NOISE AND FOURIER SERIES.

Processes where the some value such as color is assigned randomly for each region in an image are of some interest, but often one wants to have random colors which glide smoothly and continuously from one value to another across the image, i.e., the RGB values are to be smooth functions of the picture coordinates x and y.

The artist who wishes to explore these techniques should be prepared to write computer code.

This can be done in a straightforward way using separate two-dimensional Fourier series (with random coefficients) for R, G, and B. There is nothing very mysterious about this because such series have a huge literature and a wide field of application in applied mathematics and engineering. What is novel here is the use of such series ("noise functions") in color space.

Such a series is a sum over products of terms cos(kx*x+px) and cos(ky*y+py) where the kx and ky values are 2*pi/wavelength and px and py are random values of the phase lying between 0 and 2*pi. The coefficients in front of these terms have random values. The kx,ky values are to be chosen such that an even number of wavelengths fit into the width and height (respectively) of the picture. Such series are periodic.

The series used here represent "bandwidth-limited noise" where the number of wavelengths across the picture ranges from 1 up to some number N (in general N is different in the x and y directions).

The coefficients are chosen at random within some statistical distribution, and after they have been chosen it is quite useful to "normalize" them in such a way that the highest value of the summed series within the fundamental region is 1.0.

The simplest approach would be to choose all of the coefficients according to some statistical distribution which is the same for all k values. In practice this leads to a pattern where the shortest wavelengths dominate. This has been dealt with here by having the average values of the coefficients fall off according to some negative-exponent power law in k. Choice of this "roll-off" exponent can make quite visible differences in the visual appearance. Such "power-law" tailing off of the coefficients is seen in a number of kinds of naturally-occurring noise.

The use of various schemes for adjusting the coefficients is called "filtering" and has a very large literature. The simple scheme used here barely scratches the surface of the possibilities.