How do I know the covariance matrix of colored noise?

I am currently working on simulation of color noise using MATLAB. I understand that the covariance matrix of colored noise will be non-diagonal.

I also studied that passing a white-noise vector from a low pass filter will generate colored noise since individual elements become correlated.

My question is :

1) How do I generate a random color noise with a given covariance matrix (I know mvrnd method) for a 512 X 512 image? Problem is covariance matrix becomes of the size (512^4) for such an image and it's impossible to store it and generate from a mv distribution.

2) If I instead use the filter based method. How do I estimate the entry Sigma_k_l of the covariance matrix ??

The main goal is to have both colored noise and full covariance matrix of the vector.

Thanks in advance.