Infimal Convolution Coupling of First and Second Order Differences on Manifold-Valued Images

Many methods for processing scalar and vector valued images, volumes and other data in the context of inverse problems are based on variational formulations. Such formulations require appropriate regularization functionals that model expected properties of the object to reconstruct. Prominent examples of regularization functionals in a vector-space context are the total variation (TV) and the Mumford-Shah functional, as well as higher-order schemes such as total generalized variation models. Driven by applications where the signals or data live in nonlinear manifolds, there has been quite some interest in developing analogous methods for nonlinear, manifold-valued data recently. In this chapter, we consider various variational regularization methods for manifold-valued data. In particular, we consider TV minimization as well as higher order models such as total generalized variation (TGV). Also, we discuss (discrete) Mumford-Shah models and related methods for piecewise constant data. We develop discrete energies for denoising and report on algorithmic approaches to minimize them. Further, we also deal with the extension of such methods to incorporate indirect measurement terms, thus addressing the inverse problem setup. Finally, we discuss wavelet sparse regularization for manifold-valued data. arXiv:1909.08921v1 [math.NA]

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“…where u = (u i,j ) i,j ∈ (R d ) N ×M . This representation was taken in [14,15] and extended for u = (u i,j ) i,j ∈ M N ×M (up to constants) via…”

Section: Modelsmentioning

confidence: 99%

“…where v = (v i,j ) i,j and w = (w i,j ) i,j . Following [14], we here use the symbol [[v i,j , w i,j ]] instead of [v i,j , w i,j ], where we define the former to include also non-distance minimizing geodesics. In order to generalize the second order TGV functional to a manifold setting, we consider (12) for vector spaces.…”

Section: Modelsmentioning

confidence: 99%

Non-smooth Variational Regularization for Processing Manifold-Valued Data

Höller

Weinmann

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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“…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]. In the Euclidean setting, the infimal convolution is related to the total generalized variation (TGV) approach of Bredies et al [21].…”

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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“…The most difficult part in this respect is to define suitable notions in the manifold setup which have both reasonable analytic properties on the one hand and which are algorithmically realizable on the other hand. Concerning infimal-convolution type functionals such as TV-TV 2 infimal convolutions, a first effort towards a generalization to the manifold setting has been made in the recent conference proceeding [17] which has later been extended in an arXiv preprint [16]. The present manuscript [28], which was submitted to arXiv at the same time as [16], emerged independently.…”

Section: Introductionmentioning

confidence: 99%

Total Generalized Variation for Manifold-Valued Data

Bredies¹,

Höller²,

Storath³

et al. 2018

SIAM J. Imaging Sci.

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In this paper we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data in a discrete setting. We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present two possible concrete instances that fulfill the proposed axioms. We provide well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup. Further, we provide experimental results for synthetic and real data to further underpin the proposed generalization numerically and show its potential for applications with manifold-valued data.Mathematical subject classification: 94A08, 68U10, 90C90 53B99, 65K10.

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Infimal Convolution Coupling of First and Second Order Differences on Manifold-Valued Images

Cited by 8 publications

References 29 publications

Non-smooth Variational Regularization for Processing Manifold-Valued Data

Non-smooth Variational Regularization for Processing Manifold-Valued Data

Recent advances in denoising of manifold-valued images

Total Generalized Variation for Manifold-Valued Data

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