Variational approximation for detecting point-like target problems

Abstract

Abstract. The aim of this paper is to provide a rigorous variational formulation for the detection of points in 2-d biological images. To this purpose we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for which we prove the Γ-convergence to the initial one.Mathematics Subject Classification. 49J45, 49Q20.

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

“…We point out that the Γ-convergence result is not proved in this paper but only conjectured. A complete proof of the Γ-convergence result and the equicoerciveness of the sequence Φ ε , in the particular case where the vector field U is a gradient, has been provided by the first and third authors in [5].…”

“…DM p (Ω) is the space of vector fields U : Ω → R 2 whose distributional divergence is a Radon measure (see subsection 2.2 for definitions and examples). The restriction 1 < p < 2 is due to the fact that for p ≥ 2 the distributional divergence of U cannot charge isolated points (see [5]). …”

“…We stress that the Γ-convergence result is only conjectured in this paper, whose purpose is to test a new variational method from an experimental point of view. For a rigorous variational approximation in a particular case, we refer the reader to [5].…”

A new variational method for preserving point-like and curve-like singularities in 2-D images

“…Moreover in the choice of F we must take into account that the singularities to be preserved are not jump singularities. It means that, in the continous setting, u must belongs to the space ∆M p (Ω) of functions whose gradient is an L p -vector field with distributional divergence given by a Radon measure (see [1] for the precise definiton of this space). These considerations leads us to choose the laplacian as differential operator and minimize the following energy:…”

“…The restriction on p is due to the fact that when p ≥ 2 the distributional laplacian ∆u of u cannot be a measure concentrated on points (see [1,3] on this issue). Therefore it would not be anymore the right operator to restore the singularities we are interested in.…”