Shrinkage versus deconvolution

Klann, Esther; Kuhn, Michael; Lorenz, Dirk A.; Maaß, Peter; Thiele, Herbert

doi:10.1088/0266-5611/23/5/025

Cited by 21 publications

(19 citation statements)

References 20 publications

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“…This 'practical approach' proceeds by computing a wavelet-shrinkage on an appropriate wavelet decomposition followed by simply plotting the positions and amplitudes of the remaining coefficients. It has been shown, [26], that this approach is indeed equivalent to a regularized deconvolution scheme in Besov scales. Besov scales are needed in order to obtain an appropriate mathematical model for the reconstruction of such sequences of delta peaks.…”

Section: Deconvolution Of δ-Sequencesmentioning

confidence: 96%

“…The aim of the present section is to summarize the mathematical justification of this approach as given in [26] and to present reconstruction results for deconvolving 1D MALDI/TOF-data (Bruker Daltonics GmbH) and 2D LCMS spectra (HofmannLARoche AG).…”

Section: Deconvolution Of δ-Sequencesmentioning

confidence: 99%

“…This is an approximation result on the data side, which needs to be transferred to an estimate on the reconstruction side, see again [26].…”

Section: Analysis Of the Model Problemmentioning

confidence: 99%

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Multiscale Approximation

Dahlke

2010

Handbook of Geomathematics

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Section: Deconvolution Of δ-Sequencesmentioning

confidence: 96%

Section: Deconvolution Of δ-Sequencesmentioning

confidence: 99%

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Multiscale Approximation

Dahlke

2010

Handbook of Geomathematics

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“…in mass spectrometry [32] where the signal is modeled as a sum of Dirac peaks (so-called impulse trains)…”

Section: Orthogonal Matching Pursuitmentioning

confidence: 99%

Sparsity and Compressed Sensing in Inverse Problems

Herrholz

Lorenz

Teschke

et al. 2014

Extraction of Quantifiable Information From Complex Systems

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This chapter is concerned with two important topics in the context of sparse recovery in inverse and ill-posed problems. In first part we elaborate conditions for exact recovery. In particular, we describe how both 1 -minimization and matching pursuit methods can be used to regularize ill-posed problems and moreover, state conditions which guarantee exact recovery of the support in the sparse case. The focus of the second part is on the incomplete data scenario. We discuss extensions of compressed sensing for specific infinite dimensional ill-posed measurement regimes. We are able to establish recovery error estimates when adequately relating the isometry constant of the sensing operator, the ill-posedness of the underlying model operator and the regularization parameter. Finally, we very briefly sketch how projected steepest descent iterations can be applied to retrieve the sparse solution.

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“…If the forward operator F and hence the operator equation (1) is linear, then in Hilbert spaces a comprehensive and rather complete regularization theory, including a general regularization schema and a well-established collection of methods, assertions on stability, convergence, and convergence rates, is available since more than 20 years; see [10,35,[77][78][79]84]. For recent progress of regularization theory applied to linear ill-posed problems, please refer to the papers [16,22,31,57,61,80,85,86,90].…”

Section: Introductionmentioning

confidence: 99%

Regularization Methods for Ill-Posed Problems

Cheng

Hofmann

2014

Handbook of Mathematical Methods in Imaging

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In this chapter are outlined some aspects of the mathematical theory for direct regularization methods aimed at the stable approximate solution of nonlinear ill-posed inverse problems. The focus is on Tikhonov type variational regularization applied to nonlinear ill-posed operator equations formulated in Hilbert and Banach spaces. The chapter begins with the consideration of the classical approach in the Hilbert space setting with quadratic misfit and penalty terms, followed by extensions of the theory to Banach spaces and present assertions on convergence and rates concerning the variational regularization with general convex penalty terms. Recent results refer to the interplay between solution smoothness and nonlinearity conditions expressed by variational inequalities. Six examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.

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Shrinkage versus deconvolution

Cited by 21 publications

References 20 publications

Multiscale Approximation

Multiscale Approximation

Sparsity and Compressed Sensing in Inverse Problems

Regularization Methods for Ill-Posed Problems

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