Examples of how images can be built up by adding together basic forms consisting of so-called two-dimensional Walsh functions. On the top left the basic forms are given in so-called sequency order. On the top right they are reordered roughly so as to go systematically from coarser to finer. In the bottom arrays of pictures each successive picture is obtained by adding in the corresponding basic form with an appropriate weight. The basic forms shown here have the property of being orthogonal, so that the weight for each form can be deduced simply by multiplying the original image by that form. Note that the forms involve numerical values -1 and +1, corresponding to cells colored white and black. The images shown here are all rescaled so that smallest values are white and largest black. The JPEG method of image compression uses an approach similar to the one shown here, though with basic forms that have continuous levels of gray, rather than just black and white.

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