Approximation of functional depending on jumps by elliptic functional via t‐convergence

1990/12/01 by Luigi Ambrosio, Vincenzo Maria Tortorelli · 50 citations
Mathematics · Engineering · Computer Science · #Mathematical Approximation and Integration #Advanced Numerical Analysis Techniques #Medical Image Segmentation Techniques

paper · doi:10.1002/cpa.3160430805

Abstract

Abstract We show how it is possible to approximate the Mumford‐Shah (see [29]) image segmentation functional magnified image by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionals 𝒢 h ( u, z ) with z ranging between 0 and 1 and related to the set K . The minimizing z h are near to 1 in a neighborhood of the set K , and far from the neighborhood they are very small. The neighborhood shrinks as h → + ∞. For a similar approach to the problem compare Kulkarni; see [25]. The approximation of 𝒢 h to 𝒢 takes place in a variational sense, the De Giorgi F‐convergence.

Citations

• Optimal approximations by piecewise smooth functions and associated variational problems

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