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Bacak, M.* ; Bergmann, R.* ; Steidl, G.* ; Weinmann, A.

A second order non-smooth variational model for restoring manifold-valued images.

SIAM J. Sci. Comput. 38, A567-A597 (2016)
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We introduce a new nonsmooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization, and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.
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Publication type Article: Journal article
Document type Scientific Article
Keywords Manifold-valued Data ; Second Order Differences ; Tv-like Methods On Manifolds ; Nonsmooth Variational Methods ; Jacobi Fields ; Hadamard Spaces ; Proximal Mappings ; Dt-mri; Total Variation Minimization; Proximal Point Algorithm; Bounded Variation; Riemannian-manifolds; Hadamard Spaces; Metric-spaces; Harmonic Maps; Diffusion; Restoration; Regularization
ISSN (print) / ISBN 1064-8275
e-ISSN 1095-7197
Quellenangaben Volume: 38, Issue: 1, Pages: A567-A597 Article Number: , Supplement: ,
Publisher Society for Industrial and Applied Mathematics (SIAM)
Publishing Place Philadelphia
Reviewing status