Persistent Betti Numbers for a Noise Tolerant Shape-Based Approach to Image Retrieval

Frosini, Patrizio; Landi, Claudia

doi:10.1007/978-3-642-23672-3_36

Cited by 27 publications

(28 citation statements)

References 16 publications

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“…For instance, we can refer to the problem of comparing the shapes of objects represented by clouds of points belonging to a fixed compact subset B of a Euclidean space

{double-struckE}^{n}

. In this case, we can set X = B , while each filtering function

ϕ MathClass-punc: B MathClass-rel\to {double-struckR}^{k}

can describe both the distance from the given cloud and other properties (cf., e.g., ). Also in this case, just an invariant triangulation of B is required.…”

Section: Discussion and Further Researchmentioning

confidence: 99%

G -invariant persistent homology

Frosini

2014

Math. Meth. Appl. Sci.

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Classical persistent homology is not tailored to study the action of transformation groups different from the group Homeo(X) of all self-homeomorphisms of a topological space X. In order to obtain better lower bounds for the natural pseudo-distance d G associated with a group G ⊂ Homeo(X), we need to adapt persistent homology and consider G-invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper we formalize this idea, and prove the stability of G-invariant persistent homology with respect to the natural pseudo-distance d G . We also show how G-invariant persistent homology could be used in applications concerning shape comparison.

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“…For instance, we can refer to the problem of comparing the shapes of objects represented by clouds of points belonging to a fixed compact subset B of a Euclidean space

{double-struckE}^{n}

. In this case, we can set X = B , while each filtering function

ϕ MathClass-punc: B MathClass-rel\to {double-struckR}^{k}

can describe both the distance from the given cloud and other properties (cf., e.g., ). Also in this case, just an invariant triangulation of B is required.…”

Section: Discussion and Further Researchmentioning

confidence: 99%

G -invariant persistent homology

Frosini

2014

Math. Meth. Appl. Sci.

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“…The solution we adopted in an experiment is the geometrical-topological tool of Persistent Betti Numbers in degree zero (also called Size Functions) [7,8,1,3,9]. They are modular shape descriptors particularly apt to capturing qualitative aspects of images.…”

Section: Fig 2 Different Strategies In Drawing Keypicsmentioning

confidence: 99%

Graphic-Based Concept Retrieval

Ferri

2013

Availability, Reliability, and Security in Information Systems and HCI

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Abstract. Two ways of expressing concepts in the context of image retrieval are presented. One, Keypics, is on the side of an image owner, who wants the image itself to be found on the Web; the second, Trittico, is on the side of the image searcher. Both are based on the paradigm of human intermediation for overcoming the semantic gap. Both require tools capable of qualitative analysis, and have been experimented by using persistent homology.

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“…7, 22, 23, 26, 50 Some of these persistent homology algorithms have been implemented in many software packages, namely Perseus, 50, 52 JavaPlex 71 and Dionysus. In the past few years, persistent homology has been applied to image analysis, 5, 9, 58, 67 image retrieval, 30 chaotic dynamics verification, 42, 49 sensor networks, 66 complex networks, 40, 45 data analysis, 8, 47, 53, 60, 73 computer vision, 67 shape recognition 24 and computational biology. 21, 31, 43, 86 …”

Section: Introductionmentioning

confidence: 99%

Object-oriented persistent homology

Wang

Wei

2016

Journal of Computational Physics

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Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-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.

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Persistent Betti Numbers for a Noise Tolerant Shape-Based Approach to Image Retrieval

Cited by 27 publications

References 16 publications

G -invariant persistent homology

G -invariant persistent homology

Graphic-Based Concept Retrieval

Object-oriented persistent homology

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