A discrete level set approach to image segmentation

Models and algorithms in image processing are usually defined in the continuum and then applied to discrete data, that is the signal samples over a lattice. In particular, the set up in the continuum of the segmentation problem allows a fine formulation basically through either a variational approach or a moving interfaces approach. In any case, the image segmentation is obtained as the steady-state solution of a nonlinear PDE. Nevertheless the application to real data requires discretization schemes where some of the basic image geometric features have a loosemeaning. In this paper, a discrete version of the level set formulation of a modified Mumford and Shah energy functional is investigated, and the optimal image segmentation is directly obtained through a nonlinear finite difference equation. The typical characteristics of a segmentation, such as its component domains area and its boundary length, are all defined in the discrete context thus obtaining a more realistic description of the available data. The existence and uniqueness of the optimal solution is proved in the class of piece wise constant functions, but with no restrictions on the nature of the segmentation boundary multiple points. The proposed algorithm compared to a standard segmentation procedure in the continuum generally provides a more accurate segmentation, with a much lower computational cost.