Algorithm 457: finding all cliques of an undirected graph
Communications of the ACM
Hierarchical morse complexes for piecewise linear 2-manifolds
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Mean Shift: A Robust Approach Toward Feature Space Analysis
IEEE Transactions on Pattern Analysis and Machine Intelligence
Molecular shape analysis based upon the morse-smale complex and the connolly function
Proceedings of the nineteenth annual symposium on Computational geometry
Discrete & Computational Geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
A Topological Approach to Simplification of Three-Dimensional Scalar Functions
IEEE Transactions on Visualization and Computer Graphics
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
Persistence-sensitive simplification functions on 2-manifolds
Proceedings of the twenty-second annual symposium on Computational geometry
Stability and Computation of Topological Invariants of Solids in ${\Bbb R}^n$
Discrete & Computational Geometry
Reconstruction using witness complexes
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Stability of Critical Points with Interval Persistence
Discrete & Computational Geometry
Geodesic Delaunay triangulation and witness complex in the plane
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
A topological hierarchy for functions on triangulated surfaces
IEEE Transactions on Visualization and Computer Graphics
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Zigzag persistent homology and real-valued functions
Proceedings of the twenty-fifth annual symposium on Computational geometry
Geodesic delaunay triangulations in bounded planar domains
ACM Transactions on Algorithms (TALG)
Shape google: Geometric words and expressions for invariant shape retrieval
ACM Transactions on Graphics (TOG)
Persistence-based clustering in riemannian manifolds
Proceedings of the twenty-seventh annual symposium on Computational geometry
Reeb graphs: approximation and persistence
Proceedings of the twenty-seventh annual symposium on Computational geometry
Incremental-decremental algorithm for computing AT-models and persistent homology
CAIP'11 Proceedings of the 14th international conference on Computer analysis of images and patterns - Volume Part I
Failure filtrations for fenced sensor networks
International Journal of Robotics Research
SHREC'10 track: feature detection and description
EG 3DOR'10 Proceedings of the 3rd Eurographics conference on 3D Object Retrieval
wFEM heat kernel: Discretization and applications to shape analysis and retrieval
Computer Aided Geometric Design
Scalar field visualization via extraction of symmetric structures
The Visual Computer: International Journal of Computer Graphics
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Given a real-valued function f defined over some metric space X, is it possible to recover some structural information about f from the sole information of its values at a finite set L ⊆ X of sample points, whose pairwise distances in X are given? We provide a positive answer to this question. More precisely, taking advantage of recent advances on the front of stability for persistence diagrams, we introduce a novel algebraic construction, based on a pair of nested families of simplicial complexes built on top of the point cloud L, from which the persistence diagram of f can be faithfully approximated. We derive from this construction a series of algorithms for the analysis of scalar fields from point cloud data. These algorithms are simple and easy to implement, have reasonable complexities, and come with theoretical guarantees. To illustrate the generality of the approach, we present some experimental results obtained in various applications, ranging from clustering to sensor networks (see the electronic version of the paper for color pictures).