Types of noise

Looking at the residual in Figures - demonstrates another problem with satisfying the requirements for multiple realizations to work correctly. There are two different structures that we see in the residual. The first is the low spatial frequency structure that is associated with the noise that we added to the original data. This is the actual noise in our problem and what we hope to attempt to exploit when doing multiple realizations.

The second type of error has a higher frequency component, the outline of the objects in the photo are an example of this type of noise. This is a more disturbing component of the residual. It is actually caused by our problem formulation. Our regularization operator $\bf A$ makes an assumption of model smoothness that isn't valid everywhere. Our data and data fitting operator $\bf L$ indicate that the model should not be smooth at these locations. These two conflicting pieces of information result in additional structure in our residual. If we estimate our noise covariance operator directly from our data residual, the resulting noise covariance operator will attempt to represent both types of noise and we will be introducing additional unwanted structure into our noise when doing multiple realizations.