New concepts in mathematical morphology: the topographical and differential distance functions

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Abstract
If the concept of Euclidean and geodesic distance is of great importance in binary mathematical morphology (MM), the grey-level MM deals mainly with neighborhood configuration analysis. This paper presents a novel approach to grey-level MM based on the concept of the distance function relative to topographical surfaces. By introducing the notions of connection cost and deviation cost, we define the topographical and differential distances and develop a powerful theoretical framework for establishing the equivalence between the two fundamental notions of skeleton by influence zones and watershed: the SKIZ of the set of the minima of a grey-level image f with respect to the differential distance function is exactly the watershed of f. This leads to a duality between binary and grey-level images as well as new fast algorithms for computing the SKIZ and the watershed.

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