IEEE Systems, Man and Cybernetics Magazine - January 2021 - 29

applications include feature extraction via 2DPCA in specsize mn # 1 lead to an mn # mn covariance matrix.
tral clustering of video face image trajectories [22] and
Two bottlenecks in feature extraction involve calculanatural image applications [23].
tion of the covariance matrix and eigendecomposition of
the covariance matrix. If the number of sample images is
Experiments: CSA and PCA
M 1 mn , the approach outlined in [2] can alleviate sigThe first of two image data sets used in our experiments is
nificant computational burden, where eigendecomposithe MNIST handwritten digits image set [24]. We utilized a
tion is performed first on the M # M matrix (1/M ) X T X ,
subset of this image set to provide an illustration and comwhere X = 6x 1 f x M@, and the resulting M # 1 eigenvecparison of CSA and PCA, both in classification and recontors are used to compute the mn # 1 eigenvectors of the
struction. Each 28 × 28-pixel grayscale image is a
mn # mn covariance matrix (rather than directly perhandwritten digit 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. The MNIST
forming eigendecomposition on the mn # mn covarihandwritten digits image set consists of 60,000 of these
ance matrix).
images often used for training and 10,000 additional images
In the row and column versions of 2DPCA, where the
for testing. For our experiments, we took 1,000 images of
scatter matrix sizes are n # n and m # m , respectively,
each category (0-9) from the 60,000 image set, for a total of
computing the associated eigenvectors is obviously less
10,000 sample images. We repeated
computationally demanding than
the comparison of CSA and PCA in
for PCA when the number of sample
classification and reconstruction
images M exceeds n or m, respecIn 2DPCA, B2DPCA,
using a subset of the second data
tively. For example, in [3], Yang et al.
and CSA, a single
set, the Fashion-MNIST image set
showed a more than 20-fold advan[25]. Fashion-MNIST was designed
tage in feature extraction time with
nearest neighbor
as a more challenging drop-in
2DPCA over PCA in one face recogclassifier is often used
replacement for the MNIST digits
nition application. With B2DPCA,
set, with the same 28 × 28-pixel
eigendecomposition is performed
in image recognition
grayscale image format, and 10
on both the m # m and n # n scatapplications.
image categories denoted as (0)
ter matrices, so 2DPCA has a comthrough (9): (0) T-shirt/top, (1) trouputational advantage over B2DPCA
sers, (2) pullover, (3) dress, (4) coat,
in finding the U or V matrix that
(5) sandal, (6) shirt, (7) sneaker, (8) bag, and (9) ankle boot.
defines the subspace onto which each centered image is
Like the MNIST digits set, the Fashion-MNIST set has
projected to compute the feature matrix corresponding to
60,000 images typically used for training and 10,000 images
each image. B2DPCA typically has computational and memoften used for testing. In our experiments using Fashionory advantages over PCA, as outlined in [15]. Because CSA is
MNIST, we followed the identical procedure as for our
essentially an iterated version of B2DPCA, eigendecomposiMNIST digits experiments, taking 1,000 images of each cattion is performed on two scatter matrices each iteration, so
egory from the Fashion-MNIST 60,000 image set to form a
the computational demand for finding the matrices U and V
10,000 image sample set.
in Algorithm 1 has a corresponding increase over that of
B2DPCA for t iterations. Any computational advantages over
PCA depend significantly on the number M of sample images
Classification
and the number of rows m and columns n of the images. A
Using the MNIST digits and Fashion-MNIST sample
potential disadvantage of CSA is that improvements in clasimage sets described previously, each containing 10,000
sification may be small after the iterative portion of the CSA
images with 1,000 images from each of the 10 categories,
algorithm, and the additional calculation time may not sigwe followed the same procedure for both sample sets in
nificantly improve performance over B2DPCA.
our classification experiments. We utilized 10-fold cross
validation, and the training and validation sets were,
respectively, 90% and 10% of the sample set for each
A Brief Look at Some Applications
round of cross validation. With each round of cross-valiApplications of 2DPCA, B2DPCA, and CSA are diverse. In
dation, the category sizes in the training set were kept
biometrics, examples include face recognition [3] (2DPCA),
equal, and likewise for the validation set. We initialized
[4], [14] (B2DPCA), [5] (CSA); palm print identification [14]
the columns of the n # n matrix V so that the ith column
(B2DPCA), iris recognition [16] (2DPCA, B2DPCA, and
CSA), human gait recognition [17] (CSA), and finger vein
was the eigenvector corresponding to the ith largest
analysis using a variation of B2DPCA [18]. In medical
eigenvalue of the scatter matrix S U , then the first p colimaging, [19] utilizes 2DPCA in detection of early diabetic
umns were retained to form n # p matrix V0 (see footretinopathy; in [20] row 2DPCA is proposed for reducing
note * in Algorithm 1 for the case where V is initialized).
false positives in breast mass detection. Forestry and botaThe iterative portion in Algorithm 1 converges typically in
ny applications include wood species identification [13]
two to three iterations (Figure 1 shows a typical example
and plant identification via leaf structure [21]. Other
of error RMSE versus iteration number for the iterative
	

Ja nu a r y 2021

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IEEE Systems, Man and Cybernetics Magazine - January 2021

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