Journal
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
Volume 42, Issue 6, Pages 2286-2304Publisher
SIAM PUBLICATIONS
DOI: 10.1137/S0363012902411945
Keywords
oriented distance function; signed distance function; level set method; narrow-band method; extension velocity; nonlinear evolution equation; smooth sets; submanifold; shape; design; image; vision
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The central result of this paper is a new nonlinear equation which describes the evolution of the oriented distance function b(Omega) of a set Omega with thin boundary under the influence of a velocity field. We relate it to equations and constructions used in the context of level set methods. We further introduce a new moving narrow-band method which not only can be readily implemented to solve our evolution equation, but could also be used for equations of motion by curvatures. In the process we review and sharpen the characterization of smooth sets and manifolds and sets of positive reach (e.g., local semiconvexity in an extended sense of the oriented distance function of the closure of the set). For W(2,p)-Sobolev domains a new characterization and a compactness theorem are given in terms of the Laplacian of the oriented distance function rather than its whole Hessian matrix.
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