Double-sided probing by map of Asplund's distances using Logarithmic Image Processing in the framework of Mathematical Morphology

Guillaume Noyel (1) , Michel Jourlin (2, 1)

1 IPRI - International Prevention Research Institute
2 LHC - Laboratoire Hubert Curien

We establish the link between Mathematical Morphology and the map of Asplund's distances between a probe and a grey scale function, using the Logarithmic Image Processing scalar multiplication. We demonstrate that the map is the logarithm of the ratio between a dilation and an erosion of the function by a structuring function: the probe. The dilations and erosions are mappings from the lattice of the images into the lattice of the positive functions. Using a flat structuring element, the expression of the map of Asplund's distances can be simplified with a dilation and an erosion of the image; these mappings stays in the lattice of the images. We illustrate our approach by an example of pattern matching with a non-flat structuring function.

Mots clés: Map of Asplund's distances; double-sided probing; Logarithmic Image Processing; pattern recognition; Mathematical Morphology; erosion; dilation; Asplund's metric

Domaines: Vision par ordinateur et reconnaissance de formes [cs.CV]; Topologie générale [math.GN]; Analyse numérique [math.NA]; Traitement du signal et de l'image [eess.SP]

Guillaume Noyel, Michel Jourlin. Double-sided probing by map of Asplund's distances using Logarithmic Image Processing in the framework of Mathematical Morphology. 2017. ⟨hal-01435269v2⟩