A Topological Similarity Measure Between Multi-Resolution Reeb Spaces

Searching similarity between a pair of shapes or data is an important problem in data analysis and visualization. The problem of computing similarity measures using scalar topology has been studied extensively and proven useful in the shape and data matching. Even though multi-field or multivariate (consists of multiple scalar fields) topology reveals richer topological features, research on building tools for computing similarity measures using multi-field topology is still in its infancy. In the current article, we propose a novel similarity measure between two piecewise-linear multi-fields based on their multi-resolution Reeb spaces - a newly developed data-structure that captures the topology of a multi-field. Overall, our method consists of two steps: (i) building a multi-resolution Reeb space corresponding to each of the multi-fields and (ii) proposing a similarity measure between two multi-resolution Reeb spaces by computing a list of topologically consistent matching pairs (of nodes) and the similarity between them. We demonstrate the effectiveness of the proposed similarity measure in detecting topological features from real time-varying multi-field data in two application domains - one from computational physics and one from computational chemistry.