Quasi-Linear Algorithms for the Topological Watershed

The watershed transformation is an efficient tool for segmenting grayscale images. An original approach to the watershed (Bertrand, Journal of Mathematical Imaging and Vision, Vol. 22, Nos. 2/3, pp. 217–230, 2005.; Couprie and Bertrand, Proc. SPIE Vision Geometry VI, Vol. 3168, pp. 136–146, 1997.) consists in modifying the original image by lowering some points while preserving some topological properties, namely, the connectivity of each lower cross-section. Such a transformation (and its result) is called a W-thinning, a topological watershed being an ultimate W-thinning. In this paper, we study algorithms to compute topological watersheds. We propose and prove a characterization of the points that can be lowered during a W-thinning, which may be checked locally and efficiently implemented thanks to a data structure called component tree. We introduce the notion of M-watershed of an image F, which is a W-thinning of F in which the minima cannot be extended anymore without changing the connectivity of the lower cross-sections. The set of points in an M-watershed of F which do not belong to any regional minimum corresponds to a binary watershed of F. We propose quasi-linear algorithms for computing M-watersheds and topological watersheds. These algorithms are proved to give correct results with respect to the definitions, and their time complexity is analyzed.Michel Couprie received his Ingénieurs degree from the École Supérieure dIngénieurs en Électrotechnique et Électronique (Paris, France) in 1985 and the Ph.D. degree from the Pierre et Marie Curie University (Paris, France) in 1988. Since 1988 he has been working in ESIEE where he is an Associate Professor. He is a member of the Laboratoire Algorithmique et Architecture des Systémes Informatiques, ESIEE, Paris, and of the Institut Gaspard Monge, Universit é de Marne-la-Vallée. His current research interests include image analysis and discrete mathematics.Laurent Najman received his Ph.D. of applied mathematics from Paris-Dauphine university and an Ingénieurs degree from the Ecole des Mines de Paris. After earning his Ingénieurs degree, he worked in the research laboratories of Thomson-CSF for three years, before joining Animation Science in 1995, as director of research and development. In 1998, he joined OcÉ Print Logic Technolgies, as senior scientist. Since 2002, he is associate professor with the A2SI laboratory of ESIEE, Paris. His current research interest is discrete mathematical morphology.Gilles Bertrand received his Ingénieurs degree from the École Centrale des Arts et Manufactures in 1976. Until 1983 he was with the Thomson-CSF company, where he designed image processing systems for aeronautical applications. He received his Ph. from the École Centrale in 1986. He is currently teaching and doing research with the Laboratoire Algorithmique et Architecture des Systémes Informatiques, ESIEE, Paris, and with the Institut Gaspard Monge, Université de Marne-la-Vallée. His research interests are image analysis, pattern recognition, mathematical morphology and digital topology.