An Active Contour Approach for a Mumford-Shah Model in X-Ray Tomography
This paper presents an active contour approach for the simultaneous inversion and segmentation of X-ray tomography data from its Radon Transform. The optimality system is found as the necessary optimality condition for a Mumford-Shah like functional over the space of piecewise smooth densities, which may be discontinuous across the contour. In our approach the functional variable is eliminated by solving a classical variational problem for each fixed geometry. The solution is then inserted in the Mumford-Shah cost functional leading to a geometrical optimization problem for the singularity set. The resulting shape optimization problem is solved using shape sensitivity calculus and propagation of shape variables in the level-set form.
As a special feature of this paper, a new, second order accurate, finite difference method based approach for the solution of the optimality system is introduced and numerical experiments are presented.
KeywordsIntersection Point Optimality System Active Contour Taylor Series Expansion Descent Direction
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- 3.Chan, T.F., Vese, L.A.: A level set algorithm for minimizing the Mumford-Shah functional in image processing. UCLA CAM Report 00-13, University of California, Los Angeles (2000)Google Scholar
- 7.Delfour, M.C., Zolésio, J.-P.: Shapes and geometries. In: Analysis, differential calculus, and optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)Google Scholar
- 8.Hoetzl, E.: Numerical treatment of a mumford-shah model for x-ray tomography. PhD.thesis, Karl Franzens University Graz, Institute for Mathematics (2009)Google Scholar
- 11.Jehan-Besson, S., Barlaud, M., Aubert, G.: DREAM2S: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation (November 2001)Google Scholar
- 16.Paragios, N., Deriche, R.: Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision. Int. J. of Vis. Communication and Image Representation (2001) (to appear)Google Scholar