Thursday, March 15, 2007
Weighted PLSD
The PLSD can underestimate the local feature similarity when the minimum of the PLSD curve lies at the smallest scale. In the following example, the left corner of the perturbance at the top of the red shape matches locally to the top-left corner of the blue shape, and thus the PLSD (the dashed line in the figure) is just as low as for the correct match in the figures below. However, at larger neighborhood sizes, the local shapes do not match well.
Also plotted on the PLSD curve figures on the right is the weighted PLSD (solid line), which is formed by the weighted sum of Procrustes distances over each scale, where the weights are proportional to 1 at the smallest scale, 1 - 1/n at the second smallest scale, ... , and 1/n at the highest scale. (n is the number of scales in total, and weights are scaled by 2/(n+1) so as to form a probability distribution.) Clearly, the weighted PLSD, which weights smaller scales more heavily than larger scales, is more discriminitive for the features above.
Running COPAP on the two shapes (blue to red) yields:
where the regular (min) PLSD was used on the left, and the weighted PLSD is shown on the right. Spacing penalty (lambda) was zero for both figures, although matched are similar for lambda up to 12 (which is extremely high).
Matching from the red shape to the blue shape does not yield good results, however:
To handle this case (where a shape with an extra part is matched to a shape without the part), we must modify the COPAP graph--no local distance will compensate for this.
Also plotted on the PLSD curve figures on the right is the weighted PLSD (solid line), which is formed by the weighted sum of Procrustes distances over each scale, where the weights are proportional to 1 at the smallest scale, 1 - 1/n at the second smallest scale, ... , and 1/n at the highest scale. (n is the number of scales in total, and weights are scaled by 2/(n+1) so as to form a probability distribution.) Clearly, the weighted PLSD, which weights smaller scales more heavily than larger scales, is more discriminitive for the features above.
Running COPAP on the two shapes (blue to red) yields:
where the regular (min) PLSD was used on the left, and the weighted PLSD is shown on the right. Spacing penalty (lambda) was zero for both figures, although matched are similar for lambda up to 12 (which is extremely high).
Matching from the red shape to the blue shape does not yield good results, however:
To handle this case (where a shape with an extra part is matched to a shape without the part), we must modify the COPAP graph--no local distance will compensate for this.
# posted by Jared @ 10:19 PM 
