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