Inverse scale space decomposition

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards nonlinear regularization methods even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this development towards modern nonlinear regularization methods, including their analysis, applications, and issues for future research.In particular we will discuss variational methods and techniques derived from those, since they have attracted particular interest in the last years and link to other fields like image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions, and learning theory.

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“…Given (OC) and (SUB0), we can guarantee the following decomposition result, which is a direct generalization of [333,Theorem 3.14].…”

Section: Inverse Scale Spacementioning

confidence: 97%

“…We refer to [30] for more information on individual generalized singular vectors and the inverse scale space flow. For more theoretical results and analytical as well as numerical examples of ordered sets of singular vectors that satisfy (OC) and (SUB0), we refer to [333].…”

Section: Inverse Scale Spacementioning

confidence: 99%

Modern regularization methods for inverse problems

2018

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“…Since then, researchers have studied the ISS flow with applications to generalised spectral analysis in a nonlinear setting, i.e. by Burger et al [8], Gilboa et al [23], and Schmidt et al [51]. The Bregman method has been studied for 1 -regularisation and compressed sensing by Goldstein and Osher [24] and Yin et al [59], and extended to primal-dual algorithms by Zhang et al [61].…”

Section: Related Literaturementioning

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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“…The optimality condition of (3) is given via 0 ∈ ∂F (ĥ) −μ ∂G(ĥ), where ∂F (ĥ) and ∂G(ĥ) denote the subdifferential of F and G atĥ, respectively, andμ = F (ĥ)/G(ĥ). Note that the Lagrange multipliers for the constraints are zero by the same argumentation as in [18,Section 2], and can therefore be omitted.…”

Section: Numerical Implementationmentioning

confidence: 99%

Learning Filter Functions in Regularisers by Minimising Quotients

Benning

Gilboa

Grah

et al. 2017

Lecture Notes in Computer Science

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Learning approaches have recently become very popular in the field of inverse problems. A large variety of methods has been established in recent years, ranging from bi-level learning to high-dimensional machine learning techniques. Most learning approaches, however, only aim at fitting parametrised models to favourable training data whilst ignoring misfit training data completely. In this paper, we follow up on the idea of learning parametrised regularisation functions by quotient minimisation as established in [3]. We extend the model therein to include higher-dimensional filter functions to be learned and allow for fit-and misfit-training data consisting of multiple functions. We first present results resembling behaviour of wellestablished derivative-based sparse regularisers like total variation or higher-order total variation in one-dimension. Our second and main contribution is the introduction of novel families of non-derivative-based regularisers. This is accomplished by learning favourable scales and geometric properties while at the same time avoiding unfavourable ones.

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Inverse scale space decomposition

Cited by 18 publications

References 94 publications

Modern regularization methods for inverse problems

Modern regularization methods for inverse problems

Bregman Itoh–Abe Methods for Sparse Optimisation

Learning Filter Functions in Regularisers by Minimising Quotients

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