Adaptive Gaussian Filter for Image Denoising
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.