Stefan's Site

On the Vast Number of Finitely Possible Images - Part 2

I did some more digging into this subject trying to understand the complexity better. My reasoning starts with two color images and how many combinations there are depending on the number of white versus black pixels. It turns out that the distribution follows a binomial function no matter for any image dimensions. The following set of images shows the base 10 logarithms of the number of possible combinations as a function of the number of white pixels.

Combinations 2

For a 64 by 64-pixel image, there are around \(10^{1200}\) possible combinations when 2048 out of 4096 pixels are white.

The distribution is the same for more colors, but it becomes harder to visualize. For a 4 by 4-pixel image with three possible colors the number of combinations varies according to the following figure:

Combinations 2 (3D)

For a 2 by 2-pixel image with two colors all combinations can be quickly rendered:

Permutations 2x2

and all combinations for a 3 by 3-pixel image are also fast to generate.

Permutations 3x3

For 4 by 4-pixel images, it is not realistic to render all combinations anymore due to the large volume of pictures.