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Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition
The method, called Mapper, is based on the idea of partial clustering of the data guided by a set of functions defined on the data, and is not dependent on any particular clustering algorithm, i.e. any clustering algorithms may be used with Mapper.
Gromov–Wasserstein Distances and the Metric Approach to Object Matching
- F. Mémoli
- Computer ScienceFound. Comput. Math.
- 1 August 2011
This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers.
Characterization, Stability and Convergence of Hierarchical Clustering Methods
It is shown that within this framework, one can prove a theorem analogous to one of Kleinberg (2002), in which one obtains an existence and uniqueness theorem instead of a non-existence result.
Variational Problems and Partial Differential Equations on Implicit Surfaces: Bye Bye Triangulated Surfaces?
A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is described and examples in computer graphics and image processing applications are presented, including texture synthesis, flow field visualization, as well as image and vector field intrinsic regularization for datadefined on 3D surfaces.
A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data
A geometric framework for comparing manifolds given by point clouds is presented and the underlying theory is based on Gromov-Hausdorff distances, leading to isometry invariant and completely geometric comparisons.
One Point Isometric Matching with the Heat Kernel
It is shown that under mild genericity conditions, a single correspondence can be used to recover an isometry defined on entire shapes, and thus the space of all isometries can be parameterized by one correspondence between a pair of points.
Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces: 730
An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper-surfaces is introduced, and based on work on geodesics on Riemannian manifolds with boundaries, the error between the two distance functions is bound.
On the use of Gromov-Hausdorff Distances for Shape Comparison
- F. Mémoli
These reformulations render these distances more amenable to practical computations without sacrificing theoretical underpinnings, and establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms.
Gromov‐Hausdorff Stable Signatures for Shapes using Persistence
- F. Chazal, D. Cohen-Steiner, L. Guibas, F. Mémoli, S. Oudot
- MathematicsComput. Graph. Forum
- 1 July 2009
We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We…
Classifying Clustering Schemes
A framework is constructed for studying what happens when one imposes various structural conditions on the clustering schemes, under the general heading of functoriality, and it is shown that, within this framework, one can prove a theorem analogous to one of Kleinberg (Becker et al).