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}
}
  • V. Kurlin
  • Published 6 July 2015
  • Environmental Science, Computer Science
  • Computer Graphics Forum
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… 
A higher-dimensional homologically persistent skeleton
A fast approximate skeleton with guarantees for any cloud of points in a Euclidean space
TLDR
The Approximate Skeleton outperforms past skeletonization algorithms on the size and accuracy of reconstruction for a large dataset of real micelles and random clouds.
Improving the Projection of Global Structures in Data through Spanning Trees
TLDR
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.
A fast persistence-based segmentation of noisy 2D clouds with provable guarantees
  • V. Kurlin
  • Mathematics, Computer Science
    Pattern Recognit. Lett.
  • 2016
TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data
TLDR
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.
Structure and Stability of the 1-Dimensional Mapper
TLDR
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
TLDR
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
TLDR
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.
...
1
2
3
4
...

References

SHOWING 1-10 OF 20 REFERENCES
Data Skeletonization via Reeb Graphs
TLDR
This paper develops a framework to extract, as well as to simplify, a one-dimensional "skeleton" from unorganized data using the Reeb graph, which is very simple, does not require complex optimizations and can be easily applied to unorganized high-dimensional data such as point clouds or proximity graphs.
A Fast and Robust Algorithm to Count Topologically Persistent Holes in Noisy Clouds
  • V. Kurlin
  • Computer Science
    2014 IEEE Conference on Computer Vision and Pattern Recognition
  • 2014
TLDR
This work designs the algorithm to count holes that are most persistent in the filtration of offsets (neighborhoods) around given points and proves theoretical guarantees when the algorithm finds the correct number of holes of an unknown shape approximated by a cloud.
Topological Analysis of Scalar Fields with Outliers
TLDR
This work proposes a new algorithm that deals with aberrant functional values with a method inspired from the k-nearest neighbors regression and the local median filtering, while the geometric outliers are handled using the distance to a measure.
Graph induced complex on point data
TLDR
It is shown that, using the graph induced complex, one can infer the one-dimensional homology of a manifold from a very lean subsample, reconstruct a surface in three dimension from a sparse subsample without computing Delaunay triangulations, and infer the persistent homology groups of compact sets from a sufficiently dense sample.
Auto-completion of Contours in Sketches, Maps, and Sparse 2D Images Based on Topological Persistence
  • V. Kurlin
  • Computer Science
    2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
  • 2014
TLDR
This work designs a new fast algorithm to automatically complete closed contours in a finite point cloud on the plane and proves theoretical guarantees when, for a given noisy sample of a graph in the plane, the output contours geometrically approximate the original contour in the unknown graph.
Self-organizing maps for the skeletonization of sparse shapes
TLDR
The proposed method involves an iterative evolution of a piecewise-linear approximation of the shape skeleton by using a minimum spanning tree-based self-organizing map (SOM) and the adjacency relationships between regions in the shape are detected and used in the evolution of the skeleton.
A Homologically Persistent Skeleton is a Fast and Robust Descriptor of Interest Points in 2D Images
TLDR
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.
Reconstructing persistent graph structures from noisy images
TLDR
A new fast algorithm for reconstructing the original graph from the given point cloud using methods of persistent topology and machine learning tools is presented.
Piecewise Linear Skeletonization Using Principal Curves
TLDR
The results indicated that the proposed algorithm can find a smooth medial axis in the great majority of a wide variety of character templates and that it substantially improves the pixel-wise skeleton obtained by traditional thinning methods.
Gromov–Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs
TLDR
It is proved that filamentary structures that can be seen as topological metric graphs can be approximated with respect to the Gromov–Hausdorff distance by well-chosen Reeb graphs and an efficient and easy-to-implement algorithm is provided to compute such approximations in almost linear time.
...
1
2
...