1. ROBUST SUBSPACE LEARNING FOR VISION AND GRAPHICS
Fernando De la Torre Department of Signal Theory. La Salle school of Engineering. Barcelona. Spain.
Michael J. Black
Computer Science Department.
Brown University.
Providence. R.I

2. Low dimensional Linear Models?
Linear Model which captures just the illumination variations.
General Problems:
Outliers (specularities, motion tree, people moving etc..).
Missing or weighting Data.

3. Principal Component Analysis(PCA)
D=[d1 d2 … dn ]. PC are the eigenvectors of the covariance matrix
Alternatively, PCA can be associated with an energy function:
People influence.
PCA Basis

4. Sample outliers
Xu & Yuille[2] propose a RPCA based on sample outliers:
Vi is a binary random variable. Vi=1 inlier otherwise outlier.
Xu & Yuille use least square projection of the data (BTdi).
Binary outlier process, no efficient in front gaussian noise with different variances.
Basis
Threshold
Binary
outlier
process

5. Intra-Sample outliers
We introduce an intra-outlier process in the formulation.
Exploiting the relationship between outlier process and robust statistics, minimizing the previous equation is equivalent to minimize:

6. A quantitative comparison…
The second image is contamined with one outlying pixel with 10 times more energy than the sum of others image pixels.
Reconstruction from
noiseless images
Learned basis images
PCA
Xu & Yuille
Our RPCA

7. Optimization Iteratively re-weighted least-(IRWLS) squares:
Differentiating w.r.t B and C:
In the more general case (any Wi), there is NO SOLUTION in terms of Eigen-equation.

8. Problem: Local minima and high dimensional space.
Solution: Deterministic annealing algorithms (Graduated Non Convexity).
Problem: High computational cost
Solution: Alternate between calculating C in closed form and updating B incrementally.

9. The big picture Weighted subspace analysis
Robust Principal Component Analysis (RPCA)
Robust Singular Value Decomposition (RSVD)
Weighted
subspace
analysis
Robust Multilinear Models (RML)
Missing or Weighted Data (Wx)
Robust eigenvector (RE).
Robust Generalized eigenvector (RGE).

10. Learning a subspace of Illumination (RPCA)

11. Structure from motion (SFM)
Recovering 3D shape and motion from p feature correspondences across f views is the well known SFM[6].
Assuming an affine camera model, the SFM problem can be solved factorizing the data matrix D=MS ( Data= Motion * Structure ), where:
Frames
Features

12. Modeling 3D objects with 2D models
W is a predefined weighting matrix (WPCA).
Learning limbs appearance models for 3D-tracking[7].
Data=D
Mask=W

13. Conclusions /Future Work
Weighted Subspace Analysis for Vision and Graphics problems.
Adding spatial coherence to the outliers.
References and Related work
[1] F. De la Torre and M. Black. Robust Principal Component Analysis for Computer vision. ICCV 2001.
[2] L. Xu and A. Yuille. Robust principal component analysis by self-organizing rules based on statistical physics approach. IEEE Transactions on Neural Networks, 6(1): , 1995.
[3] K.I. Diamantaras. Principal Component Neural Networks. John Wiley & Sons.1986.
[4] K P. J. Huber. Robust Statistics. New York. Wiley 1981.
[5] H.Y. Shum, K. Ikeuchi and R. Reddy. Principal Component Analysis with missing Data and its application to polyhedral object modeling. PAMI . Vol 17, n 9, september 1995.
[6] C. Tomasi and T. Kanade. Shape and motion from image streams under orthography---a factorization method. International Journal on Computer Vision, 9(2): , November 1992.
[7] Sidenbladh, H., De la Torre, F., Black, M. J., A framework for modeling the appearance of 3D articulated figures, Int. Conf. on Automatic Face and Gesture Recognition, Grenoble, France, pp , March 2000.