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In this paper we exhibit a family of stationary solutions of the Mumford–Shah functional in R 3 , arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet [4] . We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set. A self-contained proof of the stationarity of the cracktip function for the Mumford–Shah problem in 2D is presented. [ABSTRACT FROM AUTHOR]

Abstract: A level-set based approach for the determination of a piecewise constant density function from data of its Radon transform is presented. Simultaneously, a segmentation of the reconstructed density is obtained. The segmenting contour and the corresponding density are found as minimizers of a Mumford–Shah like functional over the set of admissible contours and – for a fixed contour – over the space of piecewise constant densities which may be discontinuous across the contour. Shape sensitivity analysis is used to find a descent direction for the cost functional which leads to an update formula for the contour in the level-set framework. The descent direction can be chosen with respect to different metrics. The use of an L 2-type and an H 1-type metric is proposed and the corresponding steepest descent flow equations are derived. A heuristic approach for the insertion of additional components of the density is presented. The method is tested for several data sets including synthetic as well as real-world data. It is shown that the method works especially well for large data noise (∼10% noise). The choice of the H 1-metric for the determination of the descent direction is found to have positive effect on the number of level-set steps necessary for finding the optimal contours and densities. [Copyright &y& Elsevier]