A Formal $\Gamma$-Convergence Approach for the Detection of Points in 2-D Images

Abstract

We propose a new variational model to locate points in 2-dimensional biological images. To this purpose we introduce a suitable functional whose minimizers are given by the points we want to detect. In order to provide numerical experiments we replace this energy with a sequence of more treatable functionals by means of the notion of Γ-convergence.

Variational approximation for detecting point-like target problems

“…Inspired by the De Giorgi's conjecture (see [15] for the original statement) it appears natural to investigate, in the spirit of [9], the possibility of approximating the functional F by means of a sequence F ε much more convenient from a numerical point view (see [19]):…”

Parameter Selection in a Mumford–Shah Geometrical Model for the Detection of Thin Structures

“…In this paper, we have decided to use a variational model. Such models have been investigated by Aubert and al [7,8,9,10,14] especially in the detection of points in 2D images. One important tool is the capacity theory [1].…”

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

“…Hence, from a variational point of view, we have to nd a functional space whose elements are able to produce Radon measures, whose support contains the singularities we would like to preserve. Then, in line with recent works for detecting point-like target problem (see [5,14,15]), we consider the divergence as an operator able of producing Radon measures concentrated on points and curves. We deal with the space ∆M p loc (Ω) of W 1,p loc -functions whose gradient is an L p -vector eld with distributional divergence given by a Radon measure, where…”