A study of hierarchical watersheds on graphs with applications to image segmentation
Deise Santana Maia, Deise Santana, Jean Cousty, Laurent Najman, Benjamin Perret · HAL (Le Centre pour la Communication Scientifique Directe) · 2019
The wide literature on graph theory invites numerous problems to be modeled in the framework of graphs. In particular, clustering and segmentation algorithms designed this framework can be applied to solve problems in various domains, including image processing, which is the main field of application investigated in this thesis. In this work, we focus on a semi-supervised segmentation tool widely studied in mathematical morphology and used in image analysis applications, namely the watershed transform. We explore the notion of a hierarchical watershed, which is a multiscale extension of the notion of watershed allowing to describe an image or, more generally, a dataset with partitions at several detail levels. The main contributions of this study are the following : - Recognition of hierarchical watersheds : we propose a characterization of hierarchical watersheds which leads to an efficient algorithm to determine if a hierarchy is a hierarchical watershed of a given edge-weighted graph. - Watersheding operator : we introduce the watersheding operator, which, given an edge-weighted graph, maps any hierarchy of partitions into a hierarchical watershed of this edge-weighted graph. We show that this operator is idempotent and its fixed points are the hierarchical watersheds. We also propose an efficient algorithm to compute the result of this operator. - Probability of hierarchical watersheds : we propose and study a notion of probability of hierarchical watersheds, and we design an algorithm to compute the probability of a hierarchical watershed. Furthermore, we present algorithms to compute the hierarchical watersheds of maximal and minimal probabilities of a given weighted graph. - Combination of hierarchies : we investigate a family of operators to combine hierarchies of partitions and study the properties of these operators when applied to hierarchical watersheds. In particular, we prove that, under certain conditions, the family of hierarchical watersheds is closed for the combination operator. - Evaluation of hierarchies : we propose an evaluation framework of hierarchies, which is further used to assess hierarchical watersheds and combinations of hierarchies. In conclusion, this thesis reviews existing and introduces new properties and algorithms related to hierarchical watersheds, showing the theoretical richness of this framework and providing insightful view for its applications in image analysis and computer vision and, more generally, for data processing and machine learning