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Communications on Pure and Applied Analysis (CPAA)
 

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

Pages: 905 - 927, Volume 9, Issue 4, July 2010      doi:10.3934/cpaa.2010.9.905

 
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Giovanna Citti - Dipartimento di Matematica, Piazza Porta S. Donato 5, 40126 Bologna, Italy (email)
Maria Manfredini - Dipartimento di Matematica, Piazza Porta S. Donato 5, 40126 Bologna, Italy (email)
Alessandro Sarti - CREA, École Polytechnique, 32, Boulevard Victor, 75015 Paris, France (email)

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.

Keywords:  Mumford and Shah, contact manifold, Heisenberg group, model of the primary visual cortex.
Mathematics Subject Classification:  Primary: 49J45,35R03; Secondary: 49M25, 35Q92.

Received: December 2008;      Revised: March 2010;      Available Online: April 2010.