Multigrid Algorithm Based on Hybrid Smoothers for Variational and Selective Segmentation Models

TL;DR

This paper introduces a novel multigrid algorithm with a new smoother for efficiently solving PDEs in selective image segmentation, especially handling non-smooth coefficients and edges.

Contribution

It develops a multigrid method with a non-standard smoother tailored for non-smooth coefficients in variational segmentation models, ensuring convergence and improved performance.

Findings

Outperforms existing multigrid methods with better smoothing rates

Ensures convergence through a small and global smoothing rate

Numerical tests demonstrate superior efficiency and accuracy

Abstract

Automatic segmentation of an image to identify all meaningful parts is one of the most challenging as well as useful tasks in a number of application areas. This is widely studied. Selective segmentation, less studied, aims to use limited user specified information to extract one or more interesting objects (instead of all objects). Constructing a fast solver remains a challenge for both classes of model. However our primary concern is on selective segmentation. In this work, we develop an effective multigrid algorithm, based on a new non-standard smoother to deal with non-smooth coefficients, to solve the underlying partial differential equations (PDEs) of a class of variational segmentation models in the level set formulation. For such models, non-smoothness (or jumps) is typical as segmentation is only possible if edges (jumps) are present. In comparison with previous multigrid methods which were shown to produce an acceptable {\it mean} smoothing rate for related models, the new algorithm can ensure a small and {\it global} smoothing rate that is a sufficient condition for convergence. Our rate analysis is by Local Fourier Analysis and, with it, we design the corresponding iterative solver, improving on an ineffective line smoother. Numerical tests show that the new algorithm outperforms multigrid methods based on competing smoothers.