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Overview

Overview. Definition of Norms Low Rank Matrix Recovery Low Rank Approaches + Deformation Optimization Applications. Definition of Norms. L1 vs L2 Norm. L1 Norm induces sparsity. Matrix Norms. Low Rank Matrix Recovery. Low Rank Matrix Recovery. Low Rank Matrix Recovery.

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Overview

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Presentation Transcript

  1. Overview • Definition of Norms • Low Rank Matrix Recovery • Low Rank Approaches + Deformation • Optimization • Applications

  2. Definition of Norms

  3. L1 vs L2 Norm • L1 Norm induces sparsity

  4. Matrix Norms

  5. Low Rank Matrix Recovery

  6. Low Rank Matrix Recovery

  7. Low Rank Matrix Recovery

  8. Low Rank Matrix Recovery

  9. Surveillance Example Candès, Li, Ma, and W., JACM 2011.

  10. Low Rank Matrix Recovery+Deformation

  11. Problem Setting

  12. Modeling Misalignment Definitions Target Approach

  13. Iterative Linearization Definitions Optimization Problem

  14. Results – Face Alignment

  15. Results – Face Alignment

  16. Aligning Natural Faces

  17. Stabilization of faces in the video

  18. Comparison of aligning handwritten digits

  19. Aligning planar homographies

  20. OPTIMIZATION

  21. Improvement of Algorithms

  22. Drawbacks • Many SDP solvers exist but they are not very efficient for nuclear norm minimization. • Accelerated Proximal Gradient Algorithms exist but no general purpose tools

  23. Applications

  24. TILT: Transform Invariant Low-rank Textures [Zhang, Liang, Ganesh, Ma, ACCV’10]

  25. TILT: All Types of Regular Geometric Structures in Images [Zhang, Liang, Ganesh, Ma, ACCV’10]

  26. TILT: Shape from Patterns and Textures [Zhang, Liang, Ganesh, Ma, ACCV’10]

  27. TILT: Examples of Natural Objects with Bilateral Symmetry [Zhang, Liang, Ganesh, Ma, ACCV’10]

  28. TILT: Examples of Characters, Signs, and Texts [Zhang, Liang, Ganesh, Ma, ACCV’10]

  29. TILT: More Examples [Zhang, Liang, Ganesh, Ma, ACCV’10]

  30. Camera Calibration with Radial Distortion [Zhang, Matsushita, and Ma, in CVPR 2011]

  31. Camera Calibration with Radial Distortion

  32. Camera Calibration with Radial Distortion

  33. Camera Calibration with Radial Distortion

  34. Conclusions • Low rank minimization is a nice way for finding regularities within the data • Nuclear norm is an efficient (fast and scalable) and effective (good proxy for low-rank) way for low rank minimization • Impressive results for handling occlusion • Not many available tools support nuclear norm minimization

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