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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A variant of the level set method and applications to image segmentation


Authors: Johan Lie, Marius Lysaker and Xue-Cheng Tai
Journal: Math. Comp. 75 (2006), 1155-1174
MSC (2000): Primary 35G25, 65K10
Published electronically: February 22, 2006
MathSciNet review: 2219023
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Abstract: In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of $ n$ level set functions are utilized to identify up to $ 2^n$ phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If $ 2^n$ phases should be identified, the level set function must approach $ 2^n$ predetermined constants. We just need one level set function to represent $ 2^n$ unique phases, and this gains in storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the Delta and Heaviside functions. Numerical examples are given, and we also compare our method with related approaches.


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Additional Information

Johan Lie
Affiliation: Department of Mathematics, University of Bergen, Bergen, Norway
Email: johanl@mi.uib.no

Marius Lysaker
Affiliation: Department of Mathematics, University of Bergen, Bergen, Norway
Address at time of publication: Simula Research Lab, Norway
Email: mariul@simula.no

Xue-Cheng Tai
Affiliation: Department of Mathematics, University of Bergen, Bergen, Norway
Email: tai@mi.uib.no

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01835-7
PII: S 0025-5718(06)01835-7
Keywords: Level set, energy minimization, partial differential equations, segmentation.
Received by editor(s): March 12, 2004
Received by editor(s) in revised form: December 9, 2004
Published electronically: February 22, 2006
Additional Notes: This work was supported by the Norwegian Research Council
Article copyright: © Copyright 2006 American Mathematical Society