On the use of the geodesic metric in image analysis

Let X be a phase in a specimen. Given two arbitrary points x and y of X, let us define the number dx(x, y) as follows: dx(x, y) is the greatest lower bound of the lengths of the arcs in X ending at points x and y, if such arcs exist, and + CIJ if not. The function dX is a distance function, called 'geodesic distance'. Note that if x and y belong to two disjoint connected components of X, dx(x, y) = + CIJ. In other words, dx seems to be an appropriate distance function to deal with connectivity problems. In the metric space (X, dx), all the classical morphological transformations (dilation, erosion, skeletonizations, etc.) can be defined. The geodesic distance dx also provides rigourous definitions of topological transformations, which can be performed by automatic image analysers with the help of iterative algorithms. All these notions are illustrated with several examples (definition of the length of a fibre; automatic detection of cells having at least one nucleus, or having exactly a single nucleus; definitions ofthe geodesic centre and of the ends of a particle without holes, etc.). As an application, a general problem of segmentation is treated (automatic separation of balls in a polished section).