| Issue |
ESAIM: M2AN
Volume 46, Number 2, November-December 2012
|
|
|---|---|---|
| Page(s) | 389 - 410 | |
| DOI | https://doi.org/10.1051/m2an/2011043 | |
| Published online | 23 November 2011 | |
A compactness result for a second-order variational discrete model
1
Dipartimento di Matematica, Università di Roma ‘Tor
Vergata’, via della Ricerca
Scientifica, 00133
Rome,
Italy
braides@mat.uniroma2.it
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
Received:
17
January
2011
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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