Variational Image Regularization with Euler's Elastica Using a Discrete Gradient Scheme

Ringholm, Torbjørn; Lazić, Jasmina; Schönlieb, Carola-Bibiane

doi:10.1137/17m1162354

Cited by 27 publications

(23 citation statements)

References 35 publications

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“…Therefore, slight modifications are required to apply the aforementioned line search methods to the SOR method. To achieve this task, we shall directly use the approach proposed in [17].…”

Section: Adaptive Sor Methodsmentioning

confidence: 99%

Adaptive SOR methods based on the Wolfe conditions

2019

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Because the expense of estimating the optimal value of the relaxation parameter in the successive overrelaxation (SOR) method is usually prohibitive, the parameter is often adaptively controlled. In this paper, new adaptive SOR methods are presented that are applicable to a variety of symmetric positive definite linear systems and do not require additional matrix-vector products when updating the parameter. To this end, we regard the SOR method as an algorithm for minimising a certain objective function, which yields an interpretation of the relaxation parameter as the step size following a certain change of variables. This interpretation enables us to adaptively control the step size based on some line search techniques, such as the Wolfe conditions. Numerical examples demonstrate the favourable behaviour of the proposed methods. IntroductionThe successive over-relaxation (SOR) method is one of the most well-known stationary iterative methods for solving linear systemswhere A ∈ R n×n and b b b ∈ R n are given nonsingular matrix and vector, respectively. The SOR method is formulated asHere, the matrix A is expressed as the matrix sumwhere D = diag(a 11 , a 22 , . . . , a nn ), and L and U are strictly lower and upper triangular n × n matrices.The convergence performance of the SOR method depends on the relaxation parameter ω. A necessary condition for the SOR method to converge is that ω ∈ (0, 2), and this condition is also sufficient for a symmetric positive definite matrix A (for further details of SOR theory see, e.g., [4,18]). The relaxation parameter

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“…Therefore, slight modifications are required to apply the aforementioned line search methods to the SOR method. To achieve this task, we shall directly use the approach proposed in [17].…”

Section: Adaptive Sor Methodsmentioning

confidence: 99%

Adaptive SOR methods based on the Wolfe conditions

2019

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“…Ehrhardt et al [21] provided additional analysis for the methods, including convergence rates for smooth, convex problems and Polyak-Łojasiewicz functions, as well as well-posedness of the implicit equation. Ringholm et al [46] applied the Itoh-Abe discrete gradient method to non-convex image problems with Euler's elastica regularisation.…”

Section: Related Literaturementioning

confidence: 99%

“…These are methods from geometric numerical integration that preserve the aforementioned geometric structures in a general setting. In recent papers [21,26,45,46], optimisation schemes based on discretising gradient flows with discrete gradients have been analysed and implemented for various problems. Favourable properties of the discrete gradient methods include unconditional dissipation, i.e.…”

Section: Introductionmentioning

confidence: 99%

Bregman Itoh–Abe Methods for Sparse Optimisation

2020

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In this paper we propose optimisation methods for variational regularisation problems based on discretising the inverse scale space flow with discrete gradient methods. Inverse scale space flow generalises gradient flows by incorporating a generalised Bregman distance as the underlying metric. Its discrete-time counterparts, Bregman iterations and linearised Bregman iterations are popular regularisation schemes for inverse problems that incorporate a priori information without loss of contrast. Discrete gradient methods are tools from geometric numerical integration for preserving energy dissipation of dissipative differential systems. The resultant Bregman discrete gradient methods are unconditionally dissipative and achieve rapid convergence rates by exploiting structures of the problem such as sparsity. Building on previous work on discrete gradients for non-smooth, non-convex optimisation, we prove convergence guarantees for these methods in a Clarke subdifferential framework. Numerical results for convex and non-convex examples are presented. Keywords Non-convex optimisation • Non-smooth optimisation • Bregman iteration • Inverse scale space • Geometric numerical integration • Discrete gradient methods Mathematics Subject Classification 49M37 • 49Q15 • 65K10 • 90C26 This article is dedicated to Mila Nikolova whose keen interest and inspiring discussions have encouraged our research on geometric integration for optimisation. MB acknowledges support from the Leverhulme Trust Early Career Fellowship ECF-2016-611 'Learning from mistakes: a supervised feedback-loop for imaging applications'. ESR and CBS acknowledge support from CHiPS (Horizon 2020 RISE project grant) and from the Cantab Capital Institute for the Mathematics of Information. CBS acknowledges support from the Leverhulme Trust project on "Breaking the non-convexity barrier" EPSRC grant Nr EP/M00483X/1, and the EPSRC Centre Nr EP/N014588/1. Moreover, CBS acknowledges support from the RISE project NoMADS and the Alan Turing Institute.

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“…To compute the α k j at each coordinate step one can use any suitable root finder, yet to stay in line with the derivative-free nature of Algorithm 1, one may wish to use a solver like the Brent-Dekker algorithm [3]. Also worth noting is that the parallelization procedure used in [22] works for Algorithm 1 as well.…”

Section: Itoh-abe Discrete Riemannian Gradientmentioning

confidence: 99%

“…Discrete gradients were first used in optimization algorithms for image analysis in [11] and [22]. As an example of a manifold-valued imaging problem, consider Total Variation (TV) denoising of manifold valued images [30], where one wishes to minimize, based on generalizations of the L β and L γ norms:…”

Section: Manifold Valued Imagingmentioning

confidence: 99%

Dissipative Numerical Schemes on Riemannian Manifolds with Applications to Gradient Flows

Celledoni¹,

Eidnes²,

Owren³

et al. 2018

SIAM J. Sci. Comput.

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This paper concerns an extension of discrete gradient methods to finitedimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative ordinary differential equations. This includes Riemannian gradient flow systems which occur naturally in optimization problems. The Itoh-Abe discrete gradient is formulated and applied to gradient systems, yielding a derivative-free optimization algorithm. The algorithm is tested on two eigenvalue problems and two problems from manifold valued imaging: InSAR denoising and DTI denoising.

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Variational Image Regularization with Euler's Elastica Using a Discrete Gradient Scheme

Cited by 27 publications

References 35 publications

Adaptive SOR methods based on the Wolfe conditions

Adaptive SOR methods based on the Wolfe conditions

Bregman Itoh–Abe Methods for Sparse Optimisation

Dissipative Numerical Schemes on Riemannian Manifolds with Applications to Gradient Flows

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