Publication 22-CNA-004
$\Gamma$-Convergence of an Ambrosio-Tortorelli approximation scheme
for image segmentation
Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu
Lisa Maria Kreusser
Department of Mathematical Sciences
University of Bath
Bath, United Kingdom
lmk54@bath.ac.uk
Carola-Bibiane Schönlieb
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
United Kingdom
C.B.Schoenlieb@damtp.cam.ac.uk
Matthew Thorpe
Department of Mathematics
University of Manchester
matthew.thorpe-2@manchester.ac.uk
Abstract: Given an image
u0, the aim of minimising the Mumford-Shah functional is to find a decomposition of the image domain into sub-domains and a piecewise smooth approximation
u of
u0 such that
u varies smoothly within each sub-domain. Since the Mumford-Shah functional is highly non-smooth, regularizations such as the Ambrosio-Tortorelli approximation can be considered which is one of the most computationally efficient approximations of the Mumford-Shah functional for image segmentation. Our main result is the $\Gamma$-convergence of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional for piecewise smooth approximations. This requires the introduction of an appropriate function space. As a consequence of our $\Gamma$-convergence result, we can infer the convergence of minimizers of the respective functionals.
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