method

Since we have three interrelated parameters (scale, micropatch size, and micropatch origin), and a way of measuring the quality of the parameter choices without solving for a model, we can perform a global search in order to find the optimal choices for these parameters. Since these parameters are interdependent, the preferred method is to cycle through possible values for a single parameter, and then for each of the possible values generate all possible combinations of the other two parameters given the first. The most intuitive starting point appears to be micropatch size.

For each possible combination of micropatch size, micropatch grid origin, and scale, we count the number of fitting equations in each micropatch. A valid fitting equation is where all points of the filter fall on known data. Once this is done, we will have a 3N+1 dimensional hypercube, where we have N dimensions of freedom in choosing micropatch size and micropatch grid origin. Since the scaling of the data must be isotropic in order to preserve the aspect ratio of the PEF, varying scale only adds one dimension of freedom. Finally, since we are measuring the spatial variance of fitting equations, another N dimensions are added to the hypercube.

Naturally, we don't want to deal with a 3N+1 dimensional space, so we can integrate over certain dimensions, and use the helical coordinate when dealing with others. We can choose the scales with the greatest number of fitting equations (since the cost of the inversion rises by O(N) with the number of scales used) and integrate the total. We can then integrate spatially over micropatches with a weighting function that rewards a wide spatial distribution of fitting equations. Finally, we wrap micropatch size and micropatch grid origin around a helix, so the dimensionality of the problem reduces from 3N+1 to 2. However, the derivation of a weighting function for the spatial integration is still an open problem.