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Special-purpose noise function

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Aight. So what I need to implement is a discrete 2D noise function. It can have a normal distribution, a flat distribution, whatever. But what I really need is for the mean value within all circles of a particular size (say, a 5-pixel radius or so) to get as close to the mean for the entire function as possible. That is, there can't be any clumps of white pixels. I've thought about ways to accomplish this, and I have some ideas I think are workable, but I'm sure that much smarter people than I have trodden this ground before. So far, googling has not borne fruit; I just don't know what keywords to describe this with. Anyone come across this before?

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