The wavelet transform

Conceptions of the wavelet transform (WT) have been used and developed for seismic signal analysis in the field of exploration for oil. J.Morlet introduced these conceptions as wavelets in 1982. The mathematical foundations of WT were studied by A. Grossmann, Y. Meyer and S. Mallat. In 1988,I. Daubechies provided orthonormal sets of wavelets with compact support[38]. The WT is effective for the analysis of the varying signal. The frequency of signal in the localized time is characterized by using the WT. The mother wavelet h(t) produces daughter wavelets by dilation and translation. The daughter wavelets are written as:

, (6)

where a> 0 is dilation a parameter and b is a translation parameter. The WT of a signal f(t) is defined by

. (7)

This definition shows that the WT is processed by correlation between signal and dilated wavelets. The inverse WT is defined by

, (8)

where C is the normalization constant defined as

. (9)

The inverse WT requires C< . H(v) is the Fourier transform of h(t). For optical implementations and other applications, discrete wavelet transform (DWT) is useful, because only the DWT can be orthogonal. The orthogonal DWT can be made with discrete dilation and translation parameters. Substituting (a,b) by , the orthogonal discrete wavelets is given by

. (10)

On a multiresolution analysis (MRA), a signal is decomposed into linear combinations of subspaces that are composed with linear combinations of uniform-scale discrete wavelet bases. The MRA is applied for an image compression with wavelet . Many optical wavelet systems are introduced in recent years. In Ref.39, the advantages of WT over the Fourier transform and the windowed Fourier transform were suggested, and one-dimensional WT was implemented by two-dimensional optical correlator with a spatial light modulator[40],[41]. Two procedures of two dimensional optical WT is introduced by SLM of phase-only mode and thermoplastic hologram in Ref.42. The applicapability of the optical wavelet systems has been also studied in the pattern recognition[43],[44], the synthetic aperture radar[45], the image processing and so on.

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6.D. Roberge and Y. Sheng, "Optical composite wavelet-matched filters," Opt. Eng. 33, 2290-2295 (1994).
7. T. Chao, B. Lau and W. Miceli, " Optical implementation of a matching pursuit for image representations," Opt. Eng. 33, 2303-2309 (1994).
8. M. Sanghadasa, P. Erbach, C. Sung, D. Gregory and W.Friday, "Wave let transform applied to synthetic aperture radar-optical implementation and adaptive techniques," Opt. Eng. 33, 2282-2289 (1994).
9.T. Poon and K. Ho, "Real-time optical image processing using difference-of-Gaussians wavelets," Opt. Eng. 33, 2296-2302 (1994).

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