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

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Presentation on theme: "Approximate Nearest Subspace Search with applications to pattern recognition Ronen Basri Tal Hassner Lihi Zelnik-Manor Weizmann Institute Caltech."— Presentation transcript:

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

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

3 Query Nearest Subspace Search Which is the Nearest Subspace?

4 Sequential Search Sequential search: O(ndk) Too slow!! Is there a sublinear solution? Database d dimensions n subspaces k subspace dimension

5 A Related Problem: Nearest Neighbor Search d dimensions n points Sequential search: O(nd) There is a sublinear solution! Database

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

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

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

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

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

11 Basic Reduction Want: minimize  / 

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

13 Improving the Reduction

14 Final Reduction = constants

15 Can We Do Better? If  =0 Trivial mappingAdditive Constant is Inherent

16 Final Mapping Geometry

17 ANS Complexities Preprocessing: O(nkd 2 ) Linear in n Log in n Query: O(d 2 )+T ANN (n,d 2 )

18 Dimensionality May be Large Embedding in d 2 Might need to use small ε Current solution: –Use random projections (use Johnson- Lindenstrauss Lemma) –Repeat several times and select the nearest

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

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

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

22 Retiling with Patches Patch databaseQueryApprox Image Wanted

23 Retiling with Subspaces Subspace database QueryApprox Image Wanted

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 d 2 –Additive constant  Other methods? Additional applications? A lot more to be done…..

29 THANK YOU

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