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Model. Anal. Inform. Sist., 2019, Volume 26, Number 3, Pages 450–468 (Mi mais690)  

Discrete mathematics in relation to computer science

The comparison of diffeomorphic images based on the construction of persistent homology

S. N. Chukanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Omsk Branch 13 Pevtsova str., Omsk 644043, Russia

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, shift and similarity is considered.

Keywords: pattern recognition, diffeomorphic transformations, persistent (co)homology, Wasserstein distance.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-07-00526_а
18-08-01284_а
Siberian Branch of Russian Academy of Sciences I.5.1., проект № 0314-2019-0020
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.


DOI: https://doi.org/10.18255/1818-1015-450-468

Full text: PDF file (1134 kB)
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UDC: 004.932.2
Received: 11.05.2019
Revised: 29.08.2019
Accepted:31.08.2019

Citation: S. N. Chukanov, “The comparison of diffeomorphic images based on the construction of persistent homology”, Model. Anal. Inform. Sist., 26:3 (2019), 450–468

Citation in format AMSBIB
\Bibitem{Chu19}
\by S.~N.~Chukanov
\paper The comparison of diffeomorphic images based on the construction of persistent homology
\jour Model. Anal. Inform. Sist.
\yr 2019
\vol 26
\issue 3
\pages 450--468
\mathnet{http://mi.mathnet.ru/mais690}
\crossref{https://doi.org/10.18255/1818-1015-450-468}


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