Implementation of Hadamard Matrices for Image Processing

Part of the Intelligent Systems Reference Library book series (ISRL, volume 73)


The image quality influences the accuracy of obtained results. In the chapter, the application of the strip-method for noise-immune storage and transmission of images is analyzed. At the same time, before transmitting the matrix transformation of an original image has to be done, when the image fragments are mixed up and superimposed each other. The transformed image is transmitted over a communication channel, where it is distorted with a pulse noise, the latter being, for example, a possible reason for a complete loss of separate image fragments. After the signal transmission to the receiving end, an inverse transformation is performed. During this transformation, the reconstruction of the image takes place. If it is possible to provide a uniform distribution of the pulse noise over the whole area, which the image occupies without any changes of its energy, then a noticeable decrease of noise amplitude will take place and an acceptable quality of all fragments of the image are reconstructed. The tasks of the chapter are the consideration the versions of the two-sided strip-transformation of images and the choice of optimal transformation matrices. A great attention has been paid to the implementation of Hadamard matrices and matrices close to them such as Hadamard-Mersenne, Hadamard-Fermat, and Hadamard-Euler matrices.


Image quality Strip-method Image transmission Matrix transformations Pulse noise Inverse transformation Hadamard matrix Two-levels M-matrix Three-levels M-matrix Multi-levels M-matrices 


  1. 1.
    Mironovskii LA, Slaev VA (2006) The strip method of transforming signals containing redundancy. Meas Tech 49(7):631–638CrossRefGoogle Scholar
  2. 2.
    Mironovskii LA, Slaev VA (2006) The strip method of noise-immune image transformation. Meas Tech 49(8):745–754CrossRefGoogle Scholar
  3. 3.
    Mironowsky LA, Slaev VA (2011) Strip-Method for Image and Signal Transformation. De Gruyter, BerlinCrossRefGoogle Scholar
  4. 4.
    Totty RE, Clark GC (1967) Reconstruction error in waveform transmission. IEEE Trans Inf Theory 13(2):333–338CrossRefGoogle Scholar
  5. 5.
    Andrews HC (1970) Computer techniques in image processing. Academic Press, New YorkGoogle Scholar
  6. 6.
    Haar A (1955) Zur Theorie der Orthogonal Funktionen-System. Inaugural Dissertation. Math Annalen 5:17–31Google Scholar
  7. 7.
    Livak EN Algorithms Compression. Accessed 14 June 2014 (in Russian)
  8. 8.
    Mironowsky LA, Slayev VA (1975) Equalization of the variance of a nonstationary. Signal Telecom Radio Eng 29–30(5):65–72Google Scholar
  9. 9.
    Costas JP (1952) Coding with Linear Systems. Proc IRE 40(9):1101–1103Google Scholar
  10. 10.
    Lang GR (1963) Rotational transformation of signals. IEEE Trans Inf Theory 9(3):191–198CrossRefMATHGoogle Scholar
  11. 11.
    Leith EN, Upatnieks J (1962) Reconstructed wavefronts and communication theory. J Opt Soc America 52(10):1123–1133CrossRefGoogle Scholar
  12. 12.
    Medianik AI (1997) Proper simplex inscribed in a cube and hadamard matrices of the half-circulante type. Math Phys Analis Geom 4(4):458–471MATHGoogle Scholar
  13. 13.
    Mironovskii LA, Slaev VA (2002) Optimal Chebishev pre-emphasis and filtering. Meas Tech 45(2):126–136CrossRefGoogle Scholar
  14. 14.
    Paley REAC (1933) On orthogonal matrices. J Math Phys 12:311–320Google Scholar
  15. 15.
    Pierce WH (1968) Linear-real codes and coders. Bell Syst Techn J 47(6):1067–1097CrossRefMathSciNetGoogle Scholar
  16. 16.
    Rao KR, Narasimhan MA, Revuluri K (1975) Image data processing by hadamard-haar transforms. IEEE Trans Comput C-23(9):888–896Google Scholar
  17. 17.
    Votolin DS (1998) Algorithms of images compression. Moscow State University (in Russian)Google Scholar
  18. 18.
    Mironowsky LA, Slayev VA (2013) Double-sided noise-immune strip transformation and its root images. Meas Tech 55(10):1120–1127CrossRefGoogle Scholar
  19. 19.
    Selyakov IS (2005) Analysis and computer imitation of images STRIP-transformation. Master’s dissertation, Saint-Petersburg State University of Aerospace InstrumentationGoogle Scholar
  20. 20.
    Mironowsky LA, Slaev VA (2011) Root images of two-sided noise immune strip-transformation. International workshop on physics and mathematics IWPM 2011. Hangzhou, ChinaGoogle Scholar
  21. 21.
    Hadamard J (1893) Resolution d’une Question Relative aux Determinants. Bull Sci Math ser 2 17(1):240–246Google Scholar
  22. 22.
    Ruben S (1990) Methods of cipher video compression. Comput Press 10:22–30 (in Russian)Google Scholar
  23. 23.
    Umnyashkin SV (2004) Mathematical Foundations of Signal Cipher Processing and Coding. National Research University of Electronic Technology (in Russian)Google Scholar
  24. 24.
    Williamson J (1944) Hadamard’s determinant theorem and the sum of four squares. Duke J Math 11:65–81CrossRefMATHGoogle Scholar
  25. 25.
    Zalmanzon LA (1989) Transformations of Furier, Walsh, Haar and their implementation in control, communication and other areas. Science, Moscow (in Russian)Google Scholar
  26. 26.
    Shintyakov DV (2006) Algorithm for searching hadamard matrices of odd order. Techn Sci 2:207–211Google Scholar
  27. 27.
    Contemporary Design Theory: A Collection of Essays (1992) Orthogonal arrays. Wiley, New YorkGoogle Scholar
  28. 28.
    Belevitch V (1950) Theorem of 2n-networks with application to conference telephony. Electr Commun 26:231–244Google Scholar
  29. 29.
    Balonin NA, Mironovsky LA (2006) Hadamard matrices of the odd order. Inf Control Syst 22(3):46–50 (in Russian)Google Scholar
  30. 30.
    Golova EA (2013) Properties investigation of M-matrices applied for filtration. Master’s dissertation, Saint-Petersburg State University of Aerospace InstrumentationGoogle Scholar
  31. 31.
    Golub GH, van Loan CF (1989) Matrix computations, 3rd edn. John Hopkins University Press, BaltimoreMATHGoogle Scholar
  32. 32.
    Balonin NA, Sergeyev MB (2011) M-matrices. Inf Control Syst 50(1):14–21 (in Russian)Google Scholar
  33. 33.
    Balonin NA, Sergeyev MB (2011) M-matrix of the 22-th order. Inf Control Syst 54(5):87–90 (in Russian)Google Scholar
  34. 34.
    Hadamard matrices monitoring. Accessed 14 June 2014 (in Russian)
  35. 35.
    Balonin NA, Sergeyev MB, Mironovsky LA (2012) Calculation of Hadamard-Fermat matrices. Inf Control Syst 61(6):90–93 (in Russian)Google Scholar
  36. 36.
    Balonin NA, Sergeyev MB (2013) Two ways to construct Hadamard-Euler matrices. Inf Control Syst 62(1):7–10 (in Russian)Google Scholar
  37. 37.
    Balonin NA, Mironovsky LA, Sergeyev MB (2012) Computation of Hadamard-Mersenne matrices. Inf Control Syst 60(5):92–94 (in Russian)Google Scholar
  38. 38.
    Gantmacher F (1959) Matrix theory. Chelsea Publishing, New YorkGoogle Scholar
  39. 39.
    Gersho AB, Gray RM (1992) Vector quantization and signal compression. Kluwer, BostonCrossRefMATHGoogle Scholar
  40. 40.
    Golov AS (2011) Robustness Investigation of strip-method to channel noises. Bachelor’s thesis, Saint-Petersburg State University of Aerospace InstrumentationGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.St. Petersburg State University of Aerocosmic InstrumentationSt. PetersburgRussian Federation
  2. 2.D.I. Mendeleyev Research Institute for MetrologySt. PetersburgRussian Federation

Personalised recommendations