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

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July 2010, 9(4): 905-927. doi: 10.3934/cpaa.2010.9.905

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

Giovanna Citti ^1, , Maria Manfredini ^1, and Alessandro Sarti ^2,

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

Received December 2008 Revised March 2010 Published April 2010

Abstract
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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.

Keywords: model of the primary visual cortex., Mumford and Shah, contact manifold, Heisenberg group.

Mathematics Subject Classification: Primary: 49J45,35R03; Secondary: 49M25, 35Q9.

Citation: Giovanna Citti, Maria Manfredini, Alessandro Sarti. Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 905-927. doi: 10.3934/cpaa.2010.9.905

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