On the Local Behavior of Spaces of Natural Images

Carlsson, Gunnar; Ishkhanov, Tigran; Silva, Vin; Zomorodian, Afra

doi:10.1007/s11263-007-0056-x

Cited by 437 publications

(528 citation statements)

References 14 publications

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“…Bud is a set of points sampled from the surface of the Stanford Buddha in R 3 . Bro is a set of 5 × 5 high-contrast patches derived from natural images, interpreted as vectors in R 25 , from the Brown database (with parameter k = 300 and cut 30%) [12,6]. Cy8 is a set of points in R 24 , sampled from the space of conformations of the cyclo-octane molecule [14], which is the union of two intersecting surfaces.…”

Section: Methodsmentioning

confidence: 99%

The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes

Boissonnat

Maria

2014

Algorithmica

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This paper introduces a new data structure, called simplex tree, to represent abstract simplicial complexes of any dimension. All faces of the simplicial complex are explicitly stored in a trie whose nodes are in bijection with the faces of the complex. This data structure allows to efficiently implement a large range of basic operations on simplicial complexes. We provide theoretical complexity analysis as well as detailed experimental results. We more specifically study Rips and witness complexes.Key-words: simplicial complexes, data structure, flag complexes, Rips complexes, witness complexes, relaxed witness complexes, high dimensions * INRIA Sophia Antipolis-Méditerranée † ENS Cachan Une structure de données arborescente pour représenter les complexes simpliciauxRésumé : Nous définissons dans cet article une nouvelle structure de données, appelée "simplex tree", pour représenter les complexes simpliciaux abstraits de toutes dimensions. Le complexe simplicial est représenté par un arbre préfixe dont les noeuds sont en bijection avec les faces du complexe. Cette structure de données permet de calculer efficacement un grand nombre d'opérations de bases sur les complexes simpliciaux. Nous développons dans cet article une analyse théorique de la complexité de ces algorithmes, ainsi qu'une analyse expérimentale détail-lée. Nous étudions plus particulièrement la construction des complexes de Rips et des witness complexes.

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Section: Methodsmentioning

confidence: 99%

The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes

Boissonnat

Maria

2014

Algorithmica

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“…Persistent Homology [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55] is a means of topological data analysis. Now let us use an example to show how the topological data analysis methods can overcome the limitations of geometrical methods.…”

Section: Persistent Homologymentioning

confidence: 99%

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

2018

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Atom probe tomography (APT) represents a revolutionary characterization tool for materials that combine atomic imaging with a time-of-flight (TOF) mass spectrometer to provide direct space three-dimensional, atomic scale resolution images of materials with the chemical identities of hundreds of millions of atoms. It involves the controlled removal of atoms from a specimen's surface by field evaporation and then sequentially analyzing them with a position sensitive detector and TOF mass spectrometer. A paradox in APT is that while on the one hand, it provides an unprecedented level of imaging resolution in three dimensions, it is very difficult to obtain an accurate perspective of morphology or shape outlined by atoms of similar chemistry and microstructure. The origins of this problem are numerous, including incomplete detection of atoms and the complexity of the evaporation fields of atoms at or near interfaces. Hence, unlike scattering techniques such as electron microscopy, interfaces appear diffused, not sharp. This, in turn, makes it challenging to visualize and quantitatively interpret the microstructure at the "meso" scale, where one is interested in the shape and form of the interfaces and their associated chemical gradients. It is here that the application of informatics at the nanoscale and statistical learning methods plays a critical role in both defining the level of uncertainty and helping to make quantitative, statistically objective interpretations where heuristics often dominate. In this chapter, we show how the tools of Topological Data Analysis provide a new and powerful tool in the field of nanoinformatics for materials characterization.

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“…Successful examples include Prof. Robert Ghrist's DARPA project on sensor networks [12], image analysis [2], and solid model reconstruction and simplification [3]; more examples can be found in the CAT Workshop report [1]. All these applications have required new approaches to modeling the problem.…”

Section: Maturity Of Homology For Research and Applicationsmentioning

confidence: 99%

LDRD 149045 final report distinguishing documents.

Mitchell¹

2010

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This LDRD 149045 final report describes work that Sandians Scott A. Mitchell, Randall Laviolette, Shawn Martin, Warren Davis, Cindy Philips and Danny Dunlavy performed in 2010.Prof. Afra Zomorodian provided insight. This was a small late-start LDRD. Several other ongoing efforts were leveraged, including the Networks Grand Challenge LDRD, and the Computational Topology CSRF project, and the some of the leveraged work is described here.We proposed a sentence mining technique that exploited both the distribution and the order of parts-of-speech (POS) in sentences in English language documents. The ultimate goal was to be able to discover "call-to-action" framing documents hidden within a corpus of mostly expository documents, even if the documents were all on the same topic and used the same vocabulary. Using POS was novel. We also took a novel approach to analyzing POS. We used the hypothesis that English follows a dynamical system and the POS are trajectories from one state to another. We analyzed the sequences of POS using support vector machines and the cycles of POS using computational homology. We discovered that the POS were a very weak signal and did not support our hypothesis well. Our original goal appeared to be unobtainable with our original approach.We turned our attention to study an aspect of a more traditional approach to distinguishing documents. Latent Dirichlet Allocation (LDA) turns documents into bags-of-words then into mixture-model points. A distance function is used to cluster groups of points to discover relatedness between documents. We performed a geometric and algebraic analysis of the most popular distance functions and made some significant and surprising discoveries, described in a separate technical report. [9] 3 4

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On the Local Behavior of Spaces of Natural Images

Cited by 437 publications

References 14 publications

The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes

The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

LDRD 149045 final report distinguishing documents.

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