Volume 46, Number 2, November-December 2012
|Page(s)||389 - 410|
|Published online||23 November 2011|
A compactness result for a second-order variational discrete model
Dipartimento di Matematica, Università di Roma ‘Tor
Vergata’, via della Ricerca
2 Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123 Povo, Italy
3 Dipartimento di Matematica ‘F. Casorati’, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
Revised: 22 June 2011
We analyze a nonlinear discrete scheme depending on second-order finite differences. This is the two-dimensional analog of a scheme which in one dimension approximates a free-discontinuity energy proposed by Blake and Zisserman as a higher-order correction of the Mumford and Shah functional. In two dimension we give a compactness result showing that the continuous problem approximating this difference scheme is still defined on special functions with bounded hessian, and we give an upper and a lower bound in terms of the Blake and Zisserman energy. We prove a sharp bound by exhibiting the discrete-to-continuous Γ-limit for a special class of functions, showing the appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensional effect.
Mathematics Subject Classification: 49J45 / 49Q20 / 68U10 / 65D19 / 65M06
Key words: Computer vision / finite-difference schemes / gamma-convergence / free-discontinuity problems
© EDP Sciences, SMAI, 2011
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