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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 for MALDI-imaging data. We address this issue proposing a new procedure for spatial segmentation of MALDI-imaging data sets. This procedure clusters all spectra into different groups based on their similarity. This partition is represented by a segmentation map, which helps to understand the spatial structure of the sample. The core of our segmentation procedure is the edge-preserving denoising of images corresponding to specific masses that reduces pixel-to-pixel variability and improves the segmentation map significantly. Moreover, before applying denoising, we reduce the data set selecting peaks appearing in at least 1% of spectra. High dimensional discriminant clustering completes the procedure. We analyzed two data sets using the proposed pipeline. First, for a rat brain coronal section the calculated segmentation maps highlight the anatomical and functional structure of the brain. Second, a section of a neuroendocrine tumor invading the small intestine was interpreted where the tumor area was discriminated and functionally similar regions were indicated.

We generalize the notion of Bregman distance using concepts from abstract convexity in order to derive convergence rates for Tikhonov regularization with non-convex regularization terms. In particular, we study the non-convex regularization of linear operator equations on Hilbert spaces, showing that the conditions required for the application of the convergence rates results are strongly related to the standard range conditions from the convex case. Moreover, we consider the setting of sparse regularization, where we show that a rate of order δ 1/p holds, if the regularization term has a slightly faster growth at zero than |t| p .

Motivated by 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 that regularization in general is an unconstrained optimization problem, while in compressed sensing a constrained one is used. Since the two methods have been developed in two different communities, the theoretical approaches to them appear to be rather different: In compressed sensing, the restricted isometry property seems to be central for proving linear convergence rates, whereas in regularization theory range or source conditions are imposed. The paper gives a common meaning to the seemingly different conditions and puts them into perspective with the conditions from the respective other community. A particularly important observation is that the range condition together with an injectivity condition is weaker than the restricted isometry property. Under the weaker conditions, linear convergence rates can be proven for compressed sensing and for Tikhonov regularization. Thus existing results from the literature can be improved based on a unified analysis. In particular, the range condition is shown to be the weakest possible condition that permits the derivation of linear convergence rates for Tikhonov regularization with a priori parameter choice.

We consider the stable approximation of sparse solutions to nonlinear 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 the noise level δ. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to O(δ).

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