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Approximate Nearest Subspace Search with applications to pattern recognition

Approximate Nearest Subspace Search with applications to pattern recognition. Ronen Basri Tal Hassner Lihi Zelnik-Manor Weizmann Institute Caltech. Basri & Jacobs, PAMI’03. Nayar et al., IUW’96. Subspaces in Computer Vision. Illumination. Faces.

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Approximate Nearest Subspace Search with applications to pattern recognition

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  1. Approximate Nearest Subspace Searchwith applications to pattern recognition Ronen Basri Tal Hassner Lihi Zelnik-Manor Weizmann Institute Caltech

  2. Basri & Jacobs, PAMI’03 Nayar et al., IUW’96 Subspaces in Computer Vision • Illumination • Faces • Objects • Viewpoint, Motion • Dynamic textures • … Zelnik-Manor & Irani, PAMI’06

  3. Nearest Subspace Search Query Which is the Nearest Subspace?

  4. Sequential Search Database nsubspaces ddimensions ksubspace dimension Sequential search:O(ndk) Too slow!! Is there a sublinear solution?

  5. A Related Problem:Nearest Neighbor Search Database npoints ddimensions Sequential search:O(nd) There is a sublinear solution!

  6. Approximate NN • Tree search (KD-trees) • Locality Sensitive Hashing r (1+)r Query: Logarithmic Preprocessing: O(dn) Fast!!

  7. Is it possible to speed-up Nearest Subspace Search? Existing point-based methods cannot be applied LSH Tree search

  8. Sequential Our Our Suggested Approach • Reduction to points • Works for both linear and affine spaces Run time Database size

  9. Problem Definition Find Mapping Independent mappings Monotonic in distance A linear function of original distance Apply standard point ANN to u,v

  10. Finding a Reduction Feeling lucky? We are lucky !! Constants? Depends on query

  11. Basic Reduction Want: minimize /

  12. Query Lies on a cone Database Lies on a sphere and on a hyper-plane Geometry of Basic Reduction

  13. Improving the Reduction

  14. Final Reduction = constants

  15. Can We Do Better? If =0 Trivial mapping Additive Constant is Inherent

  16. Final Mapping Geometry

  17. ANS Complexities Linear in n Preprocessing:O(nkd2) Log in n Query:O(d2)+TANN(n,d2)

  18. Dimensionality May be Large • Embedding in d2 • Might need to use smallε • Current solution: • Use random projections (use Johnson-Lindenstrauss Lemma) • Repeat several times and select the nearest

  19. Synthetic Data Varying dimension Varying database size Sequential Sequential Our Our Run time Run time dimension Database size n=5000, k=4 d=60, k=4

  20. Query: New illumination Face Recognition (YaleB) Database 64 illuminations k=9 subspaces

  21. True NS Approx NS Face Recognition Result Wrong Match Wrong Person

  22. Retiling with Patches Wanted Query Patch database Approx Image

  23. Retiling with Subspaces Wanted Subspace database Query Approx Image

  24. Patches + ANN ~0.6sec

  25. Subspaces + ANS ~1.2 sec

  26. Patches + ANN ~0.6sec

  27. Subspaces + ANS ~1.2 sec

  28. Summary • Fast, approximate nearest subspace search • Reduction to point ANN • Useful applications in computer vision • Disadvantages: • Embedding in d2 • Additive constant  • Other methods? • Additional applications? A lot more to be done…..

  29. THANK YOU

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