The Stability of the Apparent Contour of an Orientable 2-Manifold

Edelsbrunner, Herbert; Morozov, Dmitriy; Patel, Amit

doi:10.1007/978-3-642-15014-2_3

Cited by 20 publications

(22 citation statements)

References 21 publications

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“…The main result of this paper implies that this measure is stable, that is, W ∞ (Dgm(U), Dgm(V)) ≤ f a − g a ∞ , where U and V are the well modules defined by the mappings f, g : M → R and the value a ∈ R 2 . As shown in [6], this implies that the apparent contours of f and of g are close. The sense in which they are close is interesting in its own right, and we refer to that paper for details.…”

Section: Apparent Contoursmentioning

confidence: 67%

“…Specifically, the stability of the diagrams mentioned above results in a stability result for the apparent contour of a smooth mapping from an orientable 2-manifold to the plane [6]. The need for these stable diagrams was indeed what triggered the development described in this paper.…”

Section: Introductionmentioning

confidence: 91%

“…As mentioned in the introduction, [6] reduces the stability of the contour of a mapping to the stability of well diagrams, the main result of this paper. We briefly review the reduction.…”

Section: Apparent Contoursmentioning

confidence: 97%

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Quantifying Transversality by Measuring the Robustness of Intersections

2011

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By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.

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Section: Apparent Contoursmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 91%

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Quantifying Transversality by Measuring the Robustness of Intersections

2011

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“…While persistence has been used successfully for scalar field visualization and simplification of topological structures such as contour trees and Jacobi sets [27], [28], [29], [30], robustness, first introduced in [31], is specifically designed for vector-valued data [32], [33]. Recent work [34] assigns robustness to critical points in both stationary and time-varying settings and obtains a structural description of the vector field.…”

Section: Related Workmentioning

confidence: 99%

Robustness-Based Simplification of 2D Steady and Unsteady Vector Fields

Škraba

Wang

Chen

et al. 2015

IEEE Trans. Visual. Comput. Graphics

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Abstract-Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices, using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness which enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them. This leads to a hierarchical simplification scheme that encodes flow magnitude in its perturbation metric. Our novel simplification algorithm is based on degree theory and has minimal boundary restrictions. Finally, we provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets. We show local and complete hierarchical simplifications for steady as well as unsteady vector fields.

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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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The Stability of the Apparent Contour of an Orientable 2-Manifold

Cited by 20 publications

References 21 publications

Quantifying Transversality by Measuring the Robustness of Intersections

Quantifying Transversality by Measuring the Robustness of Intersections

Robustness-Based Simplification of 2D Steady and Unsteady Vector Fields

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

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