Contributions to Topological Data Analysis for Scientific Visualization

Scientific visualization aims at helping users (i) abstract, (ii) interact with and (iii) analyze simulated or acquired geometrical data for interpretation, validation and communication purposes. Among existing techniques, algorithms inspired by Morse theory have demonstrated their utility to robustly and efficiently capture and summarize the structure of the studied phenomena, at multiple scales of importance.In this thesis, I review the main results of my research over the last seven years in this area, with contributions to topological data analysis for each of the topics described above (abstraction, interaction and analysis).First, I give a short tutorial on the topological analysis of scalar fields, introducing key concepts such as the Reeb graph, the Morse-Smale complex and the persistence diagram.Second, I present my contributions to:(i) data abstraction, including fast practical algorithms for the computation of the Reeb graph and the simplification of scalar fields;(ii) interaction, including interactive segmentation techniques based on the Morse-Smale complex and the Reeb graph;(iii) analysis, including specializations of topological data analysis approaches to combustion and chemistry applications.Third, I discuss practical challenges that recently arose with the ongoing development of high performance computing resources. Not only do these challenges yield data-sets of unprecedented size, but also new types of data such as multivariate or uncertain scalar fields. These difficulties are particularly exciting for the research community because of their practical importance, and also because they require complete reboots of the research effort that has been achieved in this area over the last two decades. In particular, I present new research directions, supported by recent preliminary results for the topological analysis of uncertain and bivariate scalar fields