Image analysis has experienced a boom of applications in the recent years, in particular in medical sciences which is a major topic at the University of Graz. We have looked at several image analysis problems at the START project. The first problem we looked at was segmentation of images. Mathematical image segmentation is concerned with the task of partitioning a given image into disjoint (homogeneous) regions. Among the many available paradigms, we use a generalized Mumford-Shah functional approach.

While the original approach due to Mumford and Shah admits some variation of the reconstructed image on the disjoint pieces, in the model introduced by Chan and Vese, a limitation to piecewise constant functions is proposed. In the present work we focus on the piecewise constant version of the Mumford-Shah functional. This opens up a new perspective. This corresponds to finding a topological distribution of the subregions. This approach has several benefits which ultimately result in a highly efficient algorithm for image segmentation.

> Multiphase Image Segmentation and Modulation Recovery Based on Shape and Topological Sensitivity

Radio frequency coils are used in MRI for both nuclear excitation and for signal detection. The acquisition of the data by the coil is done in the frequency space. It is possible to program the machine so that it will pick only certain frequencies. The subsampling in the frequency domain results in folded or aliased images, which must be proceeded in order to recover the original image. The goal of the subsampling is to speed-up the acquisition process. The folded nature of the obtained image is such that several images obtained with different coils are necessary for the reconstruction. Each coil produces in addition to the folding its own modulation of the original image, and the information provided by the different modulations can be used to undo the folding effect due to the subsampling. However the modulations created by the coils are not known beforehand, as they may vary depending on the positions of the coils as well as the different conductivities inside the body and additional noise.

In this paper we have considered parameterized sensitivities which depend on three space parameters. The parameterized approach is a simplification of the true sensitivities which are determined by the Biot-Savart law. When the subsampling is periodic, the folded image has a simple representation with respect to the original image and it is possible to define a particular minimization problem which takes advantage of the periodic structure of the subsampling.

> Modulation recovery and image reconstruction in MRI: a structural study by parameterization
> An image space approach to Cartesian based parallel MR imaging with total variation regularization