Wavelets in Signal and Image Analysis pp 245-280 | Cite as

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

## Abstract

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.

## Keywords

Compression Ratio Discrete Fourier Transform Spectral Component Binary Representation Orthogonal Wavelet## Preview

Unable to display preview. Download preview PDF.

## References

- 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 - Ahmed, N., Rao, K.R. (1975) . Orthogonal Transforms for Digital Signal Processing.
*Springer- verlag*. zbMATHCrossRefGoogle Scholar - Bellman, R., Dreyfus, S. (1962). Applied Dynamic Programming.
*Princeton University Press*, Princeton, NJ.zbMATHGoogle Scholar - Daubechies, I. (1992). Ten lectures on Wavelets.
*Soc. for Industrial and Applied Math.*, Philadelphia, PA.zbMATHCrossRefGoogle Scholar - Golubov, B., Yefimov, A., Skvortsov, B. (1987). Walsh Series and Transforms: Theory and Applications.
*Nauka*, Moscow, (in Russian) .zbMATHGoogle Scholar - Haar, A. (1910). Zur Theorie der Orthogonalen Funktionensysteme.
*Math. Ann.*69, pp. 331–371.MathSciNetzbMATHCrossRefGoogle Scholar - 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 - Jain, A. (1989). Fundamentals of Digital Image Processing.
*Prentice Hall*, Englewood Cliffs, NJ.zbMATHGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Rademacher, H. (1922). Einige Sätze von allgemeinen Orthogonalfunktionen.
*Math. Ann.*87, pp. 122–138.MathSciNetCrossRefGoogle Scholar - 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 - Tikhomirov, V. M. (1976). Some problems of Approximation Theory.
*Moscow State University*(in Russian).Google Scholar - Walsh, J. (1923). A Closed Set of Orthogonal Functions.
*Amer. J. of Mathematics*, 45, pp. 5–24.zbMATHCrossRefGoogle Scholar - Wickerhauser, M. (1994). Adapted Waveform Analysis: from Theory to Software.
*A K Peters, Ltd.*, Wellesley, MA.Google Scholar