Filtering

Averaging

When does it make sense to average in order to smooth out noise? Assumptions:

Images are smooth/continuous in general. That is, true value of a pixel is similar to true value of nearby pixels.
The noise is independently distributed. So averaging out will get you the mean, which is assumed to be zero
Mean of the noise is zero. See above

Treat pixels that are further away as less important.

Non-uniform is smoother.

Cross-correlation is literally just a weighted moving average.

Say you are finding the target G[i, j], the target pixel at i, j, and your source is F[i, j].

You have a parameter k which is the amount to go left and right, e.g. a k of 1 means a 3x3 mask. So you sum them together:

G[i, j] = sum of (H[u, v] * F[i + u, j + v]) for u in [-k, k] for v in [-k ,k]

That's it. e.g. H may be:

  |-1  0  1
--+--------
-1| 1  1  1
 0| 1  5  1
 1| 1  1  1

(but you'd divide by 13 to sum them to 1).

G here is known as the cross-correlation of H and F, aka G = H ⨂ F.

H is known as the mask or kernel. It's just the matrix of linear weights that is used.

This is a filter where the mask/kernel is an approximation of a Gaussian function.

In 2 dimensions the circularly symmetric Gaussian function is:

This filter is isotropic, which is just a fancy word for "circularly symmetric".

So any isotropic filter, is circularly symmetric.