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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 46, 2015 

A publication of 

 

The Italian Association 
of Chemical Engineering  
Online at www.aidic.it/cet 

Guest Editors: Peiyu Ren, Yancang Li, Huiping Song  
Copyright © 2015, AIDIC Servizi S.r.l.,  
ISBN 978-88-95608-37-2; ISSN 2283-9216                                                                                

An Improved Sparse Representation Model for Robust Image 
Denoising 

Zhi Cui*, Xianpu Cui 

School of Communication and Electronic Engineering, Hunan City University, Yiyang, 413000, China. 
zhicui@yeah.net 

Though the sparse representation has demonstrated to be a very effective tool to de-noise the images with low 
levels of noise, it usually losses the power to well preserve structural features in images with high levels of noise. 
In this paper, we propose an improved de-noising method for images with low signal-to noise ratio. Specifically, 
the proposed method takes the histogram structural similarity (HSSIM) as similarity factor to replace the 
reconstruction error as the new fidelity term, and finds the most appropriate sparse coefficients by using the 
modified orthogonal matching pursuit (OMP) algorithm which enables structures in the recons tructed image run 
as close as possible to the ideal image. In addition, the proposed method adaptively trains the initialized 
dictionary by using the K-singular value decomposition (K-SVD) algorithm based on HSSIM to assure the image 
structures can be well reconstructed under the high noise circumstance. Experiment results have shown that the 
proposed method is better than some well-known de-noising methods in terms of PSNR and edge-preserved 
index (EPI) in high noise condition. 

1. Introduction 

De-noising is a fundamental work in image processing. In the past few years, m any approaches have been 
investigated to against this problem such as curvelet transformation (Ma and Plonka (2007)), total variation 
diffusion (Easley, et al (2009)), wavelet threshholding (Donoho (1995)), quaternion wavelet (Yin, et al (2012)), 
and inter-scale orthonormal wavelet thresholding (Luisier and Blu (2008)). 
Recently, sparse and redundant representation has attracted more and more attention. It is based on the theory 
that the given non-zero vector can be linear combined by minimum column in a predefined row-full-rank matrix 
with constraint of row number is much less than column number (Mallat and Zhang (1993)). Generally, the 
row-full-rank matrix is called dictionary and the columns in this matrix are called atoms. Sparse and redundant 
representation model has broken up the limit of orthogonal basis and maximize the advantages of redundant 
basis, and has been widely used in image fusion (Li, et al (2013)), image restoration (Ning, et al (2013)), and 
image Super-resolution (Kim and Kwon (2010)).  
Application of sparse representation in image de-noising has also garnered considerable attention. Because of 
the satisfaction of sparsity and separability for signal to noise, the sparse representation method based on 
redundant dictionary can achieve better de-noised effect than the transform domain based method (Li, et al 
(2012)). One of the most successful technique is the K-SVD algorithm reported by Elad and Aharon (2006). 
This approach extends the work by reported by Aharon and Elad (2006) in many ways, and the results have 
demonstrated that a dictionary which can adaptively updated would cause improved de -noising performance. 
On the foundation of work by Elad and Aharon (2006), Z. Zhou, et al (2012) proposed a K-least mean square 
algorithm (K-LMS) for the dictionary learning and image representation, and the de-noising could be achieved 
by combine the adaptive image sparse decomposition in a learned over-complete dictionary and the threshold 
process. However, the fixed step length adopted in the sparse decomposition would cause a large level of 
steady-state error. 
From the research course of the image de-noising based on the sparse representing, it is known that most of 
the current de-noising methods consider the reconstruction error of the images before and after de-noising as 
the fidelity term. However, affected by work environment and monitoring objects, image collected in the real 
condition always contains rich details and low signal-to-noise ratio. If we continue using the above criteria, the 
noise components introduced in the reconstructing process will have a larger impact on the image 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1546030

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Cui Z., Cui X.P., 2015, An improved sparse representation model for robust image denoising, Chemical Engineering 
Transactions, 46, 175-180  DOI:10.3303/CET1546030  

175



reconstruction accuracy, and the image's structural similarity also could not be well preserved. In order to keep 
the structural features information as much as possible when the image is effectively de -noised, in this paper, 
we focus on the removal of additive Gaussian white noise from standard images and propose an im proved 
method based on the foundational work by Elad and Aharon (2006). Our method has two specific parts. The 
first is to replace the reconstructed error with the histogram structural similarity as the new fidelity term. This is 
motivated by the conclusion that HSSIM aligns to the features of visual cortex and thus tends to produce better 
results that agree with the human visual system (Wang and Bovik (2004)). The second relates to the training of 
redundant dictionary. With our method, the initialized dictionary is trained by using K-SVD with constraint of 
HSSIM so that the image structures can be well reconstructed under the high noise circumstance. 

2. Related Work 

Fang et al (2012) reported that any ideal image could be described by the model of y R
n N  in which 

the image is separated into n n  parts. According to the sparse representation, a dictionary 

matrix D R n N  is defined to represent all the im age parts as follows: 

2

0 2
ˆ arg min . .  a a Da xs t   

(1) 

where a Rn  is the sparse representation coefficient, 
0

a  is the number of the non-zero values which 

means the sparsity of a , D  is the dictionary. 
Transform the constraints to the penalty term. According to the regularization optimization, the equation (1) can 
be changed to: 

2

2 0
, , ,

ˆ ˆ{ , } argmin ( )     
a y

a y y x a Da R y
ij

ij ij ij ij ij

i j i j

   
(2) 

where   is the Lagrange multiplier, 
ij

  is a coefficient, the first component represent the integral similarity 

between the image with noise and the clear image, and it should be less than the convolution of 2C   (C is 

a constant). The second one is the sparse constraints. R
ij  

is the n N  matrix to pick up the image block in 

( , )i j  from the image of. D  is the over-completed dictionary. Finally, the denoised image y  is updated by  

22

2 2
,

ˆ argmin   
y

y y x Da R y
ij ij ij

i j

   
(3) 

3. The proposed method 

In this section, we describe the proposed method. The core of our method is to replace the reconstruction error 
with histogram structure similarity and make it as a new fidelity term. In addition, when the ways of initial 
over-complete dictionary and orthogonal matching pursuit algorithm are applied in sparse decomposition, we 
incorporate HSSIM into the constraint condition.  
In practical terms, the demonstration of the proposed method can be shown in Fig. 1. 
 

176

javascript:void(0);


Image
patches

Clustering
K-SVD

based on
HSSIM

Updated
dictionary

OMP
based on
HSSIM

The
preliminary
de-noised

image

DCT
dictionary

-

+

The
compensation

image

The final
de-noised

image

De-noising 
by using

DCT 
dictionary

The de-noised
Compensation

image

The 
observed 
Image 

Merging

 

Figure 1: Flowchart of the proposed method 

Here is the improved sparse representation de-noising model that we have proposed: 

2

2 0
, , ,

ˆ ˆ{ , } argmin (1 H( ))      
a y

a y y x a Da R y
ij

ij ij ij ij ij

i j i j

 

 

(4) 

In equation (4), the first and the second parts on the right side are the constraints, and the third one is the 
similarity factor which replaces the reconstruction error of K-SVD as the new computational fidelity term. 
HSSIM is defined as follows: 

( , ) ( , ) ( , ) ( , )H x y l x y q x y h x y  
(5) 

where ( , )l x y , ( , )q x y  and ( , )h x y  represents intensity, contrast and am biguity, respectively . 
Assume that the initial condition is y = x , the sparse coefficient of each image patch can be calculated as 
follows: 

0
ˆ argmin (1 H( ))   

a

a a Da R y
ij

ij ij ij ij ij
  

(6) 

Then the result of de-noising can be acquired by solving y  partial derivatives of equation (5): 

T 1 T

, ,

ˆˆ ( )


   y I R R x R Da（ ）ij ij ij ij
i j i j

   
(7) 

From the above analysis, the procedure of our method is summarized as follows: 
(1) We adopt the completed DCT dictionary as the initial dictionary, satisfying y = x , and cluster each sample 
into K  sub-groups. 

(2) W e divide the observed image into overlapping blocks, and take these overlapping image patches as the 
training data set for the K-SVD algorithm. Specifically, we take 1 HSSIM( ) Da R y

ij ij
 to replace 

2

2
Da R y

ij ij
 as the new fidelity term in order to well preserve the image structural features. Through 

equation (13), we can obtain all the sparse coefficients. 

(3) In this stage, we define the error matrix as T



 y d ak j j
j k

E , and define the atom used for the sparse 

separation in the dictionary as 
T

{ 1 , ( ) 0}   a
i j

i i k i , where d
j
 is the first column of dictionary 

atom, and 
T

a
j is the first row of the sparse coefficient matrix.. Then the dictionary updating is switched to the 

following model: 

177



 

,

T T
min(1 ( , )) . .



 
d a

y d a a
j j

j j j i

j k

H s t   
(8) 

Repeat the stage (3) J  times until the stop criterion. 

(4) W hen the above stages are completed, we can now obtain the preliminary de-noised image y . However, 

the details such as edge in the image might be lost. To solve it, we first obtain the compensation image y
dif

 

by calculating the difference between the preliminary de-noised image y  and the noisy image x , then obtain 

the de-noised compensation image ŷ
dif

 by using the DCT redundant dictionary, after that we add the 

de-noised compensation image ŷ
dif

 to the preliminary de-noised image y and obtain the final de-noised 

image y
t
. The solution is formulated as follows: 

ˆy y + y
t dif

 
(9)

4. Experiments 

In this section, we carry out experiments with the purpose of testing the performance of the proposed method. 
We used other three different methods for a fair comparative analysis. The parameters of the proposed method 
are set as 10J  and 8 8 n , respectively. Since in [25], Elad and Aharon set   and C  as 30   
and 1.15 respectively, we adopt the same value, but default value for the parameters in the other methods. 
Considering the objectivity of the effect evaluation, PSNR and EPI were adopted as the objective indices to 
evaluate the quality of de-noised images. All the experiments are promoted on a Core i5(R) 2.6 GHz PC with 4 
GB RAM. 
EPI is defined as follows: 

( , )









i j

i j

x x

y y

p p
E x y

p p
 

(10) 

where 
x

p  and yp  represents the gray level in observed im age and denoise im age . 

First, we compared the EPI and PSNR results obtained by the proposed method and some other methods, 
such as contourlet, K-SVD, and K-LMS. As is shown that, the tests were performed on three images: “Man”, 
“Babara” and “Boat”. AWGN is superimposed on them with  20, 30 and 40. The results are given in Tab. 1.  

Table 1: Performance of the de-noising methods by HSSIM and PSNR 

Image 
Noise 

σ 
EPI PSNR/dB 

Contourlet K-SVD K-LMS Proposed Contourlet K-SVD K-LMS Proposed 

Man 

20 0.546 0.603 0. 617 0.689 26.45 28.74 28.83 28.91 

30 0.527 0.582 0.594 0.632 25.97 28.13 28.24 28.47 

40 0.503 0.541 0.545 0.599 24.44 26.69 26.71 27.85 

Babara 

20 0.573 0.612 0.633 0.681 24.42 27.48 27.54 27.83 

30 0.554 0.593 0.605 0.624 23.57 26.60 26.63 26.97 

40 0.539 0.567 0.575 0.591 21.31 24.26 24.39 25.65 

Boat 

20 0.534 0.601 0.627 0.673 22.63 25.54 25.60 25.89 

30 0.516 0.587 0.593 0.616 21.48 24.21 24.59 24.70 

40 0.489 0.551 0.572 0.584 20.19 22.48 22.57 23.94 
 
The differences of de-noising results obtained by the proposed method and the others for the same image with 
the same   are significant. Specifically, by adding the white noise with the variance of 30 to “Man”, the EPI 

178



result obtained by the proposed method is 0.632, with the increased ratio by 19.92%%, 8.59% and 6.39% 
respectively compared with the values by using Contourlet, K-SVD, and K-LMS, and the PSNR result obtained 
by the proposed method is 28.47 dB, with increased value by 2.50dB, 0.34dB and 0.23dB respectively 
compared with the values in Contourlet, K-SVD and K-LMS. It also can be found that the results obtained by 
the proposed method are better than the others for different images with different  . For instance, the PSNR 
result obtained by the proposed method is increased by 3.31dB, 0.67dB and 0.56dB on average respectively 
than in Contourlet, K-SVD and K-LMS, and the EPI result obtained by the proposed method is increased by 
19.11%, 8.64% and 6.14% on average respectively than in Contourlet, K-SVD and K-LMS. 
The comparison of the de-noising effects obtained by the proposed method and the others for all the three 
images are given in Fig. 2, with the high noise variance of 70, because under the low noise condition 
( 20 ), the de-noising results are almost perfect, just with some slight differences. From the aspect of 
subjective visual effect, it can be found that the proposed method has some advantages in keeping the 
structure features compared with the other methods. 
 

 

Figure 2: Visual comparison of the reconstructed results on three images (Man, Babara and Boat), with σ=70. 

The first column: noise-free images. The second column: reconstructed results obtained by Contourlet. The third 

column: reconstructed results obtained by K-SVD. The fourth column: reconstructed results obtained by K-LMS. 

The fifth column: reconstructed results obtained by the proposed method. 

5. Conclusions 

In this paper, we have proposed an improved de-noising method for low-SNR images. The proposed method 
makes use of structure information in image in order to keep the de-noising result meet the human visual 
system. Traditional sparse representation based methods take the reconstruction error as fidelity term, which 
leads to the shortage of structure preserving. This regret motivated us to replace the reconstruction error with 
similarity factor as a new fidelity term. Detail compensation is also used for improve the effect of de -noising. 
Experiment results show that the proposed method outperforms Contourlet, K-SVD and K-LMS in terms of EPI 
and PSNR. In the future, we will consider the dictionary learning method. 

Acknowledgments 

This work is supported by Scientific Research Fund of Hunan Provincial Education Department under Grant of 
13C122. 

179



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