# Denoising a Grayscale Image Using Random Matrix Theory (RMT)

I am applying RMT on grayscale images. Basically for any matrix, it consists of two parts: one is pure noise and other is information. Now, to distinguish noise from information, RMT comes handy. We calculate the eigenvalues of the covariance matrix of the original matrix and by scaling them (Unfolding), the eigenvalues are compared to the probability distribution of the eigenvalues of Random Matrix.

So, we have some eigenvalues of our original matrix in the range of minimum and maximum eigenvalues of random matrix, which says that the eigenvalue and it's corresponding eigenvector is noise.

Now to remove the noise, one method is to average out all the eigenvalues which consists of noise and reconstruct the image. Another method is to remove the eigenvectors and corresponding eigenvalues and then reconstruct the image.

Which method is the best and ideal?

• Do you mean doing this for the whole image? Commented Mar 13, 2022 at 18:05
• @EricJohnson yes, for removing the noise in the image Commented Mar 14, 2022 at 4:20
• It's probably not the correct way doing so. Commented Mar 14, 2022 at 20:04
• @EricJohnson Do you have any idea, what is the correct way of doing? Can you share any papers on it? Commented Mar 16, 2022 at 3:39
• @Jaimin, The methods you describe has nothing to do with Random Matrix Theory. RMT deals with the distribution of the Eigen Values of a randomly generated matrix. Image isn't a random matrix. Moreover, we never apply the methods you described on an image as a whole but in patches.
– Royi
Commented Apr 9, 2022 at 19:10

Usually the way this is done (Look for Low Rank models for Image Denoising) is something like:

1. Decompose the image into d x d patches.
2. Cluster patches which are similar.
3. Per cluster, build a matrix of the whole patches (Each patch as a column).
4. Apply SVD on each matrix and reconstruct it with smaller number of singular values.
5. Recompose the image from the reconstructed matrix (Patches).

Usually, the more advanced models, apply the decomposition on a tensor which each patch is a slice.

• Hii, actually I am looking for methods from Random Matrix Theory. Commented Apr 7, 2022 at 13:04
• Random Matrix Theory deals with the distribution of eigen values while your questions deals with denoising data.
– Royi
Commented Apr 8, 2022 at 17:34
• @Jaimin, Could you please review my answer? If it fits you, could you mark it?
– Royi
Commented May 30, 2023 at 7:53