Denoising of Sphere- and SO(3)-Valued Data by Relaxed Tikhonov Regularization

Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, Condat (IEEE Trans. Signal Proc.) proposed a convex relaxation of the Tikhonov-regularized nonconvex problem for denoising circle-valued data. Using Schur complement arguments, we show that this variational model can be simplified while leading to the same solution. Our simplified model can be generalized to higher dimensional spheres and to SO(3)-valued data, where we rely on the quaternion representation of the latter. Standard algorithms from convex analysis can be applied to solve the resulting convex minimization problem. As proof-of-the-concept, we use the alternating direction method of minimizers to demonstrate the denoising behavior of the proposed approach

Multi-Marginal Gromov-Wasserstein Transport and Barycenters

Gromov-Wasserstein (GW) distances are combinations of Gromov-Hausdorff and Wasserstein distances that allow the comparison of two different metric measure spaces (mm-spaces). Due to their invariance under measure- and distance-preserving transformations, they are well suited for many applications in graph and shape analysis. In this paper, we introduce the concept of multi-marginal GW transport between a set of mm-spaces as well as its regularized and unbalanced versions. As a special case, we discuss multi-marginal fused variants, which combine the structure information of an mm-space with label information from an additional label space. To tackle the new formulations numerically, we consider the bi-convex relaxation of the multi-marginal GW problem, which is tight in the balanced case if the cost function is conditionally negative definite. The relaxed model can be solved by an alternating minimization, where each step can be performed by a multi-marginal Sinkhorn scheme. We show relations of our multi-marginal GW problem to (unbalanced, fused) GW barycenters and present various numerical results, which indicate the potential of the concept

Sliced Optimal Transport on the Sphere

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 the weighted semicircle transform, which integrates along all half great circles. 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.Comment: 38 pages, 6 figure