On the Optimal Choice of Quality Metric in Image Compression: a Soft Computing Approach I. Formulation of the Problem

| Images take lot of computer space; in many practical situations, we cannot store all original images, we have to use compression. Moreover, in many such situations , compression ratio provided by even the best lossless compression is not suucient, so we have to use lossy compression. In a lossy compression, the reconstructed image e I is, in general, diierent from the original image I. There exist many diierent lossy compression methods, and most of these methods have several tunable parameters. In diierent situations , diierent methods lead to diierent quality reconstruction, so it is important to select, in each situation, the best compression method. A natural idea is to select the compression method for which the average value of some metric d(I; e I) is the smallest possible. The question is then: which quality metric should we choose? In this paper, we show that under certain reasonable symmetry conditions , L p metrics d(I; e I) = R jI(x) ? e I(x)j p dx are the best, and that the optimal value of p can be selected depending on the expected relative size r of the informative part of the image. A. Image Compression Is Necessary Images tend to take up a lot of computer space, so in many applications, where we cannot store the original images, we must use image compression. Ideally , we would like to use a lossless compression, but unfortunately, there are serious limitations on how much we can compress without losing information. For a more radical compression, we must therefore use lossy compression schemes. In these schemes, some information about the image is lost; as a result , for every point x, the intensity e I(x) of reconstructed image e I at this point may be slightly different from the intensity I(x) of the original image I at this point. B. It Is Important to Select Optimal (Or At Least Good Enough) Compression Scheme There exist many diierent compression schemes, from standard ones like gif, jpg, jpg2000, etc., to specially designed ones. Most of these schemes comes with one or several parameters which we can select. One of the reasons why so many diierent schemes co-exist is that in diierent applications, diierent schemes (with diierent values of parameters) work better. It is vitally important to select an appropriate compression scheme, i.e., a scheme which provides the best compression ratio within the same accuracy. How …