# A one‐dimensional homologically persistent skeleton of an unstructured point cloud in any metric space

@article{Kurlin2015AOH, title={A one‐dimensional homologically persistent skeleton of an unstructured point cloud in any metric space}, author={V. Kurlin}, journal={Computer Graphics Forum}, year={2015}, volume={34} }

Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We define a homologically persistent skeleton, which depends only on a cloud of points and contains optimal subgraphs representing 1‐dimensional cycles in the cloud across all scales. The full skeleton is…

## 35 Citations

A higher-dimensional homologically persistent skeleton

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STAD (Spanning Trees as Approximation of Data), a dimensionality reduction method to approximate the high-dimensional structure into a graph with or without formulating prior hypotheses, is presented.

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TopoMap is introduced, a novel projection technique which provides topological guarantees during the mapping process, and performs the mapping from a high-dimensional space to a visual space, while preserving the 0-dimensional persistence diagram of the Rips filtration of the high- dimensional data.

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A theoretical framework relating the structure of the Mappers to that of the Reeb graph is proposed, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability.

A Homologically Persistent Skeleton is a Fast and Robust Descriptor of Interest Points in 2D Images

- Computer ScienceCAIP
- 2015

The skeletonization problem for such a noisy sparse cloud is to summarize the topology of a given 2D cloud across all scales in the form of a graph, which can be used for combining local features into a more powerful object-wide descriptor.

Regularization of Mixture Models for Robust Principal Graph Learning

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- 2021

A regularized version of Mixture Models is proposed to learn a principal graph from a distribution of D-dimensional data points, assuming that the underlying structure can be modeled as a graph acting like a topological prior for the Gaussian clusters turning the problem into a maximum a posteriori estimation.

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