Digital skeletons in Euclidean and geodesic spaces

Abstract This paper is an attempt to bridge the gap between two classes of skeleton algorithms widely used in mathematical morphology: the skeleton by maximal balls and the skeleton by thinnings. First, we show that the first type of skeleton, usually resulting from morphological openings according to Lantuejoul's formula, can also be obtained by non-homotopic thinnings. Moreover, these thinnings are not sequential but based on the intersection of elementary thinnings. The second step is a restrictive selection among the previous structuring elements which preserve homotopy. It leads to the definition of a connected skeleton containing the skeleton by maximal balls. This algorithm is given in the 2D Euclidean space for both the hexagonal and square grids. This skeleton combines the advantages of the two classes of skeletons and avoids the main drawbacks involved by rotating thinnings for building connected skeletons. The extension of this definition to geodesic spaces is discussed in the third part. The geodesic thinnings are introduced to define a connected skeleton containing the skeleton by maximal balls.