Adaptive Gaussian Filter for Image Denoising

Question

I am looking for methods to enhance noisy images, where:

• some pixels in the image are very noise,
• some other pixels do not contain so much noise.

My first thought is to build an adaptive Gaussian filter. This means that the Gaussian kernel will depend on the (estimated) noise status of the current pixel (large radius of the Gaussian kernel for noisy pixels and a smaller in the converse case).

Could you help me in calculating the kernel values?

Answer 1

It's not clear in your case that Gaussian kernels will help you.

Since you have a noise shape that is closer to salt-and-pepper noise instead of additive white noise, you may want to try other algorithms (that are somehow linked to an $L_1$ best fit instead of a least squares solution):

• you can try median filtering, since it is by design more robust to this noise shape,
• you can also try to minimize some TV-$L_1$ functional over your image, see Chambolle-Pock's paper on primal dual minimization for a concrete example,
• you also can use a bilateral filter. You will have to play with the exponential decay of the pixel similarity function ($\sigma_d$ in Wikipedia page) to get the desired behaviour.

Answer 2

So if I understand your description correctly your data looks something like this:

In this case you could try using a robust Gaussian smoothing. This involves an extra weight term to discard 'outliers'. There are many possible ways to define the outliers using deviation from the local mean (over the kernel) greater than $\pm 3 \sigma$ is a common way. Unfortunately I can't seem to find any decent papers or articles describing the method in more detail.