2005/12/01 by Gunnar Carlsson, Afra Zomorodian, Anne M. Collins +1 · 294 citations
Computer Science · Mathematics · Medicine · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Advanced Neuroimaging Techniques and Applications #Persistent homology #Tangent #Barcode #Mathematics #Curvature #Invariant (physics) #Tangent vector #Parameterized complexity #Topology (electrical circuits) #Tangent space #Merge (version control) #Homology (biology) #Metric (unit) #Geometry #Pure mathematics #Computer science #Algorithm #Combinatorics
paper · doi:10.1142/s0218654305000761
published in International Journal of Shape Modeling 11(02), 149-187 (World Scientific)
openalex publication_date 2005/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
In this paper, we initiate a study of shape description and classification via the application of persistent homology to tangential constructions on geometric objects. Our techniques combine the differentiating power of geometry with the classifying power of topology. The homology of our first construction, the tangent complex, can distinguish between topologically identical shapes with different “sharp” features, such as corners. To capture “soft” curvature-dependent features, we define a second complex, the filtered tangent complex, obtained by parameterizing a family of increasing subcomplexes of the tangent complex. Applying persistent homology, we obtain a shape descriptor, called a barcode, that is a finite union of intervals. We define a metric over the space of such intervals, arriving at a continuous invariant that reflects the geometric properties of shapes. We illustrate the power of our methods through a number of detailed studies of parameterized families of mathematical shapes.