Inpainting of Cyclic Data Using First and Second Order Differences

Bergmann, Ronny; Weinmann, Andreas

doi:10.1007/978-3-319-14612-6_12

Cited by 14 publications

(30 citation statements)

References 53 publications

(53 reference statements)

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“…A method which circumvents the direct work with manifold-valued data by embedding the matrix manifold in the appropriate Euclidean space and applying a back projection to the manifold was suggested in [65]. TV-like functionals on manifolds with higher order differences were handled in [8,15,16]. Finally we mention the relation to wavelet-type multiscale transforms which were handled, e.g., in [42,43,63,74].…”

Section: Introductionmentioning

confidence: 99%

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

Bergmann¹,

Persch²,

Steidl³

2016

SIAM J. Imaging Sci.

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We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation like regularizing term. To solve the convex minimization problem, we extend the Douglas-Rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the Douglas-Rachford algorithm are reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski-Mann iterations imply the convergence of the Douglas-Rachford algorithm. Unfortunately, this general results does not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions namely, several distance-like functions and an indicator functions of a special convex sets. We prove that the reflections of certain distance-like functions on Hadamard manifolds are nonexpansive which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the Douglas-Rachford algorithm converges here.Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods as the cyclic proximal point algorithm or half-quadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric which does not have a constant curvature.

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Section: Introductionmentioning

confidence: 99%

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

Bergmann¹,

Persch²,

Steidl³

2016

SIAM J. Imaging Sci.

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“…In the regularizer, the TV and TV 2 terms can appear separately or in a coupled way. Actually, their addition R(u) := β TV(u) + (1 − β) TV 2 (u), β ∈ (0, 1) was considered in [5,17]. Alternatively, couplings which generalize the infimal convolution approach [24] to the manifold-valued setting were proposed in [11,13].…”

Section: Intrinsic Variational Restoration Modelsmentioning

confidence: 99%

Recent advances in denoising of manifold-valued images

Bergmann

Laus

Persch

et al. 2019

Handbook of Numerical Analysis

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Modern signal and image acquisition systems are able to capture data that is no longer real-valued, but may take values on a manifold. However, whenever measurements are taken, no matter whether manifold-valued or not, there occur tiny inaccuracies, which result in noisy data. In this chapter, we review recent advances in denoising of manifoldvalued signals and images, where we restrict our attention to variational models and appropriate minimization algorithms. The algorithms are either classical as the subgradient algorithm or generalizations of the half-quadratic minimization method, the cyclic proximal point algorithm, and the Douglas-Rachford algorithm to manifolds. An important aspect when dealing with real-world data is the practical implementation. Here several groups provide software and toolboxes as the Manifold Optimization (Manopt) package and the manifold-valued image restoration toolbox (MVIRT).

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“…Recently several works tackled these tasks such as [18,30,75] for inpainting, or [5,13,20] for segmentation of such data. For denoising the TV approach or Rudin-Osher-Fatemi (ROF) [70] model was introduced by [58,79] and generalized to second order methods in [8,16,18,22]. Furthermore, for the ROF model half-quadratic minimization [15] and the Douglas-Rachford algorithm [17] have led to a significant increase in computational efficiency.…”

Section: Related Workmentioning

confidence: 99%

A Graph Framework for Manifold-Valued Data

Bergmann¹,

Tenbrinck²

2018

SIAM J. Imaging Sci.

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Graph-based methods have been proposed as a unified framework for discrete calculus of local and nonlocal image processing methods in the recent years. In order to translate variational models and partial differential equations to a graph, certain operators have been investigated and successfully applied to real-world applications involving graph models. So far the graph framework has been limited to real-and vector-valued functions on Euclidean domains. In this paper we generalize this model to the case of manifold-valued data. We introduce the basic calculus needed to formulate variational models and partial differential equations for manifold-valued functions and discuss the proposed graph framework for two particular families of operators, namely, the isotropic and anisotropic graph p-Laplacian operators, p ≥ 1. Based on the choice of p we are in particular able to solve optimization problems on manifold-valued functions involving total variation (p = 1) and Tikhonov (p = 2) regularization. Finally, we present numerical results from processing both synthetic as well as real-world manifold-valued data, e.g., from diffusion tensor imaging (DTI) and light detection and ranging (LiDAR) data. * Felix-Klein-Zentrum,

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Inpainting of Cyclic Data Using First and Second Order Differences

Cited by 14 publications

References 53 publications

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

Recent advances in denoising of manifold-valued images

A Graph Framework for Manifold-Valued Data

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