ISSN:
1573-7683
Keywords:
mathematical morphology
;
erosion curve
;
skeleton
;
quench function
;
granulometry
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let us associate to any binary planar shape X the erosion curve ΨX defined by ΨX: r ∈ IRX→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 ΨX = ΨY. Under some regularity conditions on X, ΨX 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 ΨX: quantifies soft shapes. We then prove, in the generic case, that the five possible cases of behavior of the second derivative ΨX ″ 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 ΨX = ΨY for a given X.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00118591
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