A convergence framework for optimal transport on the sphere

Hamfeldt, Brittany Froese; Turnquist, Axel G. R.

doi:10.1007/s00211-022-01292-1

Cited by 6 publications

(3 citation statements)

References 29 publications

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“…While W can be interpreted as the adjoint of W * : C(SO( 3)) → C(S 2 ) in ( 36), the same reasoning does not hold for W α,β by the lack of continuity of W * α,β g for continuous g. Proposition 4.7. The semicircle transforms (39) and (38) satisfy Wµ, g = µ, W * g for all g ∈ C(SO( 3))and…”

Section: Normalized Semicircle Transform Of Measuresmentioning

confidence: 99%

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Sliced optimal transport on the sphere

Quellmalz,

Beinert,

Steidl

2023

Inverse Problems

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Sliced optimal transport reduces optimal transport on multi-dimensional domains to transport on the line. More precisely, sliced optimal transport is the concatenation of the well-known Radon transform and the cumulative density transform, which analytically yields the solutions of the reduced transport problems. Inspired by this concept, we propose two adaptions for optimal transport on the 2-sphere. Firstly, as counterpart to the Radon transform, we introduce the vertical slice transform, which integrates along all circles orthogonal to a given direction. Secondly, we introduce a semicircle transform, which integrates along all half great circles with an appropriate weight function. Both transforms are generalized to arbitrary measures on the sphere. While the vertical slice transform can be combined with optimal transport on the interval and leads to a sliced Wasserstein distance restricted to even probability measures, the semicircle transform is related to optimal transport on the circle and results in a different sliced Wasserstein distance for arbitrary probability measures. The applicability of both novel sliced optimal transport concepts on the sphere is demonstrated by proof-of-concept examples dealing with the interpolation and classification of spherical probability measures. The numerical implementation relies on the singular value decompositions of both transforms and fast Fourier techniques. For the inversion with respect to probability measures, we propose the minimization of an entropy-regularized Kullback-Leibler divergence, which can be numerically realized using a primal-dual proximal splitting algorithm.

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Section: Normalized Semicircle Transform Of Measuresmentioning

confidence: 99%

“…Spherical optimal transport has been intensely studied in recent years. For instance, the problem can be solved using a Monge-Ampère type equation [38,54,88] or a variational framework [22]. The regularity of optimal maps has been investigated in [51].…”

Section: Introductionmentioning

confidence: 99%

Sliced optimal transport on the sphere

Quellmalz,

Beinert,

Steidl

2023

Inverse Problems

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“…Recently, using moving mesh methods, some authors are generally addressed the application to the field of weather prediction and climate simulation, which represent small-scale problems [46]; thus, based on the optimal transport problem the process requires the resolution of the Monge-Ampere equation on the sphere domain (earth shape), the investigation in this direction is given in the papers [53,46,21]. Moreover, the global adaptive meshes and conservation of the connection between the grid points are the most attractive properties of optimal transport problem [49,38].…”

Section: Introductionmentioning

confidence: 99%

Adaptive Isogeometric Analysis using optimal transport and their fast solvers

Bahari,

Habbal,

Ratnani

et al. 2024

Computer Methods in Applied Mechanics and Engineering

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Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group

Quellmalz,

Buecher,

Steidl

2024

J Math Imaging Vis

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Sliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on the sphere $$\mathbb {S}^{d-1}$$ S d - 1 and on the rotation group $$\textrm{SO}(3)$$ SO ( 3 ) . We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For $$\textrm{SO}(3)$$ SO ( 3 ) , we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on $$\textrm{SO}(3)$$ SO ( 3 ) . As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior for both the free- and fixed-support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.

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A convergence framework for optimal transport on the sphere

Cited by 6 publications

References 29 publications

Sliced optimal transport on the sphere

Sliced optimal transport on the sphere

Adaptive Isogeometric Analysis using optimal transport and their fast solvers

Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group

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