Assume that we have a $d\times d$ grey-scale image represented as a vector $$ \mathbf{x}=(x_1,\ldots,x_D)^T\in[0,255]^D, $$ where $D=d\times d$. We would like to import some noise concerning the image's position, in order to create a new noisy image represented by the vector $\mathbf{x}'=(x_1',\ldots,x_D')^T\in[0,255]^D$.
First question: What could be that noise? Ideally, I would like that noise to be an anisotropic $D$-dimensional Gaussian distribution with covariance matrix $\Sigma\in\mathbb{R}^{d\times d}$. How could we pollute this image with this kind of noise?
Second question: Could we estimate the covariance matrix given the two representations?
EDIT: Assume that $\mathbf{x}$ is the mean vector of the Gaussian and $\mathbf{x}'$ is some point drawn from this Gaussian. We want to express $\mathbf{x}'$ using $\mathbf{x}$ and the Covariance matrix $\Sigma$, while after that we want to estimate the covariance matrix $\Sigma$ using $\mathbf{x}$ and $\mathbf{x}'$.
Is this possible?