Geodesic methods in quantitative image analysis

Abstract Let X be a part of an image to be analysed. Given two arbitrary points x and y of X , let us define the number d x ( x , y ) as follows: d x ( x , y ) is the lower bound of the lengths of the arcs in X ending at points x and y , if such arcs exist, and + α if not. The function d x is an X -intrinsic distance function, called ‘geodesic distance’. Note that if x and y belong to two disjoint connected components of X , d x ( x , y ) = + α . In other words, d x seems to be an appropriate distance function to deal with connectivity problems. In the metric space ( X , d x ), all the classical morphological transformations (dilation, erosion, skeletonization, etc.) can be defined. The geodesic distance d x also provides rigorous definitions of topological transformations, which can be performed by automatic image analysers with the help of parallel iterative algorithms. All these notions are illustrated by several examples (definition of the length of a fibre and of an effective length factor; automatic detection of cells having at least one nucleus or having one single nucleus; definitions of the geodesic center and of the ends of an object without a hole; etc.). The corresponding algorithms are described.