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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 l.jpg

Approximate Nearest Subspace Searchwith applications to pattern recognition

Ronen Basri Tal Hassner Lihi Zelnik-Manor

Weizmann Institute Caltech


Subspaces in computer vision l.jpg

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


Nearest subspace search l.jpg
Nearest Subspace Search

Query

Which is the Nearest Subspace?


Sequential search l.jpg
Sequential Search

Database

nsubspaces

ddimensions

ksubspace

dimension

Sequential search:O(ndk)

Too slow!!

Is there a sublinear solution?


A related problem nearest neighbor search l.jpg
A Related Problem:Nearest Neighbor Search

Database

npoints

ddimensions

Sequential search:O(nd)

There is a sublinear solution!


Approximate nn l.jpg
Approximate NN

  • Tree search (KD-trees)

  • Locality Sensitive Hashing

r

(1+)r

Query: Logarithmic

Preprocessing: O(dn)

Fast!!


Is it possible to speed up nearest subspace search l.jpg
Is it possible to speed-up Nearest Subspace Search?

Existing point-based methods cannot be applied

LSH

Tree search


Our suggested approach l.jpg

Sequential

Our

Our Suggested Approach

  • Reduction to points

  • Works for both

    linear and affine spaces

Run time

Database size


Problem definition l.jpg
Problem Definition

Find Mapping

Independent mappings

Monotonic in distance

A linear function of original distance

Apply standard point ANN to u,v


Finding a reduction l.jpg
Finding a Reduction

Feeling lucky?

We are lucky !!

Constants?

Depends on query


Basic reduction l.jpg
Basic Reduction

Want: minimize /


Geometry of basic reduction l.jpg

Query

Lies on a cone

Database

Lies on a sphere

and on a hyper-plane

Geometry of Basic Reduction



Final reduction l.jpg
Final Reduction

= constants


Can we do better l.jpg
Can We Do Better?

If =0

Trivial mapping

Additive Constant is Inherent



Ans complexities l.jpg
ANS Complexities

Linear in n

Preprocessing:O(nkd2)

Log in n

Query:O(d2)+TANN(n,d2)


Dimensionality may be large l.jpg
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


Synthetic data l.jpg
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


Face recognition yaleb l.jpg

Query:

New illumination

Face Recognition (YaleB)

Database

64 illuminations

k=9 subspaces


Face recognition result l.jpg

True NS

Approx NS

Face Recognition Result

Wrong Match

Wrong Person


Retiling with patches l.jpg
Retiling with Patches

Wanted

Query

Patch database

Approx Image


Retiling with subspaces l.jpg
Retiling with Subspaces

Wanted

Subspace database

Query

Approx Image


Slide24 l.jpg

Patches

+

ANN

~0.6sec


Slide25 l.jpg

Subspaces

+

ANS

~1.2 sec


Slide26 l.jpg

Patches

+

ANN

~0.6sec


Slide27 l.jpg

Subspaces

+

ANS

~1.2 sec


Summary l.jpg
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…..



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