Non-linear optimization for rotation

Question

I had a chat with an engineer the other day and we both were stumped on a question related to bundle adjustment. For a refresher, here is a good link explaining the problem:

http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/ZISSERMAN/bundle/bundle.html

The problem requires optimization over 3n+11m parameters. The camera optimization consists of 5 intrinsic camera parameters, 3 DOF for position (x,y,z), and 3 DOF for rotation (pitch, yaw and roll).

Now, when you actually go about implementing this algorithm, a rotation matrix consists an optimization over 9 numbers. Euler's Axis Theorem says these 9 numbers are related and there are only 3 degrees of freedom overall.

Suppose you represent the rotation using a normalized quaternion. Then you have optimization over 3 numbers. Same DOF.

Is one representation more computationally efficient and better than the other? Will you have less variables to optimize using a rotation quaternion over rotation matrix?

Answer 1

You never optimize over 9 numbers! Of course this would be inefficient. One efficient representation in which you only need 3 parameters is to parametrize your rotation matrix R using the Lie algebra of the groupe SO(3). If you are not familiar with Lie algebra, here's a tutorial that explains everything in an intuitive (but sometimes oversimplified) manner. To explain it in a few short sentences, in this representation, each rotation matrix R is written as expmat(a*G_1+b*G_2+c*G_3) where expmat is the matrix exponential, and the G_i are the "generators" of the lie algebra of SO(3), i.e. the tangent space to SO(3) at the identity. Therefore, to estimate a rotation matrix, you only need to learn the three parameters a,b,c. This is roughly equivalent to decomposing your rotation matrix in three rotations around x,y,z and estimating the three angles of these rotations.