Analysis of a New Variational Model to Restore Point-Like and Curve-Like Singularities in Imaging

Abstract

The paper is concerned with the analysis of a new variational model to restore point-like and curve-like singularities in biological images. To this aim we investigate the variational properties of a suitable energy which governs these pathologies. Finally in order to realize numerical experiments we minimize, in the discrete setting, a regularized version of this functional by fast descent gradient scheme.

A Relaxation Result for an Inhomogeneous Functional Preserving Point-Like and Curve-Like Singularities in Image Processing

“…Finally u 0 is a given data and here the restriction p ≥ 2N N +2 is assumed to give sense to the discrepancy term Ω |u − u 0 | 2 dx. In [2], under suitable growth assumptions on the integrands f and g, an existence result was proven. It was also shown that functional (1.1) coincides with the lower semicontinuous envelope, with respect to W 1,p loc (Ω)-weak convergence of the following functional F (u, Ω) := Ω f (∆u)dx + Ω g(∇u)dx…”

“…In a recent paper (see [2]) the authors have introduced a new functional of calculus of variations to preserve point-like and curve-like singularities in biological images corrupted by noise. The energy, they deal with, was the following dµ a d|µ a | , dµ 0 d|µ 0 | are the Radon-Nikodym derivatives of the measures µ a and µ 0 with respect to their total variation.…”

“…The restriction p < N N −1 is needed to allow singularities on curves and points (see [3,14] on this issue). The integrands f , g are convex functions and f ∞ is the recession function (see [2] for a precise de nition of all these quantities and references on p-capacity). Finally u 0 is a given data and here the restriction p ≥ 2N N +2 is assumed to give sense to the discrepancy term Ω |u − u 0 | 2 dx.…”

“…However in this work we have limited ourselves to a pure theoretical analysis. We refer to [2] for numerical applications in the homogeneous case.…”

“…This result, under a suitable uniform lower semicontinuity condition with respect to the spatial variable (see Theorem 2.3), permits to write a convex integrand f as a supremum of functions which are split as a product of a function depending only on the spatial variable times a function only depending on the second variable. Then in order to prove lim sup inequality we can adapt the technique used in [2] for the isotropic functional (1.1). Moreover to attain the upper bound we need, as in the case of integral functional de ned on BV -space, to require a linear growth from above on the integrand f .…”