Multi-scale non-stationary PEFs

The combination of non-stationary filters and estimation on multiple scales of data introduces a new issue, that non-stationary filters are linked to the size of the data they operate on. If the dimensions of the data change, the dimensions of the PEF must change as well. This causes issues regarding the consistency of the PEF across scales, as well as limits on the type of regularization that can be applied to the filter coefficients.

In order to maintain a consistent PEF across scales, the filter must be sub-sampled so that the same spatial coordinates of the PEF correspond to the proper locations within the scaled data. Since our non-stationary PEF has micro-patches where the filter coefficients are constant, we can scale the patches to match the scaling of the data, so that the number of filter coefficients in the non-stationary filter remains constant, but the size of the micro-patches has decreased. The limiting case for this scaling is when a micro-patch reduces to a single point. Beyond this point, the patches could be sub-sampled during the scaling. This avenue has not been explored, since a need for that level of scaling has not yet been shown.

 

patchscale
Figure 2 Regridding both the data (fine grid) and the micro-patches (thick grid) simultaneously. In this case a 9*9 grid with 3*3 micro-patches is regridded to a 6*6 grid with 2*2 micro-patches. The sizes of the micro-patches remain constant.

I represent the sub-sampling of the patch table by P_i, which acts upon the non-stationary filter f in the non-stationary fitting goal shown below:  

 635#635 (260)

In addition to the above fitting goal, a set of regularization equations must also be solved, where A is our regularization operator:  

 636#636 (261)

The scaling of both the filter and the data to some extent limits the choices of regularization available. Specifically, radial micro-patches do not scale well, so applying radial regularization would have to be done over square micro-patches. Laplacian regularization of filter coefficients across rectangular micro-patches is also a reasonable approach in some cases.