ISSN 2070-7401 (Print), ISSN 2411-0280 (Online)
Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa


Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2015, Vol. 12, No. 1, pp. 131-144

Texture recognition in digital images by computational topology methods

N.G. Makarenko1,2  , F.A. Urtiev1  , I.S. Knyazeva1  , D.B. Malkova3 , I.T. Park2  , L.M. Karimova2 
1 Central Astronomical Observatory RAS, Saint-Petersburg, Russia
2 Institute of Information and Computing Technologies, Ministry of Education and Science of Kazakhstan, Almaty, Kazakhstan
3 P.G. Demidov Yaroslavl State University, Yaroslavl , Russia
In this paper we discuss the recognition of textures of digital images with the help of methods of computation topology. The main idea is to use the logic of attributes based on a topological filtration. Pixels sorted by photometric measure are scanned by ascending levels of gray. Each local minimum generates a connected component. It disappears if a similar in magnitude maximum appears in its local neighborhood. When the level increases, the clusters generated by the primary components merge. The process ends when it turns one global cluster. The number of connected components is measured by the topological invariant - Betti-zero. The life span of the components or its persistence is measured by the difference of the two levels. The first one marks appearance of the component, the second - a merging with neighboring clusters. Merging of the individual components is accompanied by the appearance of "holes" within the complex clusters. The amount of Holes is measured by Betti-1 number and the persistence by the difference between the levels of cancellation and the appearance of the Hole. We show how the distribution of persistent Betti numbers can be used for texture recognition in digital images.
Keywords: image segmentation, topological filtration, the Cech complex, persistent Betti numbers, pattern recognition
Full text


  1. Borges J.L., Analiticheskii yazyk Dzhona Uilkinsa (The Analytical Language of John Wilkins), Saint-Petersburg: Amphora, 2005, pp. 416—420.
  2. Grenander U., Lektsii po teorii obrazov. 3 Regulyarnye struktury (Regular Structures Lectures in Pattern Theory. Volume III), Moscow: Mir, 1983, 432 p.
  3. Makarenko N.G., Kruglun O.A., Makarenko I.N., Karimova L.M., Mul'tifraktal'naya segmentatsiya dannykh distantsionnogo zondirovaniya (Multifractal segmentation of data of remote sensing), Issledovanie Zemli iz kosmosa, 2008, No. 3, pp. 18-26.
  4. Makarenko N.G., Karimova L.M., Kruglun O.A., Skeilingovye svoistva tsifrovykh izobrazhenii zemnykh landshaftov (Scaling properties of digital images of Earth landscape), Sovremennye problemy distantsionnogo zondirovaniya Zemli iz kosmosa, 2014, Vol. 11, No. 2, pp. 26-37.
  5. Makarenko N.G., Malkova D.B., Machin M.L., Knyazeva I.S., Makarenko I.N., Diagnostika magnitnoi dinamiki aktivnykh oblastei Solntsa metodami vychislitel'noi topologii (Diagnostics of magnetic dynamics of active areas of the Sun by methods of computing topology), Fundamental'naya i prikladnaya matematika, 2013, Vol. 18, No. 2, pp. 79-93.
  6. Foucault M., Slova i veshchi. Arkheologiya gumanitarnykh nauk (Les mots et les choses. Une archeologie des sciences humaines), SPb. Accad., 1994, 408 p.
  7. Winkler G., Analiz izobrazhenii, sluchainye polya i dinamicheskie metody Monte-Karlo. Matematicheskie osnovy (Image Analysis, Random Fields and Dynamic Monte Carlo Methods. A Mathematical Introduction), Novosibirsk: SO RAN, Teo, 2002, 343 p.
  8. Zopf G W., In: Printsipy samoorganizatsii (Principles of self-organisation), Moscow: Mir, 1966, pp. 399-427.
  9. Bubenik P. Statistical topological data analysis using persistence landscapes, 2014, available at: [math.AT]
  10. Carlsson G., Topology and data, Bull. of the Amer. Mathem. Soc., 2009, Vol. 46(2), pp. 255-308.
  11. Carlsson E., Carlsson G., Silva Vin De, An algebraic topological method for feature identification, Intern. J. of Computational Geometry & Applications, 2006, Vol. 16 (04), pp. 291-314.
  12. Edelsbrunner H., Harer J., Computational Topology, An Introduction, American Mathematical Society, 2009, 241 p.
  13. Edelsbrunner H., Morozov D., Persistent Homology: Theory and Practice, European Congress of Mathematics, Krakow, 2-7 July, 2012, Europ. Math. Soc., 2012, pp. 31-50.
  14. Ester M., Kriegel H-P., Sander J., Xu X., A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise, Proc. 2nd Intern. Conf. Knowledge Discovery and Data Mining - KDD-96, 1996, pp. 226-231.
  15. Geman S., Geman D., Stochastic relaxation, Gibbs distributions, and Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1984, Vol. PAMI-6, No. 6, pp. 721-741.
  16. Ghrist R., Barcodes: The persistent topology of data, Bull. Amer. Math. Soc., 2008, Vol. 45(1), pp. 61-75.
  17. Kaczynski T., Mischaikow K., Mrozek M., Computational Homology, Springer, 2004, 482 p.
  18. Luxburg U. von, A tutorial on spectral clustering, Stat. Comput., 2007, Vol. 17. pp. 395–416.
  19. Li Stan Z., Markov Random Field Modeling in Image Analysis, Springer, 2009, 357 p.
  20. Mumford D., Shah J., Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 1989, No. 42, pp. 577–685.
  21. Robins V., Abernethy J., Rooney N., Bradley E., Topology and Intelligent Data Analysis, J. Intelligent Data Analysis, 2004, Vol. 8, No. 5, pp. 505–515.
  22. Rui Xu R., Wunsch II D., Survey of Clustering Algorithms., IEEE Trans. on Neural Networks, 2005, Vol. 16, No. 3, pp. 645-678.
  23. Shen J., A Stochastic-variational model for soft Munford –Shah segmentation, Intern. J. of Biomedical Imaging, 2006, Vol. 2006, Article ID 92329, pp. 1–14.
  24. Selfridge O.G., Pandemonium: A Paradigm for Learning, Mechanisation of Thought Processes. National Phys.Labor. Symp. No. 10. London: Her Majestry’s Stationery office. 1959, pp. 513-530.
  25. Serra J., Image Analysis and Mathematical Morphology, Academic Press, New-York, 1982, 610 p.
  26. Zomorodian A.J., Topology for computing, Cambridge Univ.Press, 2005, 243 p.