Computing Robustness and Persistence for Images

Bendich, Paul; Edelsbrunner, Herbert; Kerber, Michael

doi:10.1109/tvcg.2010.139

Cited by 98 publications

(74 citation statements)

References 16 publications

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“…Each k-dimensional cube in V n has a unique lowest vertex, at which it is anchored. The limit of the number of k-cubes over the number of n-cubes is thereforeā n k = n k ; see (2). In Table 3, we show the ratios of the ratio limits for small values of n and k. Table 3: The ratio of the number of k-simplices in D n over the number of k-cubes in V n , up to one decimal position.…”

Section: Uniform Cubical Subdivisionsmentioning

confidence: 99%

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Dual Complexes of Cubical Subdivisions of ℝ n

Edelsbrunner

Kerber

2011

Discrete Comput Geom

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We use a distortion to define the dual complex of a cubical subdivision of R n as an n-dimensional subcomplex of the nerve of the set of n-cubes. Motivated by the topological analysis of high-dimensional digital image data, we consider such subdivisions defined by generalizations of quad-and oct-trees to n dimensions. Assuming the subdivision is balanced, we show that mapping each vertex to the center of the corresponding n-cube gives a geometric realization of the dual complex in R n .

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Section: Uniform Cubical Subdivisionsmentioning

confidence: 99%

“…While this property generalizes to R n , it no longer implies the geometric realization of the dual complex. Following [2], we now describe a counterexample in three dimensions. We begin with two cubes, A and B, that share a common edge of length 8.…”

Section: Ratio Boundsmentioning

confidence: 99%

Dual Complexes of Cubical Subdivisions of ℝ n

Edelsbrunner

Kerber

2011

Discrete Comput Geom

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“…The persistence algorithm runs in time O(v 3 ), where v is the number of simplices [5]. However this takes roughly linear time in practice [2]. For a detailed performance analysis, see [28].…”

Section: Detecting Circular Features In a Point Cloudmentioning

confidence: 99%

Topological analysis and visualization of cyclical behavior in memory reference traces

Choudhury

Wang

Rosen

et al. 2012

2012 IEEE Pacific Visualization Symposium

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(a) (b) Figure 1: One circular parameterization of the memory behavior of bubble sort (box (a)) morphing into another (box (b)), highlighting both their similarities and differences, and giving two views of the recurrent nature of the program. ABSTRACTWe demonstrate the application of topological analysis techniques to the rather unexpected domain of software visualization. We collect a memory reference trace from a running program, recasting the linear flow of trace records as a high-dimensional point cloud in a metric space. We use topological persistence to automatically detect significant circular structures in the point cloud, which represent recurrent or cyclical runtime program behaviors. We visualize such recurrences using radial plots to display their time evolution, offering multi-scale visual insights, and detecting potential candidates for memory performance optimization. We then present several case studies to demonstrate some key insights obtained using our techniques.

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“…Persistent homology is a new tool for studying shape in application areas such as digital images [17,18], dynamical systems [19], and high-dimensional data-mining [20]. There are two main strengths of our approach:…”

Section: Introductionmentioning

confidence: 99%

Texture Analysis of Hydrophobic Polycarbonate and Polydimethylsiloxane Surfaces via Persistent Homology

et al. 2017

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Due to recent climate change-triggered, regular dust storms in the Middle East, dust mitigation has become the critical issue for solar energy harvesting devices. One of the methods to minimize and prevent dust adhesion and create self-cleaning abilities is to generate hydrophobic characteristics on surfaces. The purpose of this study is to explore the topological features of hydrophobic surfaces. We use non-standard techniques from topological data analysis to extract morphological features from the AFM images. Our method recovers most of the previous qualitative observations in a robust and quantitative way. Persistence diagrams, which is a summary of topological structures, witness quantitatively that the crystallized polycarbonate (PC) surface possesses spherulites, voids, and fibrils, and the texture height and spherulite concentration increases with the increased immersion period. The approach also shows that the polydimethylsiloxane (PDMS) exactly copied the structures at the PC surface but 80 to 90 percent of the nanofibrils were not copied at PDMS surface. We next extract a feature vector from each persistence diagram to show which experiments hold features with similar variance using principal component analysis (PCA). The K-means clustering algorithm is applied to the matrix of feature vectors to support the PCA result, grouping experiments with similar features.

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Computing Robustness and Persistence for Images

Cited by 98 publications

References 16 publications

Dual Complexes of Cubical Subdivisions of ℝ n

Dual Complexes of Cubical Subdivisions of ℝ n

Topological analysis and visualization of cyclical behavior in memory reference traces

Texture Analysis of Hydrophobic Polycarbonate and Polydimethylsiloxane Surfaces via Persistent Homology

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