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 3*N*+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 3*N*+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 3*N*+1 to 2. However, the derivation of a weighting function for the
spatial integration is still an open problem.

7/8/2003