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We now turn our attention to an alternative approach to image compression that makes use of wavelets. Wavelet image compression belongs to the transform class of methods and as such differs from fractal methods. Nevertheless, there are fundamental connections between the two approaches. Fractal methods use self-similarity across different scales to reduce stored information. Wavelet methods exploit redundancies in scale to reduce information stored in the wavelet transform domain. Hybrid methods apply fractal techniques to information in the wavelet transform domain to provide even greater compression performance. This chapter introduces wavelet analysis through the use of the simple Haar wavelets. The following chapters look at wavelet image compression and more advanced wavelet topics. The idea behind wavelet image compression, like that of other transform compression techniques, is fairly simple. One applies a wavelet transform to an image and then removes some of the coefficient data from the transformed image. Encoding may be applied to the remaining coefficients. The compressed image is reconstructed by decoding the coefficients, if necessary, and applying the inverse transform to the result. The hope is that not too much image information is lost in the process of removing transform coefficient data. Fig. 5.1.1 illustrates this process. The process of image compression through transformation and information removal provides a useful framework for understanding wavelet analysis. The literature contains a number of different approaches to the study of wavelets, some of which may seem daunting to the aspiring wavelet analyst. Mathematical references develop multiresolution analysis through the definition of continuous scaling and wavelet functions.
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