A Splitting Algorithm for Image Segmentation on Manifolds Represented by the Grid Based Particle Method

  • Authors:

  • Jun Liu;Shingyu Leung

  • Affiliations:

  • School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, People's Republic of China 100875;Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2013

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Abstract

We propose a numerical approach to solve variational problems on manifolds represented by the grid based particle method (GBPM) recently developed in Leung et al. (J. Comput. Phys. 230(7):2540---2561, 2011), Leung and Zhao (J. Comput. Phys. 228:7706---7728, 2009a, J. Comput. Phys. 228:2993---3024, 2009b, Commun. Comput. Phys. 8:758---796, 2010). In particular, we propose a splitting algorithm for image segmentation on manifolds represented by unconnected sampling particles. To develop a fast minimization algorithm, we propose a new splitting method by generalizing the augmented Lagrangian method. To efficiently implement the resulting method, we incorporate with the local polynomial approximations of the manifold in the GBPM. The resulting method is flexible for segmentation on various manifolds including closed or open or even surfaces which are not orientable.