A Novel Approach for Automatic Palmprint Recognition

A Novel Approach for Automatic Palmprint

Recognition

Murat Ekinci and Murat Aykut

Computer Vision Lab.

Department of Computer Engineering,

Karadeniz Technical University,Trabzon,Turkey

ekinci@666d2ebdf121dd36a32d8249.tr

Abstract.In this paper,we propose an e?cient palmprint recognition

scheme which has two features:1)representation of palm images by two

dimensional(2-D)wavelet subband coe?cients and2)recognition by a

modular,personalized classi?cation method based on Kernel Principal

Component Analysis(Kernel PCA).Wavelet subband coe?cients can

e?ectively capture substantial palm features while keeping computational

complexity low.We then kernel transforms to each possible training palm

samples and then mapped the high-dimensional feature space back to

input space.Weighted Euclidean linear distance based nearest neighbor

classi?er is?nally employed for recognition.We carried out extensive

experiments on PolyU Palmprint database includes7752palms from386

di?erent palms.Detailed comparisons with earlier published results are

provided and our proposed method o?ers better recognition accuracy

(99.654%).

1Introduction

Biometrics is becoming more and more popular in an increasingly automated world.Palmprint recognition is one kind of biometric technology and a rela-tively new biometric 666d2ebdf121dd36a32d8249pared with other biometrics,the palmprints has several advantages:low-resolution imaging can be employed;low-cost cap-ture devices can be used;it is di?cult to fake a palmprint;the line features of the palmprints are stable,etc.[1].It is for these reasons that palmprint recognition has recently attracted an increasing amount of attention from researchers.

There are many approaches for palmprint recognition using line-based[2][4][5], texture-based[9][5],and appearance-based methods[3][8][7][6]in various litera-ture.In the line-based approach,the features used such as principal lines,wrin-kles,delta points,minutiae,etc.are sometimes di?cult to extract directly from a given palmprint image with low resolution.The recognition rates and com-putational e?ciency are not strong enough for palmprint recognition.In the texture-based approach,the texture features are not su?cient and the extracted features are greatly a?ected by the lighting conditions.From that disadvantages, researches have developed the appearance-based approaches.The appearance-based approaches only use a small quantity of samples in each palmprint class Z.Kobti and D.Wu(Eds.):Canadian AI2007,LNAI4509,pp.122–133,2007.

c Springer-Verlag Berlin Heidelberg2007

A Novel Approach for Automatic Palmprint Recognition123 randomly selected as training samples to extract the appearance features(com-monly called algebraic features)of palmprints and form feature vector.

Eigenpalms method[8],?sherpalms method[3],and eigen-and-?sher palms [7]are presented as the appearance-based approaches for palmprint recognition in literature.Basically,their representations only encode second-order statistics, namely,the variance and the covariance.As these second order statistics pro-vide only partial information on the statistics both natural images and palm images,it might become necessary to incorporate higher order statistics as well. In other words,they are not sensitive to higher order statistics of features.A kernel?sherpalm[6]is presented as another work to resolve that problem.In addition,for palmprint recognition,the pixelwise covariance among the pixels may not be su?cient for recognition.The appearance of a palm image is also severely a?ected by illumination conditions that hinder the automatic palmprint recognition process.

Converging evidence in neurophysiology and psychology is consistent with the notion that the visual system analyses input at several spatial resolution scales [19].Thus,spatial frequency preprocessing of palms is justi?ed by what is known about early visual processing.By spatial frequency analysis,an image is repre-sented as a weighted combination of basis functions,in which high frequencies carry?nely detailed information and low frequencies carry coarse,shape-based information.Recently,there have been renewed interests in applying discrete transform techniques to solve some problems in face recognition[13][14][17],in palmprint recognition[17][18]and many real world problems.An appropriate wavelet transform can result in robust representations with regard to lighting changes and be capable of capturing substantial palm features while keeping computational complexity low.

From these all considerations,we propose to use discrete wavelet transform (DWT)to decompose palm images and choose the lowest resolution subband coe?cients for palm representation.We then apply kernel PCA as a nonlinear method to project palmprints from the high-dimensional palmprint space to a signi?cantly lower-dimensional feature space,in which the palmprints from the di?erent palms can be discriminated much more e?ciently.The main contribu-tions and novelties of the current paper are summarized as follows:

–To reliably extract palmprint representation,we adopt a template matching approach where the feature vector of a palm image is obtained through a multilevel two-dimensional discrete wavelet transform(DWT).The dimen-sionality of a palm image is greatly reduced to produce the waveletpalm.

–A nonlinear machine learning method,kernel PCA,is applied to extract palmprint features from the waveletpalm.

–The proposed algorithm is tested on a public palmprint databases.We pro-vide some quantitative comparative experiments to examine the performance of the proposed algorithm and di?erent combinations of the proposed 666d2ebdf121dd36a32d8249parison between the proposed algorithm and other recent ap-proaches is also given.

124M.Ekinci and M.Aykut

This paper is organized as follows.Section2introduces brie?y wavelet trans-form,lowest subband image representation,and fast Fourier transform(FFT) which is also implemented in this work to compare the e?ciencies on the palm-print recognition.A brief description of kernel PCA(KPCA)and similarity mea-surement used are given in Sections3and4respectively.Experimental results on the palmprint database are summarized in Section5followed by discussions and conclusions in Section6.

2Discrete Transforms

In the proposed algorithm,the palmprint is?rst transformed into the wavelet domain,then kernel PCA is applied to extract higher order relations among waveletpalms for future recognition.In order to compare the e?ciencies of the wavelet transform and discrete fast Fourier transform(FFT)is alternately em-ployed in the proposed algorithm.

2.1Discrete Wavelet Transform

The DWT was applied for di?erent applications given in the literature e.g.tex-ture classi?cation[12],image compression,face recognition[13][14],because of its powerful capability for multiresolution decomposition analysis.The wavelet transform breaks an image down into four subsampled,or decimated,images. They are subsampled by keeping every other pixel.The results consist of one image that has been high pass?ltered in both the horizontal and vertical direc-tions,one that has been high pass?ltered in the vertical and low pass?ltered in the horizontal,one that has been lowpassed in the vertical and highpassed in the horizontal,and one that has been low pass?ltered in both directions.

So,the wavelet transform is created by passing the image through a series of 2D?lter bank stages.One stage is shown in Fig.1,in which an image is?rst ?ltered in the horizontal direction.The?ltered outputs are then down sampled by a factor of2in the horizontal direction.These signals are then each?ltered by an identical?lter pair in the vertical direction.Decomposed image into4 subbands is also shown in Fig.1.Here,H and L represent the high pass and low pass?lters,respectively,and↓2denotes the subsampling by2.Second-level decomposition can then be conducted on the LL subband.Second-level structure of wavelet decomposition of an image is also shown in Fig.1.This decomposition can be repeated for n-levels.Fig.2shows one-level,two-level and three-level wavelet decomposition of a palm image.

The proposed work based DWT addresses the four-level decomposition of im-ages in the database used for experiments.Daubechies-8[11]low pass and high pass?lters are also implemented.Additionally,four-level of decompositions are produced,then32x32sub-images of128x128images in the wavelet are pro-cessed as useful features in the palmprint images.Reduce of the image resolution helps to decrease the computation load of the feature extraction process.

A Novel Approach for Automatic Palmprint Recognition

125 Fig.1.One-level2-D?lter bank for wavelet decomposition and multi-resolution struc-ture of wavelet decomposition of an

image

Fig.2.Palm images with one-level,two-level,and three-level wavelet decomposition are shown

2.22-D Discrete FFT

F(u,v)is2-D FFT coe?cients of W x H image I(x,y).The feature sequence is generated using the2D-FFT technique.The palmprint image(128x128)in the spatial domain is not divided into any overlap blocks.The FFT coe?cients for the palmprint image are?rst computed.In FFT,the coe?cients correspond to the lower frequencies than3x3,and to the higher frequencies than16x 16in FFT,were discarded by?ltering.In other words,247coe?cients((16x 16)-(3x3))correspond to the6%coe?cients in the frequency domain,(64x 64),were only processed.These data are empirically determined to achieve the best performance.Therefore,the size of the palmprint image(128x128)in the spatial domain was reduced to the very few coe?cients in the frequency domain correspond to the1.5%coe?cient.Finally,N=μxνfeatures form a vector χ∈ N,χ=(F0,0,F0,1,...Fμ,ν)for FFT.

126M.Ekinci and M.Aykut

3Kernel PCA

The kernel PCA (KPCA)is a technique for nonlinear dimension reduction of data with an underlying nonlinear spatial structure.A key insight behind KPCA is to transform the input data into a higher-dimensional feature space [10].The feature space is constructed such that a nonlinear operation can be applied in the input space by applying a linear operation in the feature space.Consequently,standard PCA can be applied in feature space to perform nonlinear PCA in the input space.

Let χ1,χ2,...,χM ∈ N be the data in the input space (the input space is 2D-DWT coe?cients in this work),and let Φbe a nonlinear mapping between the input space and the feature space 666d2ebdf121dd36a32d8249ing a map Φ: N →F ,and then performing a linear PCA in F .Note that,for kernel PCA,the nonlinear mapping,Φ,usually de?nes a kernel function [10].The most often used kernel functions are polynomial kernels,Gaussian kernels,and sigmoid kernels [10]:

k (χi ,χj )= χi ,χj d ,(1)k (χi ,χj )=exp ? χi ?χj 2

2σ2

,(2)k (χi ,χj )=tanh (κ χi ,χj +?),(3)

where d is a number in the set of natural numbers,e.g.{1,2,...},σ>0,κ>0,and ?<0.The mapped data is centered,i.e. M i =1Φ(χi )=0(for details see [10]),and let D represents the data matrix in the feature space:D =[Φ(χ1)Φ(χ2)···Φ(χM )].Let K ∈ MxM de?ne a kernel matrix by means of dot product in the feature space:

K ij =(Φ(χi )·Φ(χj )).(4)

The work in [10]shows that the eigenvalues,λ1,λ2,...,λM ,and the eigenvectors,V 1,V 2,...,V M ,of kernel PCA can be derived by solving the following eigenvalue equation:

KA =MAΛ(5)

with A =[α1,α2,...,αM ]and Λ=diag {λ1,λ2,...,λM }.A is MXM orthogonal eigenvector matrix,Λis a diagonal eigenvalue matrix with diagonal elements in decreasing order (λ1≥λ2≥···≥λM ),and M is a constant corresponds to the number of training samples.Since the eigenvalue equation is solved for α’s instead of eigenvectors,V =[V 1,V 2...V M ],of kernel PCA,?rst,A should be normalized to ensure that eigenvalues of kernel PCA have unit norm in the feature space,therefore λi αi 2=1,i =1,2,...,M .After normalization the eigenvector matrix,V ,of kernel PCA is then computed as follows:

V =DA (6)

Now let χbe a test sample whose map in the higher dimensional feature space is Φ(χ).The kernel PCA features of χare derived as follows:

F =V T Φ(χ)=A T B

(7)where B =[Φ(χ1)·Φ(χ)Φ(χ2)·Φ(χ)···Φ(χM )Φ(χ)]T .

A Novel Approach for Automatic Palmprint Recognition127 4Similarity Measurement

When a palm image is presented to the wavelet-based kernel PCA classi?er, the wavelet feature of the image is?rst calculated as detailed in Section2,and the low-dimensional wavelet-based kernel PCA features,F,are derived using the

equation7.Let M0

k ,k=1,2,..,L,be the mean of the training samples for class

w k.The classi?er applies,then,the nearest neighbor rule for classi?cation using some similarity(distance)measureδ:

δ(F,M0k)=min jδ(F,M0j)?→F∈w k,(8) The wavelet-based kernel PCA feature vector,F,is classi?ed as belong to the

class of the closest mean,M0

k ,using the similarity measureδ.

Popular similarity measures include the Weighted Euclidean Distance(WED) and Linear Euclidean Distance(LED)which are de?ned as follows:

W ED:d k=

N

i=1

(f(i)?f k(i))2

(s k)2

(9)

where f is the feature vector of the unknown palmprint,f k and s k denote the k th feature vector and its standard deviation,and N is the feature length.

LED:d ij(x)=d i(x)?d j(x)=0(10) where d i,j is the decision boundary separating class w i from w j.Thus d ij>0 for pattern of class w i and d ij<0for patterns of class w j.

d j(x)=x T m j?1

2

m T j m j,j=1,2,...M(11)

m j=

1

N j

x∈w j

x,j=1,2,...,M(12)

where M is the number of pattern classes,N j is the number of pattern vectors from class w j and the summation is taken over these vectors.

Support Vector Machines(SVMs)have recently been known to be successful

in a wide variety of applications[10][15].SVM-based and WED-based classi?er are also compared in this work.In SVM,we?rst have a training data set,

like,D={(x i,y i)|x i∈X,y i∈Y,i=1,...,m}.Where X is a vector space of dimension d and Y={+1,?1}.The basic idea of SVM consists in?rst mapping

x into a high dimension space via a function,then maximizing the margin around

the separating hyperlane between two classes,which can be formulated as the following convex quadratic programming problem:

maximize W(α)=

m

i=1

αi?

1

2

m

i,j=1

αiαj y i y j(K(x i,x j)+

1

C

δi,j)(13) subject to0≤αi≤C,?i,(14)

128M.Ekinci and M.Aykut

and

m

i

y iαi=0(15)

whereαi(≥0)are Lagrange multipliers.C is a parameter that assigns penalty cost to misclassi?cation of samples.δi,j is the Kronecker symbol and K(x i,x j)= φ(x i)·φ(x j) is the Gram matrix of the training examples.The form of decision function can be described as

f(x)= w,Φ(x) +b(16)

where,w= m

i=1

α?j y iΦ(x i),and b is a bias term.

5Experiments

The PolyU palmprint database[9]was obtained by collecting palmprint images from193individuals using a palmprint capture device.People was asked to provide about10images,each of the left and right palm.Therefore,each person provided around40images,so that this PolyU database contained a total of 7,752grayscale images from386di?erent palms.The samples were collected in two sessions,where the?rst ten samples were captured in the?rst session and other ten in the second session.The average interval between the?rst and second collection was69days.The resolution of all original palmprint images is384x 284pixels at75dpi.In addition,they changed the light source and adjusted the focus of the CCD camera so that the images collected on the?rst and second occasions could be regarded as being captured by two di?erent palmprint devices. The palmprint images collected in the second occasion were also captured under di?erent lighting conditions.

At the experiments on the database,we use the preprocessing technique de-scribed in[9]to align the palmprints.In this technique,the tangent of the two holes(they are between the fore?nger and the middle?nger,and between the ring?nger and the little?nger)are computed and used to align the palmprint. The central part of the image,which is128x128,is then cropped to represent the whole palmprint.Such preprocessing greatly reduces the translation and

Fig.3.Original palmprint and it’s cropped image

A Novel Approach for Automatic Palmprint Recognition129 rotation of the palmprints captured from the same palms.An example of the palmprint and its cropped image is shown in Figure3.

In the?rst experiment on the database,the?rst session was used as training set,second session includes3850samples of386di?erent palms was also used as testing set.In this experiment,the features are extracted by using the pro-posed kernel based eigenspace method with length50,75,100,200,and300. Weighted Euclidean distance(WED)-based matching was used to cluster those features.The matching is separately conducted and the results are listed in Table1.The numbers given in Table1correspond to the correct recognition samples in all test samples(3850).The entries in brackets also represent the corresponding recognition rate.High recognition rate93.168%was achieved for the DWT+KPCA with feature length of300.A nearest-neighbor classi?er based on WED is employed to produce recognition rates given in the Table1. The recognition rates obtained by PCA and kernel PCA based methods are comparatively illustrated in Table1.When the feature number varies from50to 300,although KPCA-based approach only achieves higher recognition rate than PCA-based with feature length of75,but DWT+KPCA based the proposed method achieved higher recognition rate then all combinations of PCA-based and FFT+KPCA-based approaches.Finally,it is evident that feature length can play an important role in the matching process.Long feature lengths lead to a high recognition rate.

666d2ebdf121dd36a32d8249parative performance evaluation for the di?erent matching schemes with di?erent feature lengths.Train is?rst session,test is second session.

Method Feature length

5075100200300 PCA3411(88.597)3477(90.311)3498(90.857)3513(91.246)3513(91.246) DWT+PCA3444(89.454)3513(91.246)3546(92.103)3570(92.727)3568(92.675) KPCA3411(88.597)3481(90.415)3498(90.857)3508(91.116)3510(91.168) FFT+KPCA2746(71.324)2933(76.181)3034(78.805)3174(82.441)3253(84.493) DWT+KPCA3457(89.792)3531(91.714)3558(92.415)3584(93.09)3587(93.168) The performance variation for WED-based nearest-neighbor(NN)and SVM classi?ers with the increase in number of features are shown in Figure4.The SVM using radial basis function was employed in the experiments and the pa-rameters of SVM were empirically selected.The training parameterγ, and C were empirically?xed at0.55,0.001,and100,respectively.As shown in Figure 4,the SVM classi?er achieved higher recognition when50features were only im-plemented.For the feature lengths longer than50,the WED-based NN classi?er has achieved better performance.

As?nal experiment and very similar to the experiments published in litera-ture,the palm images collected from the?rst session were only used to test the proposed algorithm.We use the?rst four palmprint images of each person as training samples and the remaining six palmprint images as the test samples. So,the numbers of training and test samples are1544and2316.We also test the

130M.Ekinci and M.Aykut

89.5

90

90.5

91

91.5

92

92.5

93

93.5

50

100

150 200 250 300

r e c o g n i t i o n r a t e

number of features

"SVM""WED"

Fig.4.Performance analysis of classi?er with the number of features:DWT+KPCA method using the SVM-and WED-based classi?ers

Table 2.Testing results of the eight matching schemes with di?erent feature lengths

Method Feature length 50100200300

380

PCA

LED 60.664%71.804%74.568%1723(74.395%)1717(74.136%)WED

98.747%99.179%99.093%2294(99.05%)2292(98.963%)DWT +LED 59.542%71.459%87.305%2032(87.737%)2032(87.737%)PCA WED

98.834%99.309%99.352%2301(99.352%)2302(99.395%)KPCA

LED

63.557%73.661%75.82%1730(74.697%)1712(73.92%)WED

98.877%99.222%99.05%2293(99.006%)2291(98.92%)DWT LED 83.462%86.01%86.01%2025(87.435%)2039(88.039%)KPCA WED 98.747

%

99.309%

99.568%2308(99.654%)

2308(99.654%)

8approaches against conventional PCA method using di?erent test strategies.Based on these schemes,the matching is separately conducted and the results are listed in Table 2.The meaning of LED and WED in Table 2is linear Eu-clidean discriminant and weighted Euclidean distance based nearest neighbor classi?er,respectively.The numbers given for feature lengths 300and 380in Ta-ble 2represent the number of the correct recognition samples in all 2316palms used as test samples.The entries in the brackets also represent the correspond-ing recognition rate (%).A high recognition rate (99.654%)was achieved for kernel PCA with 2D-DWT (abbreviated as DWT+KPCA)and WED classi?er approach,with feature length of 300.One of the important conclusion from Ta-ble 2is that,long feature lengths lead to a high recognition rate.However,this principle only holds to a certain point,as the experimental results summarized in Table 2show that the recognition rate remain unchanged,or even become worse,when the feature length is extended further.

A Novel Approach for Automatic Palmprint Recognition 131

Fig.5.Experimental results by the di?erent rotation and translation conditions.(Top)Some palm images in training set,(Bottom)Correctly classi?ed corresponding samples in testing set.

Fig.6.Misclassi?ed four palm samples.Top:Some palm images in training set,Bot-tom:Corresponding and misclassi?ed samples in testing set.

Typical samples in this database are shown in Figs.5in which the top images were used as training samples,the bottom images were also used as test samples.Although the rotation and translation conditions are quite di?erent from the samples used as test set,the proposed algorithm can still easily recognize the same palm.The misclassi?ed samples were only 8samples in all 2316used as testing set,and some of them are also shown in Figure 6in which the top images show the sample in training set,and corresponding bottom images were misclassi?ed samples used in test set.The other misclassi?ed four samples have not been shown because of the page limitation.

132M.Ekinci and M.Aykut

666d2ebdf121dd36a32d8249parison of di?erent palmprint recognition methods

Method

Proposed In[4]In[5]In[3]In[8]In[6]In[7]In[16]In[17]In[18]

Database

palms386310030038216010010019050 samples38603020030003056160060010003040200

Recog.Rate99.654959199.299.14997.2597.595.898.1398 Comparison has been?nally conducted among our method and other methods published in literature,and is illustrated in Table3.The databases given in the Table3are de?ned as the numbers of the di?erent palms and whole samples tested.The data represent the recognition rates given in Table3is taken from experimental results in the cited papers.In biometric systems,the recognition accuracy will decrease dramatically when the number of image classes increase [1].Although the proposed method is tested on the public database includes more di?erent palms and samples,the recognition rate of our method is more e?cient,as illustrated in Table3.

6Conclusion

This paper presents a new appearance-based non-linear feature extraction(ker-nel PCA)approach to palmprint identi?cation that uses low-resolution images. We?rst transform the palmprints into wavelet domain to decompose the origi-nal palm images.The kernel PCA method is then used to project the palmprint image from the very high-dimensional space to a signi?cantly lower-dimensional feature space,in which the palmprints from the di?erent palms can be dis-criminated much more e?ciently.WED based NN classi?er is?nally used for matching.The feasibility of the wavelet-based kernel PCA method has been suc-cessfully tested on PolyU database.The data set consists of7752images of386 subjects.Experimental results show the e?ectiveness of the proposed algorithm for palmprint recognition.

Acknowledgments.This research is partially supported by The Research Foun-dation of Karadeniz Technical University(Grant No:KTU-2004.112.009.001).The authors would like to thank to Dr.David Zhang from the Hong Kong Polytechnic University,Hung Hom,Hong Kong,for providing us with the PolyU palmprint database.

References

1.Zhan D.,Jing X.,Yang J.:Biometric Image Discrimination 666d2ebdf121dd36a32d8249-

putational Intelligence and Its Application Series,Idea Group Publishing,(2006) 80–95

2.Zhang D.,Shu W.:Two Novel Characteristics in Palmprint Veri?cation:Datum

Point Invariance and Line Feature Matching.Pattern Recognition,Vol.32,(1999) 691–702

A Novel Approach for Automatic Palmprint Recognition133

3.Wu X.,Zhang D.,Wang K.:Fisherpalms Based Palmprint Recognition.Pattern

Recognition Letters,Vol.24,Issue15,November,(2003)2829–2838.

4.Duta N.,Jain A.K.,Mardia K.V.:Matching of Palmprint.Pattern Recognition

Letters,Vol.23,No.4,(2002)477–485

5.You J.,Li W.,Zhang D.:Hierarchical Palmprint Identi?cation via Multiple Feature

Extraction.Pattern Recognition,Vol.35No.4,(2002)847–859

6.Wang Y.,Ruan Q.,Kernel Fisher Discriminant Analysis for Palmprint Recognition.

IEEE The18th Int.Conf.on Pattern Recognition,ICPR’06,(2006)457–460

7.Jiang W.,Tao J.,Wang L.:A Novel Palmprint Recognition Algorithm Based on

PCA and FLD.IEEE,Int.Conf.on Digital Telecommunications,August,(2006) 28–32

8.Lu G.,Zhang D.,Wang K.:Palmprint Recognition Using Eigenpalms Features.

Pattern Recognition Letters,Vol.24.Issue9-10,June(2003)1463–1467

9.Zhang D.,Kongi W.,You J.,Wong M.:Online Palmprint Identi?cation.IEEE

Trans.on Pattern Analysis and Machine Intelligence,Vol.25,No.9,September (2003)1041–1049

10.Scholkopf B.,Somala A.:Learning with Kernel:Support Vector Machine,Regu-

larization,Optimization and Beyond.MIT Press,(2002).

11.Daubechies I.:Ten Lectures on Wavelets.Philadelphia,,PA:SIAM,(1992).

12.Chang T.,Kuo C.J.:Texture Analysis and Classi?cation with Tree-Structured

Wavelet Transform.IEEE Trans.Image Processing,Vol.2,No.4,(1993)429–441 13.Zhang B.,Zhang H.,Sam S.:Face Recognition by Applying Wavelet Subband

Representation and Kernel Associative Memory.IEEE Transactions on Neural Net-works,Vol.15,No.1,(2004)166–177

14.Chien J.,Wu C.,Discriminant Waveletfaces and Nearest Feature Classi?er for Face

Recognition.IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.24,No.12,(2002)1644–1649.

15.Li W.,Gong W.,Yang L.,Chen W.,Gu X.:Facial Feature Selection Based on

SVMs by Regularized Risk Minimization.IEEE,The18th Conference on Pattern Recognition(ICPR’06),(2006)540–543

16.Kumar A.,Zhang D.:Personal Recognition Using Hand Shape and Texture.IEEE

Transactions on Image Processing,Vol.5,No.8,(2006)2454–2460

17.Jing X.Y.,Zhang D.:A Face and Palmprint Recognition Approach Based on Dis-

criminant DCT Feature Extraction.IEEE Transactions on Systems,Man,and Cybernetics-Part B:Cybernetics,Vol.34,No.6(2004)2405–2415

18.Zhang L.,Zhang D.:Characterization of Palmprints by Wavelet Signatures via

Directional Context Modeling.IEEE Trans.on Systems,Man,and Cybernetics-Part B:Cybernetics,Vol.34,No.3,(2004)1335–1347

19.Valentin T.:Face-space Models of Face Recognition.In Computational,Geometric,

and Process Perspectives on Facial Cognition:Context and Challenges,Hillsdale, NJ:Lawrence Erbaum,(1999)

本文来源：https://www.bwwdw.com/article/6y7q.html