Scale-invariance introduces more fitting equations

The fitting goals (3) and (4) have about double the usual number of fitting equations. Scale-invariance introduces extra equations. If the range of scale-invariance is wide, there will be more equations. Now we begin to see the view to the end of the tunnel:

1.

Refining a computational mesh improves accuracy.

2.

Refining a data mesh makes empty bins.

3.

Empty bins spoil analysis.

4.

If there are not too many empty bins we can find a PEF.

5.

With a PEF we can fill the empty bins.

6.

To get the PEF and to fill bins we need enough equations.

7.

Scale-invariance introduces more equations.

An example of these concepts is shown in Figure 2.

 

mshole90
Figure 2 Overcoming aliasing with multiscale fitting.

Additionally, when we have a PEF, often we still cannot find missing data because conjugate-direction iterations do not converge fast enough (to fill large holes). Multiscale convolutions should converge quicker because they are like mesh-refinement, which is quick. An example of these concepts is shown in Figure 3.

 

msiter90 Figure 3 Large holes are filled faster with multiscale operators.