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