Rotation averaging
Rotation averaging
Let \({}^w\mathbf{R}_i\), sometimes written as \(\mathbf{R}_i\) for brevity.
Often we we speak of rotation averaging, we are referring to the “multiple rotation averaging” problem [1]:
\[\underset{\mathbf{R_1}, ... \mathbf{R_n} \in SO(3)}{\mbox{argmin}} \sum\limits_{(i,j) \in \mathcal{N}} d({}^i\overline{\mathbf{R}}_{j}, {}^i\mathbf{R}_w {}^w\mathbf{R}_j)\]Gauss-Newton for Rotation Averaging
\[\underset{\mathbf{R} \in SO(d)^n}{\max} \sum\limits_{(i,j) \in \mathcal{E}} \kappa_{ij} \mbox{tr} ( {}^w\mathbf{R}_i {}^i\overline{\mathbf{R}}_{j} {}^j\mathbf{R}_w)\]In the paper’s notation, \(\underset{\mathbf{R} \in SO(d)^n}{\max} \sum\limits_{(i,j) \in \mathcal{E}} \kappa_{ij} \mbox{tr} ( \mathbf{R}_i \overline{\mathbf{R}}_{ij} \mathbf{R}_j^T)\)
trace is not equal to frobenius norm?
minimizing a sum of Frobenius norms:
\[\underset{ \mathbf{R} \in SO(d)^n}{\mbox{min}} \sum\limits_{(i,j) \in \mathcal{E}} \kappa_{ij} \| \mathbf{R}_j − \mathbf{R}_i {}^i\overline{\mathbf{R}}_j \|_F^2\] \[\mathbf{R}_i \leftarrow \mathbf{R}_i e^{[ \boldsymbol{\omega}_i]}\]Convex Relaxation
In Shonan averaging [2], the maximum likelihood problem is posed as:
\[f_{MLE}^* = \underset{ \mathbf{R} \in SO(d)^n}{\mathrm{min}} \mbox{tr }\Big( \overline{\mathbf{L}}\mathbf{R}^T \mathbf{R} \Big)\]where \(\mathbf{R} = (\mathbf{R}_1, \dots , \mathbf{R}_n)\) is the \(d \times dn\) matrix of rotations \(\mathbf{R}_i \in SO(d)\), and \(\overline{\mathbf{L}}\) is the connection Laplacian, a symmetric \((d \times d)\)-block-structured matrix constructed from the measurements \({}^i\overline{\mathbf{R}}_j\):
\[f_{MLE}^* = \underset{ \mathbf{R} \in SO(d)^n}{\mathrm{min}} \mbox{tr }\Bigg( \overline{\mathbf{L}} \begin{bmatrix} \mathbf{R}_1 \\ \mathbf{R}_2 \\ \vdots \\ \mathbf{R}_n \end{bmatrix} \begin{bmatrix} \mathbf{R}_1 & \mathbf{R}_2 & \cdots & \mathbf{R}_n \end{bmatrix} \Bigg)\]References
[1] Richard Hartley, Jochen Trumpf, Yuchao Dai, Hongdong Li. Rotation Averaging. IJCV 2013. PDF.
[2] Frank Dellaert, David M. Rosen, Jing Wu, Robert Mahony, Luca Carlone. Shonan Rotation Averaging: Global Optimality by Surfing \(SO(p)^n\). ECCV 2020. PDF.
Questions for frank: to say n matrices, is SO(d)^n common? trace and frobenius norm not identical?