Scale space consistency of piecewise constant least squares estimators – another look at the regressogram

Boysen, Leif; Liebscher, Volkmar; Munk, Axel; Wittich, Olaf

doi:10.1214/074921707000000274

Cited by 11 publications

(15 citation statements)

References 30 publications

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“…Then, under further suitable assumptions, the minimizers of the Potts problem converge to piecewise constant minimizer of a continuous Potts functional (or, synonymously, piecewise constant Mumford-Shah functional), as sampling gets finer [25]. This also holds in the presence of noise [25,26,43]. For related results on Potts functionals with L 1 data terms, we also refer to Weinmann et al [44].…”

Section: Theorem 21 the Potts Problem (P γ ) Has A Minimizermentioning

confidence: 92%

Iterative Potts and Blake–Zisserman minimization for the recovery of functions with discontinuities from indirect measurements

Weinmann

Storath

2015

Proc. R. Soc. A.

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Signals with discontinuities appear in many problems in the applied sciences ranging from mechanics, electrical engineering to biology and medicine. The concrete data acquired are typically discrete, indirect and noisy measurements of some quantities describing the signal under consideration. The task is to restore the signal and, in particular, the discontinuities. In this respect, classical methods perform rather poor, whereas non-convex non-smooth variational methods seem to be the correct choice. Examples are methods based on Mumford-Shah and piecewise constant Mumford-Shah functionals and discretized versions which are known as BlakeZisserman and Potts functionals. Owing to their non-convexity, minimization of such functionals is challenging. In this paper, we propose a new iterative minimization strategy for Blake-Zisserman as well as Potts functionals and a related jump-sparsity problem dealing with indirect, noisy measurements. We provide a convergence analysis and underpin our findings with numerical experiments.

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Section: Theorem 21 the Potts Problem (P γ ) Has A Minimizermentioning

confidence: 92%

Iterative Potts and Blake–Zisserman minimization for the recovery of functions with discontinuities from indirect measurements

Weinmann

Storath

2015

Proc. R. Soc. A.

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“…Note that the condition γ n · |S n |/ ln n → ∞ implies the second part of (H1.3) by (H1.1). It was used for example in F. Friedrich (2005) or L. Boysen et al (2009Boysen et al ( , 2007.…”

Section: -Convergencementioning

confidence: 99%

“…Albeit focus of this paper is on two or more dimensions, we start with one dimension. There are at least two reasons for that: illustration of the abstract results by choices of the (seemingly) most elementary example, and to generalize results like some of those in L. Boysen et al (2009Boysen et al ( , 2007 to classes of piecewise Sobolev functions.…”

Section: One Dimensional Signals -Interval Partitionsmentioning

confidence: 99%

Complexity L0-Penalized M-Estimation: Consistency in More Dimensions

Demaret

Friedrich

Liebscher

et al. 2013

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Abstract:We study the asymptotics in L 2 for complexity penalized least squares regression for the discrete approximation of finite-dimensional signals on continuous domains-e.g., images-by piecewise smooth functions. We introduce a fairly general setting, which comprises most of the presently popular partitions of signal or image domains, like interval, wedgelet or related partitions, as well as Delaunay triangulations. Then, we prove consistency and derive convergence rates. Finally, we illustrate by way of relevant examples that the abstract results are useful for many applications.

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“…[27].) Γ-convergence and convergence of minimizers in relation with the L 2 -Potts functional are topics of [4,5]. In these papers, Γ-convergence is shown for a modified L 2 -Potts functional.…”

Section: Introductionmentioning

confidence: 99%

The $L^1$-Potts Functional for Robust Jump-Sparse Reconstruction

Weinmann

Storath²,

Demaret³

2015

SIAM J. Numer. Anal.

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Abstract. We investigate the non-smooth and non-convex L 1 -Potts functional in discrete and continuous time. We show Γ-convergence of discrete L 1 -Potts functionals towards their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete L 1 -Potts problem, we introduce an O(n 2 ) time and O(n) space algorithm to compute an exact minimizer. We apply L 1 -Potts minimization to the problem of recovering piecewise constant signals from noisy measurements f. It turns out that the L 1 -Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the L 1 -Potts functional. Furthermore, for strongly blurred signals and known blurring operator, we derive an iterative reconstruction algorithm.

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Scale space consistency of piecewise constant least squares estimators – another look at the regressogram

Cited by 11 publications

References 30 publications

Iterative Potts and Blake–Zisserman minimization for the recovery of functions with discontinuities from indirect measurements

Iterative Potts and Blake–Zisserman minimization for the recovery of functions with discontinuities from indirect measurements

Complexity L0-Penalized M-Estimation: Consistency in More Dimensions

The $L^1$-Potts Functional for Robust Jump-Sparse Reconstruction

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