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- Otmar Scherzer, Markus Grasmair, Harald Grossauer, Markus Haltmeier, Frank Lenzen
- Applied mathematical sciences
- 2009

- Theodore Alexandrov, Michael F. Becker, +6 authors Peter Maass
- Journal of proteome research
- 2010

In recent years, matrix-assisted laser desorption/ionization (MALDI)-imaging mass spectrometry has become a mature technology, allowing for reproducible high-resolution measurements to localize proteins and smaller molecules. However, despite this impressive technological advance, only a few papers have been published concerned with computational methods… (More)

Motivated from the theoretical and practical results in compressed sensing, efforts have been undertaken by the inverse problems community to derive analogous results, for instance linear convergence rates, for Tikhonov regularization with `1-penalty term for the solution of ill-posed equations. Conceptually, the main difference between these two fields is… (More)

We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate O( √ δ) of the regularized solutions in dependence of… (More)

Total variation regularization and anisotropic filtering have been established as standard methods for image denoising because of their ability to detect and keep prominent edges in the data. Both methods, however, introduce artifacts: In the case of anisotropic filtering, the preservation of edges comes at the cost of the creation of additional structures… (More)

- Markus Grasmair
- 2011

The goal of this paper is the formulation of an abstract setting that can be used for the derivation of linear convergence rates for a large class of sparsity promoting regularisation functionals for the solution of ill-posed linear operator equations. Examples where the proposed setting applies include joint sparsity and group sparsity, but also (possibly… (More)

- Markus Grasmair, Markus Haltmeier, Otmar Scherzer
- Applied Mathematics and Computation
- 2011

Although the residual method, or constrained regularization, is frequently used in applications, a detailed study of its properties is still missing. In particular, the questions of stability and convergence rates have hardly been treated in the literature. This sharply contrasts the progress of the theory of Tikhonov regularization, where for instance the… (More)

- Markus Grasmair
- 2009

We study the regularising properties of Tikhonov regularisation on the sequence space l with weighted, non-quadratic penalty term acting separately on the coefficients of a given sequence. We derive sufficient conditions for the penalty term that guarantee the well-posedness of the method, and investigate to which extent the same conditions are also… (More)

In this paper, we present a method for the numerical minimization of the Mumford–Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius ε > 0 and use the number of balls, multiplied with 2ε, as an estimate for the length of the edge set. We introduce a functional… (More)

- Martin Bauer, Markus Grasmair, Clemens Kirisits
- SIAM J. Imaging Sciences
- 2015