A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data

  • Authors:

  • Ronny Ramlau;Wolfgang Ring

  • Affiliations:

  • Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria;Institut für Mathematik, Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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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.