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Digital images are claiming an increasingly larger portion of the information world. The growth of the Internet, along with more powerful and affordable computers and continuing advances in the technology of digital cameras, scanners and printers, has led to the widespread use of digital imagery. As a result, there is renewed interest in improving algorithms for the compression of image data. Compression is important both for speed of transmission and efficiency of storage. In addition to the many commercial uses of compression technology, there is also interest among military users for applications such as the downlink of image data from a missile seeker and for archival storage of image data, such as terrain data, for defense-related simulations. The problem of image compression or, more generally, image coding, has made use of, and stimulated, many different areas of engineering and mathematics. This book focuses on two relatively new areas of mathematics that have contributed to recent research in image compression: fractals and wavelets. Recognition of structure in data is a key aspect of efficiently representing and storing that data. Fractal encoding and wavelet transform methods take two different approaches to discovering structure in image data. Barnsley and Sloan (1988,1990) first recognized the potential of applying the theory of iterated function systems to the problem of image compression. They patented their idea in 1990 and 1991. Jacquin (1992) introduced a method of fractal encoding that utilizes a system of domain and range subimage blocks. This approach is the basis for most fractal encoders today. It has been enhanced by Fisher and a number of others (Fisher 1995; Jacobs, Boss, and Fisher 1995). This block fractal encoding method partitions an image into disjoint range subimages and defines a system of overlapping domain subimages. For each range, the encoding process searches for the best domain and affine transformation that maps that domain onto the range. Image structure is mapped onto the system of ranges, domains and transformations. Much of the recent research in fractal image compression has focused on reducing long encoding times. Feature extraction and classification of domains are two techniques that have proved to be successful. This book includes techniques and discusses recent results (Bogdan and Meadows 1992; Saupe 1994; Bani-Eqbal 1995; Hamzaoui 1995; Welstead 1997) for improving fractal image encoding performance. Wavelet transform approaches to image compression exploit redundancies in scale. Wavelet transform data can be organized into a subtree structure that can be efficiently coded. Hybrid fractal-wavelet techniques (Davis 1998; Hebert and Soundararajan 1998) apply the domain-range transformation idea of fractal encoding to the realm of wavelet subtrees. The result is improved compression and decoded image fidelity. This chapter provides background material on some general topics related to image coding. Following an overview of the image compression problem, we will briefly look at information theory and entropy, scalar and vector quantization, and competing compression technologies, such as those of the Joint Photographic Experts Group (JPEG).
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