# Computing Robustness and Persistence for Images

@article{Bendich2010ComputingRA, title={Computing Robustness and Persistence for Images}, author={Paul Bendich and Herbert Edelsbrunner and Michael Kerber}, journal={IEEE Transactions on Visualization and Computer Graphics}, year={2010}, volume={16}, pages={1251-1260} }

We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to a continuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbation needed to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can be visualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We…

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## References

SHOWING 1-10 OF 31 REFERENCES

Topology Based Selection and Curation of Level Sets

- Computer ScienceTopology-Based Methods in Visualization II
- 2009

This work detects topological features of an isosurface, guided by contour tree data structures, and enhances the description of these features by associating geometric attributes with them and provides a handle to them for curation of the topological anomalies.

Robust on-line computation of Reeb graphs: simplicity and speed

- Computer ScienceACM Trans. Graph.
- 2007

An on-line algorithm is introduced that reads a stream of elements and continuously maintains the Reeb graph of all elements already reed and is robust in handling non-manifold meshes and general in its applicability to input models of any dimension.

Homology and Robustness of Level and Interlevel Sets

- MathematicsArXiv
- 2011

The robustness of the homology classes under perturbations of f is quantified using well groups, and it is shown how to read the ranks of these groups from the same extended persistence diagram.

Persistence Diagrams of Cortical Surface Data

- Computer ScienceIPMI
- 2009

This work presents a novel framework for characterizing signals in images using techniques from computational algebraic topology, which uses all the local critical values in characterizing the signal and offers a completely new data reduction and analysis framework for quantifying the signal.

On the Local Behavior of Spaces of Natural Images

- MathematicsInternational Journal of Computer Vision
- 2007

A theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle and a polynomial representation is used to give coordinatization to various subspaces ofℳ.

Contour trees and small seed sets for isosurface traversal

- Computer ScienceSCG '97
- 1997

This paper gives the first methods to obtain seed sets that are provably small in size based on a variant of the contour tree (or topographic change tree), and develops a simple approximation algorithm giving a seed set of size at most twice the size of the minimum once the contours tree is known.

A topological approach to simplification of three-dimensional scalar functions

- MathematicsIEEE Transactions on Visualization and Computer Graphics
- 2006

This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The Morse-Smale complex, which provides a succinct representation of a…

Provably good mesh generation

- Computer ScienceProceedings [1990] 31st Annual Symposium on Foundations of Computer Science
- 1990

It is shown how to triangulate a planar point set or a polygonally bounded domain with triangles of bounded aspect ratio, and how to produce a linear-size Delaunay triangulation of a multidimensional point set by adding a linear number of extra points.

Quantifying Transversality by Measuring the Robustness of Intersections

- MathematicsFound. Comput. Math.
- 2011

Persistent homology assigns to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then it is proved that the robustness is stable.