IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 37, NO. 5, OCTOBER 2007
1373
Correspondence
Comments on “On Image Matrix Based
Feature Extraction Algorithms”
where
s=
Quanxue Gao, Lei Zhang, David Zhang, and Jian Yang
K m
1
akj .
K ×m
(4)
k=1 j=1
Abstract—A class of image-matrix-based feature extraction algorithms
has been discussed earlier. The correspondence argues that 2-D principal
component analysis and Fisher linear discriminant (FLD) are equivalent
to block-based PCA and FLD. In this correspondence, we point out that
this statement is not rigorous.
Index Terms—Feature extraction, 2-D linear discriminant analysis
(LDA), 2-D principal component analysis (PCA).
Comparing (2) with (4), it is easy to obtain
s = aj ,
j = 1, 2, . . . , m
(5)
and hence
Gb = Gt
I. T WO -D IMENSIONAL P RINCIPAL C OMPONENT A NALYSIS
(PCA) V ERSUS B LOCK -B ASED PCA
In the above paper [1], it is argued that 2-D PCA [2] is equivalent
to block-based PCA [3]. This statement is rigorous if we assume
that the input images have been shifted to zero mean. However, in
the actual implementations of 2-D PCA and block-based PCA, they
have different procedures of zero-mean shifting. This leads to different
results of 2-D PCA and block-based PCA.
Denote by akj ∈ R1×n and aj ∈ R1×n the jth row of Ak (k = 1,
2, . . . , K) and A, respectively, where Ak ∈ Rm×n is the training
image, K is the number of training samples, and A is the mean image
of training samples. In 2-D PCA [2], the image covariance matrix can
be expressed as
K
1 (Ak − A)T (Ak − A)
Gt =
K
k=1
=
K
m
T k
1 k
aj − a j
a j − aj
K
(1)
k=1 j=1
rather than Gb = Gt , as claimed in [1]. We have the following
comment:
Comment 1. Two-dimensional PCA is not equivalent to block-based
PCA by setting the blocks as the rows of an image, i.e., 2-D PCA is
different from row-based PCA.
II. T WO -D IMENSIONAL F ISHER L INEAR D ISCRIMINANT (FLD)
V ERSUS B LOCK -B ASED FLD
. . , K), includGiven K training samples, Ak ∈ Rm×n (k = 1, 2, .
C
ing C classes. The ith class Ci has Ki samples, and i=1 Ki = K.
i
Denote by bkl , bl , and bl the lth column of Ak , A, and Ai , respectively,
where A is the global mean image and Ai is the mean image of the
ith class Ci . In 2-D FLD [4]–[6], if the right projection matrix R is
an identity matrix, then the image between-class scatter matrix Gb and
within-class scatter matrix Gw may be expressed as
Gb =
1 k
aj ,
K
Gw =
K
j = 1, 2, . . . , m.
In general, aj1 = aj2 ∀j1 = j2 .
In block-based PCA [3], if we view each row of an image as a block,
then the image covariance matrix is
m
K
T k
1 k
aj − s
aj − s
K
C
n T
1 1 i
i
bkl − bl
bkl − bl
C
Ki
i=1
(2)
k=1
Gb =
C
n T
1 i
i
bl − bl
bl − bl
C
(7)
i=1 l=1
where
aj =
(6)
i
(8)
Ak ∈Ci l=1
K
where bl = (1/Ki ) A ∈C bkl , and bl = (1/K) k=1 bkl .
i
k
Similar to block-based PCA, if each column of an image is considered a block, i.e., bkl is viewed as a training sample in FLD, then
the image between-class scatter matrix Sb and within-class scatter
matrix Sw are
(3)
k=1 j=1
Sb =
Manuscript received December 6, 2006. This paper was recommended by
Associate Editor M. Pantic.
Q. Gao is with the School of Telecommunication Engineering, Xidian
University, Xi’an 710071, China, and also with the Department of Computing,
Hong Kong Polytechnic University, Kowloon, Hong Kong.
L. Zhang, D. Zhang, and J. Yang are with the Department of Computing,
Hong Kong Polytechnic University, Kowloon, Hong Kong.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSMCB.2007.899415
C
1 (si − s)(si − s)T
C
(9)
i=1
Sw =
C
n
T
1 1 k
bl − si bkl − si
C
Ki
i=1
(10)
Ak ∈Ci l=1
n
bk and s = (1/(K × n))
where si = (1/(Ki × n)) A ∈C
l=1 l
i
k
K n k
b are the mean vectors of the ith class and all samples,
k=1
l=1 l
respectively.
1083-4419/$25.00 © 2007 IEEE
1374
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 37, NO. 5, OCTOBER 2007
Fig. 1. (a) Recognition accuracy curves of row-based PCA and 2-D PCA. (b) Recognition accuracy curves of row-FLD and 2-D FLD.
IV. C ONCLUSION
Compare (7) with (8) and (9) with (10), it is readily seen that
Gb = Sb
Gw = Sw .
(11)
We have the following comments:
Comment 2. Two-dimensional FLD is not equivalent to columnbased linear discriminant analysis (LDA) by assuming that the right
projection matrix is identity.
Comment 3. Two-dimensional FLD is not equivalent to line-based
LDA by assuming that the left projection matrix is identity.
Comment 4. Two-dimensional FLD methods are not equivalent to
block-based FLD approaches.
This correspondence indicates that the statement in [1], i.e., 2-D
PCA and 2-D FLD are equivalent to block-based PCA and FLD, is not
rigorous. Our comments can be helpful in understanding the concepts
of 2-D PCA, 2-D FLD, block-based PCA, and FLD.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers and the
editor for their constructive comments and suggestions.
III. E XPERIMENTAL R ESULT
R EFERENCES
We use the Olivetti Research Ltd. database (http://www.cam-orl.
co.uk) to illustrate the difference between the block-based PCA and
FLD methods, and the 2-D PCA and 2-D FLD methods. In the
experiments, the last five images per class are used for training, and the
remaining images are used for testing. Fig. 1(a) plots the recognition
accuracy curves of the row-based PCA and 2-D PCA by using the
cosine distance. Fig. 1(b) plots the recognition accuracy curves of rowFLD and 2-D FLD by using Euclidean distance and assuming that the
left projection matrix is identity.
From Fig. 1(a), it can be seen that 2-D PCA and row-based PCA
have different performances. From Fig. 1(b), we see that 2-D FLD by
using identity left projection matrix obviously has better recognition
accuracy than that of row-FLD.
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