Application of Kolmogorov Complexity to Image Compression: It Is Possible to Have a Better Compression, But It Is Not Possible to Have the Best One

1 The Problem of Image Compression Image processing is important but diicult. An important part of data processing is processing images. One of the main problems with storing and processing images is that an image contains a large amount of data. A simple image on a PC contains about 1 Megabyte of data, and a good photo contain thousand times more: about 1 Gigabyte (10 9 bytes) of data. These problems are especially acute for Geographic Information Systems (GIS), where hundreds and thousands of images are stored. Some image processing algorithms are reasonably simple (e.g., detecting a certain pattern in weather analysis). These algorithms usually do not require large amounts of computations. However, there are many other algorithms, especially related to geophysical and environmental applications , where the (directly observable) data supplied by the image serves as an input to a system of partial diierential equations, which are used to determine the desired quantities. Image compression is needed. The data amount makes it very diicult to store, transmit, or process an image pixel-by-pixel. Instead, diierent image compression schemes are used that compress the data into a usually shorter le. There exist diierent image compression schemes; most of these methods are based on some numerical methods applied to the original image: e.g., instead of storing the image, we expand the image (or its parts) into Fourier (or wavelet) series and store the coeecients. Among the most widely used, we can name gif, We want a good (ideally, the best) image compression scheme. On each image, some image compression schemes compress better, some compress worse. It is desirable to have as good a compression scheme as possible. Ideally, we would like to have the best compression scheme, i.e., a scheme that compresses better than all the others. This ideal \best" compression scheme may not exit or may not be (easily) computable. In this case, we would like, at least, to nd the best among the existing schemes. In some cases, we know the best compression schemes, but they may not be the best in a general situation. For some speciic problems, the best image compression scheme is known. For example, for a problem of detecting the Surface Mounted Devices on a photo of a manufactured computer chip, wavelet coeecients provide the best image compression (see, e.g., 2, 1]; for 1-D images, see 4]). In more precise terms, this image compression is optimal …