One of the central issues in information technology is the representation of data by arrays of bits in the most efficient way possible, a never-ending quest for improvement in the representation of bits that are smaller, faster and cheaper. This is exactly the role of data compression: to convert strings of bits into shorter ones for more economical transmission, storage and processing. Abundant applications require such compression process: medical imaging, publishing, graphic arts, digital photography, wire photo transmission, and so on.

For the past few years, the Joint Photographic Experts Group (JPEG) has been working to keep an international compression standard for both, grayscale and color images. It is no surprise that strong mathematical research in this direction has been going on since then, and it is important to remark that when JPEG conducted a first selection process in 1988, they reported that a proposal based on the discrete cosine transform had produced the best picture quality. As a matter of fact, JPEG is a format for image compression based on the discrete cosine transform, which is used to reduce the file size of an image as much as possible without affecting the quality of the image as experienced by the human sensory system.

In this chapter, we present an elegant application of mathematical tools and concepts (in particular from linear algebra and numerical analysis) to the problem of image compression, and we illustrate how ...

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