Nonlinear Discrete Wavelet Transforms over Finite Sets and an Application to Binary Image Compression
- 81 Downloads
The discrete wavelet transform was originally a linear operator that works on signals that are modeled as functions from the integers into the real or complex numbers. However, many signals have discrete function values. This paper builds on two recent developments: the extension of the discrete wavelet transform to finite-valued signals and the research of nonlinear wavelet transforms triggered by the introduction of the lifting scheme by Sweldens. It defines discrete wavelet transforms as bijective, translation-invariant decompositions of signals that are functions from the integers into any finite set. Such transforms are essentially nonlinear, but they can be calculated very time efficiently since only discrete arithmetic is needed. Properties of these generalized discrete wavelet transforms are given along with an elaborate example of such a transform. In addition, the paper presents some ideas to find explicit examples of discrete wavelet transforms over finite sets. These ideas are used to show that, in case the finite set is a ring, there are much more nonlinear transforms than linear transforms. Finally, the paper exploits this increased number of transforms to do lossless compression of binary images.
Keywordssignal processing nonlinear discrete wavelet transforms over finite sets second generation wavelets perfect reconstruction filter banks
Unable to display preview. Download preview PDF.
- 3.I. Daubechies and W. Sweldens, “Factoring wavelet transforms into lifting steps,” Journal of Fourier Analysis and Applications, Vol. 4, No. 3, pp. 245–267, 1998.Google Scholar
- 5.O.N. Gerek, M.N. Gürcan, and A.E. Ĉetin, “Binary morphological subband decomposition for image coding,” in International Symposium on Time-Frequency and Time-Scale Analysis. IEEE, 1996.Google Scholar
- 7.L. Kamstra, “Discrete wavelet transforms over finite sets which are translation invariant,” Technical Report PNA-R0112, CWI, Amsterdam, The Netherlands, 2001.Google Scholar
- 9.A. Klappenecker, F.U. May, and A. Nückel, “Lossless image compression using wavelets over finite rings and related architectures,” in Akram Aldroubi, Andrew F. Laine, and Michael A. Unser, Eds. Wavelet Applications in Signal and Image Processing V, vol. 3169, SPIE, October 1997, pp. 139–147.Google Scholar
- 10.S. MacLane and G. Birkhoff, Algebra, MacMillan, New York, 2nd edition, 1979.Google Scholar
- 11.S. Mallat, A Wavelet Tour of Signal Processing, Academic Press: San Diego, 2nd edition, 1999.Google Scholar