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Image and Videl Comoression P8

During the last decade, a number of signal processing applications have emerged using wavelet theory. Among those applications, the most widespread developments have occurred in the area of data compression. Wavelet techniques have demonstrated the ability to provide not only high coding efficiency, but also spatial and quality scalability features. In this chapter, we focus on the utility of the wavelet transform for image data compression applications. 8.1 REVIEW OF THE WAVELET TRANSFORM 8.1.1 DEFINITION AND COMPARISON WITH SHORT-TIME FOURIER TRANSFORM The wavelet transform, as a specialized research field, started over a decade ago (Grossman and Morlet, 1984). To better understand the theory of wavelets, we. | Q Wavelet Transform for Image Coding During the last decade a number of signal processing applications have emerged using wavelet theory. Among those applications the most widespread developments have occurred in the area of data compression. Wavelet techniques have demonstrated the ability to provide not only high coding efficiency but also spatial and quality scalability features. In this chapter we focus on the utility of the wavelet transform for image data compression applications. 8.1 REVIEW OF THE WAVELET TRANSFORM 8.1.1 Definition and Comparison with Short-Time Fourier Transform The wavelet transform as a specialized research field started over a decade ago Grossman and Morlet 1984 . To better understand the theory of wavelets we first give a very short review of the Short-Time Fourier Transform STFT since there are some similarities between the STFT and the wavelet transform. As we know the STFT uses sinusoidal waves as its orthogonal basis and is defined as F w t I f t w t -t e jwtdt - 8.1 where w t is a time-domain windowing function the simplest of which is a rectangular window that has a unit value over a time interval and has zero elsewhere. The value t is the starting position of the window. Thus the STFT maps a function f t into a two-dimensional plane w t . The STFT is also referred to as Gabor transform Cohen 1989 . Similar to the STFT the wavelet transform also maps a time or spatial function into a two-dimensional function in a and t w and t for STFT . The wavelet transform is defined as follows. Let f t be any square integrable function i.e. it satisfies r I f t dt 8.2 - The continuous-time wavelet transform of f t with respect to a wavelet y t is defined as w - fLy dt 8.3 where a and t are real variables and denotes complex conjugation. The wavelet is defined as y J0 a 2 v 1 8.4 2000 by CRC Press LLC The above equation represents a set of functions that are generated from a single function y t by dilations and translations. The variable t .