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

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