Commun. Comput. Phys.,
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Volume 4.


Multigrid Method for the Chan-Vese Model in Variational Segmentation

Noor Badshah 1, Ke Chen 1*

1 Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK.

Received 11 July 2007; Accepted (in revised version) 14 September 2007
Available online 18 March 2008

Abstract

The Chan-Vese method of active contours without edges has been used successfully for segmentation of images. As a variational formulation, it involves the solution of a fully nonlinear partial differential equation which is usually solved by using time marching methods with semi-implicit schemes for a parabolic equation; the recent method of additive operator splitting provides an effective acceleration of such schemes for images of moderate size. However to process images of large size, urgent need exists in developing fast multilevel methods. Here we present a multigrid method to solve the Chan-Vese nonlinear elliptic partial differential equation, and demonstrate the fast convergence. We also analyze the smoothing rates of the associated smoothers. Based on our numerical tests, a surprising observation is that our multigrid method is more likely to converge to the global minimizer of the particular non-convex problem than previously unilevel methods which may get stuck at local minimizers. Numerical examples are given to show the expected gain in CPU time and the added advantage of global solutions.

AMS subject classifications: 62H35, 65N22, 65N55, 74G65, 74G75

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Key words: Active contours, energy minimization, partial differential equations, segmentation, level sets, total variation, multigrid.

*Corresponding author.
Email: noor.badshah@liv.ac.uk (N. Badshah), k.chen@liv.ac.uk (K. Chen)
 

The Global Science Journal