Departmental Papers (MEAM)

Document Type

Journal Article

Date of this Version



Suggested Citation:
G. Doğam, P. Morin, and R.H. Nochetto. (2008). "A Variational Shape Optimization Approach for Image Segmentation with a Mumford-Shah Functional." SIAM Journal on Computing Science. Vol. 30, No. 6, pp. 3028 - 3049.

© 2008 Society for Industrial and Applied Mathematics


We introduce a novel computational method for a Mumford–Shah functional, which decomposes a given image into smooth regions separated by closed curves. Casting this as a shape optimization problem, we develop a gradient descent approach at the continuous level that yields nonlinear PDE flows. We propose time discretizations that linearize the problem and space discretization by continuous piecewise linear finite elements. The method incorporates topological changes, such as splitting and merging for detection of multiple objects, space–time adaptivity, and a coarse-to-fine approach to process large images efficiently. We present several simulations that illustrate the performance of the method and investigate the model sensitivity to various parameters.


image segmentation, Mumford-Shah, shape optimization, finite element method



Date Posted: 19 January 2011

This document has been peer reviewed.