Tuesday, June 06, 2006

The Probability of Order

In the previous post, I talked about the problems I ran into tiling random noise and how it produces patterns. The first possible caveat might be a flaw in my random number generator. Maybe I should go back and get data from, but I don't believe that will make the problem go away. Another problem with all this is that it's not double-blind--I really should have someone else blindly deduce the tile dimensions of the resultant matrix.

One thing interesting regarding my own perceptions is that I see patterns in the noise, and that's the problem. I see a repeating dark spot of sorts and sort of a lighter swoosh beside repeated four times in both dimensions. If I had time, I'd circle these to spots and compare them with the general population of pixels in the tile. I'm willing to bet there will be a statistically significant difference.

Please excuse all the speculation, but raising questions regarding randomness, order, perception and how the brain works is a much greater objective here than finding answers.

My conclusion, and it very well may be wrong, is this. Given a tile of random pixels, there's a high probability of it containing some sort of perceptible order--e. g., perceptible enough to make tiling random noise an unworkable means of generating a significantly large image that appears random.

A more general (and perhaps too general) question is this:

Given randomness, what is the probability of order?

So far I've yet to run into this question or any related answer, but it may simply be the result of my own ignorance. If you stumble onto this post, and you have any good pointer, comments are most welcome.

Order seems really hard to quantify. One can calculate the entropy of a distribution. Intuitively, I suppose one could set some threshold for the entropy of outputs and calculate the probability given the distribution of the inputs.

On the way home I started thinking of a 3 x 3 grid in which each cell could be colored black or white. There are 512 ways of filling in such a matrix (2^9). If we define order as a three unit straight line, then we have 10 possibilities (3 horizontal, 3 vertical, 2 diagonal). Given all that, the probability of order given randomness is 10 / 512. That's a simple random system with some definition of order and a probability of order occurring, as it's defined.

I'm left curious and pondering how the brain processes visual information, how it finds patterns and order (even out of mathematical randomness).


Anonymous Mark VandeWettering said...

There are typically two sources of problems like this: one is random numbers that aren't very random, and the other is periodicity.

If you check out this link, you'll see images like this one, which is a problem with a hash function, which displays strong linear features. When you do it right, you get a lot nicer images.

Your problem isn't really the tiles, it's simply that your eyes are very good at finding periodic patterns. There will be small regions that are darker or lighter, and the repetition of these at regular intervals is something that your eye is very good at spotting.

10:28 PM  
Blogger metamerist said...

The fact that our eyes and brain are so adept at finding these periodic patterns is what really amazes me. I've generated a second example using random nubmers from I'll post it next.

9:35 PM  

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