From surfaces to objects: computer vision and three dimensional scene analysis
From surfaces to objects: computer vision and three dimensional scene analysis
Describing and recognizing 3-D objects using surface properties
Describing and recognizing 3-D objects using surface properties
Faster scaling algorithms for network problems
SIAM Journal on Computing
FORMS: a flexible object recognition and modeling system
International Journal of Computer Vision
Introduction to Algorithms
A barcode shape descriptor for curve point cloud data
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Local and Global Comparison of Continuous Functions
VIS '04 Proceedings of the conference on Visualization '04
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
Size functions for 3D shape retrieval
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
Geodesic Delaunay triangulation and witness complex in the plane
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Size functions for comparing 3D models
Pattern Recognition
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
A barcode shape descriptor for curve point cloud data
Computers and Graphics
A distributed topological camera network representation for tracking applications
IEEE Transactions on Image Processing - Special section on distributed camera networks: sensing, processing, communication, and implementation
Persistence-based clustering in riemannian manifolds
Proceedings of the twenty-seventh annual symposium on Computational geometry
Computing the betti numbers of arrangements in practice
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
ConTopo: non-rigid 3D object retrieval using topological information guided by conformal factors
EG 3DOR'11 Proceedings of the 4th Eurographics conference on 3D Object Retrieval
A barcode shape descriptor for curve point cloud data
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Persistence-Based Clustering in Riemannian Manifolds
Journal of the ACM (JACM)
Suspension models for testing shape similarity methods
Computer Vision and Image Understanding
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In this paper, we initiate a study of shape description and classification via the application of persistent homology to two 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 parametrizing 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 parametrized families of mathematical shapes.