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A Near-Infrared Face Detection and Recognition
System Using ASM and PCA+LDA
Peiyi Shen 1, Liang Zhang
1, Juan Song
1, Hu Xu
1, Lianjie Qin
1, Wei Wei
2, Wenzheng Zhang
3, Bin Leng
3,
and Mengqi Zeng 3
1. National school of Software, Xidian University, Xi’an 710071, P. R. China
2. School of Computer Science and Engineering, Xian University of Technology, 710048, PR China
3. Science and Tech. On Com. Security Lab, Chengdu, 610041, P. R. China
Abstract—Near-infrared (NIR) images have a lot of
advantages, which can make up the shortages of visible light
(VL) images. A novel NIR detection and recognition system
is presented in this paper. First, a NIR imaging system is
developed to provide good illumination conditions for
subsequent face detection and recognition. Then, Active
Shape Model (ASM) method is applied to detect the features
and the low-dimensional Gabor features are extracted for
recognition by combining Principal Component Analysis
(PCA) and Linear Discriminant Analysis (LDA). The
recognition rate of the system is up to 90% in a few
milliseconds. Finally, by using such real-time NIR face
recognition system, comparative results are provided, from
that we can seen that this system can work robustly and
effectively.
Index Terms—NIR; ASM; Gabor; PCA; LDA
I. INTRODUCTION
Most current face recognition systems are based on
face images captured under visible light (VL) condition,
which cannot provide accurate recognition due to changes
of environmental illumination [1]. As the Near-infrared
(NIR) images have a lot of advantages such as strong
anti-interference and independence to VL source, great attentions have been paid to face recognition for NIR
images.
Accurate face position is important to feature
extraction for face recognition. During past 20 years,
scholars have carried out a lot of research. Active Shape
Model (ASM) proposed by Cootes [2] has been a popular
object detection method. It can restrict the adjustment of
parameters according to the training data, thereby limiting the shape change in a reasonable range.
Today, the methods of feature extraction and
description in face recognition are divided into two
categories [3]: geometrical characteristics based and
statistical characteristics based. Recent years, most of the
proposed methods are based on statistical characteristics,
for example, Principal Components Analysis (PCA),
Linear Discriminant Analysis (LDA), Support Vector Machine (SVM) and Artificial Neural Network (ANN)
and etc. Among them, PCA and LDA methods based on
subspace are widely used. However, PCA is less sensitive
to the classification information of different training
samples, and LDA is fully considerate of the
classification information, but the calculation process of
LDA is too complex to ensure the accuracy.
To achieve illumination invariant face recognition, we present a novel solution using NIR imaging techniques in
this paper. The solution consists of NIR imaging
hardware and NIR-based face detection and recognition
algorithms.
The remainder of this paper is organized as follows.
Section 2 introduces the presented imaging hardware
system for producing face images of a good illumination
condition. Section 3 describes the method of face detection and recognition. Experiments and results are
provided and analyzed in Section 4, prior to discussion
and conclusion in Section 5.
II. NIR IMAGING SYSTEM
The NIR imaging system is developed to overcome the
problem arising from uncontrolled environmental lights
so as to produce face images of a good illumination
condition for face detection and recognition. The key problems to capturing NIR images with good
illumination condition are 1) the design of camera
module and 2) the arrangement of the LED lamps.
To solve the first problem, we choose LED lamps
which can emit 940nm NIR lights as active radiation
source, which are strong enough to produce a clear
frontal-lighted face image without causing disturbance to
the human eyes. As shown in Figure 1(b), we place a optical filter on the camera to cut off the visible
environmental lights, while allowing most of the 940nm
NIR light to pass.
Cameras
A: the lamp panel
Cameras
A: the lamp panel
DD
C: IC/ID card readerC: IC/ID card reader
B: the touch screenB: the touch screen
PCB
PCB
Baffle plateBaffle plate
NIR camera module
NIR camera module
Color camera module
Color camera module
Lamp panel
Optical filter
Lens shadespacer
(1) (2)
Figure 1. The structure of the NIR imaging system. In the figure (1),
the front view of the system. In the figure (2), the side view of the
system
2728 JOURNAL OF NETWORKS, VOL. 9, NO. 10, OCTOBER 2014
© 2014 ACADEMY PUBLISHERdoi:10.4304/jnw.9.10.2728-2733
For the purpose of providing homogeneous light on the
face, it is important to carefully design the layout of the
LEDs. After simulation experiments, the solution of the
second problem is to choose the number of LEDs as 24,
to arrange LEDs around the cameras as shown in Figure
2(a), and to fix each LED lamp with an angle ϕ as shown
in Figure 2(b). With the design above, there exists an image area with homogeneous illumination about 60-
80cm distance from the camera.
(1)
(2)
Figure 2. The architecture of the NIR lighting equipment. In the figure
(1), 24 LEDs which can emit 940nm NIT are placed in the way that the
NIR light can be projected onto a person’s face equally. In the figure (2),
each LED lamp has a fixed angle ϕ
After analysis of the problems, we design the hardware
structure of the NIR lighting system including: the NIR
lighting equipment, cameras, image sensor (CMOS
OV7725) and CPU (OMAP3530). The NIR lighting equipment can provide a stable infrared illumination
condition. There are two cameras to collect both the VL
and NIR images, which can be transmitted to the CPU
through an image sensor and processed using algorithms
presented in Section 3. The structure of the system is
shown in Figure 1.
The NIR lighting equipment can provide a maximum
frame number of 30 per second and produce480 640 images.
(1) The color images
(2) The NIR images
Figure 3. The VL images and their corresponding NIR images are
taken by the present NIR imaging system. The influence of the
environmental lighting on the VL images is obvious, oppositely, the
NIR images can minimum the effect of the bad lighting condition
Figure 3 shows example images of one person taken by
the color camera and the NIR camera with different
direction of environmental illumination. It can be seen
that the VL images are intensely influenced by the
lighting condition, which is adverse to the further face
recognition; while the NIR images are robust to the
changes of illumination with the NIR LEDs lighting in the front and the filter to cut off most of the visible lights.
In this way, the NIR images produced by the equipment
can provide a good foundation of face recognition.
III. DESCRIPTION OF ALGORITHMS
A. ASM Method
The primary task of developing a real-time face
detection and recognition system is to find a suitable face
detection algorithm. Active Shape Model (ASM), an
image searching method based on statistical model,
includes four steps [4]: labeling and aligning the training
set, building the global shape model and the local grey
model, and searching the profile of the object iteratively. The advantage of ASM method is that it can restrict the
adjustment of parameters according to the training data,
thereby limiting the shape change in a reasonable range.
The detailed procedures are described in Figure 4.
Training set
Training set
Labeling and aligning the training set
Labeling and aligning the training set
Building the global shape
model
Building the global shape
model
Building the local grey
model
Building the local grey
model
Testing image
The best face profile
Finding the approximate face location
Finding the approximate face location
Initializing the shape model
Initializing the shape model
Searching the best fit for each
feature point
Searching the best fit for each
feature point
Updating the model
parameters
Updating the model
parameters
Converge?Converge?
Y
N
Figure 4. The process of ASM method
The shape model of 2-Dimensional contains 68 feature
points to represent the Point Distribution Model (PDM)
of human face. After labeling the training samples
manually, we align all the samples through similarity
transformation, and then get the average shape. One disadvantage of ASM is that it is more dependent
on the initial point set, so the initial position of the model
is very important. It is usual to get an approximate face
location through some other image processing methods.
A Haar-feature cascade classifier [5] provided by
OpenCV is used to find a rough face location in this
paper. As shown in Figure 5, the initial shape is obtained
after similarity transformation from the rough face location. In the meantime, the person should always stand
just before the image acquisition equipment.
Figure 5. The model's initialization. Left: the red box shows the rough
face location. Right: the initial shape
JOURNAL OF NETWORKS, VOL. 9, NO. 10, OCTOBER 2014 2729
© 2014 ACADEMY PUBLISHER
Another disadvantage of ASM is that it is easy to fall
into local extremum during the searching process. To
avoid this, we do some improvements for searching
method. For each feature point, we select the candidate
points along its normal direction. Through calculating the
Mahalanobis Distance between the candidate points and
feature points, we can find the best fit to the feature points. The Mahalanobis Distance can be described as [6]:
1( ) ( ( ) ) ( ( ) )T
j j j jf i y i y s y i y (1)
where, jy and js are the local grey model of the feature
point j, and y(i) is the normalized grey differential vector
of the candidate point i calculated by the same way to
the grey model. This distance reflects the similarity of the
sample point to the feature point.
Figure 6 shows an iterative process for ASM to search
the best fit to the feature points. After face detecting
using ASM method, we can get an accurate face location as the red box shown in Figure 6(f), for the further use of
face recognition.
(1) (2) (3)
(4) (5) (6)
Figure 6. An Iterative Process of improved ASM: 1) initial location, 2)
after 2 iterations, 3) after 4 iterations, 4) after 8 iterations, 5) after 16
iterations, 6) the detecting result of ASM method
B. PCA+LDA Method Based on Gabor Feature
Face recognition is a process to compare the target face
image with the given face in database, whose purpose is
to reach an identification conclusion. The core of this
process is extracting suitable feature and choosing proper
matching strategy.
On the foundation of the research of VL image based
face recognition and with the consideration of the calculation capability of the presented embedded system,
we put forward a new recognition method suitable to NIR
image based face recognition. The first step is to extract
the Gabor feature of the face image, and the following is
to identify the human face with the combinative method
of PCA and LDA.
The process of face recognition is shown in Figure 7.
The NIR images in face database which is under constructing are all collected through the NIR lighting
equipment developed by our lab. At the current stage, the
database includes 80 persons, and each one has at least 80
representative pictures, for example, different expressions,
face rotating and wearing eyeglasses etc.
Training
Recognizing
The result
Extracting Feature and reducing dimensionality
Pattern recognition
Gabor transform
Gabor transform
Gabor transform and PCA
Gabor transform and PCA
Project on subspace
Project on subspace
PCA and LDA
PCA and LDA
Classify by the nearest-neighbour
rule
Classify by the nearest-neighbour
rule
FaceDatabase
Testing image
Figure 7. The flow chart of face recognition
C. Gabor Feature Extraction
Gabor feature extraction is a method based on Gabor
Filter, which is widely applied in computer vision and
image processing fields because of the high resolution in
time and frequency domain.
The kernel function of Gabor Filter is defined as
follows [7]:
2 2 2 2
, ,
2
, ( /2 ) ( /2)
, 2[ ]
k z izkk
e e e
(2)
where and define the orientation and scale of the
Gabor kernels, ( , )z x y , denotes the norm operator,
and the definition of the wave vector ,k is
,
cos
sin
kk
k
. Where k = maxk / f and / 8 .
maxk is the maximum frequency, and f is the interval
factor between kernel functions in frequency domain.
We choose 5 different scales {0,1,2,3,4} and 8
different directions μ {0,1,2,3,4,5,6,7}, and let σ=2π,
maxk = /2, f = 2 . Substitute the value of these
parameters into the equation, and then we can get a set of
kernel functions , ( )z corresponding to different
direction and scale factors.
Given a face image ( , )I x y , the Gabor features are
extracted by convolving ( , )I x y with , ( )z as follows
[8]:
, ,( ) ( ) ( )O z I z z (3)
, ( )O z is the high-dimensional Gabor features of face
image ( , )I x y after Gabor filtering.
D. Face Recognition Using PCA+LDA
Because the calculation cost of the high-dimensional
data is adverse to the performance of the real-time system,
it is necessary to reduce the dimensionality of the Gabor features obtained above, simultaneously retaining most of
the original feature information.
Principal Components Analysis (PCA) and Linear
Discriminant Analysis (LDA) are two common
dimensionality reduction methods. By the analysis of the
advantages and disadvantages of PCA and LDA [9], we
can conclude that PCA method is less sensitive to the
classification information of different training samples, while LDA is fully considerate of the classification
information, but its calculation process is too complex to
2730 JOURNAL OF NETWORKS, VOL. 9, NO. 10, OCTOBER 2014
© 2014 ACADEMY PUBLISHER
ensure the accuracy. Therefore, a mixed method
combining PCA and LDA is adopted in this paper.
The steps of PCA+LDA [10] method are as follows:
1) Reduce the dimensionality of original data by PCA
to achieve a low-dimensional feature space of PCA;
2) Calculate the within-class scatter matrix and the
between-class scatter matrix of LDA in PCA subspace; 3) Calculate the best classification subspace of LDA,
and fuse the subspace of LDA and PCA;
4) Project training samples and testing images
separately onto the fusion subspace, get the recognition
features, and accomplish face recognition using the
Nearest Neighbor (NN) Rule.
The NN Rule is based on Cosine Similarity, which is
defined as [11]:
cos ( , )
Tx yx y
x y
(4)
where, vector x and y represent the two classes
remaining to be compared. The smaller cos ( , )x y is, the
more similar the two classes are.
(1) The original images
(2) The matching faces in database
(3) The matching scores
Figure 8. The recognition result: 1) the green frames show the detected
faces by ASM, 2) the best matching faces of the upper images, 3) the
matching scores between upper and lower images in the same column
Figure 8 shows the recognition results using the
PCA+LDA method. It can be seen from the matching
scores that the NN Rule based on Cosine Similarity can
separate among the different classes well. The
recognition rate of the algorithm can reach up to 90%
with different indoor light conditions and training faces
of 33 40 size.
IV. EXPERIMENTS AND RESULTS
Based on the presented hardware and algorithms, a
real-time NIR based face recognition system is built. In
the following, we evaluate the performance of the system
with three experiments.
1) Face detection--ASM method
As shown in Figure 9, the presented ASM could adapt
to various situations, such as rotating, shaping, different
expressions and wearing glasses. The average detecting time in a 480 640 image for ASM is 65ms. Then it can
be concluded that ASM is a suitable face detection
algorithm for NIR images.
Figure 9. Face detection results of ASM: the red blocks show face
rectangle and the green gridding shows feature points on the profile of
the whole face
2) Face recognition--algorithm evaluation on self-built
database
In experiment 2, we set up the subset of 50 persons from the whole face database we have build, with the
training number of each person increasing from 1 to 7
images and the training face images normalized into
33 40 size. The testing set has 2,500 images in total. No
images in the testing set are included in the training set.
Based on these, a comparison among PCA, LDA and the
presented PCA+LDA is done.
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
50 100 150 200 250 300 350
PCA
PCA+LDA
LDA
The
re
cogn
itio
n r
ate
The numbler of training samples
Figure 10. Performance with different number of training samples.
As shown in Figure 10, no matter how many the
training samples are, the recognition rate of PCA+LDA is
higher than that of PCA or LDA. Along with the increase
of the training samples, the recognition rate of all the algorithms is on the rise. When the training number of
each person increases to 6, the recognition rate of
presented PCA+LDA reaches the highest recognition rate
in this experiment.
3) Face recognition--algorithm evaluation on PolyU
Near-Infrared face database (PolyU-NIRFD)
In order to evaluate the performance of the face
recognition algorithms on a much bigger subset than experiment 2, we do experiment 3 using the PolyU Near-
Infrared face database (PolyU-NIRFD) [12], which was
constructed by the Hong Kong Polytechnic University.
The database contains images from 350 subjects, each
contributing about 100 samples with variations of pose,
expression, focus, scale and time, etc. Figure 11 shows
sample images of one person in the database [13-17].
JOURNAL OF NETWORKS, VOL. 9, NO. 10, OCTOBER 2014 2731
© 2014 ACADEMY PUBLISHER
Figure 11. Parts of NIR face images of a subject in PolyU-NIRFD
In this experiment, we extract 200 subjects from
PolyU-NIRFD, with 6 training samples of each subject.
The testing set contains 80 images of each person, in a
total of 16,000.
As shown in Figure 12, the recognition rate of the
PCA+LDA method in experiment 3 is 89%, which proves that the presented algorithm works well on other NIR
face database.
The recognition rate
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PCA LDA PCA+LDA
Figure 12. Performance of the algorithms on PolyU-NIRFD
4) The performance of the real-time system--
comparison among different illumination conditions
(1) Case A: training images captured by color camera
(2) training images captured by color camera and eliminating the
influence of VL
(3) training images captured by NIR camera
Figure 13. Training samples under different conditions: images in
training set are all captured by the presented hardware device, and
furthermore, we conduct homomorphic filtering and histogram
equalization on images in Case B
Figure 13 shows parts of training images in three cases
to compare the performance of the system under
conditions of VL, eliminating influence of VL and NIR
light. It can be seen that unfavorable lighting is obvious
under VL condition, while the changes of environmental
illumination is almost eliminated under NIR light
condition [18-25].
The training set has 30 persons, in which, each person
has 6 images of 33 40 size for training. We set up the real-time system in the lab and do experiments under
three different cases respectively. During the experiment,
people in the training set stand before the image
acquisition equipment in order. The experimental result is
shown in Table 1. We can see that the presented
algorithms under NIR condition can well satisfy the
requests of real-time system.
TABLE I. ANALYSIS ON EFFECTS OF DIFFERENT ILLUMINATION
CONDITIONS
Situations Detection time recognition time recognition rate
Case A
Case B
Case C
60ms
73ms
65ms
78ms
90ms
75ms
78%
87%
90%
V. DISCUSSION AND CONCLUSION
In this paper, a real-time NIR face recognition system
using ASM for detection and PCA+LDA for recognition
has been proposed. The novel NIR lighting system is
developed to solve the problem of uncontrolled environmental lighting. An accurate face location is
obtained by the ASM method. By combining PCA and
LDA, suitable Gabor features are extracted, thus the
calculation cost is reduced and the face recognition rate is
increased. Experimental results show the robustness and
effectiveness of the presented system.
The proposed system still exist some shortages.
Specular reflection on eyeglasses under NIR condition is a limitation for face recognition. The algorithms
presented are practical but not optimal. Our further work
is to solve these problems.
ACKNOWLEDGMENT
This work is partially supported by the Open Projects
Program of National Laboratory of Pattern Recognition.
The Project is partially supported by Natural Science
Basic Research Plan in Shaanxi Province of China (Program No. 2010JM8005). This program is also
supported by Scientific Research Program Funded by
Shaanxi Provincial Education Department (Program No.
2013JK1139) and Supported by China Postdoctoral
Science Foundation (No. 2013M542370) and supported
by the Specialized Research Fund for the Doctoral
Program of Higher Education of China (Grant No.
20136118120010).And this project is also supported by NSFC Grant (No. 61305109, No.61072105 No.
11226173, No.11301414, No. 61272283, No. 11301414,
No. 61172018), by 863 Program (2013AA014601), by
shannxi Scientific research plan (2013K06-09).
REFERENCES
[1] Stan Z. Li, Ru Feng Chu, “Illumination Invariant Face Recognition Using Near-Infrared Images”, IEEE
2732 JOURNAL OF NETWORKS, VOL. 9, NO. 10, OCTOBER 2014
© 2014 ACADEMY PUBLISHER
transaction on pattern analysis and machine intelligence, vol. 29, no. 4, pp. 627-639, 2007.
[2] T. F. Cootes, C. J. Taylor, “Active Shape Models--Their Training and Application”, Computer Vision and Image Understanding, vol. 61, no. 1, pp. 38-59, 1995.
[3] Qing Shan Liu, Han Qing Lu, “A Survey: Subspace Analysis for Face Recognition”, Acta Automatica Sinica, vol. 29, no. 6, pp. 900-911, 2003.
[4] Jie Zhu, Zhen Min Tang, “ASM and Color Gabor Features for Facial Feature Extraction”, Computer Science, vol. 37, no. 4, pp. 265-268, 2010.
[5] Chao Yan, Yuan Qing Wang, “Infrared face detection based on real Adaboost algorithm and Cascade structure”,
Laser and Infrared, vol. 39, no. 11, pp. 1246-1250, 2009. [6] Yan Peng Sun, Rong Fu, “Face feature points detection
based on refined ASM”, Computer Engineering and Applications, vol. 44, no. 10, pp. 163-165, 2008.
[7] Chen g Jun Liu, Harry Wechsler, “Gabor Feature Based Classification Using the Enhanced Fisher Linear Discriminant Model for Face Recognition”, IEEE Transactions on Image Processing, vol. 11, no. 4, pp. 467-476, 2002.
[8] Dong Li, Xu Dong Xie, “Gabor Boost Linear Discriminant Analysis for Face Recognition”, ICME, pp. 1050-1053, 2009.
[9] Aleix M. Martinez, Avinash C. Kak, “PCA versus LDA”, IEEE transaction on pattern analysis and machine intelligence, vol. 23, no. 2, pp. 228-233, 2001.
[10] Shun Gang Hua, Yu Zhou, “Thermal Infrared Face Image Recognition Based on PCA and LDA”, Pattern Recognition and Artificial Intelligence, vol. 21, no. 2, pp. 160-164, 2008.
[11] Rui Guo, Shu Ling Zhang, “Study of Feature Extraction and Similarity Match Method on Face Recognition”, Computer Engineering, vol. 32, no. 11, pp. 225-247, 2006.
[12] Baochang Zhang, Lei Zhang, “Directional Binary Code
with Application to PolyU Near-Infrared Face Database”, Pattern Recognition Letters, vol. 31, no. 14, pp. 2337-2344, 2010.
[13] Wei Wei, Bin Zhou, “Features Detection Based on a Variational Model in Sensornets” International Journal of Digital Content Technology and its Applications, ISSN:1975-9339, DOI: 10. 4156/jdcta. vol4. issue7. 11, Volume 4, Number 7, Octber 2010, On page(s): 115-127.
[14] Wei Wei, Hui Yang, etc, “Queuing Schedule for Location Based on Wireless Ad-hoc Networks with D-Cover Algorithm”, International Journal of Digital Content
Technology and its Applications, ISSN:1975-9339, DOI: 10. 4156/jdcta. vol5. issue1. 37, 5(1), January 2011 on page(s): 356-363.
[15] Jiang Jinlin, Wei Wei, “Many-facet Rasch Model’s Application in the Evaluation of Test Validity”
International Journal of Digital Content Technology and its Applications, ISSN: 1975-9339.
[16] Wei Wei, Liang Jun, “The Integration of p-Laplace Model Certifiable Protocols with ID-based Group Key in WSNs“, JDCTA, 2012.
[17] FENG Lei, Wei Wei, "Research of PSO/Genetic Algorithms and Development of its Hybrid Algorithm", JDCTA, 2012 (EI Journal).
[18] Yunji Wang, Hai-Chao Han, Jack Y. Yang, Merry L Lindsey, Yufang Jin. A conceptual cellular interaction model of left ventricular remodelling post-MI: dynamic network with exit-entry competition strategy. BMC Systems Biology, 2010, 4 (Suppl 1): 1-10
[19] Yunji Wang, Tianyi Yang, Yonggang Ma, Ganesh V Halade, Jianqiu Zhang, Merry L Lindsey, Yu-Fang Jin. Mathematical modeling and stability analysis of macrophage activation in left ventricular remodeling post-myocardial infarction. BMC Genomics, 2012, 13: 1-8.
[20] Tianyi Yang, Ying A Chiao, Yunji Wang, Andrew Voorhees, Hai-Chao Han, Merry L Lindsey, Yu-Fang Jin. Mathematical modeling of left ventricular dimensional changes in mice during aging. BMC Systems Biology, 2012, 6: 1-12.
[21] Yunji Wang, Philip Chen, Yufang Jin, Trajectory planning for an unmanned ground vehicle group using augmented
particle swarm optimization in a dynamic environment, in: IEEE International Conference on Systems, Man, and Cybernetics, San Antonio, TX, USA, 11-14 Oct, 2009, San Antonio, pp. 4341-4346.
[22] Yunji Wang, Yufang Jin, Yonggang Ma, Halade V Halade, Merry L Lindsey, Mathematical modeling of macrophage activation in left ventricular remodeling post-myocardial infarction, in: Genomic Signal Processing and Statistics (GENSIPS), 2011 IEEE International Workshop on: San Antonio, TX, USA, 4-6 Dec, 2011, pp. 202-205.
[23] Nguyen Nguyen., Xiaolin Zhang, Yunji Wang, Hai-Chao
Han, Yufang Jin, Galen Schmidt, Richard A Lange, Robert J Chilton, Merry L Lindsey, Targeting myocardial infarction-specific protein interaction network using computational analyses, in: Genomic Signal Processing and Statistics (GENSIPS), 2011 IEEE International Workshop on: San Antonio, TX, USA, 4-6 Dec, 2011, pp. 198-201.
[24] Wenting Zha, Junyong Zhai, Shuming Fei, Yunji Wang. Finite-time stabilization for a class of stochastic nonlinear systems via output feedback. ISA transactions, 2014 (In Press)
[25] Yi Wang, Cheng Gong, Baoku Su, Yunji Wang. Delay-dependent robust stability of uncertain TS fuzzy systems with time-varying delay. International Journal of Innovative Computing, Information and Control, 2009, 5(9): 2665-2674.
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