A Convergent Iterative Support Shrinking Algorithm for Non-Lipschitz Multi-phase Image Labeling Model - Yang, Yijie; Li, Yutong; Wu, Chunlin; Duan, Yuping (Journal of scientific computing, 2023)
The non-Lipschitz piecewise constant Mumford–Shah model has been shown effective for image labeling and segmentation problems [ 33 ], where the non-Lipschitz isotropic ℓ p ( 0 < p < 1 ) regularization term can possess strong abilities to maintain sharp edges. However, the Alternating Direction Method of Multiplier (ADMM)-based algorithm used in [ 33 ] lacks the convergence guarantee. In this work, we propose an iterative support shrinking algorithm with proximal linearization for multi-phase image labeling problems, which is theoretically proven to be globally convergent. A key step is that we prove a lower bound theory for the nonzero entries of the gradient of the iterative sequence when both box constraint and simplex constraint are involved in the target energy minimization problem. To the best of our knowledge, this is the first theoretical attempt at the non-Lipschitz piecewise constant Mumford–Shah model. Numerical experiments are conducted on both two-phase and multi-phase labeling problems to indicate the efficiency and effectiveness of the proposed algorithm.