Local maximal margin discriminant embedding for face
recognition
perspective是什么意思
Pu Huang a ,⇑,Zhenmin Tang a ,Caikou Chen b ,Zhangjing Yang a
a School of Computer Science and Engineering,Nanjing University of Science and Technology,Nanjing 210094,China b
College of Information Engineering,Yangzhou University,Yangzhou 225009,China
a r t i c l e i n f o Article history:
Received 10November 2012Accepted 24November 2013
Available online 7December 2013Keywords:
Local maximal margin discriminant embedding
Locality prerving projection Maximum margin criterion Small sample size problem Local structure Appearance-bad
Dimensionality reduction Manifold learning Face recognition
a b s t r a c t
In this paper,a manifold learning bad method named local maximal margin discriminant embedding (LMMDE)is developed for feature extraction.The propod algorithm LMMDE and other manifold learn-ing bad approaches have a point in common that the locality is prerved.Moreover,
LMMDE takes con-sideration of intra-class compactness and inter-class parability of samples lying in each manifold.More concretely,for each data point,it pulls its neighboring data points with the same class label towards it as near as possible,while simultaneously pushing its neighboring data points with different class labels away from it as far as possible under the constraint of locality prerving.Compared to most of the up-to-date manifold learning bad methods,this trick makes contribution to pattern classification from two aspects.On the one hand,the local structure in each manifold is still kept in the embedding space;one the other hand,the discriminant information in each manifold can be explored.Experimental results on the ORL,Yale and FERET face databas show the effectiveness of the propod method.
Ó2013Elvier Inc.All rights rerved.
1.Introduction
Face recognition has attracted wide attention of the rearchers in the fields of pattern recognition and computer vision becau of its immen application potential.Many face recognition methods have been developed over the past few decades.One of the most successful and well-studied techniques to face recognition is the appearance-bad method.In an appearance-bad technique,
a two-dimensional face image of size w by h pixels is reprented by a vector in a w Âh -dimensional space.In practice,however,the w Âh -dimensional spaces are too large to allow robust and fast recognition.A common way to attempt to resolve this problem is to u dimensionality reduction techniques.Two of the most popular dimensionality reduction methods are principal compo-nent analysis (PCA)[1]and linear discriminant analysis (LDA)[2].PCA is a classical dimensionality reduction and data reprenta-tion technique widely ud in pattern classification and visualiza-tion tasks.PCA is an unsupervid method,which aims to find a linear mapping that prerves the total variance by maximizing the trace of feature variance.The optimal mapping is the leading eigenvectors corresponding to the largest eigenvalues of the covariance matrix for data of all class.
LDA produces an optimally discriminative projection for certain cas.LDA arches for the transformation that maximizes the be-tween-class scatter and at the same time minimizes the within-class scatter.Different from PCA which is completely unsupervid with regard to the class information of the data,LDA takes full con-sideration of the class labels and it is generally believed that LDA is able to enhance class parability.Despite the success of the LDA algorithm in many applications,its effectiveness is still limited since,in theory,the number of available projection direction
s is lower than the class number.Furthermore,class discrimination in LDA is bad upon within-class and between-class scatters,which is optimal only in cas where the data of each class is approximately Gaussian distributed,a property that cannot always be satisfied in real-world applications.At the same time,LDA can-not be applied directly to small sample size problem [3]becau the within-class scatter matrix is singular [2].To avoid the singu-larity problem of LDA,Li et al.[4]ud the difference of both be-tween-class scatter and within-class scatter as discriminant criterion,called maximum margin criterion (MMC).MMC has the advantages of effectiveness and simplicity.
Recent studies [5–7]have shown that the high-dimensional data possibly resides on a nonlinear sub-manifold.However,both PCA and LDA effectively e only the global Euclidean structure.When they are applied to face recognition,they fail to discover the underlying structure,if the face images lie on a nonlinear sub-manifold hidden in the image space.Some nonlinear
1047-3203/$-e front matter Ó2013Elvier Inc.All rights rerved.dx.doi/10.1016/j.jvcir.2013.11.007
Corresponding author.
E-mail address: (P.Huang).
techniques have been propod to discover the nonlinear structure of the manifold.The basic assumption of manifold learning is that the input data lie on a smooth low-dimensional manifold.Each manifold learning bad method attempts to prerve a different geometrical property of the underlying manifold.The reprenta-tive ones include Isomap[5],LLE[6],Laplacian Eigenmap[7]and local tangent space alignment(LSTA)[8].The nonlinear methods do yield impressive results on some benchmark artificial data ts. However,they yield maps that defined only on the training data points and how to evaluate the maps on novel test data points re-mains unclear.To overcome this limitation,He ded Laplacian Eigenmap to its linearized locality prerving projection(LPP)[9–13]for an explicit map.LPP attempts to con-struct a nearest neighbor graph and then evaluate the low-dimen-sional embedding to best prerve local structure of the data t.
Although LPP is effective in many domains,it is unsupervid and its unsupervid nature restricts its discriminating capability. To consider class label information in LPP,veral supervid LPP methods[14–21]have been developed.Local discriminant embed-ding(LDE)[15]and marginalfisher analysis(MFA)[16],who objective functions are very similar,can also be viewed as super-vid LPP methods.This is becau their training phas both ex-ploit the class label information of samples.Th
ey are derived by using a motivation partially similar to LPP and each of them is bad on an eigen-equation formally similar to the eigen-equation of LPP.On the other hand,since LDE and MFA partially borrow the idea of discriminant analysis and try to produce satisfactory linear parability,their ideas are also somewhat different from the idea of prerving the local structure of LPP.LDE and MFA can be viewed as two combinations of the locality prerving technique and the linear discriminant analysis[22].Compared with LDA,both LDE and MFA do not depend on the assumption that the data of each class is Gaussian distributed and can obtain more available projection directions and better characterize the parability of dif-ferent class.
The purpo of LPP is to prerve the proximity relationship of the input data.In LPP,by applying k nearest neighbor(k-NN)crite-rion,any point and its k nearest neighbors are viewed as located on a super-plane,where all the descriptions in linear space can be per-formed.A common problem with the classical LPP and veral supervid LPP methods[14,17,18]is that they might not necessar-ily discover the most discriminative manifold for pattern classifica-tion tasks becau the manifold learning is originally modeled bad on a characterization of‘‘locality’’,a model that has no direct con-nection to classification.This is unproblematic for existing LPP algo-rithms as they ek to model a simple manifold,for example,to recover an embedding of one person’s face images.In face recogni-ti
on each person forms his or her own manifold in the feature space [23].If one person’s face images do exist on a manifold,different persons’face images could lie on different manifolds.If the images needed to be classified reside on multi-manifolds and two or more models have a common axis,then the locality prerving algorithms of manifold learning may result in overlapped embedding belonging to different class becau to recognize faces it would be necessary to distinguish between images from different manifolds.This prob-lem is referred to as‘‘overlearning of locality’’[24].
In order to solve the problem of‘‘overlearning of locality’’,Yang et al.propod an unsupervid discriminant projection(UDP)[25] method,which can be viewed as simplified LPP on the assumption that the local density is uniform[26].In the propod method, locality and non-locality are discusd in detail,where locality means the sum of the squared distance between the points in k nearest neighbors,and the non-locality denotes the sum of the squared distance between two points not belonging to any k near-est neighbors.In order to achieve a discriminative map,UDP aims tofind a linear transformation that maximizes the ratio of the non-locality to the locality.In the literature[27],there is another algorithm named locally prerving and globally discriminant pro-jection with prior information(LPGDP)introduced to address this problem.The LPGDP method utilizes prior misclassification rate of between-class in the training data for the global discriminant measure
while using class labels for prerving locality.Besides, Li et al.propod a linear multi-manifolds learning bad approach called constrained maximum variance mapping(CMVM)[28]. CMVM aims at globally maximizing the distances between differ-ent manifolds.After the local scatters have been characterized, the CMVM algorithm focus on developing a linear transforma-tion that maximizes the dissimilarities between all the manifolds under the constraint of locality prerving.
As discusd above,when LPP is ud to map the high-dimen-sional data into a low-dimensional feature space,it may produce high between-class overlaps becau of the‘‘overlearning of local-ity’’.To solve this problem,the methods including UDP,LPGDP and CMVM ek tofind a transformation that parates different man-ifolds after the local structure has been characterized.It is unprob-lematic for the methods to effectively parate different class when the data distributed on a manifold have the same label.How-ever,in practice,the local scatter is usually constructed according to the k-NN criterion,which will bring another problem.It is that, when there is large variation within the same class,the within-class variation may be larger than the between-class variation, which means that the neighbor relationship measured by the k-NN criterion may be distorted.In other words,data samples resid-ing on a manifold possibly have different labels.In this ca,the methods may not work well becau of their common assumption that the data distributed on a manifold have the same label.
In this paper,we propo an effective supervid manifold learning algorithm,called local maximal margin discriminant embedding(LMMDE)for feature extraction and recognition.The propod algorithm LMMDE incorporates LPP and MMC for data analysis.Similar to MFA,LMMDE characterizes intra-class com-pactness and inter-class parability to maximize the margins be-tween different class.One difference between MFA and the propod method lies that MFA neglects the local structure bad on the overall samples which may be helpful for classification.In addition,both CMVM and LMMDE have the common purpo that is to take class label information into account bad on the prop-erty of locality prerving,but they are esntially different be-cau:(1)CMVM is originally designed to parate different manifolds bad on the assumption that the data distributed on a manifold have the same label,while LMMDE is designed to re-duce the between-class overlaps bad on the assumption that the data distributed on a manifold may have different labels and (2)CMVM characterizes only the inter-class parability in a global way,while LMMDE measures both the inter-class parability and the intra-class compactness in a local way like MFA.
The rest of this paper is structured as follows:In Section2,the PCA,LDA,LPP are briefly reviewed.Section3describes the pro-pod algorithm in detail.In Section4the propod algorithm is e
xamined on three data ts and the experimental results are of-fered.Section5finishes this paper with some conclusions.
2.Outline of PCA,LDA,LPP
Let us consider a t of n samples{x1,...,x n}takes values in an N-dimensional image space,and assume that each image belongs to one of C class.Let us also consider a linear transformation that maps the original N-dimensional space into a d-dimensional fea-ture space,where N>d.The new feature vectors in the d-dimen-sional space are defined by the following linear transformation: y
k
¼A T x k;k¼1;...;nð1Þ
P.Huang et al./J.Vis.Commun.Image R.25(2014)296–305297
where A e R NÂd is a transformation matrix.
2.1.Principal component analysis(PCA)
PCA eks tofind a transformation matrix such that the global scatter is maximized after the projection of samples.Let S T be the total scatter matrix:
S T¼
X n
i¼1
ðx iÀmÞðx iÀmÞTð2Þ
where m is the mean of total training samples.The PCA transforma-tion matrix is defined as:
A PCA¼arg max
A
½trðA T S T AÞ ð3Þ
where tr(Á)denotes the trace of a matrix.
Then the transformation matrix A that maximizes the objective function is obtained by solving the following generalized eigen-value problem,
S T A¼k Að4Þ2.2.Linear discriminant analysis(LDA)
LDA is a supervid algorithm,which eks tofind a transforma-tion matrix such that thefisher he ratio of the be-tween-class scatter to the within-class scatter)is maximized after projection of samples.The between-class and within-class scatter matrices S B and S W are defined by:
S B¼
X C
i¼1
n iðm iÀmÞðm iÀmÞTð5Þ
S W¼
X C
i¼1X n i
j¼1
ðx i
j
Àm iÞðx i
j
Àm iÞTð6Þ
where C denotes the total class number and n i denotes the number of training samples in the i th class;m i is the mean vector of the i th class samples and m is the mean vector of total training samples;x i
j is the j th sample in the i th class.
The LDA transformation matrix is defined as:
A LDA¼arg max
A trðA T S
B AÞ
trðA S W AÞ
ð7Þ
The optimal transformation matrix that maximizes the objec-tive function is compod of eigenvectors associated with d top eigenvalues of the following generalized eigenvalue eigen-equation,
S B A¼k S W Að8ÞNote that there are at most CÀ1non-zero(or available)general-ized eigenvalues.
2.3.Locality prerving projection(LPP)
LPP aims atfinding a transformation that prerves local struc-ture of the he neighbor relationship between samples so that samples that were originally in clo proximity in the ori-ginal s
pace remain so in the new space.Firstly an adjacency graph G={V,E}is constructed using the k-NN criterion,where G denotes the graph,V is the node t and E is the edge t.Then an adjacency matrix W is defined,who elements ud to characterize the like-lihood of two points are given by using the heat kernel weight below:
W ij¼
expðÀk x iÀx j k2=tÞ;if j2N kðiÞor i2N kðjÞ
0otherwi
(
ð9Þ
or simply0–1way,
W ij¼
1;if j2N kðiÞor i2N kðjÞ
0otherwi
&
ð10Þ
where t>0is an adjustable parameter,N k(i)is the t of k nearest
neighbors of x i.In fact,the0–1way is a special ca of(9)when
t=+1.
Due to introducing the adjacency matrix W,the local scatter
matrix S L can be expresd to:
S L¼
1
2
X n
i¼1
X n
j¼1
W ijðx iÀx jÞðx iÀx jÞT¼XðDÀWÞX T¼XLX Tð11Þ
where L=DÀW is the Laplacian matrix and D is a diagonal matrix
who entries are column(or row,sin W is symmetric)sum of W,
P n
j¼1
W ij.
To prerve local scatter of the manifold,LPP eks an optimal
linear subspace to minimize the following constrained objective
function:
A LPP¼arg min
A T XDX T A¼I
trðA T XLX T AÞð12Þ
The transformation matrix A that minimizes the objective func-
tion are given by the minimum eigenvalue solutions to the follow-
ing generalized eigenvalue problem,
XLX T A¼k XDX T Að13Þ
3.Local maximal margin discriminant embedding(LMMDE)
3.1.Motivation
The k-NN criterion is a common way to construct a local neigh-
borhood graph to model a manifold.Given an appropriate neigh-
borhood size k,define a graph G with the data points as the
vertices by the means of k-NN method.For the training data,each
point is connected to its nearest neighbors in the training t.
Apparently,the nearest neighbor approach cannot guarantee a
connected graph.At this step,veral disconnected graph compo-
nents may be obtained and each graph component can be consid-
ered as a data manifold[23].If two points A and B reside on two
manifolds respectively,we can get that A and B are not neighbors
of each A R N k(B)and B R N k(A).When LPP is ud to pro-
ject the data onto a feature space so that the neighbor relationship
of the data t is prerved,it may produce high between-class
overlaps.As described above,one reason may be that data points
from different class are evaluated to distribute on a manifold
by using the k-NN criterion,and then they get mapped clo to-
gether in the feature space.Fig.1illustrates an example of three
class(class1(h),class2(4),class3(s)).
From Fig.1a,we can e that:(1)two disconnected graph com-
ponents(data manifolds)are formed and(2)x i and its neighbors
not only from Class2but also from Class3reside on a manifold.
From Fig.1b,we can e that:(1)after LPP projection,data points
distributed on a manifold are clustered together and(2)Class2
and3are partially overlapped due to prerving the neighbor rela-
tionship of data points in a manifold.This limitation may be over-
come by developing a criterion that characterizes intra-class
compactness and inter-class parability of data points in the man-
ifold.Motivated by this,we propo a new algorithm,called local
maximal margin discriminant embedding(LMMDE).After projec-
tion by LMMDE as shown in Fig.1c,x i and its neighbors are still liv-
ing on a manifold,but data points from different class in the
manifold have been well parated.
298P.Huang et al./J.Vis.Commun.Image R.25(2014)296–305
3.2.Formulation of the between-class neighborhood scatter
In order to characterize the inter-class parability,for each data point,we need to push its neighboring data points from differ-ent class away from it as far as possible.Let l i e{1,...,C}denote the class label of x i.As mentioned in Section2.3,if j e N k(i)or
i e N k(j),then x j is thought to belong to the neighborhood of x i.Note that,the neighborhood of x i possibly contains the data points hav-ing the same label as x i or having different class labels from x i.
Thus,the between-class neighborhood N b
k
(i)of x i is defined as:
N b
k ðiÞ¼x j if j2N kðiÞor i2N kðjÞ;l i–l j;i;j¼1;...;n
ÈÉ
ð14ÞTo parate x i from its neighboring data points with different
class labels,we consider enlarging the distance between x i and the mean of its between-class neighborhood in the projected space, i.e.
y i À
1如何学习韩语
j N b
k
ðiÞj
X
j2N b
k
ðiÞ;j N b
k
ðiÞj–0
y
j
2
ð15Þ
where jÁj reprents the cardinality of a t.
Then the total between-class neighborhood scatter(e the der-ivation in Appendix A)can be defined as:
S0 b ¼
X
i
y
i
À
1
j N b
k
mdacðiÞj
X
j2N b
k
ðiÞ;j N b
k
ðiÞj–0
y
j
2
圣诞节的ppt
¼trðA T S b AÞð16Þ
where S b is called the between-class neighborhood scatter matrix which is calculated as:
S b¼
X
i x iÀ
1
j N b
k
ðiÞj
X
j2N b
k
ðiÞ;j N b
k
ðiÞj–0
x j
0 @1
A x iÀ1
j N b
k
ðiÞj
X
j2N b
k
ðiÞ;j N b
k
ðiÞj–0
x j
@
1
A
T
ð17Þ
3.3.Formulation of the within-class neighborhood scatter
In order to characterize the intra-class compactness,for each data point,we need to pull its neighboring data points of the same class toward it as near as possible.Similarly,the within-class neighborhood N w
k
(i)can be defined as:
N w
k ðiÞ¼x j if j2N kðiÞor i2N kðjÞ;l i¼l j;i–j;i;j¼1;...;n
ÈÉ
ð18ÞTo compact x i and its neighboring data points having the same
class label as it,we focus on reducing the distance between x i and the mean of its within-class
y i À
1
j N w
k
ðiÞj
X
j2N w
k
ðiÞ;j N w
k
ðiÞj–0
y
j
2
ð19Þ
Then the total within-class neighborhood scatter(e the deri-
vation in Appendix A)can be formulated as:
S0
w
¼
X
i
y
i
À
1
j N w
k
ðiÞj
X
j2N w
k
ðiÞ;j N w
k
ðiÞj–0
y
j
2
¼trðA T S w AÞð20Þ
where S w is called the within-class neighborhood scatter matrix
which is computed as:
S w¼
X
i
x iÀ
1
j N w
k
ðiÞj
X
j2N w
k
ðiÞ;j N w
k
ðiÞj–0
x j
@
1
A x iÀ1
j N w
k
ðiÞj
X
j2N w
k
ðiÞ;j N w
k
ðiÞj–0
x j
@
1
A
T
ð21Þ
3.4.The objective function and the algorithm of LMMDE
The objective function of LMMDE is constructed from two as-
pects:(1)characterizing intra-class compactness and inter-class
parability of data points in the manifold and(2)prerving the
local scatter.Therefore the transformation matrix A can be ob-
tained by solving the following objective functions:
arg max trðA T S b AÀA T S w AÞ¼arg max trðA TðS bÀS wÞAÞð22Þ
and
arg min trðA T S T
L
AÞð23Þ
To eliminate the freedom that we can multiply A with some
nonzero scalar,we add the constraint,
A T A¼I
where I is an identity matrix.
Thus the goal of LMMDE algorithm is just to solve the following
optimization problem:
arg max trðð1ÀlÞA TðS bÀS wÞAÀl A T S L AÞ
s:t:A T A¼I
ð24Þ
where0<l<1is a non-negative constant to balance the two terms
walking in the airof the objective function.Note that,both the formulations(22)and
(24)are developed bad on the MMC[4]to avoid the singularity
problem.
Using the Lagrangian method,we can easilyfind that the opti-
mal projection vectors a1,...,a d can be lected as the d eigenvec-
tors corresponding to thefirst d largest eigenvalues of the英汉翻译在线
following generalized eigenvalue problem:
ðð1ÀlÞðS bÀS wÞÀl S LÞA¼k Að25Þ
As the previous description,the propod LMMDE algorithmic
procedure can be summarized as
follows: illustration of LPP and LMMDE:(a)samples in original space;(b)samples projected by LPP;(c)samples projected
1.For a high-dimensional application,wefirst project a data t
f x i
g n
i¼1
into an m dimensional PCA subspace to reduce noi by retaining a certain portion of energy.For simplicity,we still u{x i}to denote the data projected in the PCA subspace in the following steps,let W PCA e R NÂm denote the transformation matrix of PCA.
2.Construct the adjacency matrix W using Eq.(10)and then com-
pute the local scatter matrix S L using Eq.(11),i.e.
S L¼1
2
X n
i¼1
X n
j¼1
W ijðx iÀx jÞðx iÀx jÞT¼XðDÀWÞX T¼XLX T
3.U Eq.(14),i.e.
N b
k ðiÞ¼f x j if j2N kðiÞor i2N kðjÞ;l i–l j;i;j¼1;...;n
g,to deter-
mine the between-class neighborhood of each data point,and u Eq.(17),i.e.
S b¼
X
i x iÀ
1
j N
k
ðiÞj
X
j2N b
k
ðiÞ;j N b
k
ðiÞj–0
x j
0 @1
A x iÀ1
j N
在线 一对一
k
ðiÞj
X
j2N b
k
ðiÞ;j N b
k
ðiÞj–0
x j
@
1
A
T
to compute the between-class neighborhood scatter matrix S b.
4.U Eq.(18),i.e.
元旦快乐用英语怎么说N w
kazuo ishigurok ðiÞ¼f x j if j2N kðiÞor i2N kðjÞ;l i¼l j;i–j;i;j¼1;...;n
g,to
determine the within-class neighborhood of each data point, and u Eq.(21),i.e.
S w¼
X
i
x iÀ
1
j N w
k
ðiÞj
X
j2N w
k
ðiÞ;j N w
k
ðiÞj–0
x j
@
1
A x iÀ1
j N w
k
ðiÞj
X
j2N w
k
ðiÞ;j N w
k
ðiÞj–0
x j
@
1
A
T
to compute the within-class neighborhood scatter matrix S w.
5.Solve the eigenvalue problemðð1ÀlÞðS bÀS wÞÀl S LÞA¼k A.
Let k1>k2>...>k d be the d largest eigenvalues of(1Àl)(S b-ÀS w)Àl S L and a1,...,a d be the associated eigenvectors.
6.Thefinal projection matrix is A=A PCA A LMMDE,where
我的生日派对
A LMMDE=[a1,...,a d].
4.Experiments and results
In this ction,we will conduct some experiments to systemat-ically evaluate the performance of the propod algorithm LMMDE and some other algorithms such as MMC[4],LPP[13],MFA[16], UDP[25]and CMVM[28]on the real-work facial databas such as ORL,Yale and FERET face data.It must be noticed that PCA is firstly adopted to preprocess the data before implementing MMC,4.1.Experiments on ORL databa
The ORL[29]face databa contains images from40individuals, each providing10different images.The facial expressions and fa-cial details(glass or no glass)also vary.The images were taken with a tolerance for some tilting and rotation of the face of up to 20°.Moreover,there is also some variation in the scale of up to about10%.In our experiments,two kinds of ORL databas with different resolutions are ud to show the impact of resolution on the performance of the compared methods.Fig.2shows sample images of one person from the ORL face databa.
On ORL databa,thefirst l(=2,3,4)images of each person are lected to form the training sample t,and the rest10-l are ud to form the testing t.Note that all compared methods involve a PCA p
ha for data preprocess.In this pha,nearly88%image en-ergy is kept.The parameters of each method are t as follows:for LMMDE,the parameter l is empirically t to0.1and the neighbor-hood size k varies from1to10;for LPP and UDP,the neighborhood size k varies from1to10;for MFA,the two parameters k1and k2 are t to lÀ1and(lÀ1)ÃC(C is the number of class), respectively.
Figs.3–5show the recognition performance of different meth-ods corresponding to dimensions when different trains are ud. Shown in Table1are the maximal recognition rates of MMC,LPP, MFA,UDP,CMVM and LMMDE and the corresponding dimensions (in the parenthes)when thefirst2,3,4images per class are ud for training and the remaining for testing.From the experimental results,it can be found that LMMDE performs better than the other methods no matter what the resolution of facial images is,and in particular,when the training sample number is small,the LMMDE algorithm significantly outperforms the other methods.
4.2.Experiments on Yale databa
The Yale[30]face databa contains165grayscale images of15 individuals.There are11images per subject,one per different fa-cial expression or configuration:center/left/right-light,w/wo glass,happy,normal,sad,sleepy,surprid,and winking.Two kinds of Yale databas are ud to ob
rve the impact of the res-olution on the performance of different methods.Fig.6shows sam-ples images of one person from the Yale face databa.
In the experiments,thefirst4images of each person are ud for training,and the remaining7images are ud for testing.Note that,the PCA method isfirstly ud as a preprocessing,by which the original face images are projected into a subspace where98% image energy is kept.Tofind how the neighborhood size k affects
Fig.2.Sample images of one person from the ORL databa,(a)32Â32pixels;(b)64Â64pixels. 300P.Huang et al./J.Vis.Commun.Image R.25(2014)296–305
P.Huang et al./J.Vis.Commun.Image R.25(2014)296–305301
