[MP 1] Version "MultiLevel6"

Collective Relaxation

• In each relaxation step, all nodes are now moving SIMULTANEOUSLY along there negative gradient vectors.
• The step width of each node is proportional to the magnitude of its gradient, i.e. nodes move different amounts.
• In my experience, this strategy avoids problems connected with the wrong order of successive node movements.
• The numerical effort of the collective optimization is the same as with successive optimization.

Naive Optimization Strategy

• The global gradient ( = derivatives of the mismatch potential with respect to all node coordinates) is computed.
• All nodes move simultaneously a small step.
• The global gradient is computed again and so on.
• Normally, line search is done in a different way: First walk in one direction until no more improvement is possible, and only then choose a new direction.
• For some reason, the naive approach seems to work better. Don't really understand why.

New Graphics Function

• The original intensity distribution is plotted in red.
• The momentary fit is plotted in blue.
• The goal intensity distribution is plotted in green.
• All colors are plotted on top of each other, with slightly different pixel sizes.
• So, in the end, the blue fit should coincede with the green goal.
• Graphics windows can be advanced by clicking with the mouse somewhere inside.

Performance of Multilevel-Version-6

• Amazingly, many test problems created with GenerateRndImages() or GenerateFullRndImages() (see [7]) are solved nicely, without any elastic forces.
• However, the amount by which the mismatch potential can be relaxed, depends on the individual case.
• Here, also the exact setting of the step width in the relaxation routine has an influence.
• The real cell images with LoadRealImages() are not working at all (- so far..) .
• Elastic forces don't seem to improve the results.

Elastic Forces

• Are implemented.
• Pure elastic relaxation on single grid level has been tested.
• Combination of mismatch and elastic forces don't really seem to work well yet.

Random Test Images

• GenerateRndImages(128,100,0.0,1.5) : Generates an orig and goal picture with resolution 128x128 pixels. Pixels are dark by default, but 100 random pixels have a finite random brightness. In the goal image, those pixels (more precisely: the underlying nodes) are shifted into a random direction by a random amount between 0.0 and 1.5 pixel sizes.

• GenerateFullRndImages(32,0.0,1.5): Similar to above, but here ALL 32x32 pixels will have some random brightnesses.

• To generate a new random case, change the integer variable SIT in DefParameters().

Idea: Separate Registration and Traction Computation

• If the mismatch potential can really be relaxed without any elastic forces, this would be numerically effecient for images with many dark pixels.

• Then, after we know where each bright node has moved to, the elastic problem and force calculation could be done as a second step.

Simplified Point Spread Function

• Since the new minimization strategy does not need second derivatives, it was possible to go back to the lower-order, triangle-like PSF, leading to a second order B-function.

• COI based Coarsification and Refinement

• Lolo's idea is used, i.e. in the coarsification step each low-level supernode is placed at the center of intensity (COI) of its associated high-level component nodes.
  

• In the refinement step, the group of component nodes is shifted, as a whole, by the same amount their supernode has been shifted in the grid below.

• Coarsification, low-level relaxation and refinement together now comprise a true correction: Low-level and high-level information are combined.

• It is ensured by the COI method that mere rigid shifts between the orig and goal images are treated exactly.

• With the COI method, low-level grids are not uniform any longer. Thus, elastic springs have different rest lengths.

• It is not clear yet how to implement a given Poisson ratio (requiring different spring constants for vertical/horizontal and diagonal springs) on such a non-uniform grid.

Beyond Powers of 2

• Ben had the idea to use more flexible schemes for the multigrid method coarsification, so that the linear picture sizes doent need to be powers of two each time.

• For example, one could also use 3x3 groupings instead of 2x2.

• Or, one could be satisfied with a lowest grid resolution of, say, 5x5, instead of 1x1, but still use the normal 2x2 refinement.