Double-sided probing by map of Asplund's distances using Logarithmic Image Processing in the framework of Mathematical Morphology
This work provides a theoretical connection for researchers in image processing and mathematical morphology, but it appears incremental as it builds on existing frameworks.
The paper links Mathematical Morphology to Asplund's distance maps using Logarithmic Image Processing, showing the map equals the logarithm of the ratio between dilation and erosion by a probe, and demonstrates this with a pattern-matching example.
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.