Finite difference approximation  of the Mumford and Shah functional  in a contact manifold of theHeisenberg space

Giovanna Citti; Maria Manfredini; Alessandro Sarti

doi:10.3934/cpaa.2010.9.905

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2010, Volume 9, Issue 4: 905-927. Doi: 10.3934/cpaa.2010.9.905

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Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space

1.
Dipartimento di Matematica, Piazza Porta S. Donato 5, 40126 Bologna
2.
CREA, École Polytechnique, 32, Boulevard Victor, 75015 Paris

Received: November 30, 2008

Revised: February 28, 2010

Published: June 2010

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Abstract

The functionality of the visual cortex has been described in [63] and in [50] as a contact manifold of dimension three and in [62] the Mumford and Shah functional has been proposed to segment lifting of an image in the three dimensional cortical space. Hence, we study here this functional and we provide a constructive approach to the problem, extending to the sub- Riemannian setting an approximation technique proposed by De Giorgi in the Euclidean case.

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Mathematics Subject Classification: Primary: 49J45,35R03; Secondary: 49M25, 35Q92.

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Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space

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