AF: Small: Collaborative Research: Reeb graph flows: Metrics, Drawings, and Analysis

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Shape data such as images and 3d scans come from a wide variety of sources, including medical imaging, biological analysis of plants and animals, digital preservation of artifacts, computer graphics and animation, and many others. To distill this data to its essence, this project uses tools from computational topology, a field which combines tools from mathematics and computer science to find efficient and practical ways to simplify, store, and analyze these images. One of these tools, the Reeb graph, provides a skeleton of the structure which is useful for visualization. This project creates tools to quantify, visualize, and analyze these skeletons in a time-varying fashion, i.e. when the input image data is actually a movie. Given how often multiple images or scans are collected from an object in motion or over time, the results of this project have broad applications in many fields. These results will be disseminated through publications, presentations, open-source software, and participation in a variety of workshops and activities, all of which continue collaborative and interdisciplinary work in a larger network of shape-analysis and computational-topology researchers. In addition to core theoretical development of new tools, both PIs are dedicated to broadening participation in mathematics and computer science, and as part of the project will focus extensively on teaching and mentoring students and junior researchers, including active leadership and mentoring in societies that focus primarily on underrepresented groups.In numerous application fields, there is an increasing need to analyze topological and geometric information about shapes that are generated by scanning some 3d structure. However, when given a time-varying shape, there are few strategies that utilize prior scans to quickly update and maintain a continuous family of visualizations with desirable properties. In this project, the PIs will collaborate to develop improved visualization toolkits for one of the major ways to analyze this type of data, the Reeb graph, which is heavily used in computational topology, shape analysis, and visualization. These skeletons give compressed and accurate ways to store the shape and compute shape invariants and statistics, but can nonetheless be large and difficult to visualize. In particular, the project is focused on the analysis of a time-varying Reeb graph, known as a Reeb graph flow. Over the course of the project, the PIs will investigate properties of such Reeb flows coming from a variety of natural input domains, and determine which settings allow faster and more accurate visualization and analysis. Results from this project will include a range of theoretical and applied publications, open-source software development, and will be disseminated to a larger network of researchers interested in shape analysis and the use of topology for algorithms and analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.