SUMMARYLetXbe a phase in a specimen. Given two arbitrary pointsxandyofX, let us define the numberdx(x, y) as follows:dx(x, y) is the greatest lower bound of the lengths of the arcs inXending at pointsxandy, if such arcs exist, and + ∞ if not. The functiondXis a distance function, called ‘geodesic distance’. Note that ifxandybelong to two disjoint connected components ofX, dx(x, y) = + ∞. In other words,dxseems 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 distancedxalso 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 of the 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 s