The grand tour: a tool for viewing multidimensional data
SIAM Journal on Scientific and Statistical Computing
Algorithms for clustering data
Algorithms for clustering data
Topology matching for fully automatic similarity estimation of 3D shapes
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
ACM Transactions on Graphics (TOG)
Comparing Images Using the Hausdorff Distance
IEEE Transactions on Pattern Analysis and Machine Intelligence
Shape Matching and Object Recognition Using Shape Contexts
IEEE Transactions on Pattern Analysis and Machine Intelligence
3D Shape Histograms for Similarity Search and Classification in Spatial Databases
SSD '99 Proceedings of the 6th International Symposium on Advances in Spatial Databases
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Robust Feature Detection and Local Classification for Surfaces Based on Moment Analysis
IEEE Transactions on Visualization and Computer Graphics
Parallel Coordinates: Visual Multidimensional Geometry and Its Applications
Parallel Coordinates: Visual Multidimensional Geometry and Its Applications
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Laplace-spectra as fingerprints for shape matching
Proceedings of the 2005 ACM symposium on Solid and physical modeling
Using the Inner-Distance for Classification of Articulated Shapes
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 2 - Volume 02
A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data
Foundations of Computational Mathematics
Feature-based similarity search in 3D object databases
ACM Computing Surveys (CSUR)
Integral Invariants for Shape Matching
IEEE Transactions on Pattern Analysis and Machine Intelligence
Efficient Computation of Isometry-Invariant Distances Between Surfaces
SIAM Journal on Scientific Computing
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
Laplace-Beltrami eigenfunctions for deformation invariant shape representation
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
Integral invariants for robust geometry processing
Computer Aided Geometric Design
Topology-Invariant Similarity of Nonrigid Shapes
International Journal of Computer Vision
Partial Similarity of Objects, or How to Compare a Centaur to a Horse
International Journal of Computer Vision
Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids
Computer-Aided Design
Non-rigid registration under isometric deformations
SGP '08 Proceedings of the Symposium on Geometry Processing
A concise and provably informative multi-scale signature based on heat diffusion
SGP '09 Proceedings of the Symposium on Geometry Processing
Gromov-Hausdorff stable signatures for shapes using persistence
SGP '09 Proceedings of the Symposium on Geometry Processing
Characterization, Stability and Convergence of Hierarchical Clustering Methods
The Journal of Machine Learning Research
Gromov–Wasserstein Distances and the Metric Approach to Object Matching
Foundations of Computational Mathematics
Calculus of Nonrigid Surfaces for Geometry and Texture Manipulation
IEEE Transactions on Visualization and Computer Graphics
On bending invariant signatures for surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Distance sets for shape filters and shape recognition
IEEE Transactions on Image Processing
Isometry-invariant matching of point set surfaces
EG 3DOR'08 Proceedings of the 1st Eurographics conference on 3D Object Retrieval
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Several methods in data and shape analysis can be regarded as transformations between metric spaces. Examples are hierarchical clustering methods, the higher order constructions of computational persistent topology, and several computational techniques that operate within the context of data/shape matching under invariances. Metric geometry, and in particular different variants of the Gromov-Hausdorff distance provide a point of view which is applicable in different scenarios. The underlying idea is to regard datasets as metric spaces, or metric measure spaces (a.k.a. mm-spaces, which are metric spaces enriched with probability measures), and then, crucially, at the same time regard the collection of all datasets as a metric space in itself. Variations of this point of view give rise to different taxonomies that include several methods for extracting information from datasets. Imposing metric structures on the collection of all datasets could be regarded as a "soft" construction. The classification of algorithms, or the axiomatic characterization of them, could be achieved by imposing the more "rigid" category structures on the collection of all finite metric spaces and demanding functoriality of the algorithms. In this case, one would hope to single out all the algorithms that satisfy certain natural conditions, which would clarify the landscape of available methods. We describe how using this formalism leads to an axiomatic description of many clustering algorithms, both flat and hierarchical.