Transform Coding of Signals with Bounded Finite Differences: from Fourier to Walsh, to Wavelets

  • Arthur Petrosian
Part of the Computational Imaging and Vision book series (CIVI, volume 19)


The digital spectral transform method is an important compression tool in signal and image processing applications. The fast Fourier transform algorithms, developed in the 60-s, facilitated the use of transform coding methods for redundancy elimination and efficient data representation. In order to determine the optimal zonal sampling method for a given transform, it is necessary to derive estimates of the transform spectra on a class of input signals. We present a unified approach for deriving upper bounds of spectra of orthogonal transforms on classes of input signals with bounded first and second order finite differences. Based on this approach we obtain estimates of spectra for classical discrete Fourier, Hartley, cosine, sine, as well as Walsh, Haar, and other wavelet transforms. These estimates allow one not only to select the significant transform coefficient packets a priori, but also to compute the maxima of mean-square errors of reconstruction for a given compression ratio and to compare the efficacy of different transforms based on that criterion.


Compression Ratio Discrete Fourier Transform Spectral Component Binary Representation Orthogonal Wavelet 
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  1. Agaian, S., and Petrosian, A. (1991). On the Relationship Between Compression Coefficient, Precision of Reconstruction and Complexity of Calculations. Journal of Information Processing and Cybernetics EIK, 27(91) 7, Berlin, Germany, pp. 335–345.zbMATHGoogle Scholar
  2. Ahmed, N., Rao, K.R. (1975) . Orthogonal Transforms for Digital Signal Processing. Springer- verlag. zbMATHCrossRefGoogle Scholar
  3. Bellman, R., Dreyfus, S. (1962). Applied Dynamic Programming. Princeton University Press, Princeton, NJ.zbMATHGoogle Scholar
  4. Daubechies, I. (1992). Ten lectures on Wavelets. Soc. for Industrial and Applied Math., Philadelphia, PA.zbMATHCrossRefGoogle Scholar
  5. Golubov, B., Yefimov, A., Skvortsov, B. (1987). Walsh Series and Transforms: Theory and Applications. Nauka, Moscow, (in Russian) .zbMATHGoogle Scholar
  6. Haar, A. (1910). Zur Theorie der Orthogonalen Funktionensysteme. Math. Ann. 69, pp. 331–371.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Istepanian, R., and Petrosian, A. (2000) . Optimal Zonal Wavelet-based ECG Data Compression For a Mobile Telecardiology System. IEEE Transactions on Information Technology in Biomedicine, Vol. 4, N.3, pp. 200–211.CrossRefGoogle Scholar
  8. Jain, A. (1989). Fundamentals of Digital Image Processing. Prentice Hall, Englewood Cliffs, NJ.zbMATHGoogle Scholar
  9. Mallat, S. (1989). A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 11, No. 7, pp. 674–693.zbMATHCrossRefGoogle Scholar
  10. Petrosian, A. (1988). Estimates of Spectra of Discrete Fourier Transforms. Reports of the Academy of Sciences of Armenian SSR, Math. Ser., Vol. 87, N. 5, pp. 203–206 (in Russian) .MathSciNetGoogle Scholar
  11. Petrosian, A. (1989). Optimal Method of Zonal Selection for Data Compression by Discrete Hartley Transform. Proceedings of the Academy of Sciences of Arm. SSR, Tech. Sci. Ser., Vol. 42, N. 4, pp. 193–195 (in Russian).Google Scholar
  12. Petrosian, A. (1991). Optimal Zonal Coding of Digital Signals with Transform. Problemi Peredachi Informatsii, Vol. 27, N. 2, pp. 46–58 (in Russian. Translated into English in: Problems of Information Transmission, 27–2, New York, Plenum Publishing Corp., pp. 128–140).Google Scholar
  13. Petrosian, A. (1993). On Digital Signal Zonal Coding by Spectral Transform. In the book: Recent Developments in Abstract Harmonic Analysis With Applications in Signal Processing, Edited by M.R. Stojić, and R.S. Stanković, Belgrade University, Yugoslavia.Google Scholar
  14. Petrosian, A. (1996). Upper Bounds of Wavelet Spectra on the Class of Lipschitzian Signals. SPIE conference on Wavelet Applications in Signal and Image Processing, Vol. 2825, Denver, CO, pp. 834–843.Google Scholar
  15. Petrosian, A. (1998). Optimal Zonal Compression of Signals with Bounded First and Second Order Finite Differences. 8th IEEE Digital Signal Processing Workshop, Bryce Canyon, UT, pp. 234–237.Google Scholar
  16. Petrosian, A. (2000). Wavelet Zonal Compression of 1-D and 2-D Digital Signals. IASTED International Conference on Signal Processing and Communications (SPC-2000), Marbella, Spain, pp. 377–385.Google Scholar
  17. Rademacher, H. (1922). Einige Sätze von allgemeinen Orthogonalfunktionen. Math. Ann. 87, pp. 122–138.MathSciNetCrossRefGoogle Scholar
  18. Shapiro, J. (1993). Embedded Image Coding Using Zerotrees of Wavelet Coefficients, IEEE Transactions on Signal Processing, Vol. 41, No. 12, pp. 3445–3462.zbMATHCrossRefGoogle Scholar
  19. Tikhomirov, V. M. (1976). Some problems of Approximation Theory. Moscow State University (in Russian).Google Scholar
  20. Walsh, J. (1923). A Closed Set of Orthogonal Functions. Amer. J. of Mathematics, 45, pp. 5–24.zbMATHCrossRefGoogle Scholar
  21. Wickerhauser, M. (1994). Adapted Waveform Analysis: from Theory to Software. A K Peters, Ltd., Wellesley, MA.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Arthur Petrosian
    • 1
  1. 1.Department of Electrical EngineeringTexas Tech UniversityLubbockUSA

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