarities and dissimilarities among the spherical surface representations, regardless of the mapping method employed. This capability translates into good performance in both clustering and nearest neighbor retrieval tasks. The consistency of LSSOT's effectiveness under different mapping approaches underscores its robustness as metric for analyzing spherical distributions and comparing surface geometries.
GRADIENT FLOW WITH RESPECT TO LSSOT IN THE SPHERICAL SPACE
I.1 BACKGROUND: INTRODUCTION TO STIEFEL MANIFOLD
In this section, we briefly introduce the gradient flow and related concepts in spherical space. First, we introduce the Riemannian structure of the Stiefel manifold, which is general case of spherical space. Next, we specify the related concepts in the spherical space.
Given positive integers
I d, the Stiefel manifold given by
V d. V. (V dxkk V Ix).
is a dk - (k + ( dimensional manifold.
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Given X E Va,k, the tangent space is given by:
Tx (Va,)) (Z E Baxk ZX + X =}.
(53)
Given matrix G Rdx the projection of G onto Tx (Va,)) is given by,
PTx(Fa.) == ---yymm X).
(54)
where sym
( G 2 x X is the symmetrie matrix obtained from G
Given differentiable function F Va,5 - R, the following steps describe the gradient of F in the manifold. In particular, choose X E Va,e:
Step I: compute Euclidean gradient Vg(F(X))
Step 2: the Stiefel manifold gradient is defined as:
VxF(X)= = (T.VaaVV
(55)
We consider the gradient flow problem:
0,X, = - vsF( E VAN-
Let 17 denote the step size (i.e., the learning rate for the machine learning community), the discrete gradient descent step of the above problem becomes:
X + X - VFF() = X (VEF((X) -
X)).
(56)
1.2 GRADIENT FLOW IN SPHERICAL SPACE
In spherical space, we have k (in this case, we use convention
0). Thus, the tangent space
(53
becomes:
To = span(x)* == f2 E Rd = 0},
(57)
where szan(z) is the ID line spaned by z. The corresponding projection
54 becomes
Rd g + (T)( g x(g x) (S
(58)
Note, the above projection is essentially the Euclidean projection of into the hyperplane -t.
Given X 1eI....,In, E Rdx= Y [y,...., :.. E Rdxm where Y is fixed. Let
Ei pro.,, 1/2 Ex=1 Og We consider the following function:
C(X) = LSSOT(H,
(C) (0:)
(59)
T
where each 0, E Sd-1 59 is the Monte Carlo approximation.
We select one and consider the 1-dimensional L.COT problem
LCOT((0)**) #11, (0)
1 12) = =f"² h Fogun (s E(0 + } -I (8 E2
2 ds
where ((r) = min(r, r).
Note, the above quantity implicitly defines transportation plan between l and v. In particular, suppose 7/ E RRxxm, with
reli.s] [ss Foj. 1(0 F₁, Føgi (s El0 ky
(60)
where |. is the uniform measure (Lebesgue measure) in [o, 1). a = means a mod 1 = b. is straightforward to verify to ((00 Og U2). and thus is transportation plan between 01 0z U2.
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Therefore, explicitly write 59 in term of X [x1,... kn]:
59
h00 2, 0, v..re.t..))
(61)
where each no, is defined via (60). For each 24, the Euclidean gradient is given by
T
VEC(xi) = T
Σ
O2
Σ
2(0 x, -0, 1o,
1. x,
6f 1//51/2
E
2(0 22 - 02 yソ 25)ho. 12.3
(62)
3:10
of 1/121/2
Based on this, we derive the gradient descent step for the following gradient flow problem.
(sc(()=-VD) =
That is,
X - (4XX-NV((C(M)) =
= x -X--XX (XVExX)
(63)
where (=(C( given by for each
1.3 IMPLEMENTATION OF GRADIENT FLOW
Riemannian Gradient Descent. Let μux = 1 En=1 Sx,, be the source distribution with mass N uniformly distributed at X' 1 C Sd-1, and vy - M Em=1 EM Sz.. be the target distribution with y {ym) C 5dl11 For each E X, let (bk) the location of x at k-th iteration, then at (le + 1)-th iteration, the update should be the following (Absil et al. 2008 Boumal et al. 2019):
(k++)) (2 (x
(64)
where 7 is the learning rate k-th iteration, the exponential map (x) = x cos ||u|| गল sin ||e/]|
and V sa=11((((()) is the projection of the Euclidean gradient Vz.F(z(()) onto the tangent space at (R). We implement this gradient descent method using the exponential function and projection function in the Geoopt library Kochurov et al. 2020
Spherical Coordinate Gradient Descent. A more straightforward gradient descent method is to work in the spherical coordinate system. By the following correspondences (in $2 for example) between (x, y, 2) S and (0,τ) Φ)
cos 0 sin ø
y sin @ sin Ø
cos
and
0 = arctan y
arccos
we can calculate the gradient of the loss function with respect to (0, Φ). Note here aretan must be suitably defined according the quadrant.
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1.4 ADDITIONAL POINT CLOUD INTERPOLATION VISUALIZATIONS
T=0/4
T=1/4
T=2/4
T=3/4
T=4/4
LSSOT
SSW
Spherical OT
Figure 13: Gradient flow interpolations between two tables.
T=0/4
T=1/4
T=2/4
T=3/4
T=4/4
LSSOT
b
SSW
Spherical OT
b
b
Figure 14: Gradient flow interpolations between two toilets.
T=0
T=1/4
2/4
T=3/4
4/4
LSSOT
SSW
Spherical
Figure 15: Gradient flow interpolations from a range hood plant.
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T=0/4
T=1/4
T=2/4
T=3/4
T=4/4
USS0
SSW
Spherical
Figure 16: Gradient flow interpolations from TV stand bookshelf.
T=0/4
T=1/4
T=2/4
T=4/4
LSSOT
ASS
Spherical OT
Figure 17: Gradient flow interpolations from table piano.
T=0/4
T=1/4
T=2/4
T=4/
LSS01
55W
Sphonical OT
Figure Gradient flow interpolations from guitar an airplane.
I.5 GRADIENT FLOW MOLLWEIDE VIEW VISUALIZATIONS
We provide more visualizations for the gradient flow of spherical distributions with respect to LSSOT metric. SSW, and spherical optimal transport distance (spherical OT). At first, source distribution and a target distribution are generated, both which are discrete distributions with uniform mass at all points. Then we perform gradient descent on the mass locations of the source distribution with respect
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to the above three objectives, driving the source toward the target distribution by minimizing these losses. We employ two methods of gradient descent on the sphere. See Figure t9 Figure 330 for the Mollweide views of the gradient flows between two different pairs of source-target distributions.
LSSOT Gradient Flow (Riemannian Gradient Descent)
1-3111
t>4/11
t~5/11
1-/11
t=/11
1-8/11
1-9111
t-10/11
|+
Figure 19: LSSOT gradient flow using Riemannian gradient descent.
SSW Gradient Flow (Riemannian Gradient Descent)
t=1/11
1=2/11
r-/111
t=4/11
1-5/11
116111
1-7/11
[+8/11
t-10/11
Figure 20: SSW gradient flow using Riemannian gradient descent.
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(Rlemeenian
110
1-1/11
12/11
D-311
144/11
H6111
P-1/11
Figure 21: Spherical OT gradient flow using Riemannian gradient descent.
USSOT Gractent Flow SSpherical Coorcinate Gradient Descent)
://11
28111
1-10/1
Figure 22: LSSOT gradient flow using spherical coordinate gradient descent.
111111
1+2/11
2-111
Figure 23: SSW gradient flow using spherical coordinate gradient descent.
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(lpherico
Desoent)
140
1-1/11
1-2/11
D14/1
111/1
D-8h11
110/11
Figure 24: Spherical OT gradient flow using spherical coordinate gradient descent.
LSSOT Gradient Flow (Riemannian Gradient Descent)
t=0
1=1/8
t=:2/B
t-5/8
t-5/8
t-7/8
[-1
Figure 25: LSSOT gradient flow using Riemannian gradient descent.
SSW Gradient Flow (Riemannian Gradient Descent)
t-2/8
t=3/8
t=4/8
t-5/8
t-6/8
t-7/8
[-]
Figure 26: SSW gradient flow using Riemannian gradient descent.
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Spherical OT Gradient Flow (Riemannian Gradient Descent)
[-0
t-1/8
1-2/8
t=3/8
t-4/8
T-5/8
t-6/8
t-
t-1
Figure 27: Spherical OT gradient flow using Riemannian gradient descent.
LSSOT Gradient Flow (Spherical Coordinate Gradient Descent)
t=1/8
t=2/8
t-3/8
t-4/8
t-5/8
t-6/8
t-7/8
[-1
Figure 28: LSSOT gradient flow using spherical coordinate gradient descent.
SSW Gradient Flow (Spherical Coordinate Gradient Descent)
t-1/8
t-2/8
t-3/8
t-4/8
t-5/8
t-6/8
t-7/8
[-1
Figure 29: SSW gradient flow using spherical coordinate gradient descent.
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Spherical OT Gradient Flow (Spherical Coordinate Gradient Descent)
I-D
t-1/8
L-2/
t=3/8
t=4/8
t=5/8
t-6/8
t-7/8
Figure 30: Spherical OT gradient flow using spherical coordinate gradient descent.
COMPUTATION COMPLEXITY
We provide the complexities for d-dimensional discrete distributions with support of size N:
Method
Complexity
Pairwise Distance Calculation Complexity
LSSOT SSW
( log N + 1))
O(KLN + log N + 1+ K))
(L + log N + log()))
O(K + log + log( )))) O(K2N2(N log N + d)
OT
ON log N + d))
Sinkhorn
O(N2( loR + d))
O(K2 (2 + +d) OK3 log N)) OKK log N))
S3W
OLLN l))
SWD
O(LN(d+ N))
Table Computation complexities for one pair of distributions (middle column) for pairwise distance computation among set of K distributions. The e the complexity of SSW arises from approximating the solution with precision in binary search. Similarly, the in the complexity of Sinkhorn refers to finding an e-approximation of the OT distance by Sinkhorn's 's algorithm.
Of note, all methods except SWD are under the spherical geometry, whereas the underlying ground distance for SWD is the Euclidean distance. For one pair distributions, SWD and S3W are marginally faster than LSSOT. However, when K is large enough in the pairwise comparison, the linear complexity with respect to K will overrule that slight advantage (as shown in Figure 2) making LSSOT the fastest of all the methods.
K INFLUENCE OF NUMBER OF PROJECTIONS AND REFERENCE SIZE
As the number of slices and reference size are the hyperparameters of the LSSOT method, we plot the evolution of LSSOT with respect to the number of slices in Figure 31 and the evolution with respect to reference size in Figure 32 for different dimensions d and VMF distributions with different k. With more than 100 slices, LSSOT performs robustly as the projector varies. LSSOT also behaves consistently as reference size changes.
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d=100
K=1
K=
10
d=500
100
d= 50
d=1000
LSSOT
LSSOT
2
10
10'
107
10a
10
Number of projections
Number of projections
Figure 31: Evolution of LSSOT with respect to number of slices. The sample sizes are fixed to be 500.
d
d=100
d= 10
d=500
==1
==
d
d=100
==10
== 100
LSSOT
LSSOT
3
Reference Size
Reference Size
Figure 32: Evolution of I.SSOT with respect reference size. Reference is sampled from the uniform distribution on 1).
L INVARIANCE TO ROTATIONS
By definition of LSSOT as an integration over the uniform distribution a (Equation 10). it invariant rotations. To empirically test the invariance, fix pair of distributions and generate 20 random rotations that are applied on the distributions. The LSSOT distances for all 20 rotated distributions plotted in Figure 33 with mean and standard deviation indicated. All 20 values fall closely with each other, with slight deviation due to the discrete approximation of the uniform distribution over the rank-2 projectors.
9.0
=std
8.5
8.0
LSSOT
7.5
7.0
6.5
9 10
12
13
14 15
17 18
Rotations
Figure LSSOT distances for pair of distributions and their rotated variants.
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M IDENTIFYING INCORRECTLY LABELED SUBJECTS
With metric properties, LSSOT can be applied to identify the same distribution. One scenario in neuroimaging analysis is when the same subject is labeled different subjects in the datasets. We design simplified experiment to demonstrate the effectiveness of LSSOT in identifying the same subjects in the ADNI dataset. We randomly select 25 subjects from the ADNI dataset and duplicate one of them (the 8-th subject) as the test subject with an incorrect label. Figure depiicss that e SSOT stands out at subject 8 with value I, as the LSSOT distance between them arbitrarily close to Such behavior makes LSSOT a very convenient tool in correcting the labels in the preprocessing of the datasets.
0.8
0.6
0.4
0.2
0123
3
8 9:011:222413337718222222222
Training Subjects
Figure 34: Exponential of the negative LSSOT distance from test subject to a set of 25 training subjects.
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Reviews:
[{"review_id": "KJiLMpWQk2", "reviewer": "Reviewer_vEru", "summary": "The authors propose the linear spherical sliced optimal transport framework (LSSOT) as an efficient method for comparing probability distributions on spherical manifolds. The authors build on several valuable past works that aim to linearize or speed-up the optimal transport problem by restricting to specific domains. In their case, the spherical domain has extensive applications in computer vision and especially in neuroscience for problems of cortical surface registration. They demonstrate results on several applications including point cloud interpolation and cortical surface registration.", "questions": "see weaknesses section above.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper has several strengths including the clarity of the writing, problem formulation, and valuable contribution to problem of spherical data analysis. I appreciated the authors placing their work in the context of past work in sliced optimal transport and linear optimal transport. They elegantly combine aspects from both to develop the spherical OT framework. There is theoretical justification for the use of the LSSOT as a distance metric, and an efficient implementation for speed. The authors also demonstrate the utility of their method on several experiments, notably cortical surface registration, where working in a spherical domain is essential.", "weaknesses": "I have several questions about the experimental evaluation of this work, specifically as it pertains to cortical surface registration. I appreciate the use of multiple datasets and outlining the problem of surface-to-atlas registration. The authors use extensive evaluation metrics. However, I have some concerns with the experimental setup and results.\n\nThe main purpose of these experiments was to quantify the improvement in performance and speed in using their metric over conventional ones. The authors state that they also use a Dice loss and regularization term. Did the authors perform new hyperparameter tuning on these losses when switching the image similarity metric? Failure to do so may bias results.\n\nWhy was only the sulcal depth used as a registration feature? Curvature is typically used and also offers a lot of information for registration.\n\nThe results on registration are puzzling. The authors' method performs considerably worse on SWD than the MSE loss for the NKI (Left hemisphere) and in MAE for NKI (Right hemisphere). Can the authors comment why? \n\nIn general, registration is an ill-posed problem, and one can modify the hyperparameters appropriately to either obtain very close matchings or well behaved fields. Can the authors discuss this and describe how this tradeoff was achieved. \n\nHow do these results compare to classical freesurfer registration? Which although is slow, represents a SOA in performance.", "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": "xVVik6QXMf", "reviewer": "Reviewer_uY8P", "summary": "This paper introduces Linear Spherical Sliced Optimal Transport (LSSOT), a framework for embedding spherical probability distributions into L2 space while maintaining spherical geometry. Extending previous work on circular distributions, LSSOT offers a computationally efficient metric for spherical data comparison, proving to be both a valid metric and faster than baseline approaches. The effectiveness of LSSOT is demonstrated through applications in cortical surface registration, 3D point cloud analysis, and shape embedding.", "questions": "The suggestions are mentioned in Weaknesses(2,3,4).", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tAccording to time complexity and runtime analysis and the results in Table 1 in this paper, the proposed method achieves significant computational efficiency while preserving the spherical geometry, making it suitable for large-scale applications.\n\n2.\tThe authors prove that the proposed LSSOT satisfies metric properties, strengthening the approach's theoretical validity.\n\n3.\tThis work performed extensive experimental validation. The effectiveness and efficiency of LSSOT are evaluated across multiple applications, including cortical surface registration, 3D point cloud analysis, and shape embedding.\n\n4.\tThe cortical surface registration results indicate that LSSOT achieves high Dice scores and minimizes area/edge distortions in the deformation field. This may provide biologically meaningful insights into neuroimaging applications.", "weaknesses": "1.\tFrom a technical point of view, the extension of the proposed method to the existing LOCT is limited. Still, it should be noted that it is useful in spherical signal analysis applications.\n\n2.\tLSSOT is compared to several baseline metrics, but the paper lacks a more detailed statistical analysis to quantify performance improvements (e.g., significance testing on Dice scores and area distortions). A statistical evaluation would help substantiate LSSOT’s claimed advantages over other metrics.\n\n3.\tAlthough LSSOT demonstrates effective interpolation performance, there is little information on its computational complexity compared to SSW and Spherical OT in point cloud analysis. A complexity analysis of interpolation tasks would better illustrate LSSOT's efficiency benefits.\n\n4.\tHyperparameter settings, such as λ, the number of slices L, and reference size M in LSSOT, are briefly mentioned but not systematically analyzed. A discussion on the sensitivity of these parameters and their impact on performance would be beneficial, particularly for users looking to optimize LSSOT for specific applications.", "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": "1Ar67Z2nAt", "reviewer": "Reviewer_Pf7Y", "summary": "This paper combines ideas from sliced optimal transport theory and linear optimal transport and introduces the idea of linear spherical sliced optimal transport. Specifically, they extend a recently proposed work (Martin et al. 2023) that introduced the linear circular optimal transport for embedding  circular distributions into a L^2 space. In this paper, the authors propose a new embedding method for spherical probability distributions. \n\nThe authors present experimental results for the method of linear spherical sliced optimal transport (LSSOT) in applications such as point cloud registration and cortical surface matching.", "questions": "This question pertains to the original circular optimal transport construction, and the authors do not have to answer it here, as they do something different in this paper. However, out of interest, why is the COT distance defined in terms of c(x, y) uses min{|x−y|,1 −|x−y|}? Can some other distance be used here? For e.g. min(arccos(<x, y>)). Is that the main difference between this earlier work and  the spherical sliced Wasserstein (SSW) approach? \n\n\nThe computational efficiency of this approach also naturally follows from the idea of slicing (projections on the great circle). Further efficiency is achieved by replacing the circular optimal transport distance by the linear version. However, can the authors provide slightly  more details on why the complexity of spherical sliced Wasserstein better than sliced Wasserstein? The projections on spheres should be more costly than lines. \n\nCan the authors comment on why LSSOT has better accuracy than SSW? Should SSW in fact perform as well as or better than LSSOT? \nFor SSW, the COT distance measures the discrepancy between the push-forward maps of the spherical measures, while LSSOT uses the LCOT distance. So there is yet one more embedding ( transport a reference measure to a target measure) performed. Does this result in loss of accuracy? Is this dataset dependent? Can the authors provide more detailed analysis or any ablation studies to explain this result? \n\nCan the authors expand more upon the injectivity of R? This is related to the results in theorems D.2 and D.3 and is inspired by the theorem in Bonet et al. 2024.  For e.g. the map R: L^1(S^{d−1}) → L^1(V_2(R^d) ×S^1) is not injective everywhere, although it is true for absolutely continuous measures on the sphere. What are the implications when the operator R is applied in practice? This could be discussed more for the benefit of the reader.", "limitations": "", "rating": 10, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "This paper presents an elegant construction of the spherical sliced embedding in a linear sense. \n\nThe main strength of the paper is the clever extension of the sliced optimal transport and linear circular optimal transport to the case of spherical distributions. This is novel.\n\nThere exist other methods for computing distances between spherical distributions, for e.g. spherical sliced-Wasserstein, Stereographic sliced-Wasserstein which also allows to efficiently compute distances. Other works have achieved further computational efficiency due to the linear distance between the circular transport embeddings. In this paper, the authors first obtain an embedding of the spherical sliced measures P(S^{d-1}) and then define a distance between the measures as the linear circular optimal transport distance between the pushforward maps of the spherical measures. While this paper borrows and combines ideas from two earlier works, the novelty of this approach is that in this instance it is achieved for embeddings of spherical distributions. \n\nThe LSSOT distance is a pseudo-metric on P(S^{d−1}) as proven in an earlier work (spherical sliced Wasserstein), since they use the same notion of the distance. In this work the authors show that it is a well-defined metric for probability measures. Also see weakness. \n\nThe results of Theorems D.2 and D.3 and the corollary B.7 are important as they ensure that the pushforward maps can be written in terms of the Radpm transform operator R. \n\nThe paper is written in great detail and provides appropriate background for the concepts that are need to understand it. This is a strength. \n\nThe experimental results are somewhat comprehensive  and the authors do a good job of comparing their method with other spherical slicing methods (Wasserstein etc.). \n\nThe authors also provide a time complexity and runtime analysis and compare them to those with other related methods.", "weaknesses": "This is a minor weakness, but as mentioned in the strengths above, the LSSOT is a metric only for probability measures. Thus the data has to be first converted to distributions, which is both a lossy transformation (which the authors acknowledge). \n\nBy definition, the projection is not invertible. That is one cannot recover the original signal/shape. Again, this drawback exists for all the other methods, including those that the authors build upon. \n\nBecause of the above 2 points, when the authors present the interpolation results, they resort to an auto-encoder that projects the original data to the nearest point in the spherical latent space. Then to reconstruct the intermediate points along the interpolation path between the two signal, they need a decoder. This is similar to the backprojection approaches, which are not ideal. But again this is not a major weakness, but currently, this is the most convenient approach. \n\nIn addition to the above loss of information, the projection step (slicing) will also introduce further discrepancy. However, this is a drawback for all the slicing based methods (including spherical) that are mentioned in the paper. In this paper, the authors use the same idea as Bonet et al. 2023 for performing the projection. \n\nWhile the authors do provide algorithms for computing flows using Riemannian gradient descent, this is technically not a geodesic, but an approximation. \n\nThe notation for P^C (Equation 6) and P^U (Equation 7) gets confusing. The first one is a geodesic distance, while the second one is a projection.", "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": "CCiTCweLXD", "reviewer": "Reviewer_qMUW", "summary": "A fast metric is proposed for spherical data, based on sliced optimal transport.", "questions": "To discuss:\n-  Invariance of this metric to nuisances such as rotation. \n-  Is this a metric or pseudo-metric, if a pseudo-metric (as in proposition D1?) why not state this in the title?\n-  Discuss invertible spherical harmonics, filter banks, which can be used to form a metric and are used on spherical brain data [1].\n-  the order of computational complexity of this metric in a practical spherical data retrieval scenario. Apparently it would require O(N) search time in N the number of data, other fast methods (non spherical) used on the same ADNI brain data require O(log N) [2].\n\nPlease demonstrate the utility by identifying the incorrectly labelled ADNI subjects[1], these will immediately stand out as unusually close according to a valid metric as they are the same subjects incorrectly labeled as different[2].\n\nThe paper states “the Dice loss to impose biological validity”, whereas Dice and other tissue overlap measures specifically do not enforce biological validity as they permit arbitrary biologically implausible mappings[3].\n\n[1] Yeo, B. T. T., Ou, W., & Golland, P. (2008). On the construction of invertible filter banks on the 2-sphere. IEEE Transactions on Image Processing, 17(3), 283-300.\n\n[2] Chauvin, Laurent, et al. \"Neuroimage signature from salient keypoints is highly specific to individuals and shared by close relatives.\" NeuroImage 204 (2020): 116208.\n\n[3] Rohlfing, Torsten. \"Image similarity and tissue overlaps as surrogates for image registration accuracy: widely used but unreliable.\" Medical Imaging, IEEE Transactions on 31.2 (2012): 153-163.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The math appears interesting and valid. Spherical data are of interest in various contexts.", "weaknesses": "While the metric appears compariatively fast in comparisons of pairwise distance calculation, it does not address the order of complexity of performing many distance calculations given a dataset of N points, which is often the limiting factor in practical contexts. For example here nearest neighbor search would be O(N), whereas other methods particularly indexing in the context of brain data similary using the same ADNI data offer O(logN) performance.", "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 (Spotlight)