# Probability to locate linked pixels

Say I have an image, with pixels that can be either $0$ or $1$. For simpleness, think it's a $2D$ image (though I would certainly want a $3D$ remedy too ).

A pixel has $8$ next-door neighbors (if that's also difficult, we can go down to $4$-connectedness ). 2 bordering pixels with value $1$ are taken into consideration to be attached.

If I recognize the probability $p$ that a specific pixel is $1$, and also if I can think that all pixels are independent, the amount of teams of at the very least $k$ linked pixels should I anticipate to locate in a photo of dimension $n\times n$?

What I actually require is an excellent way of computing the probability of $k$ pixels being attached offered the specific pixel chances. I have actually begun to list a tree to cover all the opportunities approximately $k=3$, yet also after that, it comes to be actually hideous actually quickly. Exists an extra brilliant means to deal with this?

This looks a little bit like percolation concept to me. In the 4 - neighbor instance, if you consider the twin of the photo, the opportunity that a side is attached (runs in between 2 pixels of the very same colour) is `1-2p+2p^2`

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I do not assume you can get wonderful shut - kind solution for your inquiry, yet possibly a computer system can aid with some Monte Carlo simulation?

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