Inverse Problems for Granulometries by Erosion

Abstract

Let us associate to any binary planar shape X the erosion curve Ψ _X defined by Ψ _X : r ∈ IR⁺ → A(X⊖rB), where A(X) stands for the surface area of X and X ⊖ rB for the eroded set of X with respect to the ball rB of size r. Note the analogy to shape quantification by granulometry. This paper describes the relationship between sets X and Y verifyingΨ _X =Ψ _Y . Under some regularity conditions on X, Ψ _X is expressed as an integral on its skeleton of the quench function qx (distance to the boundary of X). We first prove that a bending of arcs of the skeleton of X does not affectΨ _X : Ψ _X quantifies soft shapes. We then prove, in the generic case, that the five possible cases of behavior of the second derivative Ψ” _X characterize five different situations on the skeleton Sk(X) and on the quench function qx: simple points of Sk(X) where qx is a local minimum, a local maximum, or neither, multiple points of Sk(X) where qx is a local maximum or not. Finally, we give infinitesimal generators of the reconstruction process for the entire family of shapes Y verifyingΨ _X =Ψ _Y for a given X.