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wavelet and ridgelet transform, the second continuouscurvelet transform is also fallen into the category ofsparseness theory. And it can be used to representsparsely signal or function by applying the inner productof basis function and signal or function. Then, the

curvelet transform can be expressed by

kljfkljc ,,,:),,( = (1)

whereklj ,, denotes curvelet function, and j, l and k

denotes the variable of scale, orientation, and positionrespectively.

In the frequency domain, the Curvelet transform canbe implemented with by means of the window functionU. Defining a pair of windows W(r) (a radial window)and V(t) (an angular window) as the followings:

= =j

j

rrW )2/3,4/3(,1)2(2

(2)

=

=l

ttV )2/1,2/1(,1)1(2 (3)

where variables Was a frequency domain variable, and rand as polar coordinates in the frequency domain.

For each 0jj , Uj is defined in the Fourier domainby

=

2

2)2(2),(

]2/[4/3

jjj

j vrwrU (4)

where [j/2] denotes the integer part ofj/2.A polar wedge represented by Uj , shown in the

shadow region of Fig. 1, is supported by Wand V, theradial and angular windows. Fig. 1 shows the division ofwedges of the Fourier frequency plane. The wedges are

the result of partitioning the Fourier plane in radial(concentric circles) and angular divisions. Concentric

circles are responsible for decomposition of the image inmultiple scales (used for bandpassing the image) andangular divisions corresponding to different angles ororientation. So, to address a particular wedge one needsto define the scale and angle first.

Figure 1. Curvelet representation in the frequency domain

Let )()( jj U= , and j at the scale j, then the

curvelet can be obtained by rotating and shiftingj at

the other scale 2j.

Defines:

(1) Uniform rotation angle serial 20,,1,0,22 ]2/[ == l

j

l ll L

(2) Shift parameter 221),( Zkkk =

According to the above ideas, the curvelet can bedefined as a function ofx =(x1,x2) at scale 2

j, orientation

l , and position),( lj

kx by

))(()(),(

,,

lj

kjklj xxRx l = (5)

where )2,2( 2/211),( jjlj

k kkRx l = , and lR is the

rotation in radians.Then the continuous curvelet transform can be defined

by

dxxxffkljcR

kljklj == 2 )()(,:),,( ,,,, (6)

Based on Plancherel Therory, the following formula

can be deduced from the above,

deRUf

dxfkljc

jik

l

xj

j

klj

,

2

,,2

),(

)()(2

1

)()(2

1:),,(

=

=(7)

Set the inputf[t1,t2] (0t1, t2

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and the classifier takes the following form:

])([)( bxsignxyT += (9)

where )( : nhn RR is the mapping to the high

dimensional and potentially infinite dimensional featurespace.

In the primal weight space, the optimization problembecomes:

libxyts

J

ii

T

i

l

i

i

T

b

,,2,1,1])([..

2

1

2

1),(min

1

2

,,

L==+

+= =

(10)

where 0> denotes a real constant used to control the

punishment degree for misclassification. Because wbecomes infinite dimensional, this primal problem cannotdirectly be solved. Therefore, let us proceed by

constructing the following Lagrangian:

{ }iiTil

i

i bxyJbL ++= =

1])([),();,,(1

(11)

where thei

values are Lagrange multipliers, which can

be positive or negative now due to the equalityconstraints.

The conditions for optimality are given by

=++=

==

==

==

=

=

01])([0

,0

0,0

)(,0

1

1

ik

T

i

i

ii

i

i

iii

l

i

iii

bxyL

L

yb

L

xyL

(12)

Defining ])(;;)([ 11 lT

l

TT yxyxZ L= ,

],1;;1;1[1 =v ];;;[ 21 lyyyy = , ];;;[ 21 l = ,

];;;[ 21 l = , and eliminating the variables

and , one obtains the following linear KKT system:

=

+ v

T

l

b

Iy

y 0

/

0

(13)

where llRI denotes a unit matrix, ZZT= , and

the kernel trick can be applied within the matrix,namely

ljixxKyyxxyy jijijT

ijiij ,,2,1,),,()()( === (14)

Solving the above equations, then the resulting

LS-SVM model for classification becomes

+= =

l

i

jii bxxKysignxy1

),()( (15)

Based on the above classification model, the unknown

data can be gotten to classify. The main kernel functionsused to LS-SVM have polynomial kernel, RBF kernel,and sigmoid kernel. In this paper, RBF kernel is appliedas the following:

)2/exp(),(22

ii xxxxK = (16)

IV. EXPERIMENTS AND RESULTSBased on ORL face database, the above curvelet

transform combined with LS-SVM is applied torecognize faces. In this method, curvelet transform is

used to extract features from facial images first, and thenLS-SVM is used to classify facial images based onfeatures. The ORL database contains 400 images of 40persons (10 images per person). Some images werecaptured at different times and have different variationsincluding expression (open or closed eyes, smiling or

non-smiling) and facial details (glasses or no glasses).The images were taken with a tolerance for some tiltingand rotation of the faces up to 20

. A part of the samples

images are displayed in Fig. 2. In the experiments, thefront five samples per person (total 200 images) are usedfor training LS-SVM, and then the remaining fivesamples per person (total 200 images) are used for testing

LS-SVM. To reduce the dimensionality of facial imagefed to LS-SVM, curvelet transform is applied to

decompose the image and extract curvelet coefficients asa representative set of facial features at scales 4 (coarse,detail1, detail2, fine). For example, Fig. 3 shows a resultof curvelet transform at scales 4 for a facial image.

Because five curvelet coefficients (energy, maximum,minimum, mean and variance) are extracted at each scale,each facial image has twenty features to be extracted atscales 4 by curvelet transform. Set these features as the

inputs of above LS-SVM, and corresponding personnumber as the outputs of above LS-SVM. Adopting themethod, after training LS-SVM by the front five samplesper person (total 200 images), the LS-SVM is applied torecognize the remaining five samples per person (total200 images). The results show that the correct recognitionrate is up to 96%, and the computational speed is faster.

Figure 2. Part images from ORL face database

Original image Coarse Detail1 Detail2 Fine

Figure 3. A result of curvelet transform for a facial image

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V. CONCLUSIONAs a means of biometric technique, face recognition

has been studied diligently for more than 30 years nowand has emerged as one of the most successful

applications of image analysis. Feature extraction is a keystep prior to face recognition. To reduce the featuresdimensionality and better represent the main features offacial image, many methods have been developed. As alatest multiresolution analysis method, curvelet transformhas improved directional elements with anisotropy andbetter ability to represent sparsely edges and other

singularities along curves. But, not much work has beendone to explore the potential of curvelet transform tosolve pattern recognition problems. In the domain ofpattern recognition, Support Vector Machine (SVM) is anideal nonlinear classification tool nowadays. In this paper,a face recognition system based on curvelet transform

and Least Square Support Vector Machine (LS-SVM) hasbeen developed, which uses curvelet transform to extract

features from facial images first, and then uses LS-SVMto classify facial images based on features. The proposedmethod has been evaluated by carrying out experimentson the well-known ORL databases. The results show thatthe correct recognition rate is up to 96%, and thecomputational speed is faster.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the support of thescience and technology research program of JiangxiProvincial Department of Education, China (Project No.

[2007]277).

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