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^X→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 verifying Ψ_X = Ψ_Y. Under some regularity conditions on X, Ψ_X is expressed as an integral on its skeleton of the quench function q _X(distance to the boundary of X). We first prove that a bending of arcs of the skeleton of X does not affect 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 q _X: simple points of Sk(X) where q _Xis a local minimum, a local maximum, or neither, multiple points of Sk(X) where q _Xis a local maximum or not. Finally, we give infinitesimal generators of the reconstruction process for the entire family of shapes Y verifying Ψ_X = Ψ_Y for a given X.