A binary wavelet decomposition of binary images

We construct a theory of binary wavelet decompositions of finite binary images. The new binary wavelet transform uses simple module-2 operations. It shares many of the important characteristics of the real wavelet transform. In particular, it yields an output similar to the thresholded output of a real wavelet transform operating on the underlying binary image. We begin by introducing a new binary field transform to use as an alternative to the discrete Fourier transform over GF(2). The corresponding concept of sequence spectra over GF(2) is defined. Using this transform, a theory of binary wavelets is developed in terms of two-band perfect reconstruction filter banks in GF(2). By generalizing the corresponding real field constraints of bandwidth, vanishing moments, and spectral content in the filters, we construct a perfect reconstruction wavelet decomposition. We also demonstrate the potential use of the binary wavelet decomposition in lossless image coding.

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

[2]  Robert L. Grossman,et al.  Wavelet transforms associated with finite cyclic groups , 1992, [1992] Conference Record of the Twenty-Sixth Asilomar Conference on Signals, Systems & Computers.

[3]  Henning F. Harmuth,et al.  Transmission of information by orthogonal functions , 1969 .

[4]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[5]  D. L. Donoho,et al.  Ideal spacial adaptation via wavelet shrinkage , 1994 .

[6]  Joseph A. O'Sullivan,et al.  A method of sieves for multiresolution spectrum estimation and radar imaging , 1992, IEEE Trans. Inf. Theory.

[7]  Srinath Hosur,et al.  Recent progress in the application of wavelets in surveillance systems , 1994 .

[8]  Pierre Moulin.  Wavelet thresholding techniques for power spectrum estimation , 1994, IEEE Trans. Signal Process..

[9]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[10]  Kannan Ramchandran,et al.  Tilings of the time-frequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms , 1993, IEEE Trans. Signal Process..

[11]  T. W. Parks,et al.  Time-frequency concentrated basis functions , 1990, International Conference on Acoustics, Speech, and Signal Processing.

[12]  Rolf Johannesson,et al.  Algebraic methods for signal processing and communications coding , 1995 .

[13]  Andrew F. Laine,et al.  Wavelet descriptors for multiresolution recognition of handprinted characters , 1995, Pattern Recognit..

[14]  Stéphane Mallat,et al.  Characterization of Signals from Multiscale Edges , 2011, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  Adrian S. Lewis,et al.  Image compression using the 2-D wavelet transform , 1992, IEEE Trans. Image Process..

[16]  P. P. Vaidyanathan,et al.  Unitary and paraunitary systems in finite fields , 1990, IEEE International Symposium on Circuits and Systems.

[17]  Martin Vetterli,et al.  Hartley Transforms over Finite Fields , 1990, 1990 Conference Record Twenty-Fourth Asilomar Conference on Signals, Systems and Computers, 1990..

[18]  P. Vaidyanathan.  Multirate Systems And Filter Banks , 1992 .

[19]  Jerome M. Shapiro,et al.  Embedded image coding using zerotrees of wavelet coefficients , 1993, IEEE Trans. Signal Process..

[20]  Srinath Hosur,et al.  Recent progress in the application of wavelets in surveillance systems , 1994, Defense, Security, and Sensing.

[21]  Ronald R. Coifman,et al.  Entropy-based algorithms for best basis selection , 1992, IEEE Trans. Inf. Theory.

[22]  Gregory W. Wornell,et al.  Wavelet-based representations for a class of self-similar signals with application to fractal modulation , 1992, IEEE Trans. Inf. Theory.

[23]  Tariq S. Durrani,et al.  IFS fractals and the wavelet transform , 1990, International Conference on Acoustics, Speech, and Signal Processing.

[24]  P. P. Vaidyanathan,et al.  Paraunitary filter banks over finite fields , 1997, IEEE Trans. Signal Process..

[25]  Dennis M. Healy,et al.  Wavelet transform domain filters: a spatially selective noise filtration technique , 1994, IEEE Trans. Image Process..