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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

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Approximate Nearest Subspace Searchwith 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

  • Objects

  • Viewpoint, Motion

  • Dynamic textures

Zelnik-Manor & Irani, PAMI’06


Nearest Subspace Search

Query

Which is the Nearest Subspace?


Sequential Search

Database

nsubspaces

ddimensions

ksubspace

dimension

Sequential search:O(ndk)

Too slow!!

Is there a sublinear solution?


A Related Problem:Nearest Neighbor Search

Database

npoints

ddimensions

Sequential search:O(nd)

There is a sublinear solution!


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?

Existing point-based methods cannot be applied

LSH

Tree search


Sequential

Our

Our Suggested Approach

  • Reduction to points

  • Works for both

    linear and affine spaces

Run time

Database size


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

Feeling lucky?

We are lucky !!

Constants?

Depends on query


Basic Reduction

Want: minimize /


Query

Lies on a cone

Database

Lies on a sphere

and on a hyper-plane

Geometry of Basic Reduction


Improving the Reduction


Final Reduction

= constants


Can We Do Better?

If =0

Trivial mapping

Additive Constant is Inherent


Final Mapping Geometry


ANS Complexities

Linear in n

Preprocessing:O(nkd2)

Log in n

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


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

Varying dimension

Varying database size

Sequential

Sequential

Our

Our

Run time

Run time

dimension

Database size

n=5000, k=4

d=60, k=4


Query:

New illumination

Face Recognition (YaleB)

Database

64 illuminations

k=9 subspaces


True NS

Approx NS

Face Recognition Result

Wrong Match

Wrong Person


Retiling with Patches

Wanted

Query

Patch database

Approx Image


Retiling with Subspaces

Wanted

Subspace database

Query

Approx Image


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 d2

    • Additive constant 

  • Other methods?

  • Additional applications?

    A lot more to be done…..


THANK YOU


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