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

Guillaume Noyel (1, 2, 3) , Michel Jourlin (4, 1)

1 IPRI - International Prevention Research Institute
2 SIGPH@iPRI - University of Strathclyde Institute of Global Public Health
3 University of Strathclyde [Glasgow]
4 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: Logarithmic Image Processing; Asplund's metric; double-sided probing; Erosion; Mathematical Morphology; Pattern recognition; Map of Asplund's distances; Dilation

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. 13th International Symposium on Mathematical Morphology (ISMM 2017), May 2017, Fontainebleau, France. pp.408-420, ⟨10.1007/978-3-319-57240-6_33⟩. ⟨hal-01435269v5⟩