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

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Presentation transcript:

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

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 …

Query Nearest Subspace Search Which is the Nearest Subspace?

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

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

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

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

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

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

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

Basic Reduction Want: minimize  / 

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

Improving the Reduction

Final Reduction = constants

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

Final Mapping Geometry

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

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

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

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

Face Recognition Result Wrong Match Wrong Person True NS Approx NS

Retiling with Patches Patch databaseQueryApprox Image Wanted

Retiling with Subspaces Subspace database QueryApprox Image Wanted

Patches + ANN ~0.6sec

Subspaces + ANS ~1.2 sec

Patches + ANN ~0.6sec

Subspaces + ANS ~1.2 sec

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…..

THANK YOU