## The BALM

MATLAB code available — Please check the code page.

We are presenting the BALM — a general computational framework for optimising various bilinear problems! We hope this algorithm will ease the life of many researchers when optimising several classes of bilinear problems such as rigid and non-rigid structure from motion, photometric stereo, image registration, learning by factorization and many more. The computational core is based on Augmented Lagrange Multipliers method and the main feature is that it can deal with specific manifold constraints on one of the bilinear component. This is the problem we optimise:

$\text{minimize } \left\| Y - S M \right\|^2 \\ \\\text{subject to } M_i \in {\mathcal M}, \quad i = 1, \ldots, f,$

where $Y$ is a matrix containing our measured data (e.g. image point trajectories in SfM or image pixels variations for Photometric Stereo). The matrices $M$ and $S$ represents our bilinear components to estimate. The matrix $M$ is formed as:

$M = \begin{bmatrix} M_1 & \cdots & M_i & \cdots & M_f \end{bmatrix} \in {\mathbb R}^{r \times m}, \quad M_i \in {\mathbb R}^{r \times p}.$

Each of the sub-matrices $M_i$ lies on a specific manifold ${\mathcal M}$ (i.e. its values are not arbitrary but constrained). When dealing with SfM the manifolds are usually Stiefel while with Photometric Stereo are mostly spherical.

Now a video showing the resilience of BALM to missing data in structure from motion and photometric stereo problems:

At the moment we have found more than 12 problems that can be explained with bilinear models and thus solved with BALM. Some examples: 2D-3D registration of rigid/articulated/non-rigid models, Structure from Sound, BRDFs factorisation for Computer Graphics, Modelling pose/expression/identity, gait analysis and many more. More details and the code will appear soon in a dedicated research area.

• A. Del Bue, J. Xavier, L. Agapito, and M. Paladini, "Bilinear Factorization via Augmented Lagrange Multipliers," in 11th European Conference on Computer Vision (ECCV 2010), Crete, Greece, 2010, pp. 283-296.
@inproceedings{DelBue:etal:2010,   author = {A. {Del Bue} and J. Xavier and L. Agapito and M. Paladini},   title = {Bilinear Factorization via Augmented Lagrange Multipliers},   editor = {Kostas Daniilidis and Petros Maragos and Nikos Paragios},   booktitle = {11th European Conference on Computer Vision (ECCV 2010), Crete, Greece},   publisher = {Springer},   location = {Heidelberg},   series = {Lecture Notes in Computer Science},   volume = {6314},   year = {2010},   isbn = {978-3-642-15560-4},   pages = {283--296} }
• A. Del Bue, J. Xavier, L. Agapito, and M. Paladini, "Bilinear Modeling via Augmented Lagrange Multipliers (BALM)," Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. 34, iss. 8, pp. 1496-1508, 2012.
@Article{DelBue:etal:PAMI2012,   author = {A. {Del Bue} and J. Xavier and L. Agapito and M. Paladini},   title={Bilinear Modeling via Augmented Lagrange Multipliers (BALM)},   journal={Pattern Analysis and Machine Intelligence, IEEE Transactions on},   year={2012},   month={August},   volume={34},   number={8},   pages={1496 -1508},   doi={10.1109/TPAMI.2011.238},   ISSN={0162-8828},   url = {http://users.isr.ist.utl.pt/~adb/publications/2012_PAMI_Del_Bue.pdf} }