Relation between variance of gaussian noise and intensity variance in Bilateral filter

Question

So I implemented the code for bilateral filter from here and ran it for various values of spatial sigma and intensity sigma. I noticed that when I add gaussian noise to an image with variance say 0.02, and then try to filter it with bilateral filter, and if I enter value of spatial sigma say 18 and of intensity sigma as 0.02, the filtered image looks apparently same as the noisy one.

I am guessing that 'intensity sigma' (or sigma r as mentioned in this paper) tells us the difference in intensity levels between neighbouring pixels that we are allowing (though I am not sure if this is correct). So when we are smoothing with sigma r value same as variance of gaussian noise, essentially it makes no change since the intensity difference in the noisy image is within the same bounds as allowed by our bilateral filter.

It smooths the picture nicely with sigma r = 0.25.

Answer 1

You may and should read 2 questions about the bilateral filter:

• Understanding the Parameters of the Bilateral Filter.
• Understanding the Bilateral Filter - Neighbors and Sigma.

Once you cover that, let's address your specific question.
Indeed in the course notes A Gentle Introduction to Bilateral Filtering and its Applications by Sylvain Paris, Pierre Kornprobst, Jack Tumblin and Fredo Durand they suggest, in section 4.2 a simple linear connection between the local noise standard deviation to the local range standard deviation $ {\sigma}_{r} = 1.95 {\sigma}_{n} $.

A practical trick to have relatively small value yet iterate the same filter several iterations as shown in in Analysis of the Bilateral Filter by Michael Elad. He analyzes the effects on piece wise constant images, something you may replicate easily and tune the parameters.

At the end, there is no closed form solution here. It is a balance hence a pretty subjective decision.

Remark: Basically, what you are asking is how to tune the denoising filter by the noise level. Something many commercial algorithms did during the previous decade or 2. Specifically with Bilateral Filter which was popular (Mostly where the time and hardware are limited: Digital Cameras).