Communications in Mathematical Sciences

Volume 13 (2015)

Number 6

A convex and selective variational model for image segmentation

Pages: 1453 – 1472



Jack Spencer (Centre for Mathematical Imaging Techniques and Department of Mathematical Sciences, University of Liverpool, United Kingdom)

Ke Chen (Mathematical Sciences Department, University of Liverpool, United Kingdom)


Selective image segmentation is the task of extracting one object of interest from an image, based on minimal user input. Recent level set based variational models have shown to be effective and reliable, although they can be sensitive to initialization due to the minimization problems being nonconvex. This sometimes means that successful segmentation relies too heavily on user input or a solution found is only a local minimizer, i.e. not the correct solution. The same principle applies to variational models that extract all objects in an image (global segmentation); however, in recent years, some have been successfully reformulated as convex optimization problems, allowing global minimizers to be found.

There are, however, problems associated with extending the convex formulation to the current selective models, which provides the motivation for the proposal of a new selective model. In this paper we propose a new selective segmentation model, combining ideas from global segmentation, that can be reformulated in a convex way such that a global minimizer can be found independently of initialization. Numerical results are given that demonstrate its reliability in terms of removing the sensitivity to initialization present in previous models, and its robustness to user input.


image processing, variational segmentation, level set function, edge detection, convex functional, Euler–Lagrange equation, AOS

2010 Mathematics Subject Classification

35A15, 62H35, 65C20, 65N22, 68U10, 74G65, 74G75

Published 13 May 2015