Adaptive wavelet collocation methods for image segmentation using TV–Allen–Cahn type models

Abstract

An adaptive wavelet-based method is proposed for solving TV(total variation)–Allen–Cahn type models for multi-phase image segmentation. The adaptive algorithm integrates (i) grid adaptation based on a threshold of the sparse wavelet representation of the locally-structured solution; and (ii) effective finite difference on irregular stencils. The compactly supported interpolating-type wavelets enjoy very fast wavelet transforms, and act as a piecewise constant function filter. These lead to fairly sparse computational grids, and relax the stiffness of the nonlinear PDEs. Equipped with this algorithm, the proposed sharp interface model becomes very effective for multi-phase image segmentation. This method is also applied to image restoration and similar advantages are observed.