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--- rotations [2016/12/06 22:39] – awf
+++ rotations [2017/11/11 13:39] (current) – awf
@@ Line 2: / Line 2: @@
 If finding a solution using optimization, e.g.
-  R* =argmin_(R∈SO(3) )⁡ norm(f(R))^2
+$$
+  R^\ast = \operatorname*{argmin}_{R \in SO(3)} \|f(R)\|^2
+$$
 Then given the function quat2mat which converts to SO(3), I optimize the function
@@ Line 12: / Line 14: @@
 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]_× )
+  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
@@ Line 18: / Line 20: @@
 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.
-Update: http://awful.codeplex.com/SourceControl/latest#examples/optimization_tutorial/test_rodrigues.m plots some graphs that suggest rebased exponential map does reduce average iteration counts.
+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.