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Recent advances in denoising of manifold-valued images. (English) Zbl 1446.94004

Kimmel, Ron (ed.) et al., Processing, analyzing and learning of images, shapes, and forms. Part 2. Amsterdam: Elsevier/North Holland. Handb. Numer. Anal. 20, 553-578 (2019).
Summary: Modern signal and image acquisition systems are able to capture data that are 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 manifold-valued 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).
For the entire collection see [Zbl 1428.94001].

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
65K10 Numerical optimization and variational techniques
49M37 Numerical methods based on nonlinear programming
49Q05 Minimal surfaces and optimization
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65K05 Numerical mathematical programming methods
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
62H35 Image analysis in multivariate analysis

Software:

Manopt; MVIRT; MTEX; Camino
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