Simultaneous Compaction and Factorization of Sparse Image Motion Matrices
(ECCV 2012)

Matrices that collect the image coordinates of point features tracked through video - one column per feature - often have low rank, either exactly or approximately. This observation has led to many matrix factorization methods for 3D reconstruction, motion segmentation, or regularization of feature trajectories. However, temporary occlusions, image noise, and variations in lighting, pose, or object geometry often confound trackers. A feature that reappears after a temporary tracking failure - whatever the cause - is regarded as a new feature by typical tracking systems, resulting in very spase matrices with many columns and rendering factorization problematic.

We propose a method to simultaneoulsy factor and compact such a matrix by merging groups of columns that correspond to the same feature into single columns. Combining compaction and factorization makes the imputation of missing matrix entries - and therefore matrix factorization - significantly more reliable.

S. Ricco and C. Tomasi. Simultaneous Compaction and Factorization of Sparse Image Motion Matrices.
12th European Conference on Computer Vision (ECCV 2012). Florence, Italy, October 2012. [PDF]
(Note: This version corrects a typo from Section 4 of the published version. The objective function of (4) is quartic, not quadratic.
The results in the paper are unaffected.)