We show Euler equations fulfilled by strong minimizers of Blake and Zisserman functional. We prove an Almansi-type decomposition and provide explicit coefficients of asymptotic expansion for bi-harmonic functions in a disk with a cut from center to boundary. We deduce the stress intensity factor and modes coefficients of the leading term in the expansion around crack-tip for any locally minimizing triplet of the main part of Blake and Zisserman functional in the strong formulation. We exhibit explicitly a non-trivial candidate for minimality which has a crack-tip and fulfills all integral and geometric conditions of extremality.

In this article we prove the existence of strong minimizers for Blake & Zisserman’s functional under Dirichlet boundary condition. The result is obtained by showing partial regularity of weak solutions up to the boundary through blow-up techniques and decay properties for bi-harmonic functions in the half-disk. This research is also motivated by the possibility of studying variational formulation of the inpainting problem for 2-dimensional images which are locally damaged and its approximation via the variational convergence introduced and studied by E. De Giorgi.

The aim of this work is to provide a concise survey of results about the Blake-Zisserman functional for image segmentation and inpainting. Moreover a refinement of the Almansi decomposition is shown for biharmonic functions in 2-dimensional open disks with crack-tip at the origin.

We present in a unified context several Poincaré type inequalities involving median: the list includes also inequalities which hold true for functions lacking summability properties. Such inequalities take the form of L^p estimates of suitable truncations of u in terms of L^q norm of the absolutely continuous part of the derivatives. These inequalities can be applied to several free discontinuity problems, mainly in image segmentation and inpainting and in continuum mechanics.

In questa presentazione esponiamo alcuni problemi con discontinuità libere relativi alla segmentazione d’immagini, introducendo in particolare lo studio, dal punto di vista analitico e numerico, dei funzionali di Mumford & Shah e di Blake & Zisserman.

In a previous paper, focused on the analysis of Blake & Zisserman functional in image segmentation, we showed an Almansi-type decomposition and explicit coefficients of asymptotic expansion for bi-harmonic functions in a disk with a cut from center to boundary. The real form expansions and their subsequent applications are correct, but the auxiliary analysis of complex form expansions is imprecise. Here we wish to make precise this point.

We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space BV of functions of bounded variation, whose derivatives are not functions but measures and the space SBV, say the space of bounded variation functions whose derivative has no Cantor part. We prove that SBV is included in W^{s,1} for every s ∈ (0, 1) while the result remains open for BV. We study examples and address open questions.

We introduce and study a formulation of inpainting problem for 2-dimensional images which are locally damaged. This formulation is based on the regularization of the solution of a second order variational problem with Dirichlet boundary condition. A variational approximation algorithm is proposed.

We introduce and study a formulation of inpainting problem for 2-dimensional images which are locally damaged. This formulation is based on the regularization of the solution of a second order variational problem with Dirichlet boundary condition. A variational approximation algorithm is proposed.

We study the variational approximation of an inpainting model for 2-dimensional images which are locally damaged. The scheme provides Gamma€-convergence of elliptic functionals to a Dirichlet problem with free discontinuity and free gradient discontinuity.

We prove density estimates and elimination properties for minimizing triplets of functionals which are related to contour detection in image segmentation and depend on free discontinuities, free gradient discontinuities and second order derivatives. All the estimates concern optimal segmentation under Dirichlet boundary conditions and are uniform in the image domain up to the boundary.