Numerical inversion of the Funk transform on the rotation group

Hielscher, Ralf

doi:10.1088/0266-5611/29/12/125014

Cited by 2 publications

(2 citation statements)

References 42 publications

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“…W * α,β g ∈ L ∞ (S 2 ), the integral is no dual pairing in the measure/continuous function sense since W * α,β g is discontinuous at the zenith Φ(α, β) in general. Therefore, unlike in proposition 3.5 for the vertical slice transform, equation (40) does not constitute a proper definition of W α,β for measures via the dual pairing.…”

Section: Normalized Semicircle Transform Of Measuresmentioning

confidence: 95%

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: 95%

Sliced optimal transport on the sphere

Quellmalz,

Beinert,

Steidl

2023

Inverse Problems

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“…As the Radon transform on the rotation group allows for a similar singular value decomposition as the spherical Radon transform [20] there are many algorithms for the solution of the inverse problem, cf. [9,20,5,19], that uses polynomials on the rotation group as approximation of the ODF.…”

Section: The Rotation Groupmentioning

confidence: 99%

Fast Global Optimization on the Torus, the Sphere, and the Rotation Group

Gräf¹,

Hielscher²

2015

SIAM J. Optim.

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Detecting all local extrema or the global extremum of a polynomial on the torus, the sphere or the rotation group is a tough yet often requested numerical problem. We present a heuristic approach that applies common descent methods like nonlinear conjugated gradients or Newtons methods simultaneously to a large number of starting points. The corner stone of our approach are FFT like algorithms, i.e., algorithms that scale almost linearly with respect to the sum of the dimension of the polynomial space and the number of evaluation points. These FFT like algorithms allow us to compute one step of a descent method simultaneously for all staring points at almost the same cost as for one single starting point. The effectiveness of the proposed algorithms is demonstrated in various applications. In particular, we apply it to the Radon transform of a spherical function which allows us to detect lines in spherical patterns.

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Numerical inversion of the Funk transform on the rotation group

Cited by 2 publications

References 42 publications

Sliced optimal transport on the sphere

Sliced optimal transport on the sphere

Fast Global Optimization on the Torus, the Sphere, and the Rotation Group

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