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The Comparison of Diffeomorphic Images based on the Construction of Persistent Homology

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Abstract

An object shape analysis is a problem that is related to such areas as geometry, topology, image processing and machine learning. For analyzing the form, the deformation between the source and terminal form of the object is estimated. The most used form analysis model is the Large Deformation Diffeomorphic Metric Mapping (LDDMM) model. The LDDMM model can be supplemented with functional non-geometric information about objects (volume, color, formation time). The paper considers algorithms for constructing sets of barcodes for comparing diffeomorphic images, which are real values taken by persistent homology. A distinctive feature of the use of persistent homology with respect to methods of algebraic topology is to obtain more information about the shape of the object. An important direction of the application of persistent homology is the study invariants of big data. A method based on persistent cohomology is proposed that combines persistent homology technologies with embedded non-geometric information presented as functions of simplicial complexes. The proposed structure of extended barcodes using cohomology increases the effectiveness of persistent homology methods. A modification of the Wasserstein method for finding the distance between images by introducing non-geometric information was proposed. The possibility of the formation of barcodes of images invariant to transformations of rotation, translation and similarity is considered.

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Funding

The work was funded by RFBR according to the research projects 18-07-00526 and 18-08-01284. The work was funded by the program of fundamental scientific researches of the SB RAS I.5.1, project 0314-2019-0020.

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  1. Omsk Branch, Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, 644043, Omsk, Russia

    S. N. Chukanov

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  1. S. N. Chukanov
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Correspondence to S. N. Chukanov.

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Sergey N. Chukanov, orcid.org/0000-0002-8106-9813, PhD.

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Chukanov, S.N. The Comparison of Diffeomorphic Images based on the Construction of Persistent Homology. Aut. Control Comp. Sci. 54, 758–771 (2020). https://doi.org/10.3103/S0146411620070056

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  • DOI: https://doi.org/10.3103/S0146411620070056

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