The Robustness of Level Sets

Bendich, Paul; Edelsbrunner, Herbert; Morozov, Dmitriy; Patel, Amit

doi:10.1007/978-3-642-15775-2_1

Cited by 12 publications

(13 citation statements)

References 9 publications

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“…The vertical arrows H * (−1) (X) → H * (−1) (CA) are trivial as both spaces are contractible and H * (−1) (A) → H * (−1) (A) are identities. By the five-lemma [37,11 Note that x…”

Section: A Secondary Obstructionmentioning

confidence: 99%

Solving equations and optimization problems with uncertainty

Franek

Krčál

Wagner

2017

J Appl. and Comput. Topology

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We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the theoretical work of Franek and Krčál (J ACM 62(4):26:1-26:19, 2015) with explicit algorithms and experiments with an implementation (https://bitbucket.org/robsatteam/rob-sat). Further, we show how to use the algorithm for approximating worst-case optima in optimization problems in which the feasible domain is defined by the zero set of a function f : X ! R n which is only known approximately. The algorithm first identifies a subdomain A where the function f is provably non-zero, a simplicial approximation f 0 : A ! S nÀ1 of f/|f|, and then verifies non-extendability of f 0 to X to certify a zero. Deciding extendability is based on computing the cohomological obstructions and their persistence. We describe an explicit algorithm for the primary and secondary obstruction, two stages of a sequence of algorithms with increasing complexity. Using elements and techniques of persistent homology, we quantify the persistence of these obstructions and hence of the robustness of zero. We provide experimental evidence that for random Gaussian fields, the primary obstruction-a much less computationally demanding test than the secondary obstruction-is typically sufficient for approximating robustness of zero.

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Section: A Secondary Obstructionmentioning

confidence: 99%

Solving equations and optimization problems with uncertainty

Franek

Krčál

Wagner

2017

J Appl. and Comput. Topology

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“…Well groups, introduced Edelsbrunner, Morozov and Bendich [1,43,45], quantify acceptable noise in the sublevel set approach to f : X → R, but have yet to find a large range of computable situations.…”

Section: Shapes Of Theories Future Directionsmentioning

confidence: 99%

Sketches of a platypus: a survey of persistent homology and its algebraic foundations

Vejdemo-Johansson¹

2014

Algebraic Topology: Applications and New Directions

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Abstract. The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are rooted in choosing a particular algebraic foundational theory to describe persistent homology, and applying results from that theory to prove useful and important results.In this survey paper, we shall examine the various choices in use, and what they allow us to prove. We shall also discuss the inherent differences between the choices people use, and speculate on potential directions of research to resolve these differences.Johnstone [51] named his book on topos theory "Sketches of an elephant", referencing a joke: three blind wise men encounter an elephant. They each try to describe it to each other. The wise man who caught hold of the elephant's trunk says "An elephant is like a snake."; the wise man holding the ear says "An elephant is like a palm leaf."; and the wise man holding its leg says "An elephant is like a tree.".The joke is highly relevant to topos theory; which has its roots in logic, in geometry, and in topology, with the three perspectives being fundamentally different and enriching each other in surprising ways.The title of this paper is similar, but different. The platypus is well-known to be a hybrid of an animal: sharing traits both with the phylum of birds and with the phylum of mammals. The field of persistent homology is in a similar situation to the platypus: there are at least two different viewpoints of what persistent homology should be, and they interact in sometimes unexpected ways.

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“…For the case of holes, the creators and destroyers are saddle points and maximal points respectively. An algorithm to compute (c α , d α ), and thus ρ φ (α), is devised in [2]. Robustness was used to enforce topological constraints in curve and surface evolusion [6].…”

Section: The Algorithmmentioning

confidence: 99%

“…We first compute the robustness of all components of O using the algorithm from [2]. We keep the β 0 most robust components and eliminate the rest.…”

Section: Toposimp: Remove Topology Noise Of An Objectmentioning

confidence: 99%

Enforcing topological constraints in random field image segmentation

2011

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We introduce TopoCut: a new way to integrate knowledge about topological properties (TPs) into random field image segmentation model. Instead of including TPs as additional constraints during minimization of the energy function, we devise an efficient algorithm for modifying the unary potentials such that the resulting segmentation is guaranteed with the desired properties. Our method is more flexible in the sense that it handles more topology constraints than previous methods, which were only able to enforce pairwise or global connectivity. In particular, our method is very fast, making it for the first time possible to enforce global topological properties in practical image segmentation tasks.

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The Robustness of Level Sets

Cited by 12 publications

References 9 publications

Solving equations and optimization problems with uncertainty

Solving equations and optimization problems with uncertainty

Sketches of a platypus: a survey of persistent homology and its algebraic foundations

Enforcing topological constraints in random field image segmentation

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