# Differences

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 rotations [2013/04/08 17:06]awf created rotations [2017/11/11 13:39] (current)awf 2017/11/11 13:39 awf 2017/11/11 13:39 awf 2017/11/11 13:37 awf 2017/11/11 09:22 awf 2016/12/06 22:56 awf 2016/12/06 22:39 awf 2013/04/08 17:06 awf created 2017/11/11 13:39 awf 2017/11/11 13:39 awf 2017/11/11 13:37 awf 2017/11/11 09:22 awf 2016/12/06 22:56 awf 2016/12/06 22:39 awf 2013/04/08 17:06 awf created Line 2: Line 2: If finding a solution using optimization,​ e.g. If finding a solution using optimization,​ e.g. - R^*=argmin_(R∈SO(3) )⁡〖f(R)〗 + $$+ ​R^\ast = \operatorname*{argmin}_{R \in SO(3)} \|f(R)\|^2 +$$ - Then given the function ​ + Then given the function quat2mat ​which converts to SO(3), I optimize the function - quat2mat(q)=(■(q_00^2-…&&​@&&​@&&​)) + - I optimize the function + - g(q)=f(quat2mat(q/​(\|q\|))) + - The division by norm doesn'​t actually make the derivatives much worse, and is much better than optimization subject to \|q\|=1.  I tend not to worry too much about the gauge freedom g(q) = g(λq), but one should renormalize q after each descent step/​linesearch. + g(q) = f(quat2mat(q/​norm(q))) + + The division by norm doesn'​t actually make the derivatives much worse, and is much better than optimization subject to |q|=1. ​ I tend not to worry too much about the gauge freedom g(q) = g(λq), but one should renormalize q after each descent step/​linesearch. If I believe I have a decent initial estimate R_init (sounds like your problem), then I tend to use the exponential map If I believe I have a decent initial estimate R_init (sounds like your problem), then I tend to use the exponential map - R=R_init ​ exp⁡([a]_× ) + - where exp is the matrix exponential,​ [v]_×is the 3x3 matrix which effects cross-product by v, and a is, oddly, the rotation axis multiplied by the angle of rotation. ​ http://​en.wikipedia.org/​wiki/​Rodrigues%27_rotation_formula + ​R(a) = R_init ​* exp([a]_× ) + + where exp is the matrix exponential,​ [v]_× is the 3x3 matrix which effects cross-product by v, and a is, oddly, the rotation axis multiplied by the angle of rotation. ​ http://​en.wikipedia.org/​wiki/​Rodrigues%27_rotation_formula It has pleasant derivatives,​ its singularities live far from the origin, and it’s 3 parameters, so you don’t need to get inside the optimizer to renormalize. It has pleasant derivatives,​ its singularities live far from the origin, and it’s 3 parameters, so you don’t need to get inside the optimizer to renormalize. + But of course, for best behaviour, you do need to rebase inside the optimizer to ensure that each iteration computes an update in the tangent plane (you'​ll need to read up for this, starting at http://​lear.inrialpes.fr/​pubs/​2000/​TMHF00/​Triggs-va99.pdf perhaps.) + + Update: https://​github.com/​awf/​awful/​blob/​master/​examples/​optimization_tutorial/​test_rodrigues.m plots some graphs that suggest rebased exponential map does reduce average iteration counts.