A New Method Based on MDA to Enhance the Face Recognition Performance

Aref Shams Baboli, Seyyedeh Maryam Hosseyni nia, Ali Akbar Shams Baboli & Gholamali Rezai rad
International Journal of Image Processing (IJIP), Volume (5) : Issue (1) 69
A New Method Based on MDA to Enhance the Face Recognition
Performance
Aref Shams Baboli Aref.shams2222@gmail.com
BSc Student at Department of Electrical and
computer Engineering, Noshirvani University
of Technology, mazandaran, Iran
Seyyedeh Maryam Hosseyni Nia maryamhosseini16@yahoo.com
MSc Student at Department of Biomedical
Engineering and Medical Physics Shahid
Beheshti University, Tehran, Iran
Ali Akbar Shams Baboli ali.shams2222@gmail.com
MSc Student at Department of Electrical
Engineering, Iran University of Science
and Technology, Tehran, Iran
Gholamali Rezai Rad rezai@iust.ac.ir
Associate professor at Department of
Electrical Engineering, Iran University of
Science and Technology, Tehran, Iran
Abstract
A novel tensor based method is prepared to solve the supervised dimensionality reduction
problem. In this paper a multilinear principal component analysis (MPCA) is utilized to reduce the
tensor object dimension then a multilinear discriminant analysis (MDA), is applied to find the best
subspaces. Because the number of possible subspace dimensions for any kind of tensor objects
is extremely high, so testing all of them for finding the best one is not feasible. So this paper also
presented a method to solve that problem, the main criterion of algorithm is not similar to
Sequential mode truncation (SMT) and full projection is used to initialize the iterative solution and
find the best dimension for MDA. This paper is saving the extra times that we should spend to
find the best dimension. So the execution time will be decreasing so much. It should be noted that
both of the algorithms work with tensor objects with the same order so the structure of the objects
has been never broken. Therefore the performance of this method is getting better. The
advantage of these algorithms is avoiding the curse of dimensionality and having a better
performance in the cases with small sample sizes. Finally, some experiments on ORL and
CMPU-PIE databases are provided.
Keywords: Dimensionality Reduction, HOSVD, Subspace Learning, Multilinear Principal
Component Analysis, Multilinear Discriminant Analysis.
1. INTRODUCTION
A typical tensor object in machine vision or pattern recognition applications is actually in a high-
dimensional tensor space. In reality, the extracted features of an object often has some specific
structures that are in the form of second or even higher order tensors [1]. Most previous works
transform the input image data into a 1-D vector, which ignores the underlying data structure so
these methods suffer from curse of dimensionality and small sample size problem. Subspace
learning is one of the most important directions in computer vision research [2], [3]. Most
traditional algorithms, such as LDA [4] input an image object as a 1-D vector. It is well understood
that reshaping breaks the natural structure and correlation in the original data.

Some recent works have started to consider an object as a 2-D matrix rather than vectors for
subspace learning. A 2-D PCA algorithm is proposed in [5] where gets the input images as a
matrix and compute a covariance matrix. As we mentioned before, in this paper a method that
utilized the MDA after MPCA algorithms has been proposed in which both of those algorithms
work with tensor objects that give us the better results.
It should be noted that recently there are many developments in the analysis of higher order.
Reference [6] used a MPCA method based on HOSVD [7]. There is also a recent work on
multilinear discriminant analysis (MDA) in [8] which is used for maximizing a tensor-based
discriminant criterion. Previously, we proposed MPCA+MDA [9] for face recognition. In that paper
we use MPCA algorithm for tensor object feature extraction. MPCA is a multilinear algorithm
reducing dimension in all tensor modes to find those bases in each mode that allows projected
tensors to achieve most of the original tensors variation. Then these bases are applied on
samples and a new data set with a new dimension will be generated. This new data set will be
the inputs of our MDA algorithm. MDA uses a novel criterion for dimensionality reduction,
discriminant tensor criterion (DTC), which maximizes the interclass scatter and simultaneously
time minimizes the intraclass scatter. In that paper we should give the goal dimension for
reduction manually. As we know, the number of possible subspace dimensions for tensor objects
is extremely high, so testing all of them to find the best one is not feasible. To solve that problem,
a method is used to find the best dimension that gives us the best accuracy. Our method is a little
similar to SMT that is used in MPCA algorithm [5]. To start the algorithm like SMT we need to
initialization the subspaces. So this paper is used full projection to initialize the iterative solution
for MDA [6]. The main idea of this paper is saving the extra times that we should spend to find the
best dimension and of course the final dimension in MDA that we find practically is not optimal.
But with our improvement we are decreasing the execution time so much.
MPCA+Improved MDA can avoid the curse of dimensionality dilemma by using higher order
tensor for objects and n-mode optimization approach. Due to using the MDA after applying the
MPCA, this method is performed in a much lower-dimension feature space than MDA and the
traditional vector-based methods, such as LDA and PCA do. Also because of the structure of
MDA, it can overcome the small sample size problem. As we know, the available feature
dimension of LDA is theoretically limited by the number of classes in the data set but in our
algorithm it is not limited. So it would give us the better recognition accuracy. As a result of all the
above characteristics, we expect this novel method to be a better choice than LDA and PCA
algorithms and more general than MDA for the pattern classification problems in image analysis
and also overcome the small sample sizes and curse of dimensionality dilemma.
The rest of this paper is organized as follows. Section 2 introduces basic multilinear algebra
notations and concepts. In Section 3, the Initialization procedures of MPCA and introducing the
DTC and n-mode optimization that are used in MDA is discussed after that we will see our
proposed method for finding the best subspaces dimension. Then, in Section 4, we present the
face recognition experiments by encoding the image objects as second or third-order tensors and
compare them to traditional subspace learning algorithms and MDA algorithm. Finally, in Section
5, the major point of this paper and the future work is summarized.
2. MULTILINEAR NOTATIONS AND BASIC ALGEBRA
This section briefly will be reviewed some basic multilinear concepts used in our framework and
see an example for n–mode unfolding of a tensor. Here, vectors are denoted by lowercase
boldface letters, such as, x, y. The bold uppercase symbols are used for representing matrices,
such as U, S, and tensors by calligraphic letters, e.g. ࣛ. An Nth-order tensor is denoted as
1 2
.NI I I× ×…×
∈RA It is addressed by N indices In, n = 1, … , N and each In addresses the n-mode of
ࣛ. The n-mode product of a tensor ࣛ by a matrix U , is
( )( ) ( )1 1 1 1, , , , , , , , . ( , )
n
n n n n N N n n
i
i i j i i i i j i− +× … … = …∑U UA A (1)

The scalar product of two tensors 1 2
, NI I I× ×…×
∈RA B is defined as
1 2
,
i i iN
= …∑∑ ∑p fA B
( )1 1, , . ( , , )N Ni i i i… …A B and the Frobenius norm of ℬ is defined as ,F = p fB B B [7].
Unfolding along the n-mode is denoted as
( )1 1 1
( )
n n n NI I i i i
n
− +× ×…× × ×…×
∈A R . The column vectors of
( )nA are the n-mode vectors of A . Fig. 1 illustrates three ways to unfold a third-order tensor. For
unfolding along the first-mode, a tensor is unfolded into a matrix along the I1 axis, and the matrix
width direction is indexed by searching index I2 and I3 index iteratively. In the second-mode, the
tensor is unfolded along the I2 axis and the same trend afterwards.
Following standard multilinear algebra, tensor A can be expressed as the
product
( ) ( ) ( )1 2
1 2
N
N= × × ×…×U U UA S . Where
( ) ( ) ( )1 2T T T
1 2
N
N= × × ×…×U U US A and we
call S core tensor that will be used for HOSVD and
( ) ( ) ( ) ( )
( )1 2 n
n n n n
I= …U u u u is an orthogonal
n nI I× matrix. The relationship between unfolded tensor ( )nA and its decomposition core tensor
( )nS is
( )
( )
( )
( ) ( )
( ) ( ) ( ) ( )
1 2
1 2 1 T
. .(
)
n n n
n n
N n
+ +
−
= ⊗ ⊗…
⊗ ⊗ ⊗ ⊗…⊗
A U U U
U U U U
S
(2)
Where ⊗ means the Kronecker product [7].
The projection of an n-mode vector of A by
( )Tn
U is computed as the inner product between the
n-mode vector and the rows of
( )Tn
U . For example in Fig. 2, a third-order tensor 1 2 3I I I× ×
∈RA is
projected in the 1-mode vector space by a projection matrix
( ) 1 11 T m I×
∈RB , the projected tensor
is
( ) 1 2 31 T
1
m I I× ×
× ∈RA B . In the 1- mode projection, each 1-mode vector of length 1I is projected
by
( )1 T
B to obtain a vector of length 1m .
FIGURE 1: Illustration of the n-mode unfolding of a third–order tensor.

FIGURE 2: Illustration of multilinear projection in the mode 1
3. MULTILINEAR PRINCIPAL COMPONENT ANALYSIS & MULTILINEAR
DISCRIMINANT ANALYSIS
Some previous approaches to subspace learning, such as PCA and LDA, consider an object as a
1-D vector so the learning algorithms should be applied on a very high dimension feature space.
So these methods suffer from the problem of curse of dimensionality. Most of the objects in
computer vision are more naturally represented as second or higher order tensors. For example,
the image matrix in Fig. 3(a) is a second-order tensor and the filtered Gabor image in Fig. 3(b) is
a third-order tensor.
In this section, first we see, how the MPCA solution for tensor objects is working and then we will
see the DTC and n-mode optimization that is used in MDA for tensor objects. A set of M tensor
objects 1 2{ , ,..., }MX X X is available for training. Each tensor object 1 2 NI I I
m
× ×…×
∈RX assumes
values in a tensor space 1 2
,NII I
⊗ …⊗R R R where nI is the n-mode dimension of the tensor.
The MPCA defines a multilinear transformation that maps the original tensor space into a tensor
subspace. In other words, the MPCA objective is the determination of the projection matrices
( )
{ }, 1, ,n nn I P
n N×
∈ = …U R that maximize the total tensor scatter, yΨ
( )
( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 y
, , , , , ,
{ , 1, , } argarg max max ΨN N
n
n N
… …
= … =
U U U U U U
U (3)
Where
1
y
1
2
, 1Ψ ( )
M
F
M
m
mm
mm ==
−= =∑ ∑A A A A .
FIGURE 3: Second- and third-order Tensor representations samples
3.1 MPCA Algorithm
There is no optimal solution for optimizing the N projection matrices simultaneously. An Nth-order
tensor consists of N projections with N matrix, so N optimization subproblems can be solved by
finding the
( )n
U that maximizes the scatter in the n-mode vector subspace. If
( )
{ }, 1, ,n
n N= …U be the answer of (3) and
( ) ( ) ( ) ( )1 1 1
, , , , ,
n n N− +
… …U U U U be all the other

known projection matrices, the matrix
( )n
U consists of the nP eigenvectors corresponding to the
largest eigenvalues of the matrix
( )
Φ n
( )( ) ( )( ) ( )
( )
( ) ( )( )Φ Φ
1
Φ . . .n n
M T
n T
n nm nm n
m=
= − −∑ X X U U X X (4)
Where
( ) ( ) ( ) ( ) ( ) ( )
( )
1 2 1 2 1
Φ n
n n N n+ + −
= ⊗ ⊗…⊗ ⊗ ⊗ ⊗…U U U U U U U .
The proof of (4) is given in [6].
Since
( )
Φ n
depends on all the other projection matrices, there is no closed-form solution for this
maximization problem. Instead, reference [6] introduce an iterative procedure that can be utilized
to solve (4). For initialization, MPCA used full projection. The term full projection refers to the
multilinear projection for MPCA with Pn=In for n= 1, …, N. There is no dimensionality reduction
through this full projection. The optimal is obtained without any iteration, and the total scatter in
the original data is fully captured. After finding the projection matrices,
( )
, 1, , ,
n
n N= …U we
applied those matrices to the training set. At this point, we provide a set of tensors with the new
dimension that would be the new training set for MDA algorithm.
3.2 Multilinear Discriminant Analysis
Here, the DTC is introduced which is used in MDA algorithm. The DTC is designed to provide
multiple interrelated projection matrices, which maximize the interclass scatter and at the same
time minimize the intraclass scatter. That is
( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1
1 1 2
1 1*
1 1 21
1 1
arg max .N
n
i
n
n n
N cc n nn c
n nn ci n ni
n
=
=
× …× − × …×
=
× …× − × …×
∑
∑U
U U U U
U
U U U U
X X
X X
(5)
Where cX is the average tensor of class c samples, X is the total average tensor of all the
samples, and cn is sample number of class c. We could optimize that function by using n-mode
optimization approach that is proved in [8]. The optimization problem can be reformulated as
follows:
( )
( )
( ) ( )
( ) ( )
* ( )
arg max
( )n
n nT
n B
n nT
W
Tr S
Tr S
=
U
U U
U
U U
(6)
( )( ) ( ) ( )
1 1
, ( )( )
o
o n c
m
N
j j j jj j T
c n n c n nB B B c
j c
S S S n
≠
= =
∏
= = − −∑ ∑ X X X X
( ) ( ) ( ) ( )
1 1
, ( )( )
o
o n c
i i
m
N
j j j jj j T
i n c n i n c nW W W
j i
S S S
≠
= =
∏
= = − −∑ ∑ X X X X
Where, ( )
j
i nX is the jth column vector of matrix ( )
j
i nX which is the n-mode unfolded matrix from
sample tensor iX . ( )
j
c nX and ( )
j
nX are defined in the same way as ( )i nX with respect to tensors
cX and X and the proofs are given in [8]. To utilizing n-mode optimization, first the input tensors
(that are the outputs of MPCA) should be projected with all the other modes matrices and then all

the new tensors are unfolded into a matrix along the nth-mode. Therefore, the optimization
problem in (5) can be reformulated as a special discriminant analysis problem, and it can be
solved in the same way for the traditional LDA algorithm [8]. Since DTC has no closed form the
projection matrices can be iteratively optimized.
3.3 Determination of the Tensor Subspace Dimensionality
The target dimensionality Pn has to be determined. So the objective MPCA function should be
revised to include a constraint on the favorite dimensionality reduction. The revised objective
function is as follows [6]:
( )
{ }, , 1, ,n
nP n N= …U
( ) ( ) ( )
( ) ( )
( ) ( )1 2
1 2, , , , , , ,
( )
arg max
( )N
N
n nT
B
n nT
P P P
W
Tr S
Tr S… …
=
U U U
U U
U U
1
1
subject to
N
nn
N
nn
P
I
=
=
<
∏
∏
(7)
Where the ratio between the reduced dimensionality and the original tensor space dimensionality
is utilized to measure the amount of dimensionality reduction, and is a threshold to be
specified by user.
The proposed tensor subspace dimensionality determination solution is Starting with Pn=In for all
n at t=0, at each subsequent step t=t+1, this algorithm truncates, in a selected mode n, the Pn th
n-mode eigenvector of the reconstructed input tensors. The truncation can be interpreted as the
elimination of the corresponding Pn th n-mode slice of the total scatter tensor. For the specific
mode selection, the scatter loss rate
( )n
tδ due to the truncation of its Pn th eigenvector is
calculated for each mode.
( )n
tδ is defined as follows [6]:
( )
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( )t t 1
1, 1,
( ) ( )
( ) ( )
. ( 1).
n n n nT T
B B
n n n nT T
W Wn
t N N
n j n jj j n j j n
Tr S Tr S
Tr S Tr S
P P P P
δ
−
= ≠ = ≠
−
=
   − −
   ∏ ∏
U U U U
U U U UY Y
( )
1,
n
n
P
N
jj j n
P
λ
= ≠
=
∏ (8)
Where
( ) ( ) ( ) ( )
(t)( ) ( )n n n nT T
B WTr S Tr SU U U U Y is maximizing the between class scatter and
at the same time minimizing the within class scatter at step t,
1,
N
j j n jP
= ≠∏ is the amount of
dimensionality reduction achieved, and
( )
n
n
Pλ , which is the corresponding nP th n-mode eigenvalue,
is the loss due to truncating the nP th n-mode eigenvector. The mode with the smallest
( )n
tδ is
selected for the step-t truncation. For the selected mode n, nP is decreased by 1: 1n nP P= −
and
1 1
N N
n nn n
P I= =∏ ∏ is tested. The truncation stops when
1 1
N N
n nn n
P I= =∏ ∏ is satisfied. The
term full projection refers to the multilinear projection for MDA with Pn=In for n= 1, … , N for
starting the algorithm. There is no dimensionality reduction through this full projection [5]. The

optimal is obtained without any iteration. As we know, if all eigenvalues (per mode) are distinct,
the full projection matrices are also distinct. Therefore, the full projection is unique [6].
4. EXPERIMENTS
In this section, two standard face databases ORL [10], CMU PIE [11] were used to evaluate the
effectiveness of our proposed algorithm, MPCA+Improved MDA, in face recognition accuracy.
These algorithms were compared with the popular Eigenface, Fisherface and MDA/2-1, MDA/2-2,
MDA/3-3 and the MPCA + MDA algorithms. In this work, we report the best result on different test
and for the fisherface on different feature dimensions in the LDA step, in all the experiments, the
training and test data were both transformed into lower dimensional tensors or vectors via the
learned subspaces, and we use the nearest neighbour classifier for final classification. The
performances on the cases with different number of training samples were also evaluated to
illustrate their robustness in the small sample size problems.
4.1 ORL Database
The ORL database includes 400 images of 40 persons. These images were captured at different
times and have different expression such as open or closed eyes, smiling or nonsmiling and facial
details like: glasses or no glasses. All images were in grayscale and centered with the resolution
of 112*92 pixels. Ten sample images of one person in the ORL database are displayed in Figure
FIGURE 4: Ten samples of one person in the ORL face database
Four sets of experiments were managed to compare the performance of our algorithm with
Eigenface, Fisherface, and MDA/2-1, MDA/2-2. In each experiment, the image set was
partitioned into the test and train set with different numbers. Table 1 shows the best face
recognition accuracies of all the algorithms in our experiments with different train and test set
partitions. The results show that our algorithm outperforms Eigenface, Fisherface, MDA/2-1,
MDA/2-2 and MPCA+MDA on all four sets of experiments, especially in the cases with a small
number of training samples and also we can see the performance of MPCA+Improved MDA is
the same as MPCA+MDA or even better than that. It means we provide the same performance
without spending the spare time to find the best dimension. So we can say our very new
proposed algorithm has the best performance and also save the spare times.
TABLE 1: Recognition Accuracy (%) Comparsion of MDA+MPCA, Eigenface, Fisherface, MDA/2-1,
MDA/2-2, MPCA+MDA on ORL database
Algorithms
Train-Test
5-5 4-6 3-7 2-8
Eigenface 97.0 91.25 87.50 81.56
Fisherface 93.0 85.83 87.50 79.68
MDA/2-1 97.5 96.25 94.28 88.13
MDA/2-2 99.0 97.91 95.00 90.31
MPCA + MDA 99.0 98.75 96.43 91.56
MPCA + Improved MDA 99.0 98.75 96.78 91.87

4.2 CMU PIE Database
The CMU PIE database contains more than 40,000 facial images of 68 people. The images were
obtained over different poses, under variable illumination conditions and with different facial
expressions. In our experiment, two sub-databases were used to evaluate our methods. In the
first sub-database, PIE-1, five near frontal poses (C27, C05, C29, C09 and C07) and illumination
indexed as 08 and 11 were used. The data set was randomly divided into training and test sets;
and two samples per person was used for training. We extracted 40 Gabor features. Table II
shows the detailed face recognition accuracies. The results clearly demonstrate that
MPCA+Improved MDA is superior to all other algorithms. As we knew, this database is really
hard for algorithms and most of them had a problem with that. As we can see, our algorithms
perform a really good job here and have the most accuracy and also work faster than the others,
especially the Improved algorithm, MPCA+Improved MDA, because it eliminates the extra time
that we should spend to find the best dimension.
TABLE 2: Recognition Accuracy (%) Comparison of Eigenface, Fisherface, MDA, MPCA+MDA, of MPCA+
Improved MDA with tensors of different orders on PIE-1 Database
Algorithms Accuracy
Eigenface (Grey) 57.2
Eigenface (Gabor) 70.5
Fisherface (Grey) 67.9
Fisherface (Gabor) 76
MDA/2-1 (Grey) 72.9
MDA/2-2 (Grey) 80.4
MDA/3-3 (Gabor) 83.6
MPCA+MDA 87.2
MPCA + Improved MDA 87.5
Another sub-database PIE-2 consists of the same five poses as in PIE-1, but the illumination
indexed as 10 and 13 were also used. Therefore, the PIE-2 database is more difficult for
classification. We conducted three sets of experiments on this sub-database. As we can see in
Table 3, in all the three experiments, MPCA + Improved MDA performs the best and the
eigenface has the worst performance. Especially in the cases with a small number of training
samples. Also for gaining that performance from our algorgorithm we don’t have to spend much
time that we use for MPCA+MDA and because of that privilege, our algorithm became a great
algorithm to choose.
TABLE 3: Recognition Accuracy (%) Comparison of MPCA+ Improved MDA, MPCA+MDA, Eigenface,
Fisherface, MDA/2-1 and MDA/2-2 on the PIE-2 Database
Algorithms
Test-Train
4-6 3-7 4-6
Eigenface 39.3 Eigenface 39.3
Fisherface 79.9 Fisherface 79.9
MDA/2-1 74.1 MDA/2-1 74.1
MDA/2-2 81.9 MDA/2-2 81.9
MPCA + MDA 84.1 MPCA + MDA 84.1
MPCA + Improved MDA 84.5 MPCA + Improved MDA 84.5

5.CONCLUSION
In this paper, we improve the performance of MPCA + MDA algorithm by optimizing the
subspaces dimension and full projection. Full projection is utilized for initialization the changed
SMT and the changed SMT is used to find the optimal subspaces dimension. After that, MDA has
been applied for supervised dimensionality reduction. Compared with traditional algorithms, such
as PCA and LDA, our proposed algorithm effectively avoids the curse of dimensionality dilemma
and overcome the small sample size problem and the advantage of this work is finding the
subspaces dimension Because in MDA algorithm the number of possible subspace dimensions
for tensor objects is extremely high, comprehensive testing for determination of parameters is not
feasible so with this work we save that amount of time. We are eager to apply this algorithm for
video-based (fourth order tensor) face recognition and we want to explore this work in our future
researches.
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