Differential Equations Solutions 62

Differential Equations Solutions 62 - matrices Q...

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72 Chapter 12. Solutions: Case Study: Achieving a Common Viewpoint Therefore, t = 1 n n X j =1 b j 1 n Qa j = c B Qc A . This very nice observation was made by Hanson and Norris [1]. CHALLENGE 12.6. The results are shown in Figure 12.2. With no pertur- bation, the errors in the angles, the error in the matrix Q , and the RMSD are all less than 10 15 . With perturbation in each element uniformly distributed between 10 3 and 10 3 , the errors rise to about 10 4 . Comparison of the SVD method with other methods can be found in [2] and [3], although none of these authors knew that the method was due to Hanson and Norris. CHALLENGE 12.7. (a) Yes. Since in this case the rank of matrix A is 1, we have two singular values σ 2 = σ 3 = 0. Therefore we only need z 11 = 1 in equation (12.1) and we don’t care about the values of z 22 or z 33 . (b) Degenerate cases result from unfortunate choices of the points in A and B .I f all of the points in A
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Unformatted text preview: matrices Q . Additionally, if two singular values of the matrix B T A are nonzero but equal, then small perturbations in the data can create large changes in the matrix Q . See [1]. (c) A degenerate case and a case of gymbal lock are illustrated on the website. [1] Richard J. Hanson and Michael J. Norris, “Analysis of measurements based on the singular value decomposition,” SIAM J. Scientifc and Statistical Computing , 2(3):363-373, 1981. [2] Kenichi Kanatani, “Analysis of 3-d rotation ±tting,” IEEE Transactions on Pattern Analysis and Machine Intelligence , 16(5):543-549, May 1994. [3] D.W. Eggert and A. Lorusso and R.B. Fisher, “Estimating 3-d rigid body trans-formations: a comparison of four major algorithms,” Machine Learning and Appli-cations , 9:272-290, 1997....
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