Averaged alternating reflections in geodesic spaces

Fernández-León, Aurora; Nicolae, Adriana

doi:10.1016/j.jmaa.2013.01.060

Cited by 3 publications

(7 citation statements)

References 27 publications

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“…Unfortunately, the well-known result in the Euclidean setting that reflections of convex lower semicontinuous functions are nonexpansive does not carry over to the Hadamard manifold setting, see [19,32]. Up to now it was only proved that indicator functions of closed convex sets in Hadamard manifolds with constant curvature possess nonexpansive reflections [32]. In this paper, we prove that certain distance-like functions have nonexpansive reflections on general symmetric Hadamard manifolds.…”

Section: Introductionmentioning

confidence: 88%

“…Next we deal with the reflection operator R ι D , i.e., the reflection operator which corresponds to the orthogonal projection operator onto the convex set D. Unfortunately, in symmetric Hadamard manifolds, reflections corresponding to orthogonal projections onto convex sets are in general not nonexpansive. Counterexamples can be found in [19,32]. Unfortunately this is also true for our special set D as the following example with symmetric positive definite 2 × 2 matrices P(2) shows.…”

Section: Reflections On Convex Setsmentioning

confidence: 93%

“…Then the following statements hold Proof. The case κ < 0 was proved in [32]. For κ = 0 the assumption follows from the Hilbert space setting.…”

Section: Reflections On Convex Setsmentioning

confidence: 99%

“…If these reflections are nonexpansive, then it follows from general convergence results of Krasnoselski-Mann iterations in CAT(0) spaces [48] that the sequence produced by the algorithm converges. Unfortunately, the well-known result in the Euclidean setting that reflections of convex lower semicontinuous functions are nonexpansive does not carry over to the Hadamard manifold setting, see [19,32]. Up to now it was only proved that indicator functions of closed convex sets in Hadamard manifolds with constant curvature possess nonexpansive reflections [32].…”

Section: Introductionmentioning

confidence: 99%

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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: 88%

Section: Reflections On Convex Setsmentioning

confidence: 93%

“…Then the following statements hold Proof. The case κ < 0 was proved in [32]. For κ = 0 the assumption follows from the Hilbert space setting.…”

Section: Reflections On Convex Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Here those reflections are nonexpansive, see [30,51]. So, in summary, although the parallel DR algorithm showed very good numerical performance on general Hadamard manifolds in [16], theoretical convergence results remain up to now limited to manifolds with constant non-positive curvature.…”

Section: Now In Order To Minimize Argminmentioning

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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