Approximating rotation matrix of arbitrary objective function

Question

I want to approximate the rotation in SO(3) (ie. 3D rotation) that minimizes some objective function.

I'm looking for a SO(3) rotation representation that lends itself to energy minimization. I'm writing a framework that performs minimization on shape parameters, where the goodness of the fit is given solely based on some arbitrary objective function which is a function of a rotation matrix.

One example solution would be to convert euler angles to matrix representation, minimize the objective function as a function of the angle parameters, and convert back to a matrix. This effectively to reparameterizing the objective function into some other rotation representation. However, one problem with euler angles is that the representation does not preserve uniformity and is prone gimbal locking problems. Thus I am afraid it may give biased results or create local minima.

I have heard it is possible to optimize based on quaternion or axis angle representations of 3D rotations, but I have been unable to find a good source or description.

(Added) More specifically. As suggested by Choward in the comments, a solution using quaternions would formaly be: Find some quaternion,$q^*$, that satisfies: $q^*=\arg \min f(R(q))$, where $R(q)$ is a rotation matrix as a function of the quaternion, and $f(\cdot)$ is some cost function being minimized as a function of the rotation matrix.

Edit: typo

Edit2: Added clarification based on comment.

Answer 1

So your problem statement is to find some quaternion, $q^*$, such that it satisfies the following:

$$ q^* = \text{arg} \min f(R(q))$$

where $f(\cdot)$ is some cost function, and $R(q)$ is a matrix created from some quaternion $q$. Now, this problem can be viewed as a more general problem, such as:

$$ \textbf{x}^* = \text{arg} \min J(\textbf{x})$$

where $\textbf{x} \in \mathbb{R}^4$, and $J(\cdot)$ is a more general cost function. In your case, you can tackle it in a variety of ways. If you can compute $\nabla J$ with respect to $\textbf{x}$, then you can use various gradient based methods. If not, then doing a gradient-free approach can work. Note that you will need to ensure you normalize the input $\textbf{x}$ within $J(\cdot)$ before you feed it into your original $f(R(\cdot))$ cost function.

Answer 2

I agree that quaternions are the way to go, as stated in other answers, but let me add a brief note on the axis-angle representation.

Given a vector \begin{equation} \mathbf{v} = \begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix} \end{equation} let \begin{align} \theta & = \lvert \mathbf{v} \rvert & \mathbf{e} & = \frac{\mathbf{v}}{\lvert \mathbf{v} \rvert} & [\mathbf{v}]_\times & = \begin{bmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & -v_1 \\ -v_2 & v_1 & 0 \end{bmatrix} \end{align} and \begin{equation} \mathbf{R} = \exp([\mathbf{v}]_\times) \end{equation} where $\exp$ is the matrix exponential.

It can be shown that $\mathbf R$ is a rotation matrix corresponding to a rotation of magnitude $\theta$ about axis $\mathbf e$.

An alternative expression of $\mathbf{R}$ is the Rodrigues' rotation formula: \begin{equation} \mathbf{R} = \mathbf{I} + \sin\theta \, [\mathbf{e}]_\times + (1-\cos\theta) \,[\mathbf{e}]_\times^2 \end{equation}

The advantage of this representation is that you can optimise with respect to the three independent parameters $(v_1, v_2, v_3)$. The disadvantage of course is that you have to evaluate a matrix exponential (!) or $\sin(\lvert \mathbf v \rvert)$ and $\cos(\lvert \mathbf v \rvert)$. However a quick and dirty implementation using this representation is almost trivial, especially in a high level language that, like scipy, has a matrix exponential.

Answer 3

Basically the quaternions are the four dimensional Euclidean space together with a division algebra structure. The three dimensional space is the same as the imaginary part of the quaternions. A rotation of three space is given by a unit quarternion $q$ which acts on an imaginary quaternion $x$ by the formula $x \mapsto qx\bar{q}$. Now, one way you can implement rotations is the following: fix a vector in three space, say $\hat{a}=(a_1,a_2,a_3)^T$, write it as an imaginary quaternion $a=a_1 i + a_2 j + a_3 k$ and then a rotation around the vector $\hat{a}$ by angle $\theta$ is written as a unit quaternion in the form $$q = ( \cos{\theta}) + (\sin{\theta}) a.$$ Its inverse is its conjugat $$\bar{q}= ( \cos{\theta}) - (\sin{\theta}) a.$$ Thus, your cost function depends on four parameters $J(\theta,a_1,a_2,a_3) = J(q,\bar{q})$.

Edit: $ \hat{a}$ needs to be a unit vector so that $a$ is a unit imaginary quaternion.

Edit: In general, for any nonzero $q$, a transformation $x \mapsto q x q^{-1}$ of an imaginary quaternion $x$, is also a rotation so normalizing quaternions is not really necessary. This is true for example because $q^{-1} = \frac{1}{|q|^2} \bar{q}$, so $$q x q^{-1} = q x \left( \frac{\bar{q}}{|q|^2} \right) = \left(\frac{q}{|q|}\right) x \left(\frac{\bar{q}}{|q|}\right).$$ So it seem like you can end up with optimizing a function $J(q,\bar{q})$ without constraints.