2021/07/09 by Rehab Alharbi, Alharbi, Rehab, Erin Wolf Chambers +3
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Computational Geometry (cs.CG) #Data Management and Algorithms #Data Visualization and Analytics #FOS: Computer and information sciences #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2107.04654
openalex publication_date 2021/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
Reeb graphs are widely used in a range of fields for the purposes of\nanalyzing and comparing complex spaces via a simpler combinatorial object.\nFurther, they are closely related to extended persistence diagrams, which\nlargely but not completely encode the information of the Reeb graph. In this\npaper, we investigate the effect on the persistence diagram of a particular\ncontinuous operation on Reeb graphs; namely the (truncated) smoothing\noperation. This construction arises in the context of the Reeb graph\ninterleaving distance, but separately from that viewpoint provides a\nsimplification of the Reeb graph which continuously shrinks small loops. We\nthen use this characterization to initiate the study of inverse problems for\nReeb graphs using smoothing by showing which paths in persistence diagram space\n(commonly known as vineyards) can be realized by a path in the space of Reeb\ngraphs via these simple operations. This allows us to solve the inverse problem\non a certain family of piecewise linear vineyards when fixing an initial Reeb\ngraph.\n