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By: Hauke (haukeheibel) - 2007-06-29 01:05 |
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Hi Matrin, I took a look at the discussion about re-orthogonalization matrices. I think that the Taylor series expansion will never work since you are omitting a term (the part converging toward zero) which will always keep you from really being in SO3, i.e. det(R)=1. I recently stumbled over the following way to do it. It is much shorter and will yield an orthogonal matrix with determinant one (so no reflection) for sure. Assume R' is a matrix which is meant to be a rotation matrix but due to numerical error is not. If we apply an SVD to that matrix R' we yield U*S*V^T where U and V^T are both orthogonal having a determinant of +- 1 (the SVD may yield reflections for U and V). If R' were from O3 det(S) = +-1 following S being a diagonal matrix consisting only of +-1 on the diagonal. Since we want det(R')=1 we can now simply set S=diag([1 1 det(U*V^T)]). The last value can not be fixed to 1 but should be either +1 or -1, depending on the determinants of U and V. So finally your SO3 matrix R = U*diag([1 1 det(U*V^T])*V^T. Regards, Hauke |
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By: Martin Baker (martinbaker  ) - 2007-06-29 10:26 |
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Hi Hauke, Thanks very much for this, I was suspicious of the Taylor series without knowing why. SVD is new to me, I guess I need to find out more about SVD, so that I can understand this better. I did a quick search: http://en.wikipedia.org/wiki/Singular_value_decomposition http://www.uwlax.edu/faculty/will/svd/svd/index.html I'll do some reading, in the meantime I'll put your message on the webpage if that's alright with you. Cheers, Martin |
By: Hauke (haukeheibel) - 2007-06-30 03:03 |
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Hi Martin, Feel free to put the procedure on the web-site. You should just note that it is absolutely not new... ;) For the web-sites I think the wiki site is quite good. I did recently take a look at an existence proof but in general I don't care since I usually just "use" the SVD as a tool and that's that. I am no mathematician. If you want to point users to existing implementations you might find the following links of interest: http://math.nist.gov/tnt/index.html http://sourceforge.net/projects/opencvlibrary/ Of course there are much more implementations around. But these are the ones I used to use. Regards, Hauke |
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Hi Hauke, Yes, I can't say I understood the existence proof, but it would be nice to find an algorithm to explain what this code is doing. It seems to me that a lot of the complexity in this code comes because SVD works on a non-square matrix. I read somewhere that for a square matrix then SVD is equivalent to eigen-decomposition. In that case : R' = U * S * U^T In other words the same as SVD but with V=U where: S is a diagonal matrix made up from the eigenvalues. U is a matrix made up from the eigenvectors. Unfortunately I think there are some conditions for this to be true so possibly this does not help? Anyway it would be nice to relate this to eigenvalues and eigenvectors. I'll continue investigating this, its not only useful for the reorthogonalisation page, I think the theory is also interesting for the physics pages about inertia matrix. Thanks, Martin |
By: Hauke (haukeheibel) - 2007-07-02 00:49 |
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Hi Matrin, For some reason my last message ended up in the nirvana... I think the SVD code by itself is even for square matrices a little bit more complex. Actually I would suggest to not put it on the web-site. As said before, it exists in many libraries. As far as I know, the singular values correspond to the square-roots of the eigenvalues. And regarding the orthogonal matrices, you have A = U*S*V^T and A*A^T = V*S^T*S*V^T. So what you have written holds only for the eigenvalue decomposition of A*A^T. Furthermore I have little left to contribute. Just some applications the SVD can be used for: - computation of the determinant (product of the singular values, probably there are easier ways to do it...) - computation of the condition number (quotient of biggest and smallest singular value) - minimization of |Ax|=0 with subject to |x|=1 (L2 norm) - computation of a pseudo-inverse ... Probably the list can be made much longer. These are the main tasks where I use the SVD. Actually there is one thing I was always looking for which is a nice geometric explanation of the SVD. The wiki explanation might be a good start but unfortunately I did not yet find (resp. take) the time to dig deeper into the topic. Regards, Hauke |
By: Martin Baker (martinbaker ) - 2007-07-03 01:11 |
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Hauke, It would be really good to add some pages about matrix factoring containing these topics and cross linking to the various applications. I think I'll try that. I agree it would be good to have a better geometric explanation of the SVD. I think perhaps eigenvalue decomposition has a reasonable geometric interpretation as follows: Since the full matrix represents the scaling by the eigenvalues in the direction of the corresponding eigenvectors, this can be factored into the following sequence: 1) rotate so that the eigenvectors align with the x,y and z coordinates. 2) diagonal matrix scales in the direction of the x,y and z coordinates. 3) rotate back to the original alignment. Although the SVD is more general than this, the diagonal matrix still seems to provide the scaleing in the direction of the x,y and z coordinates, I guess this is as close as we will get to being able to factor a general matrix into: rotation, scaling, shear, reflect, etc. Thanks, Martin |
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