Spatial aliasing and scale invariance

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 big picture.

1. Refining a model mesh improves accuracy.
2. Refining a model 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.

mshole
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.

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

Spatial aliasing and scale invariance