Coding properties of time-warped polynomial transforms

Abstract This paper describes a general class of orthogonal transforms, which includes the well-known DCT. Each transforms is completely determined by two functions: a warping and a weight function. Therefore, the transforms can be described very efficiently and this makes them very useful in adaptive transform coding. The base vectors of the new transforms are samples of time-warped polynomials. The latter approximately equal amplitude- and frequency-modulated sine waves. The paper describes how the structure of the base vectors and the coding properties of the transforms depend on the warping and weight functions. In particular, the minimum number of coefficients for almost error-free coding is determined and the effects of quantization are investigated. ECG, image line and speech coding examples illustrate the theoretical results. Finally, the time-warped polynomial coding method is shown to have some advantages over another approach that uses a fixed base to represent a time-warped version of the signal.

[1]  F. Hlawatsch,et al.  Linear and quadratic time-frequency signal representations , 1992, IEEE Signal Processing Magazine.

[2]  Thomas Engelhardt,et al.  Coding of arbitrarily shaped image segments based on a generalized orthogonal transform , 1989, Signal Process. Image Commun..

[3]  Zoran Cvetkovic,et al.  New fast recursive algorithms for the computation of discrete cosine and sine transforms , 1992, IEEE Trans. Signal Process..

[4]  Benoit M. Macq,et al.  Signal-adapted multiresolution transform for image coding , 1992, IEEE Trans. Inf. Theory.

[5]  Norman Morrison,et al.  Introduction to Sequential Smoothing and Prediction , 1969 .

[6]  Zhongde Wang.  Fast algorithms for the discrete W transform and for the discrete Fourier transform , 1984 .

[7]  Jimmie D. Eggerton,et al.  A Visually Weighted Quantization Scheme for Image Bandwidth Compression at Low Data Rates , 1986, IEEE Trans. Commun..

[8]  O. Rioul,et al.  Wavelets and signal processing , 1991, IEEE Signal Processing Magazine.

[9]  D. Slepian,et al.  On bandwidth , 1976, Proceedings of the IEEE.

[10]  W. Philips,et al.  Data compression of ECG's by high-degree polynomial approximation , 1992, IEEE Transactions on Biomedical Engineering.

[11]  L. C. Ludeman.  Tailored orthonormal sets for expansion of specific classes of functions , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[12]  R. R. Clarke.  Transform coding of images , 1985 .

[13]  James J. Clark,et al.  A transformation method for the reconstruction of functions from nonuniformly spaced samples , 1985, IEEE Trans. Acoust. Speech Signal Process..

[14]  W. A. Coberly,et al.  ECG data compression techniques-a unified approach , 1990, IEEE Transactions on Biomedical Engineering.

[15]  A. Sandman,et al.  Third order polynomial—its use in data compression , 1988 .

[16]  N. Garguir.  Comparative Performance of SVD and Adaptive Cosine Transform in Coding Images , 1979, IEEE Trans. Commun..

[17]  Jean-Bernard Martens,et al.  The Hermite transform-theory , 1990, IEEE Trans. Acoust. Speech Signal Process..

[18]  Oleh John Tretiak.  Rate Distortion Theory: A Mathematical Basis for Data Compression, Toby Berger. Prentice-Hall, Urbana, IL (1971) , 1974 .

[19]  W. Philips,et al.  ECG data compression with time-warped polynomials , 1993, IEEE Transactions on Biomedical Engineering.

[20]  Anil K. Jain,et al.  Recursive Block Coding - A New Approach to Transform Coding , 1986, IEEE Transactions on Communications.

[21]  Hsieh S. Hou.  A fast recursive algorithm for computing the discrete cosine transform , 1987, IEEE Trans. Acoust. Speech Signal Process..

[22]  The Approximate Approach to Orthogonal Polynomials for Weights On ( , 1990 .

[23]  Jean-Bernard Martens.  The Hermite transform-applications , 1990, IEEE Trans. Acoust. Speech Signal Process..

[24]  Peter No,et al.  Digital Coding of Waveforms , 1986 .