How can I find the 2D rotation that best aligns (in a least-squares sense) two sets of corresponding 2D points?
Background
Wahba's problem seeks to find a 3x3 rotation matrix that best aligns two 3D point clouds $\{\mathbf{a}_i\in\mathbb{R}^3\}$ and $\{\mathbf{b}_i\in\mathbb{R}^3\}$ of corresponding points. The solution can be found simply by singular value decomposition of the matrix $\sum_i\mathbf{a}_i\mathbf{b}_i^T$ into the form $USV^T$, then
$$R=U\begin{bmatrix}1&0&0\\0&1&0\\0&0&\det(U)\det(V)\end{bmatrix}V^T$$
Question
What is the corresponding solution if $\{\mathbf{a}_i\in\mathbb{R}^2\}$ and $\{\mathbf{b}_i\in\mathbb{R}^2\}$ and we are seeking a 2x2 rotation matrix (or simply a rotation angle $\theta$) to align them?