Wavelet Data Compression

There are two kinds of compression—“lossless ”and“lossy. ”In this chapter, we describe what these terms mean, and then describe in more detail examples of lossy compression algorithms where the discrete wavelet transform plays a critical role. The basic idea of compression is to use fewer bits to represent the same information as some given representation (lossless compression) or to use fewer bits to represent the given data approximately. The localization of the wavelet transform allows most of the energy of a signal to be concentrated among a small subset of the wavelet transform coefficients for structured signals, such as images which can be recognized by a human being. The use of either type of compression can become an economic factor for either archiving or transmitting data, and the explosive growth of the Internet and graphical representations of information is causing these issues to become more and more important. In this chapter we discuss the role of wavelets as applied to lossy compression. It is evolving into an important technology for image manipulation. Another type of lossy compression goes under the label “denoising. ” This type of specialized compression takes a given signal with noise in it and throws away some of the bits but just the right amount and in such a way that the reconstructed image is “essentially free ” of the noise present in the original signal. An important paradigm for denoising using the discrete wavelet transform, developed by Donoho and collaborators, is discussed briefly in the final section of this chapter with references to current literature.