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Bug 247 - Umeyama transform doesn't support weighting
Summary: Umeyama transform doesn't support weighting
Status: NEW
Alias: None
Product: Eigen
Classification: Unclassified
Component: Geometry (show other bugs)
Version: 3.0
Hardware: All All
: --- enhancement
Assignee: Nobody
Depends on:
Reported: 2011-04-11 15:54 UTC by Semen
Modified: 2019-12-04 10:38 UTC (History)
4 users (show)


Description Semen 2011-04-11 15:54:54 UTC
In real life Umeyama transform is typically used in molecular simulations to fit two molecules together, which means that the positions of their atoms (geometrical points) must be mass-weighted. Umeyama in Eigen does not support any weighting, which makes it very limited in real life usage. In order to be really useful this function should accept additional optional parameter - the vector of weights.
Comment 1 Gael Guennebaud 2011-04-19 14:21:22 UTC
Yes I agree that weighting is important. So, patch welcome ;)
Comment 2 Semen 2011-04-19 18:17:59 UTC
(In reply to comment #1)
> Yes I agree that weighting is important. So, patch welcome ;)

Unfortunately I'm not familiar with this algorithm. I'm not even sure that it can be formulated for weighted data. The only thing I know is that all molecular analysis programs use weighted implementations of such fitting.
Comment 3 Hauke Heibel 2011-04-25 12:51:40 UTC

The algorithm [1] is minimizing

  e^2(R,t,c) = 1/n sum_{i=1}^n |\ y_i - (cRx_i + t) |\^2

where x_i and y_i are corresponding points which should be fit to each other and you would like to change this to

  e^2(R,t,c,W) = 1/n sum_{i=1}^n w_i |\ y_i - (cRx_i + t) |\^2

if I am not completely wrong. Right now, I don't have the time to go through the derivation and verify if such a weighting can be easily integrated and how the minimization would be affected but maybe the paper is enough information for you.

You might also be wanting to take a look at the ICP (iterative closest point) algorithm where the point correspondences x_i and y_j are not known a priori.

- Hauke

[1] Least Squares Estimation of Transformation Parameters Between Two Point Pattern, Shinji Umeyama, IEEE PAMI, 1991
Comment 4 Charles Karney 2014-03-23 15:46:18 UTC
For 3d points, this capability is included in the enhancement offered in Bug 771.
Comment 5 Charles Karney 2014-04-22 13:33:54 UTC
The enhancement offered in Bug 771 now offers weighted fits of n-d points.
Comment 6 Nobody 2019-12-04 10:38:11 UTC
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