Skip to main content


On a Class of Parametric Transforms and Its Application to Image Compression

Article metrics

  • 889 Accesses


A class of parametric transforms that are based on unified representation of transform matrices in the form of sparse matrix products is described. Different families of transforms are defined within the introduced class. All transforms of one family can be computed with fast algorithms similar in structure to each other. In particular, the family of Haar-like transforms consists of discrete orthogonal transforms of arbitrary order such that they all may be computed with a fast algorithm that is in structure similar to classical fast Haar transform. A method for parameter selection is proposed that allows synthesizing specific transforms with matrices containing predefined row(s). The potential of the proposed class of Haar-like parametric transforms to improve the performance of fixed block transforms in image compression is investigated. With this purpose, two image compression schemes are proposed where a number of Haar-like transforms are synthesized each adapted to a certain set of blocks within an image.The nature of the proposed schemes is such that their performance (in terms of PSNR versus compression ratio) cannot be worse than a scheme based on classical discrete cosine transform (DCT). Simulations show that a significant performance improvement can be achieved for certain types of images such as medical X-ray images and compound images.


  1. 1.

    Pennebaker WB, Mitchell JL: JPEG Still Image Data Compression Standard. Van Nostrand Reinhold, New York, NY, USA; 1993.

  2. 2.

    Agaian SS: Optimal algorithms of fast orthogonal transforms and their implementation on computers. In Kibernetika I Vichislitelnaya Tekhnika, issue 2. Nauka, Moscow, Russia; 1986:231-319.

  3. 3.

    Agaian SS, Astola J, Egiazarian K: Binary Polynomial Transforms and Non-linear Digital Filters. Marcel Dekker, New York, NY, USA; 1995.

  4. 4.

    Agaian SS, Matevosian AK: Generalized Haar transforms and automation systems for testing quality of circuits. Acta Cybernetica 1981, 5: 345–362.

  5. 5.

    Ahmed NU, Rao KR: Orthogonal Transforms for Digital Signal Processing. Springer, Secaucus, NJ, USA; 1975.

  6. 6.

    Ersoy OK: A comparative review of real and complex Fourier-related transforms. Proceedings of the IEEE 1994,82(3):429-447. 10.1109/5.272147

  7. 7.

    Jain AK: Fundamentals of Digital Image Processing. Prentice-Hall, Englewood Cliffs, NJ, USA; 1989.

  8. 8.

    Poularikas AD (Ed): The Transforms and Applications Handbook. CRC Press, Boca Raton, Fla, USA; 1996.

  9. 9.

    Malvar HS: Signal Processing with Lapped Transforms. Artech House, Norwood, Mass, USA; 1992.

  10. 10.

    Wickerhauser MV: Adapted Wavelet Analysis from Theory to Software. IEEE Press, A. K. Peters, Wellesley, Mass, USA; 1994.

  11. 11.

    Labunets VG: A unified approach to fast transformation algorithms. In Primeneniye Ortogonalnix Metodov pri Obrabotke Signalov i Analize System. UPI, Sverdlovsk, Russia; 1980:4-14.

  12. 12.

    Traxtman M, Traxtman VA: Osnovi Teorii Discretnix Signalov na Konechnix Intervalax. Sovetskoye Radio, Moscow, Russia; 1975.

  13. 13.

    Yaroslavskiy LP: Some questions of the theory of discrete orthogonal transforms of signals. In Cifrovaya Obrabotka Signalov I ee Primeneniya. Nauka, Moscow, Russia; 1981:33-71.

  14. 14.

    Christopoulos C, Skodras A, Ebrahimi T: The JPEG 2000 still image coding system: an overview. IEEE Transactions on Consumer Electronics 2000,46(4):1103-1127. 10.1109/30.920468

  15. 15.

    Foos DH, Muka E, Slone RM, et al.: JPEG 2000 compression of medical imagery. Medical Imaging 2000: PACS Design and Evaluation: Engineering and Clinical Issues, February 2000, San Diego, Calif, USA, Proceedings of SPIE 3980: 85–96.

  16. 16.

    Ponomarenko N, Lukin V, Egiazarian K, Astola J: DCT based high quality image compression. Proceedings of the 14th Scandinavian Conference on Image Analysis (SCIA '05), June 2005, Joensuu, Finland 1177–1185.

  17. 17.

    Solodovnikov AI, Kanatov II, Spivakovskii AM: Synthesis of orthogonal bases from a generalized spectral kernel. In Voprosy Teorii Sistem Avtomaticheskogo Upravleniya. Volume 2. LGU, Leningrad, Russia; 1978:99-112.

  18. 18.

    Solodovnikov AI: Synthesis of complete orthonormal systems of functions having fast transform algorithm. In Voprosy Teorii System Avtomaticheskogo Upravleniya. Volume 4. LGU, Leningrad, Russia; 1978:94-105.

  19. 19.

    Andrews HC, Caspary KL: A generalized technique for spectral analysis. IEEE Transactions on Computers 1970,19(1):16-25.

  20. 20.

    Agaian SS, Gevorkian DZ: Complexity and parallel algorithms of discrete orthogonal transforms. In Kibernetika I Vichislitelnaya Tekhnika, issue 4. Nauka, Moscow, Russia; 1988:124-169.

  21. 21.

    Agaian SS, Gevorkian D: Synthesis of a class of orthogonal transforms: parallel SIMD-algorithms and specialized processors. Pattern Recognition and Image Analysis 1992,2(4):394-408.

  22. 22.

    Minasyan S, Guevorkian D, Sarukhanyan H: On parameterized fast Haar- and Hadamard-like transforms of arbitrary order. Proceedings of the 3rd International Conference on Computer Science and Information Technologies (CSIT '01), September 2001, Yerevan, Armenia 294–298.

  23. 23.

    Minasyan S, Guevorkian D, Agaian SS, Sarukhanyan H: On "slant-like" fast orthogonal transforms of arbitrary order. Proceedings of the 4th EURASIP IEEE Region and International Symposium on Video/Image Processing and Multimedia Communications (VIPromCom '02), June 2002, Zadar, Croatia 309–314.

  24. 24.

    Minasyan S, Astola J, Guevorkian D: An image compression scheme based on parametric Haar-like transform. Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS '05), May 2005, Kobe, Japan 3: 2088–2091.

  25. 25.

    Astola J, Minasyan S, Guevorkian D: Multiple transform based image compression technique. Proceedings of the 5th IASTED International Conference on Visualization, Imaging, and Image Processing, September 2005, Benidorm, Spain

  26. 26.

    Soderquist P, Leeser M: Division and square root: choosing the right implementation. IEEE Micro 1997,17(4):56-66. 10.1109/40.612224

Download references

Author information

Correspondence to Susanna Minasyan.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Minasyan, S., Astola, J. & Guevorkian, D. On a Class of Parametric Transforms and Its Application to Image Compression. EURASIP J. Adv. Signal Process. 2007, 058416 (2007) doi:10.1155/2007/58416

Download citation


  • Compression Ratio
  • Discrete Cosine Transform
  • Unify Representation
  • Fast Algorithm
  • Matrix Product