Multidimensional Image Models and Processing

  • Victor KrasheninnikovEmail author
  • Konstantin Vasil’ev
Part of the Intelligent Systems Reference Library book series (ISRL, volume 135)


The problems of developing mathematical models and statistical algorithms for processing of multidimensional images and their sequences are presented in this chapter. Different types of random fields are taken for the basic mathematical image model. This implies two main problems associated with image modeling, namely, model analysis and synthesis. The main attention is paid to the correlation aspect, i.e. evaluation of the correlation function of a random field generated by a given model and, vice versa, development of a model generating a random field with a predetermined correlation function. For this purpose, new models (tensor and wave) and new versions of autoregressive models (with multiple roots) are suggested. The problems of image simulation on the curved surfaces are considered. The suggested models are used to synthesize the algorithms of multidimensional image processing and their sequences. The tensor filtration of imaging sequences and recursive filtration of multidimensional images, as well as the asymptotic characteristics of efficiency of random field filtration on grids of arbitrary dimension are suggested. The problem of object and anomaly detection on the background of interfering images is considered for the images of any dimension, e.g. for multi-zone data. It is shown that four equivalent forms of the optimal decision rule, which reflect various aspects of detection procedure, exist. Potential efficiency of anomaly detection is analyzed. The problems of alignment and estimation of parameters for interframe geometric image transformations are considered for multidimensional image sequences. A tensor procedure of simultaneous filtration of multidimensional image sequence and their interframe displacements are suggested. A method based on a fixed point of a complex geometric image transformation was investigated in order to evaluate large interframe displacements. Options for adaptive image processing algorithms are also discussed in this chapter. In this context, pseudo-gradient procedures are taken as a basis, as they do not require preliminary evaluation of any characteristics of the processed data. This allows to develop the high-performance algorithms that can be implemented in real-time systems.


Multidimensional image model Autoregressive model Tensor model Wave model Curved surface Processing Potential efficiency Prediction Filtration Anomaly detection Recognition Adaptive algorithm Pseudo-gradient algorithm 



The reported study was funded by the Russian Fund for Basic Researches according to the research projects № 16-41-732041 and № 16-41-732027.


  1. 1.
    Shalygin, A.S., Palagin, Y.I.: Applied Methods of Statistical Modeling. Mechanical Engineering Leningrad: Mashinostroenie (1986)Google Scholar
  2. 2.
    Habibi, A.: Two-dimensional Bayesian estimate of images. Proc. IEEE 60(7), 878–883 (1972)Google Scholar
  3. 3.
    Gimel’farb, G.L.: Image Textures and Gibbs Random Fields. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  4. 4.
    Woods, J.W.: Two-dimensional Kalman filtering. In: Huang, T.S. (ed.) Two-Dimensional Digital Signal Processing I: Linear Filters. TAP, vol. 42 pp. 155–205 Springer, Berlin, Heidelberg, New York (1981)Google Scholar
  5. 5.
    Yaroslavsky, L.: Digital Picture Processing. An Introduction. Springer, Berlin, Heidelberg (1985)CrossRefGoogle Scholar
  6. 6.
    Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley-Interscience, New York (2000)zbMATHGoogle Scholar
  7. 7.
    Dudgeon, D.E., Mersereau, R.M.: Multidimensional Digital Signal Processing. Signal Processing Series. Prentice-Hall, Englewood Cliffs, New York (1984)zbMATHGoogle Scholar
  8. 8.
    Favorskaya, M.N., Levtin, K.: Early smoke detection in outdoor space by spatio-temporal clustering using a single video camera. In: Tweedale, J.W., Jain, L.C. (eds.) Recent Advances in Knowledge-Based Paradigms and Applications. AISC, vol. 234, pp. 43–56. Springer International Publishing, Switzerland (2014)Google Scholar
  9. 9.
    Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 4th edn. Pearson/Prentice-Hall, New York (2017)Google Scholar
  10. 10.
    Serra, J. (ed.): Image Analysis and Mathematical Morphology. Vol 2: Theoretical Advances. Academic Press, London (1988)Google Scholar
  11. 11.
    Vizilter, Y.V., Pyt’ev, Y.P., Chulichkov, A.I., Mestetskiy, L.M.: Morphological image analysis for computer vision applications. In: Favorskaya, M.N., Jain, L.C. (eds.) Computer Vision in Control Systems-1, ISRL, vol. 73, pp. 9–58. Springer International Publishing, Switzerland (2015)Google Scholar
  12. 12.
    Gruzman, I.C., Kirichuk, V.P., Kosikh, G.I., Peretryagin, G.I., Spector, A.A.: Digital Image Processing in Informative Systems. Novosibirsk State Technical University (2000) (in Russian)Google Scholar
  13. 13.
    Huang, T.S. (ed.): Image Sequence Analysis. Springer, Berlin, Heidelberg, New York (1981)zbMATHGoogle Scholar
  14. 14.
    Huang, T.S. (ed.): Image Sequence Processing and Dynamic Scene Analysis. Springer, New York (1983)Google Scholar
  15. 15.
    Soifer, V.A. (ed.): Computer Image Processing. Part I: Basic Concepts and Theory. VDM Verlag Dr. Muller E.K. (2009)Google Scholar
  16. 16.
    Vasil’ev, K.K., Krasheninnikov, V.R.: Statistical Analysis of Images. Ulyanovsk State Technical University (2015) (in Russian)Google Scholar
  17. 17.
    Vasil’ev, K.K., Dement’ev, V.E., Andriyanov, N.A.: Doubly stochastic models of images. Pattern Recognit. Image Anal. 25(1), 105–110 (2015)CrossRefGoogle Scholar
  18. 18.
    Vasil’ev, K.K., Popov, O.V.: Autoregression models of random fields with multiple roots. Pattern Recognit. Image Anal. 9(2), 327–328 (1999)Google Scholar
  19. 19.
    Vasil’ev, K.K., Dement’ev, V.E., Andriyanov, N.A.: Application of mixed models for solving the problem on restoring and estimating image parameters. Pattern Recognit. Image Anal. 26(1), 240–247 (2016)CrossRefGoogle Scholar
  20. 20.
    Krasheninnikov, V.R.: Correlation analysis and synthesis of random field wave models. Pattern Recognit. Image Anal. 25(1), 41–46 (2015)CrossRefGoogle Scholar
  21. 21.
    Krasheninnikov, V.R., Kalinov, D.V., Pankratov, YuG: Spiral autoregressive model of a quasi-periodic signal. Pattern Recognit. Image Anal. 8(1), 211–213 (2001)Google Scholar
  22. 22.
    Krasheninnikov, V.R., Mikeev, R.R., Kuzmin, M.V.: The model and algorithm for simulation of planets relief as surfaces image. Radioengineering 175, 192–194 (2012). (in Russian)Google Scholar
  23. 23.
    Dikshit, S.: A recursive Kalman window approach to image restoration. IEEE Trans Acoust. Speech Signal Process. 30(2), 125–140 (1982)Google Scholar
  24. 24.
    Jähne, B.: Digital Image Processing, 6th edn. Springer, Berlin, Heidelberg (2005)zbMATHGoogle Scholar
  25. 25.
    Pratt, W.K.: Digital Image Processing. PIKS Inside. 3rd ed. Wiley, New York (2001)Google Scholar
  26. 26.
    Prewitt, J.M.S.: Object enhancement and extraction. In: Lipkin, B.S., Rosenfeld, A. (eds.) Picture Processing and Psychopictorics, pp. 75–149. Academic Press, New York (1970)Google Scholar
  27. 27.
    Zhuravlev, Yu.I.: An algebraic approach to recognition or classifications problems. Pattern Recognit. Image Anal. 8(1), 59–100 (1998)Google Scholar
  28. 28.
    Favorskaya, M., Jain, L.C., Buryachenko, V.: Digital video stabilization in static and dynamic scenes. In: Favorskaya, M.N., Jain, L.C. (eds.) Computer Vision in Control Systems-1, ISRL, vol. 73, pp. 261–309 Springer International Publishing, Switzerland (2015)Google Scholar
  29. 29.
    Krasheninnikov, V.R., Potapov, M.A.: A way to detect the straight line trajectory of an immovable point for estimating parameters of geometrical transformation of 3D images. Pattern Recognit. Image Anal. 21(2), 280–284 (2011)Google Scholar
  30. 30.
    Krasheninnikov, V.R., Potapov, M.A.: Estimation of parameters of geometric transformation of images by fixed point method. Pattern Recognit. Image Anal. 22(2), 303–317 (2012)CrossRefGoogle Scholar
  31. 31.
    Polyak, B.T., YaZ, Tsypkin: Optimal pseudogradient adaptation procedure. Autom. Remote Control 8, 74–84 (1980)Google Scholar
  32. 32.
    Widrow, B., Stearns, S.D.: Adaptive Signal Processing. Prentice-Hall Inc., Englewood, Cliffs, NJ (1985)zbMATHGoogle Scholar
  33. 33.
    Vasil’ev, K.K.: Statistical analysis of multidimensional images. Pattern Recognit. Image Anal. 9(4), 732–748 (1999)Google Scholar
  34. 34.
    Krasheninnikov, V.R.: Wave image models on the surfaces. In: 8th Open German-Russian Workshop on Pattern Recognition and Image Understanding Nizhny, Novgorod, pp. 154–157 (2011)Google Scholar
  35. 35.
    Krasheninnikov, V.R., Kuznetsov, V.V., Lebedeva, E.Y., Krasheninnikova, N.A.: Optimization of dictionary and model library for recognition of speech commands based on cross-correlation portraits. Pattern Recognit. Image Anal. 23(1), 80–86 (2013)CrossRefGoogle Scholar
  36. 36.
    Krasheninnikov, V.R., Kopylova, A.S.: Algorithms for automated processing images of blood serum facies. Pattern Recognit. Image Anal. 22(4), 583–592 (2012)CrossRefGoogle Scholar
  37. 37.
    Vasil’ev, K.K., Dement’ev, V.E., Luchkov, N.V.: Analysis of efficiency of detecting extended signals on multidimensional grids. Pattern Recognit. Image Anal. 22(2), 400–408 (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Ulyanovsk State Technical UniversityUlyanovskRussian Federation

Personalised recommendations