Say I have an image, with pixels that can be either 0 or 1. For simplicity, assume it's a 2D image (though I'd be interested in a 3D solution as well).
A pixel has 8 neighbors (if that's too complicated, we can drop to 4-connectedness). Two neighbouring pixels with value 1 are considered to be connected.
If I know the probability
p that an individual pixel is 1, and if I can assume that all pixels are independent, how many groups of at least
k connected pixels should I expect to find in an image of size
What I really need is a good way of calculating the probability of
k pixels being connected given the individual pixel probabilities. I have started to write down a tree to cover all the possibilities up to
k=3, but even then, it becomes really ugly really fast. Is there a more clever way to go about this?