Exploiting the Circulant - PowerPoint Presentation

Slide1

Exploiting the Circulant Structure of Tracking-by-detection with Kernels

Seunghoon Hong

CV Lab.

POSTECHSlide2

MotivationTracking-by-detection

A classifier is trained

in on-line using examples (patches) obtained during trackingSlide3

MotivationSampling strategy

Sparse sampling for computational efficiencySlide4

MotivationRedundancy in samplingA set of dense patches has extreme redundanciesSlide5

MotivationKey ideaRepresenting a set of dense samples by

“

circulant

structure”

Training and testing can be extremely efficient by exploiting such structureSlide6

Circulant matrix

Circulant matrix

Concatenation of all possible cyclic shifts of

 Slide7

Circulant matrix

Benefit of

circulant

matrix

Only a base vector

need to be stored

Inner product can be computed very efficiently

Convolution between

and

 Slide8

Circulant matrix

Benefit of

circulant

matrix

Only a base vector

need to be stored

Inner product can be computed very efficiently

Convolution between

and

Efficiently computed by

Fast Fourier transform

!Slide9

Circulant matrix

Benefit of

circulant

matrix

Only a base vector

need to be stored

Inner product can be computed very efficiently

It

can be generated by permutation matrix

row of

is obtained by

 Slide10

Circulant matrix

Representing dense samples by

circulant

matrix

Given a base sample

,

generate a (virtual) dense sample matrix

by

shifting

direction

 Slide11

Learning with dense sampling

Regularized risk minimization

Given a set of sample

,

optimize a classifier

by

where

is loss function and

is regularization parameter

 Slide12

Learning with dense sampling

Regularized risk minimization

Kernel trick

: map inputs to feature space

by

Representer

theorem

: the solution can be obtained by linear combination of inputs by

Close form solution of KRLS

Kernel matrixSlide13

Learning with dense sampling

Kernel matrix with dense sampling

Kernel matrix

is

circulant

if

is unitarily invariant kernel

(e.g. linear, RBF, polynomial)

is unitarily invariant if

for any unitarily matrix

=

 Slide14

Learning with dense sampling

Efficient kernel regularized RLS solution

Define kernel vector

with elements

where

Solution of KRLS is rewritten as

 Slide15

Learning with dense sampling

Fast detection

Given a test image

, a testing is performed by

Then it can be re-written by

, where

Applying property of

circulant

matrix,

 Slide16

Learning with dense sampling

Applying different kernels

Let

then

 Slide17

Learning with dense sampling

Applying different kernels

Linear kernel

 Slide18

Learning with dense sampling

Applying different kernels

Linear kernel

Polynomial kernel

 Slide19

Learning with dense sampling

Applying different kernels

Linear kernel

Polynomial kernel

RBF kernel

 Slide20

Overall algorithmSlide21

Experiments

Preprocessing

To reduce boundary effects.

Band original patch

with a cosine window.

Original image

After preprocessingSlide22

ExperimentsTraining label

Smooth regression output by Gaussian kernel

Smoothed output

 Slide23

ExperimentsSlide24

Q & A