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what happens if one divides an image into a collection of nested squares, but imposes a lower limit on the size of these squares. And what one sees is that as the lower limit is increased, the amount of compression increases rapidly—though at the cost of a correspondingly rapid decrease in the quality of the image.

So can one do better at maintaining the quality of the image? Various schemes are used in practice, and almost all of them are based on the idea from traditional mathematics that by viewing data in terms of numbers it becomes possible to decompose the data into sums of fixed basic forms—some of which can be dropped in order to achieve compression.

The pictures on the previous page show an example of how this works. On the top left is a set of basic forms which have the property that any two-dimensional image can be built up simply by adding together these forms with appropriate weights. On the top right these forms are then ranked roughly from coarsest to finest. And given this ranking, the arrays of pictures at the bottom show how two different images can be built up by progressively adding in more and more of the basic forms.

If all the basic forms are included, then the original image is faithfully reproduced. But if one drops some of the later forms—thereby reducing the number of weights that have to be specified—one gets only an approximation to the image. The facing page shows what happens to a variety of images when different fractions of the forms are kept.

Images that are sufficiently simple can already be recognized even when only a very small fraction of the forms are included—corresponding to a very high level of compression. But most other images typically require more forms to be included—and thus do not allow such high levels of compression.

Indeed the situation is very much what one would expect from the definition of complexity that I gave two sections ago. The relevant features of both simple and completely random images can readily be recognized even at quite high levels of compression. But images that one would normally consider complex tend to have features that cannot be recognized except at significantly lower levels of compression.

All the pictures on the facing page, however, were generated from the specific ordering of basic forms shown on the previous page. And

From Stephen Wolfram: A New Kind of Science [citation]