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… 
The Impact of Changes in Resolution on the Persistent Homology of Images
TLDR
The impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales, is studied.
Efficient Computation of Persistent Homology for Cubical Data
TLDR
This paper presents an efficient framework for computation of persistent homology of cubical data in arbitrary dimensions, and presents a data-structure designed to compactly store and quickly manipulate cubical complexes.
Memory-Efficient Computation of Persistent Homology for 3D Images Using Discrete Morse Theory
TLDR
A novel algorithm is proposed that efficiently extracts the Morse-Smale complex based on algorithms from discrete Morse theory and is thereby optimal with a computational complexity of O(n2).
Persistent Homology in Image Processing
TLDR
The main thesis is that persistent homology is a useful method to quantify and summarize topological information, building a bridge that connects algebraic topology with applications.
Stable Morse Decompositions for Piecewise Constant Vector Fields on Surfaces
  • A. Szymczak
  • Mathematics, Computer Science
    Comput. Graph. Forum
  • 2011
TLDR
A technique to compute topological features of user‐prescribed stability with respect to perturbation of the input vector field using a super‐transition graph based on a common graph representation of all PC vector fields.
Persistent Homology – State of the art and challenges
A recurring task in mathematics, statistics, and computer science is understanding the connectivity information, or equivalently, the topological properties of a given object. For concreteness, we
Persistent Homology Computation with a Twist
TLDR
An output-sensitive complexity analysis is given for the new algorithm which yields to sub-cubic asymptotic bounds under certain assumptions and completely avoids reduction on roughly half of the columns.
Objective-oriented Persistent Homology
TLDR
The present work offers the first example to design objective-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.
Approximation algorithms for Vietoris-Rips and Čech filtrations
TLDR
A lower bound result is provided: a point cloud is constructed that requires super-polynomial complexity for a high-quality approximation of the persistence, and it is shown that polynomial complexity is achievable for rough approximation, but impossible for sufficiently fine approximations.
A Study of Monodromy in the Computation of Multidimensional Persistence
TLDR
This contribution has identified instances of nontrivial monodromy over loops in nonsingular parameter space and given an example of a filtration in which it can be shown to be nontrivials.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 31 REFERENCES
Topology Based Selection and Curation of Level Sets
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
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
TLDR
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.
A survey of the marching cubes algorithm
Quantifying Transversality by Measuring the Robustness of Intersections
TLDR
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.
...
1
2
3
4
...