Improving the Dictionary Construction in Sparse Representation

using PCANet for Face Recognition

Peiyu Kang, Yonggang Lu, Diqi Pan and Wenjie Guo

College of Information Science and Technology, Lanzhou University, Gansu, China

Keywords: Face Recognition, Sparse Representation, PCANet, Feature Learning.

Abstract: Recently, sparse representation has attracted increasing interest in computer vision. Sparse representation

based methods, such as sparse representation classification (SRC), have produced promising results in face

recognition, while the dictionary used for sparse representation plays a key role in it. How to improve the

dictionary construction in sparse representation is still an open question. Principal component analysis

network (PCANet), as a newly proposed deep learning method, has the advantage of simple network

architecture and competitive performance for feature learning. In this paper, we have studied how to use the

PCANet to improve the dictionary construction in sparse representation, and proposed a new method for

face recognition. The PCANet is used to learn new features from face images, and the learned features are

used as dictionary atoms to code the query face images, and then the reconstruction errors after sparse

coding are used to classify the face images. It is shown that the proposed method can achieve better

performance than the other five state-of-art methods for face recognition.

1 INTRODUCTION

Face recognition technology has been developed for

a long time and a variety of methods have been

proposed (Turk and Pentland, 1991; Zhang, Chen,

and Zhou, 2005; Maksimov et al., 2006; Liu et al.,

2001). Due to the wide range of face recognition

applications, there are still many researchers

dedicated to face recognition in recent years. Facial

similarity, shape instability and facial expressions,

gestures, age and other diversity, light conditions,

facial occlusion and many factors of the outside

world increase the difficulty in face recognition

(Ghiass et al., 2012; Chen and Su, 2017).

Sparse representation (Wright et al., 2010) is a

method that commonly used for signal compression

and encoding. Sparse representation based methods,

such as sparse representation classification (SRC)

(Wright et al., 2009), have already been applied in

image recognition and led to promising results. It is

found that applying sparse representation to image

classification can both reduce the computational

complexity brought by high-dimensional data, and

improve the robustness of the method (Zhang et al.,

2010; Elad and Aharon, 2006; Mairal, Elad, and

Sapiro, 2008; Lu et al., 2015; Zhou, 2012). The

sparse representation based classification has two

steps: coding and classification. First, the query

image is coded over the features which have strong

discriminative properties between objects to be

characterized

. Then classification can be carried out

by computing the reconstruction errors using the

coding coefficirnts and the selected features. The

general form of the sparse coding model is as

follows:

min . .

α st y Dαε





(1)

where y is the query image, D is the dictionary

which is constructed by the selected features, α is the

encoding sparse vector of y on the dictionary D, and

ε (ε > 0) is a constant (Yang et al., 2011a).

In the SRC method, training samples are directly

used as dictionary atoms for coefficient encoding. It

classifies the query images by evaluating which

class leads to the minimal

reconstruction error. The

method is simple and easy to understand, but a large

amount of class information among the training

samples is not used. Another classical sparse

representation method is K-SVD (Aharon, Elad, and

Bruckstein, 2006), which is an iterative method that

alternates between sparse coding of the query

images based on the current dictionary and a process

Kang, P., Lu, Y., Pan, D. and Guo, W.

Improving the Dictionary Construction in Sparse Representation using PCANet for Face Recognition.

DOI: 10.5220/0007368105170523

In Proceedings of the 8th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2019), pages 517-523

ISBN: 978-989-758-351-3

517

of updating the dictionary atoms to better fit the

data. The update of the dictionary columns is

combined with an update of the sparse

representations, thereby accelerating the

convergence. Since sparse coding problem is

equivalent to Lasso's problem (Tibshirani, 2011),

Yang et al. propose to perform face recognition by

solving Lasso problem in Robust Sparse Coding

(RSC) (Yang et al., 2011a). The coding coefficients

are calculated by iterative improvement, and a

weight matrix is added to the face image, which

gives a very small weight to pixels with occlusion or

noisy interference. All of these methods are common

in learning a public dictionary

shared by all classes.

However, the methods of public dictionary learning

do not make full use of the relationship between

sample labels and dictionary atoms, and hence

performing classification based on the reconstruction

error associated with each class is not allowed.

Different from these works, Yang et al. proposes a

sub-dictionary learning method (Yang et al., 2011b),

which learns a structured dictionary related to class

labels. It performs classification using class-related

reconstruction errors and produces better results than

SRC.

Although sparse representation based methods

have been successfully applied in face recognition,

they depend heavily on the selected dictionary D. So

it is important to study how to improve the

dictionary construction in sparse representation.

Feature learning has been widely used in

machine learning and a variety of feature learning

methods have been proposed in recent years (Bengio

et al., 2007; Learnedmiller, Lee, and Huang, 2012).

Principal component analysis network (PCANet) is a

novel deep learning algorithm for feature learning

with the simple network architecture and parameter

settings, which can be trained very efficiently. It was

proposed by Chan et al. (Chan et al., 2015), and is a

combination of principal component analysis (PCA)

and a convolutional neural network (CNN). PCANet

uses the most basic and simple operations to

simulate the processing layers in a typical neural

network: the data adaptive convolution filter bank in

each stage is selected as the most basic PCA filter;

the nonlinear layer is set to be the simplest binary

quantization (hashing); for the feature pooling layer,

it uses only the block-by-block histogram of the

binary code, which is considered to be the final

output feature of the network. It has been shown that

even the very basic PCANet has competitive or even

better performance compared with other methods in

image classification tasks (Chan et al., 2015).

In this paper we have studied how to improve

dictionary construction in sparse representation

using PCANet, and then proposed a new method for

face recognition. As mentioned above, using the

original training samples as the dictionary atoms

could not fully exploit the discriminative

information hidden in the training samples. So

PCANet is used to learn features from original face

images, and the learned features are used as

dictionary atoms in sparse representation instead.

With the proposed improved sparse representation

based on PCANet, the reconstruction error becomes

more discriminative, which leads to a better face

recognition method.

The rest of this paper is organized as follows.

Section 2 briefly reviews some related work, Section

3 presents the proposed face recognition method

based on sparse representation and PCANet, Section

4 describes the experimental results, and Section 5

concludes the paper.

2 RELATED WORK

2.1 Sparse Representation based

Classification for Face Recognition

Wright et al. propose the sparse representation based

classification (SRC) method for face recognition

(Wright et al., 2009). Based on the assumption that

the same class training samples lie on a linear

subspace, SRC searches the representative elements

from the training sample dictionary to sparsely

represent a test sample (Yang et al., 2011a). Suppose

that there are c classes samples, and let





, ,...,

DD D

be the set of the training

samples, where

is the sub-set of the training

samples from the i-th class. A given unknown image

y can be represented by the linear combination of the

training samples associated with the i-th class as:

yDα



(2)

where

,1 ,2 ,

, ,...,

iii ip

ααα α











is the representation

coefficients, it is a column vector, and p

is the

number of the i-th training samples.

Because there are c classes samples, the linear

representation y can also be rewritten in terms of all

the training samples as follows:

yDα



(3)

ICPRAM 2019 - 8th International Conference on Pattern Recognition Applications and Methods

518

where D is the dictionary and





;...; ;...;

αα α α

,1 , 2 ,

0,...,0, , ,..., ,0,...,0

ii ip

αα α







is the coefficients

vector whose entries are zero except those associated

with the i-th class. If the number of training samples

is large enough, the non-zero coefficients are sparse

relative to the length of the coefficient vector.

The coefficient vector can be estimated by

sparsely coding y on D via l1-minimization problem:

arg min . . ααst y Dαε

(4)

Then the classification can be done via:





() argmin

identity y error

(5)

where

,1,2,...

iii

error y D α ic 

, and

is the

coefficients vector associated with class i. The

implementation details of SRC can be found in

(Yang et al., 2011b).

2.2 Structures of the PCANet

The PCANet used in the experiments has three

layers and two stages. Suppose there are N training

images



| 1, 2,...,

Si N

, the size of each image is

mn and the filter size of each layer is k

k

. Figure

1 shows a detailed block diagram of the two-stage

PCANet. Only the PCA filter core needs to be

learned from the training images. That is why the

PCANet can be designed and trained easily and

efficiently.

Figure 1: Detailed block diagram of a two-stage PCANet.

2.2.1 The First Stage PCA

For each pixel, a block image of size k

k

is located

around the pixel, then all the image blocks are

collected for cascading as the representation of the i-

th image

,1 ,2 ,

, ,...,

iii imn

yyy y











, where





2mm k







2nn k



. We then subtract the

block mean from each block and obtain

,1 ,2 ,

, ,...,

iii imn

Yyy y











, where

,ij

is a mean-

removed block. For all input images, the mean of the

image are subtracted to produce the matrix:

, ,...,

kk Nmn

YYY Y













(6)

Supposing that the number of filters in the i-th

layer is L

, the purpose of the PCA is to minimize the

reconstruction error by finding a series of standard

orthogonal matrices:

12 1

min , . .

kk L

YUUY stUU I











(7)

where U is the filter bank and

is identity matrix

of size L



. In PCANet, just the L

primary

eigenvectors of YY

are obtained. So the PCA filter

is expressed as follows:









lkkl

W matrics eig YY





(8)

where l=1,2,…L





,kk

matrics vector is a function

that map

12kk

vector



  to a matrix



  , and





eig YY

represents the l-th principal eigenvector

of YY

. Then, the PCA mapping output of the first

layer is calculated by:

, 1, 2,...,

iil

SSWi N 

(9)

where the operation * represents the convolution of

two dimensions.

2.2.2 The Second Stage PCA

The mapping process of the second layer is basically

the same as the mapping mechanism of the first

layer. As with the blocking operation done in the

first layer, block sampling, cascading, and zero-

averaging are also performed on the input matrix

(the mapped output of the first layer) in the second

layer. The above operation is performed for each

input matrix, and finally the block sampling form of

the second layer input data is obtained:

112

, ,...,

kk Nmn

ZZZ Z













(10)

where Z represents the outputs of all the images after

convolving with

Then the eigenvectors of ZZ

is computed and L

principle eigenvectors are selected as PCA filters of

Input layer

Mean

removal

PCA filters

convolution

Mean removal

PCA filters

convolution

First stage

Second stage



Output layer

Binary quantization and

histogram generation

Improving the Dictionary Construction in Sparse Representation using PCANet for Face Recognition

519

the second stage. So the PCA mapping output of the

second layer is:

, 1, 2,...

iil

RSWl L 

(11)

We can see that the first layer and the second

layer are very similar in structure, so it is easy to

expand PCANet into a deep network structure

containing more layers.

2.2.3 The Output Stage Hashing and

Histogram

The binary processing is performed on each output

matrix of the second layer,



Binarify S W l

1, 2,..., }L

, where Binarify(x) is a binarization

function. If the x is a positive value, the function

value is 1. Otherwise, the function value is 0. In the

same pixel position of the L

outputs, the L

binary

bits are viewed as a decimal number. This converts

the L

outputs into a single integer-valued “image”:



Γ 2

ll l

iil

Binarify S W









(12)

After the above processing, each pixel value is

encoded as an integer within

0, 2 1







For each output matrix of the second layer, we

divide it into C blocks of size b

b

, calculate the

histogram information of each block, and then

cascade the histogram features of each block to

finally obtain the block extended histogram features:







Γ ,..., Γ

ii i

fChist Chist









(13)

where



Chist

represents the concatenated

histogram features of C blocks in decimal value map

When the local blocks are selected, the blocks

can be either overlapping or not. Experiments show

that non-overlapping blocks are suitable for face

recognition.

3 THE PROPOSED METHOD

Assume there are N training samples

Ι [Ι , Ι ,



..., Ι ]

. First PCANet is used to learn features from

the face images. In the proposed method, a two-stge

PCANet is used to learn features from the face

images. As mentioned above, only the PCA filter

core need to be learned from the training samples.

We need just one face dataset to learn PCA filters in

PCANet, and then such trained network can be

applied to learn features from new subjects in the

other datasets. Let





, ,...,

be the set of the

features learned using PCANet from original

training samples. The dictionary in sparse

representation is constructed by





, ,...,

ff f

Then the sparse representation is used to code

the query face images. Using the method of

Lagrange multiplier, equation (4) is converted to the

following equivalent problem:





min

α Aα y λα

(14)

where λ is the Lagrange multiplier. It’s a l1-

regularized least squares problem. In our

experiments, the l1_ls interior-point method (Koh,

Kim, and Boyd, 2007) for l1-regularized least

squares is used to solve the problem.

Once the coding coefficients are obtained, the

reconstruction error can be computed with respect to

the test sample as follows:

, 1, 2,...,

iii

error y A α ic 

(15)

Finally, the identity of y is the class corresponding

to the minimal reconstruction error, as given in (5).

Algorithm 1 summarizes the above procedure for the

proposed method.

Algorithm 1: Improving the Dictionary Construction in

Sparse Representation using PCANet for Face

Recognition.

Input: Training samples A

, testing samples B

, filter

size k

, number of filters L

, block size

, regularization parameter λ.

Output: Identity of test samples.

Step1: Learn PCA filters in PCANet using one face

dataset.

Step2: Produce new training samples A and test

samples B with features learning from A

, B

using PCANet.

Step3: Let A be the dictionary, using l1_ls to

compute the coding coefficients of y

(the i-th

sample in B) on A,





min

α Aα y λα

Step4: Compute the reconstruction error：

,1,2,...,

jijj

error y A α

c  .

Step5: Output the identity of y









argmin , 1, 2,...,

identity y error j c.

Step6: Return to step3 until all samples in B are

classified.

ICPRAM 2019 - 8th International Conference on Pattern Recognition Applications and Methods

520

4 EXPERIMENTS

The proposed method is verified on three publicly

available face datasets: AR, Extended Yale B, and

FERET. Experiments are conducted on computer

with Intel Core i7 CPU(3.60GHz). The proposed

method is compared with 1-nearest-neighbor (1NN),

the sparse representation based classification (SRC),

robust sparse coding (RSC), fisher discrimination

dictionary learning (FDDL), and PCANet (classify

by cosine distance). In all experiments, Principal

Component Analysis (PCA) is applied to reduce the

dimensionality.

4.1 Parameter Selection

In our experiments, a two-stage PCANet is used.

The MultiPIE

(Gross, Matthews, and Baker, 2008)

dataset has the most face images, so it is used to

learn PCA filters in PCANet, and then apply such

trained PACNet to construct dictionaries of new

subjects in the AR, Extended Yale B, and FERET

datasets for face recognition. The important

parameters in PCANet are the filter size k

, k

, the

number of filters L

, L

, and the block size b

, b

. In

order to determine the values of k

, k

, L

, b

we conduct experiments by changing the values of

, k

, L

, b

from 1 to 15 on the MultiPIE

face dataset. It is found that k

=5, L

=8, and

=8 is a good choice. We set λ=0.001 in all

experiments. For the AR and FERET datasets, all

the images from one dataset are put together and

then the 5-fold cross-validation is used. The initial

samples are segmented into 5 parts, a single part is

retained as data for testing, and the other 4 parts are

used for training. Cross-validation is repeated 5

times, each part is tested once, and the average of 5-

time results is used to finally obtain a single estimate.

4.2 The AR Dataset

The AR dataset (Martinez, 1998) consists of over

4,000 images from 126 individuals (70 males and 56

females), which varies in illumination, expression

and accessories like scarves and sunglasses blocking

some part of the face. A subset containing 1,400

images of 100 subjects with 50 males and 50

females without accessory are chosen in the

experiment. Sample images of the first person are

illustrated in Figure 2. All images are resized into

6043. Dimensionality of the features is reduced to

300 by PCA for all experiments on the AR dataset.

Figure 2: Sample images of the first subject from AR

dataset.

Table 1 shows the results of 1NN, SRC, RSC,

FDDL, PCANet and the proposed method on the AR

dataset. The proposed method achieves best among

all the methods. It is at least 0.43% higher than

others.

Table 1: The classification accuracy on the AR dataset.

Methods Accuracy (%)

1NN 77.16±3.54

SRC 96.50±1.14

RSC 99.27±0.32

FDDL 76.07±7.69

PCANet 99.36±0.42

The proposed method 99.79±0.28

4.3 The Extended Yale B Dataset

The Extended Yale B dataset (Georghiades,

Belhumeur, and Kriegman, 2001) consists of 2,414

images of 38 individuals captured under various

lighting conditions controlled in laboratory. Figure 3

shows sample images of the first person under

various lighting conditions. For each subject, the

frontal illumination images (the first 6 images) are

selected as the training images and the rest for

testing. All images are resized into 54

48.

Dimensionality of the features is reduced to 200 by

PCA for all experiments on the Extended Yale B

dataset.

Figure 3: Sample images of the first subject from

Extended Yale B dataset.

Table 2 shows the classification accuracies of

1NN, SRC, RSC, FDDL, PCANet, and the proposed

method on the Extended Yale B dataset. The

proposed method has the highest classification

accuracy: 97.99%, which is at least 8.73% higher

than others.

Improving the Dictionary Construction in Sparse Representation using PCANet for Face Recognition

521

Table 2: The classification accuracy on the Extended Yale

B dataset.

Methods Accuracy (%)

1NN 42.32

SRC 48.67

RSC 53.43

FDDL 54.20

PCANet 89.26

The proposed method 97.99

4.4 The FERET Dataset

The FERET dataset (Phillips, 2000) consists of

14,051 images with different poses, illuminations

and expressions. We choose a subset containing

frontal images marked with “ba”, “bj”, and “bk”, of

which there 600 images from 200 individuals. Such

images from the subsets are given in Figure 4. All

images are resized to 7060. Dimensionality of the

features is reduced to 400 by PCA for all

experiments on the FERET dataset.

Table 3 shows the results of 1NN, SRC, RSC,

PCANet, and the proposed method on the FERET

dataset. The proposed method produces the second

highest classification accuracy: 89%, while the

PCANet produces the best result.

Figure 4: Sample images from FERET dataset.

Table 3: The classification accuracy on the FERET dataset.

Methods Accuracy (%)

1NN 35.33±3.01

SRC 65.33±4.40

RSC 50.33±4.03

PCANet 90.22±3.71

The proposed method 89±2.76

Comparing with other face datasets, the FERET

dataset is a small dataset. It has 200 individuals, but

one subject only has 3 images. So, another

experiment is done on the extended FERET dataset

by using both the original images and the mirror face

images of original samples.

According to (Xu et al., 2017), for original face

image x, its mirror face image is defined as:









,,1

xpq xpQq





(16)

where

1,..., ; 1,...,pPqQ



, P and Q denote the

number of rows and columns of the face image

matrix.

Table 4: The classification accuracy on the extended

FERET dataset.

Methods Accuracy (%)

1NN 56.17±1.45

SRC 76.5±1.43

RSC 61±2.27

PCANet 90.91±1.10

The proposed method 94.33±1.11

The results of 1NN, SRC, RSC, PCANet and the

proposed method on the extended FERET dataset

are showed in Table 4. The classification accuracies

of all the methods for the extended FERET dataset

become higher compared to the corresponding

results for the original FERET dataset. And the

proposed method has produced the highest

classification accuracy: 94.33%, which is at least

3.42% higher than the other methods.

In the experiments, the proposed method

produces the highest classification accuracy within

the 6 methods on AR and Extended Yale B datasets.

For the FERET dataset in which the size of the

training data in each class is very small, the

proposed method only produces the second best

result. And after increasing the size of the training

data in the FERET dataset with the mirror face

images, the proposed method can also produce the

highest classification accuracy of 94.33% on the

extended FERET dataset.

5 CONCLUSIONS

To improve the classification accuracy in sparse

representation for face recognition, in this paper, we

have proposed an improved dictionary construction

method in sparse representation using PCANet.

Extensive experiments demonstrate that the

proposed method outperforms some previous state-

of-art methods for face recognition. It is found that

that the dictionary construction is crucial for sparse

representation. Compared to the original images, the

features learned by PCANet from the images can

serve as better dictionary atoms for sparse

representation in face recognition. One disadvantage

of the method is that when the size of the training

data in each class is too small, the proposed method

does not perform satisfactorily. As shown in the

ICPRAM 2019 - 8th International Conference on Pattern Recognition Applications and Methods

522

experiments, this problem can be solved by

increasing the size of the training data with the

mirror face images. Since the process of sparse

coding is very time-consuming, we will work on

improving the efficiency of the proposed method in

the future work.

ACKNOWLEDGEMENTS

This work is supported by the National Key R&D

Program of China (Grants No. 2017YFE0111900,

2018YFB1003205), and the Lanzhou Talents

Program for Innovation and Entrepreneurship

(Grants No. 2016-RC-93).

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