# Group Sparsity Residual Constraint for Image Denoising

Zhiyuan Zha, Xinggan Zhang, Qiong Wang, Lan Tang and Xin Liu Z. Zha, X. Zhang
and Q. Wang are with the department of Electronic Science and Engineering,
Nanjing University, Nanjing 210023, China. E-mail<EMAIL_ADDRESS>Tang is with the department of Electronic Science and Engineering, Nanjing
University, and National Mobile Commun. Research Lab., Southeast University,
Nanjing 210023, China.L. Xin is with the Center for Machine Vision and Signal
Analysis, University of Oulu, 90014, Finland.

###### Abstract

Group-based sparse representation has shown great potential in image
denoising. However, most existing methods only consider the nonlocal self-
similarity (NSS) prior of noisy input image. That is, the similar patches are
collected only from degraded input, which makes the quality of image denoising
largely depend on the input itself. However, such methods often suffer from a
common drawback that the denoising performance may degrade quickly with
increasing noise levels. In this paper we propose a new prior model, called
group sparsity residual constraint (GSRC). Unlike the conventional group-based
sparse representation denoising methods, two kinds of prior, namely, the NSS
priors of noisy and pre-filtered images, are used in GSRC. In particular, we
integrate these two NSS priors through the mechanism of sparsity residual, and
thus, the task of image denoising is converted to the problem of reducing the
group sparsity residual. To this end, we first obtain a good estimation of the
group sparse coefficients of the original image by pre-filtering, and then the
group sparse coefficients of the noisy image are used to approximate this
estimation. To improve the accuracy of the nonlocal similar patch selection,
an adaptive patch search scheme is designed. Furthermore, to fuse these two
NSS prior better, an effective iterative shrinkage algorithm is developed to
solve the proposed GSRC model. Experimental results demonstrate that the
proposed GSRC modeling outperforms many state-of-the-art denoising methods in
terms of the objective and the perceptual metrics.

###### Index Terms:

Image denoising, group sparsity residual constraint, nonlocal self-similarity,
adaptive patch search, iterative shrinkage algorithm.

##  I Introduction

As a classical problem in low level vision, image denoising has been widely
studied over the last half century due to its practical significance. The goal
of image denoising is to estimate the clean image X from its noisy observation
$\textbf{\emph{Y}}=\textbf{\emph{X}}+\textbf{\emph{V}}$, where V is additive
white Gaussian noise (AWGN). In the past three decades, extensive studies have
been conducted on developing various methods for image denoising [1, 2, 3, 4,
5, 67, 6, 7, 8, 64, 9, 10, 11, 40, 60, 35, 36, 61, 63, 68]. Due to the ill-
posed nature of image denoising, it has been widely recognized that the prior
knowledge of images plays a key role in enhancing the performance of image
denoising methods. A variety of image prior models have been developed, such
as transform based [1, 2, 3], total variation based [4, 5], sparse
representation based [6, 7] and nonlocal self-similarity based ones [8, 9, 10,
64].

Transform based methods assume that natural images can be sparsely represented
by some fixed basis (e.g., wavelet). Motivated by the fact, many wavelet
shrinkage based methods have been proposed [1, 2, 3]. For instance, Chang
$\emph{et al}.$ [1] proposed a method called Bayes shrink algorithm to model
the wavelet transform coefficients as a generalized Gaussian distribution.
Remenyi $\emph{et al}.$ [3] attempted to use 2D scale mixing complex-value
wavelet transform to improve denoising performance. In the total variation
based methods [4, 5], the image gradient is modeled as Laplacian distribution
for image denoising.

Instead of modeling image statistics in some transform domain (e.g., gradient
domain, wavelet domain), sparse representation based prior assumes that image
patch can be precisely modeled as a sparse linear combination of basic
elements. These elements, called atoms, compose a dictionary [6, 11, 12, 13].
The seminal work of KSVD dictionary [11] has not only confirmed promising
denoising performance, but also been successfully used in various image
processing and computer vision tasks [14, 15, 16, 65]. Nonetheless, there
exist two main issues for typical patch-based sparse representation methods.
First, it is computationally expensive to learn an off-the-shelf dictionary;
second, this kind of sparse representation model usually neglects the
correlation between sparsely-coded patches.

Recently, a flurry of methods have exploited nonlocal self-similarity (NSS)
prior based on the fact that natural images contain a large number of mutually
similar patches at different locations. The seminal work of nonlocal means
(NLM) [8] utilized the NSS prior to implement a form of the weighted filtering
for image denoising. Since then, a flurry of nonlocal regularization methods
were proposed to solve various image inverse problems [17, 18, 19, 20, 21]. By
contrast with the local regularization based methods (e.g., total variation
method [4]), nonlocal regularization based methods can effectively generate
sharper image edges and preserve more image details. However, there are still
lots of image details and structures that cannot be accurately recovered. One
important reason is that the above nonlocal regularization terms rely on the
weighted graph [22], and thus it is unavoidable that the weighted manner leads
to disturbance and inaccuracy [23].

Inspired by the success of the NSS prior, recent studies [10, 9, 24, 25, 26,
27, 71, 72, 40, 59, 56, 63, 64] have revealed that structured or group
sparsity can provide more promising performance for noise removal. For
instance, Dabov $\emph{et al}.$ [9] proposed block matching and 3-D (BM3D)
method to combine NSS prior and transform domain filtering, which is still one
of the state-of-the-art denoising methods. Marial $\emph{et al}.$ [10] further
advanced the idea of NSS by group sparse coding. Some other methods [26, 59,
27, 71, 72] also have achieved highly competitive denoising results based on
low rank property of the matrix formed by nonlocal similar patches in a
natural image.

Though group sparsity has verified its great success in image denoising, most
existing group-based sparse representation methods only consider the NSS prior
of the noisy input. For example, LPG-PCA [40] utilized nonlocal noisy similar
patches as data samples to estimate statistical parameters for PCA training.
NLGBT [63] extracted the nonlocal similar patches from a noisy image and
performed an iterative theresholding procedure to enforce group sparsity in
the graph-based transform (GBT) domain. In SSC-GSM [25], the nonlocal similar
patches are extracted from a noisy image by simultaneous sparse coding (SSC)
and the group sparsity meets Gaussian scale mixture (GSM). However, such
methods often suffer from a common drawback that the denoising performance may
degrade quickly with increasing noise levels.

With the above question kept in mind, this paper proposes a new prior model
for image denoising, called group sparse residual constraint (GSRC). Different
from the previous group-based sparse representation denoising methods that
only consider the single NSS prior of the noisy input, two kinds of NSS prior
are (i.e., NSS prior of noisy and pre-filtered images) exploited for image
denoising. The contribution of this paper is as follows. First, to enhance the
performance of group-based sparse representation denoising methods, the group
sparsity residual is proposed, and thus the problem of image denoising is
transformed into one that reduces the group sparsity residual. Second, to
reduce the residual, we first obtain some good estimation of the group sparse
coefficients of the original image by pre-filtering and then the group sparse
coefficients of the noisy image are used to approximate this estimation.
Third, we design an adaptive patch search scheme to improve the accuracy of
the nonlocal similar patch selection. Fourth, to fuse these two NSS priors
better, we present an effective iterative shrinkage algorithm to solve the
proposed GSRC model. Experimental results show that the proposed GSRC modeling
outperforms many current state-of-the-art schemes such as BM3D [9] and WNNM
[27].

The reminder of this paper is organized as follows. Section II provides a
brief survey of the related work. Section III presents the modeling of group
sparsity residual constraint (GSRC), adaptive patch search scheme, and
discusses the main difference among the proposed GSRC method, the BM3D method
[9], the NCSR method [19] and most existing NSS prior-based denoising methods.
Section IV introduces the iterative shrinkage algorithm for solving the GSRC
model. Section V presents the experimental results. Finally, Section VI
concludes this paper. The preliminary work has appeared in [39].

##  II Related Work

Image denoising is a classical ill-posed inverse problem where the goal is to
restore a latent clean image from its noisy observation. It has been widely
recognized that the statistical modeling of natural image priors is crucial to
the success of image denoising. Many image prior models have been developed in
literature to characterize the statistical feature of natural images.

Early models mainly consider the prior on level of pixels, such as the local
structures used in Tikhonov regularization [28] and total variation (TV)
regularization [4, 5]. These methods are effective in removing the noise
artifacts but smear out details and tend to over-smooth the images.

Another popular prior is based on image patch, which has shown promising
performance in image denoising [2, 6, 7, 11]. The well-known work is sparse
representation based model, which has been successfully exploited for image
denoising [6, 7, 11]. Sparse representation based model assumes that each
patch of an image can be precisely represented by a sparse coefficient vector
whose entries are mostly zero or close to zero based on a basis set called a
dictionary. The dictionary is usually learned from a natural image dataset and
the representative dictionary learning (DL) based methods (e.g., ODL [12] and
task driven DL [13]) have been proposed and applied to image denoising and
other image processing tasks.

Image patches that have similar patterns can be spatially far from each other
and thus can be gathered in the whole image. The nonlocal self-similarity
(NSS) prior characterizes the repetitiveness of textures and structures
reflected by natural images within nonlocal regions, which can be exploited to
retain the edges and the sharpness effectively. The seminal work of nonlocal
means (NLM) denoising [8] has motivated a wide range of studies on NSS and a
flurry of NSS methods (e.g., BM3D [9], LSSC [10] and NCSR [19]) have been
proposed and applied to image denoising tasks.

Low rank modeling based methods have been widely used and achieved great
success in image or video denoising [71, 72, 26, 59, 27]. A representative
work was proposed by Ji $\emph{et al}.$ [26], to remove the flaws (e.g.,
noise, scratches and lines) in a video, the damaged pixels are first detected
and demarcated as missing. The similar patches are grouped, satisfying that
the patches in each group have similar underlying structure and carry out a
low rank matrix approximately for each group. Finally, the matrix completion
is conducted by each group to restore the image. Since the traditional low
rank models tend to over-shrink the rank components and treat different rank
components equally, Gu $\emph{et al}.$ [59, 27] proposed the weighted nuclear
norm minimization (WNNM) model for image denoising, which can achieve state-
of-the-art denoising performance.

Recently, deep learning based techniques for image denoising have been
attracting considerable attentions due to its favorable denoising performance
[32, 34, 37, 38, 58, 62]. For instance, Jain $\emph{et al}.$ [32] proposed to
use convolutional neural networks (CNNs) for image denoising and claimed that
CNNs have similar or even better representation power than Markov random field
(MRF) model [33]. In [34], the multi-layer perceptron (MLP) was successfully
exploited for image denoising. Chen $\emph{et al}.$ [62] proposed a trainable
nonlinear reaction diffusion (TNRD) model for image denoising, which learned a
modified fields of experts [66] image prior by unfolding a fixed number of
gradient descent inference steps. Zhang $\emph{et al}.$ [37] investigated the
construction of feed-forward denoising convolutional neural networks (Dn-CNN)
to embrace the progress in very deep architecture, learning algorithm and
regularization method into image denoising. Liu $\emph{et al}.$ [38]
considered the denoising problem as recursive image filtering via a hybrid
neural network.

##  III Modeling of group sparsity residual constraint

###  III-A Group-based Sparse Representation

Recent studies have revealed that structured or group sparsity can offer more
promising performance for image restoration [10, 9, 24, 25, 26, 59, 70, 27,
56, 57, 69]. Since the unit of our proposed sparse representation model is
group, this subsection will give briefs to introduce how to construct the
groups. To be concrete, image X with size N is divided into n overlapped
patches $\textbf{\emph{x}}_{i}$ of size $\sqrt{b}\times\sqrt{b},i=1,2,...,n$.
Then for each exemplar patch $\textbf{\emph{x}}_{i}$, its most similar $k$
patches are selected from an $L\times L$ sized searching window to form a set
${\textbf{\emph{S}}}_{i}$ (For the details of similar patch selection
operator, please see subsection III-D ). After this, all the patches in
${\textbf{\emph{S}}}_{i}$ are stacked into a matrix
${\textbf{\emph{X}}}_{i}\in\Re^{{b}\times{k}}$, which contains every element
of ${\textbf{\emph{S}}}_{i}$ as its column, i.e.,
${\textbf{\emph{X}}}_{i}=\\{{\textbf{\emph{x}}}_{i,1},{\textbf{\emph{x}}}_{i,2},...,{\textbf{\emph{x}}}_{i,k}\\}$.
The matrix ${\textbf{\emph{X}}}_{i}$ consisting of all the patches with
similar structures is called as a group, where ${\textbf{\emph{x}}_{i,k}}$
denotes the $k$-th similar patch (column form) of the $i$-th group. Finally,
similar to patch-based sparse representation [6, 7, 11], given a dictionary
${\textbf{\emph{D}}}_{i}$, which is often learned from each group, such as
DCT, PCA-based dictionary, each group ${\textbf{\emph{X}}}_{i}$ can be
sparsely represented as
${\textbf{\emph{B}}}_{i}={{\textbf{\emph{D}}}_{i}}^{-1}\textbf{\emph{X}}_{i}$
and solved by the following $\ell_{p}$-norm minimization problem,

${{\textbf{\emph{B}}}_{i}}=\arg\min_{{\textbf{\emph{B}}}_{i}}\\{||{\textbf{\emph{X}}}_{i}-{\textbf{\emph{D}}}_{i}{{\textbf{\emph{B}}}_{i}}||_{F}^{2}+\lambda_{i}||{{\textbf{\emph{B}}}_{i}}||_{p}\\}$
(1)

where $||\bullet||_{F}^{2}$ denotes the Frobenious norm, $\lambda_{i}$ is the
regularization parameter, and $p$ characterizes the sparsity of
${{\textbf{\emph{B}}}_{i}}$. Then the whole image X can be represented by the
set of group sparse codes ${{\textbf{\emph{B}}}_{i}}$. Fig. 1 shows the
difference between sparsity and group sparsity.

In image denoising, the goal is to exploit group-based sparse representaion
model to recover ${\textbf{\emph{X}}}_{i}$ from noisy observation
${\textbf{\emph{Y}}}_{i}$ and solve the following minimization problem,

${{\textbf{\emph{A}}}_{i}}=\arg\min_{{\textbf{\emph{A}}}_{i}}\\{||{\textbf{\emph{Y}}}_{i}-{\textbf{\emph{D}}}_{i}{{\textbf{\emph{A}}}_{i}}||_{F}^{2}+\lambda_{i}||{{\textbf{\emph{A}}}_{i}}||_{p}\\}$
(2)

Once all group sparse codes ${{\textbf{\emph{A}}}_{i}}$ are achieved, the
latent clean image can be reconstructed as
$\hat{\textbf{\emph{X}}}={\textbf{\emph{D}}}{\textbf{\emph{A}}}$, where A
includes the set of group sparse codes ${{\textbf{\emph{A}}}_{i}}$. Although
group sparsity has demonstrated its effectiveness in image denoising, most
existing group-based sparse representation denoising methods only use the NSS
prior of noisy image for noise removal (e.g., Eq. (2)), What’s more, the
denoising performance may degrade quickly with increasing noise levels, making
it challenging to recover the latent clean image directly from its noisy
observation.

Figure 1: Comparison between sparsity (where columns are sparse, but do not
alignment) and group sparsity (where columns are sparse and aligned).

Figure 2: Flowchart of image denoising by group sparsity residual constraint
(GSRC) model.

###  III-B Group Sparsity Residual Constraint

Let us revisit Eq. (1) and Eq. (2), due to the influence of noise, it is very
difficult to estimate the true group sparse code B from noisy image Y. In
other words, the group sparse code A obtained by solving Eq. (2) is expected
to be close enough to the true group sparse code B of the original image X in
Eq. (1). As a consequence, the quality of image denoising largely depends on
the level of the group sparsity residual, which is defined as the difference
between group sparse code A and true group sparse code B,

${{\textbf{\emph{R}}}}={{\textbf{\emph{A}}}}-{{\textbf{\emph{B}}}}$ (3)

Therefore, to reduce the group sparsity residual R and boost the accuracy of
A, we propose a new prior model to image denoising, called group sparse
residual constraint (GSRC) [39], and thus Eq. (2) can be rewritten as

${{\textbf{\emph{A}}}_{i}}=\arg\min_{{\textbf{\emph{A}}}_{i}}\\{||{\textbf{\emph{Y}}}_{i}-{\textbf{\emph{D}}}_{i}{{\textbf{\emph{A}}}_{i}}||_{F}^{2}+\lambda_{i}||{{\textbf{\emph{A}}}_{i}}-{{\textbf{\emph{B}}}_{i}}||_{p}\\}$
(4)

However, it can be seen that the true group sparse code B and $p$ are unknown
since the original image X is not available. Therefore, we will discuss how to
obtain B and $p$. In addition, one important issue of the proposed GSRC based
image denoising is the selection of the dictionary. To adapt to the local
image structures, instead of learning an over-complete dictionary for each
group ${{\textbf{\emph{Y}}}}_{i}$ as in [10], we learn the principle component
analysis (PCA) based dictionary [19] for each group
${{\textbf{\emph{Y}}}}_{i}$.

###  III-C Estimation of the Unknown Group Sparse Code

Eq. (3) shows that by reducing the group sparsity residual R, we could improve
the performance of image denoising. In general, the original image X is not
available in practice, and thus the true group sparse code B is unknown.
However, the true group sparse code B can be estimated based on prior
knowledge of the original image X we have. For example, if we have many
example images similar to the original image X, then a good estimation of B
could be learned from the example image set. However, under many practical
situations, the example image set is simply and unsuitable.

The strategy of pre-filtering is a popular means to image denoising. The basic
idea is similar to many denoising algorithms such as BM3D [9] where a first
stage pilot denoising is exploited before going to the second stage of the
actual denoising. In past few years, a variety of image denoising methods
based on pre-filtering have been developed, such as LPG-PCA [40], TID [41],
SOS [42], and aGMM [43] methods, etc.

Based on the above analysis, we first apply pre-filtering (e.g., BM3D [9],
EPLL [44]) to noisy image Y, and then the initialization result of pre-
filtering is defined as Z. Since the pre-filtering has an ideal denoising
performance, Z could be regarded as a good approximation of the original image
X. Therefore, in this paper the group sparse code B is achieved by the pre-
filtering Z. The flowchart of the proposed GSRC is illustrated in Fig. 2.
Specifically, to reduce the group sparsity residual, we first obtain a good
estimation of the group sparse coefficients of the original image by pre-
filtering Z and then the group sparse coefficients of noisy input image are
used to approximate this estimate.

###  III-D Adaptive Patch Search Scheme

$k$ Nearest Neighbors ($k$NN) method [45] has been widely used to nonlocal
similar patch selection. Given a noisy reference patch and a target dataset,
the aim of $k$NN is to find the $k$ most similar patches. However, since the
given reference patch is noisy, $k$NN has a drawback that some of the $k$
selected patches may not be truly similar to given reference patch. For
instance, the noisy similar patches via $k$NN and the clean patches matched
with these noisy similar patch indexes are shown in Fig. 3(a) and Fig. 3(b),
respectively. It can be seen that the 7-th patch (red box) is obviously
deviating from given reference patch (green box) in Fig. 3(b). Since the pre-
filtered image is regarded as a good estimation of the original image, in this
paper we first adopt pre-filtering result as the target image to fetch the $k$
most similar patch indexes. Fig. 3(c) shows the similar patches of BM3D-based
pre-filtered image searched by $k$NN and Fig. 3(d) shows the clean patches
matched with the pre-filtered image similar patch indexes. It can be seen that
the similar patch selection of the pre-filtered image is more accurate than
that of the noisy image. Therefore, to obtain an effective similar patch
indexes via $k$NN, an adaptive patch search scheme is designed. We define the
following formula,

$\partial={\rm
SSIM}({{\textbf{\emph{Z}}}},{\hat{{\textbf{\emph{X}}}}}^{\ell+1})-{\rm
SSIM}({{\textbf{\emph{Z}}}},{\hat{{\textbf{\emph{X}}}}}^{\ell})$ (5)

where SSIM represents structural similarity [46] and
${\hat{{\textbf{\emph{X}}}}}^{\ell}$ represents the $\ell$-th iteration
denoising result. We empirically define that if $\partial<\tau$,
${\hat{{\textbf{\emph{X}}}}}^{\ell+1}$ is regarded as target image to fetch
the $k$ similar patch indexes, otherwise Z is regarded as target image. Z is
the pre-filtered image and $\tau$ is a small constant.

Figure 3: Patch selection between noisy image and pre-filtered image based on
BM3D via $k$NN method (where green box represents the reference patch).

###  III-E Discussion

This subsection will provide detailed discussion about the main difference
among the proposed GSRC method, the BM3D mehod[9], the NCSR method [19] and
most existing NSS prior-based denoising methods.

  * •

It can be seen that the proposed GSRC method is similar to BM3D method, both
of them are two stage-based denoising methods. Nonetheless, the BM3D is based
on filtering method (e.g., DST, DCT), while the proposed GSRC is based on
sparse coding method, along with dictionary learning. Compared with the
analytically designed dictionaries (e.g., DCT/wavelet dictionary),
dictionaries learned from image patch/group have an advantage of being better
adapted to image local structures [6, 11], and thus could enhance the sparsity
which leads to better performance. Moreover, in the second stage of BM3D, they
are mixing up the pre-filtered image group with the noisy image group
unmannerly, and the Winner filtering is utilized. However, we propose a new
prior model, called group sparsity residual constraint (GSRC). Unlike the BM3D
method, we do not mix up the pre-filtering image group with the noisy image
group, instead, we adopt an iterative shrinkage algorithm [49] to solve the
proposed GSRC model, which can better integrate these two NSS priors of noisy
and pre-filtered images (Section IV for more details).

  * •

Natural images often possess similar repetitive patterns, i.e., a large number
of nonlocal redundancies [8]. By searching many nonlocal patches similar to
given reference patch, NCSR [19] first obtained good estimates of the sparse
coding coefficients of the original image by the principle of NLM, and then
centralized the sparse coding coefficients of the observed image to those
estimates to improve the performance of denoising. However, due to the fact
that NLM depends on the weighted graph [22], it is unavoidable that the
weighted manner leads to disturbance and inaccuracy [23]. It is worth
mentioning that the proposed GSRC model does not involve in the weighted
graph. In addition, NCSR is actually a patch-based sparse representation
method, which usually neglects the relationships among similar patches [24,
72].

  * •

NSS prior has shown great success in image denoising. Most existing denoising
methods only exploit the NSS prior of noisy image [10, 19, 25, 26, 59, 27,
47], and few methods use the NSS prior from natural images [48]. Actually,
different from the most existing NSS prior-based denoising methods, in this
work we consider two kinds of NSS prior, i.e., NSS priors of noisy and pre-
filtered images. Experimental results show that the proposed GSRC scheme
outperforms many state-of-the-art methods, such as BM3D [9] and WNNM [27] (See
Section V for more details).

Figure 4: The distribution of the group sparsity residual R for image lena
with $\sigma$=30 and fitting Gaussian, Laplacian and hyper-Laplacian
distribution in (a) linear and (b) log domain, respectively (pre-filtering
based on BM3D [9]).

Figure 5: The distribution of the group sparsity residual R for image House
with $\sigma$=50 and fitting Gaussian, Laplacian and hyper-Laplacian
distribution in (a) linear and (b) log domain, respectively (pre-filtering
based on EPLL [44]).

Figure 6: The 12 test images for denoising experiments. (a) Barbara; (b)
Elaine; (c) flower; (d) foreman; (e) Hill; (f) House; (g) lena; (h) lin; (i)
Monarch; (j) Parrot; (k) pentagon; (l) peppers.

##  IV Algorithm of GSRC

###  IV-A Setting of the Parameter $p$

In Eq. (4), except for estimating B, we also need to set the value of $p$.
Here we perform some experiments to investigate the statistical property of
the group sparsity residual R, where R represents the set of
${{\textbf{\emph{R}}}}_{i}={{\textbf{\emph{A}}}}_{i}-{{\textbf{\emph{B}}}}_{i}$.
In these experiments, two images lena and House are used as examples, where
Gaussian white noise is added to the images lena and House with standard
deviation $\sigma$= 30 (pre-filtering based on BM3D) and $\sigma$= 50 (pre-
filtering based on EPLL), respectively. We plot the histogram of R as well as
the fitting Gaussian, Laplacian and hyper-Laplacian distribution of R in Fig.
4(a) and Fig. 5(a). To better observe the fitting of the tails, we also plot
these distributions in the log domain in Fig. 4(b) and Fig. 5(b). It can be
seen that the histogram of R can be well characterized by the Laplacian
distribution. Therefore, we set $p=1$ and the $\ell_{1}$-norm is adopted to
regularize each group sparsity residual ${{\textbf{\emph{R}}}_{i}}$, and Eq.
(4) can be rewritten as

$\displaystyle{{\textbf{\emph{A}}}_{i}}$
$\displaystyle=\arg\min_{{\textbf{\emph{A}}}_{i}}\\{||{\textbf{\emph{Y}}}_{i}-{\textbf{\emph{D}}}_{i}{{\textbf{\emph{A}}}_{i}}||_{F}^{2}+\lambda_{i}||{{\textbf{\emph{A}}}_{i}}-{{\textbf{\emph{B}}}_{i}}||_{1}\\}$
(6)
$\displaystyle=\arg\min_{\tilde{\boldsymbol{\alpha}}_{i}}\\{||\tilde{{\textbf{\emph{y}}}}_{i}-\tilde{{\textbf{\emph{D}}}}_{i}{\tilde{\boldsymbol{\alpha}}_{i}}||_{2}^{2}+\lambda_{i}||{\tilde{\boldsymbol{\alpha}}_{i}}-{\tilde{\boldsymbol{\beta}}_{i}}||_{1}\\}$

where $\tilde{{\textbf{\emph{y}}}}_{i},{\tilde{\boldsymbol{\alpha}}_{i}}$, and
${\tilde{\boldsymbol{\beta}}_{i}}$ denote the vectorization of the matrix
${{\textbf{\emph{Y}}}_{i}},{{\textbf{\emph{A}}}_{i}}$ and
${{\textbf{\emph{B}}}_{i}}$, respectively. Each column
$\tilde{{\textbf{\emph{d}}}}_{j}$ of the matrix
$\tilde{{\textbf{\emph{D}}}}_{i}=[\tilde{{\textbf{\emph{d}}}}_{1},\tilde{{\textbf{\emph{d}}}}_{2},...,\tilde{{\textbf{\emph{d}}}}_{J}]$
denotes the vectorization of the rank-one matrix, where $J$ denotes the number
of dictionary atoms.

###  IV-B Iterative Shrinkage Algorithm to Solve the Proposed GSRC Model

For fixed ${\tilde{\boldsymbol{\beta}}_{i}},\lambda_{i}$, Eq. (6) is convex
and can be solved efficiently. We adopt an iterative shrinkage algorithm in
[49] to solve Eq. (6). In the $\ell+1$-iteration, the proposed shrinkage
operator can be calculated as

${\tilde{\boldsymbol{\alpha}}_{i}}^{\ell+1}={{{\textbf{\emph{S}}}}_{\lambda_{i}}}({\tilde{{\textbf{\emph{D}}}}_{i}}^{-1}{{\hat{\tilde{{\textbf{\emph{x}}}}}}_{i}}^{\ell}-{{\tilde{\boldsymbol{\beta}}_{i}}})+{{\tilde{\boldsymbol{\beta}}_{i}}}$
(7)

where ${{{\textbf{\emph{S}}}}_{\lambda_{i}}}(\cdot)$ is the soft-thresholding
operator, ${{\hat{\tilde{{\textbf{\emph{x}}}}}}_{i}}$ represents the
vectorization of the $i$-th reconstructed group
${{\hat{{\textbf{\emph{X}}}}}_{i}}$. In fact, according to Eq. (7), one can
observe that these two NSS priors can be better integrated into this surrogate
algorithm. The above shrinkage operator follows the standard surrogate
algorithm, from which more details can be seen in [49].

###  IV-C Adaptive Group Sparsity Regularization Parameter Setting

The parameter $\lambda_{i}$ for each group that balances the fidelity term and
the regularization term should be adaptively determined for better denoising
performance. In this subsection, inspired by [1], we propose a more robust
method for computing $\lambda_{i}$ of Eq. (6) by formulating the group sparse
estimation as a Maximum A-Posterior (MAP) estimation problem. For a given
${{\textbf{\emph{B}}}_{i}}$, the optimal solution of Eq. (6) is
${\hat{{{\textbf{\emph{R}}}}}}_{i}=\arg\max\limits_{{\textbf{\emph{R}}}_{\emph{i}}}\rm
log\
\emph{P}({{\textbf{\emph{R}}}_{\emph{i}}}|{\textbf{\emph{Y}}}_{\emph{i}})$. By
Bayes’ formula, it is equivalent to

$\displaystyle{\hat{{{\textbf{\emph{R}}}}}}_{i}$
$\displaystyle=\arg\max_{{\textbf{\emph{R}}}_{\emph{i}}}\\{\rm log\
\emph{P}({{\textbf{\emph{R}}}_{\emph{i}}}|{\textbf{\emph{Y}}}_{\emph{i}})\\}$
(8) $\displaystyle=\arg\min\limits_{{\textbf{\emph{R}}}_{\emph{i}}}\\{-\rm
log\
\emph{P}({\textbf{\emph{Y}}}_{\emph{i}}|{{\textbf{\emph{R}}}_{\emph{i}}})-\rm
log\ \emph{P}({{\textbf{\emph{R}}}_{\emph{i}}})\\}$

The log-likelihood term $\rm log\
\emph{P}({{\textbf{\emph{R}}}_{\emph{i}}}|{\textbf{\emph{Y}}}_{\emph{i}})$ is
characterized by the statistics of noise V, which is assumed to be additive
white Gaussian noise with standard deviation $\sigma$, and thus we have

$\displaystyle\emph{P}({\textbf{\emph{Y}}}_{\emph{i}}|{{\textbf{\emph{R}}}_{\emph{i}}})=\emph{P}({\textbf{\emph{Y}}}_{\emph{i}}|{{\textbf{\emph{A}}}_{\emph{i}}},{{\textbf{\emph{B}}}_{\emph{i}}})$
$\displaystyle={\rm
exp}(-\frac{1}{2{\sigma}^{2}}||{\textbf{\emph{Y}}}_{\emph{i}}-{{\textbf{\emph{D}}}_{\emph{i}}}{{\textbf{\emph{A}}}_{\emph{i}}}||_{F}^{2})$
(9)

where ${{\textbf{\emph{R}}}_{\emph{i}}}$ and
${{\textbf{\emph{B}}}_{\emph{i}}}$ are assumed to be independent. Since the
group sparsity residual ${{\textbf{\emph{R}}}_{\emph{i}}}$ can be well
characterized by the Laplacian distribution from Fig. 4 and Fig. 5. Thus, the
prior distribution $\emph{P}({{\textbf{\emph{R}}}_{\emph{i}}})$ is
characterized by an i.i.d Laplacian distribution,

$\emph{P}({{\textbf{\emph{R}}}_{\emph{i}}})=\frac{c}{\sqrt{2}\sigma_{\emph{i}}}\rm
exp(-\frac{\emph{c}\sqrt{2}|{{\textbf{\emph{R}}}_{\emph{i}}}|}{\sigma_{\emph{i}}})$
(10)

Then we substitute Eq. (9) and Eq. (10) into Eq. (8), and thus we can readily
derive the desired regularization parameter $\lambda_{i}$ for each group,

$\lambda_{i}=\frac{c*2\sqrt{2}{\sigma}^{2}}{\sigma_{i}}$ (11)

where $\sigma_{i}$ denotes the estimated variance of each group sparsity
residual ${\textbf{\emph{R}}}_{i}$, and $c$ is a small constant.

TABLE I: Group sparsity residual constraint for image denoising. Input: Noisy
image Y.
---
$\rm\textbf{Initialization:}\
{\hat{{\textbf{\emph{X}}}}}={{\textbf{\emph{Y}}}},{{\textbf{\emph{Z}}}},\emph{c},\emph{k},\emph{b},\emph{L},\sigma,\tau,\gamma,\delta$;
For $\ell=1,2,...,K$ do
Iterative regularization
${{\textbf{\emph{Y}}}}^{\ell+1}={\hat{\textbf{\emph{X}}}}^{\ell}+\delta({{\textbf{\emph{Y}}}}-{\hat{\textbf{\emph{X}}}}^{\ell})$;
Re-estimate $\sigma^{\ell+1}$ computing by Eq. (12);
If $\ell=1$
Similar patch indexes selection based on Z.
Else
If ${\rm SSIM}({{\textbf{\emph{Y}}}}^{\ell+1},{{\textbf{\emph{Z}}}})-{\rm
SSIM}({{\textbf{\emph{Y}}}}^{\ell},{{\textbf{\emph{Z}}}})<\tau$
Similar patch indexes selection based on ${{\textbf{\emph{Y}}}}^{\ell+1}$.
Else
Similar patch indexes selection based on Z.
End if
End if
For each patch ${{\textbf{\emph{y}}}}_{i}$ and ${{\textbf{\emph{z}}}}_{i}$ do
Find a group ${{{\textbf{\emph{Y}}}}_{i}}^{\ell+1}$ via $k$NN.
Find a group ${{{\textbf{\emph{Z}}}}_{i}}^{\ell+1}$ via $k$NN.
Constructing dictionary ${{{\textbf{\emph{D}}}}_{i}}^{\ell+1}$ by
${{{\textbf{\emph{Y}}}}_{i}}^{\ell+1}$ by PCA operator.
Update ${{{\textbf{\emph{B}}}}_{i}}^{\ell+1}$ computing by
${{\textbf{\emph{B}}}}_{i}={{{\textbf{\emph{D}}}}_{i}}^{-1}{{\textbf{\emph{Z}}}}_{i}$.
Update ${\lambda_{i}}^{t+1}$ computing by Eq. (11).
Update ${{{\textbf{\emph{A}}}}_{i}}^{\ell+1}$ computing by Eq. (7).
Get the estimation ${{{\textbf{\emph{X}}}}_{i}}^{\ell+1}$
=${{{\textbf{\emph{D}}}}_{i}}^{\ell+1}$${{{\textbf{\emph{A}}}}_{i}}^{\ell+1}$.
End for
Aggregate ${{{\textbf{\emph{X}}}}_{i}}^{\ell+1}$ to form the recovered image
${\hat{{\textbf{\emph{X}}}}}^{\ell+1}$.
End for
Output: ${\hat{\textbf{\emph{X}}}}^{\ell+1}$.

With the solution ${{\textbf{\emph{A}}}_{i}}$ in Eq. (7), the clean group
${{\textbf{\emph{X}}}_{i}}$ can be reconstructed as
${{\hat{\textbf{\emph{X}}}}_{i}}={{\textbf{\emph{D}}}_{i}}{{\textbf{\emph{A}}}_{i}}$.
Then the latent clean image ${{\hat{\textbf{\emph{X}}}}}$ can be reconstructed
by aggregating all the groups $\\{{{\textbf{\emph{X}}}_{i}}\\}$. In practical,
we could perform the above denoising procedures for better results by several
iterations. In the $\ell$-th iteration, the iterative regularization strategy
[50] is used to update the estimation of noise variance. Then the standard
deviation of noise in $\ell$-th iteration is adjusted as

${\sigma^{\ell}}=\gamma*\sqrt{({\sigma^{2}-||{{\textbf{\emph{Y}}}}-{\hat{{\textbf{\emph{X}}}}}^{\ell}||_{2}^{2}})}$
(12)

where $\gamma$ is a constant. The complete description of the proposed method
for image denoising based on GSRC model is exhibited in Table I.

##  V Experimental Results

In this section, extensive experimental results are presented to evaluate the
denoising performance of the proposed GSRC. For the test images, we use two
different test datasets for thorough evaluation. One is a test dataset
containing 200 natural images from Berkeley segmentation dataset (BSD200) [51]
and the other one contains 12 images which are shown in Fig. 6. We consider
two versions of pre-filtering: (1) a pre-filtered image Z generated by the
BM3D method [9], denoted as GSRC-BM3D; (2) a pre-filtered image Z generated by
the EPLL method [44], denoted as GSRC-EPLL. To evaluate the quality of
denoised image, both PSNR and SSIM [46] metrics are used.

###  V-A Parameter Setting

Parameters used in the algorithm are empirically chosen in consideration of
the noise levels in order to achieve relatively good performance. The basic
parameter setting is as follows: the searching window $L\times L$ is set to be
$30\times 30$. The size of patch $\sqrt{b}\times\sqrt{b}$ is set to be
$6\times 6$, $7\times 7$, $8\times 8$ and $9\times 9$ for $\sigma\leq 20$,
$20<\sigma\leq 50$, $50<\sigma\leq 75$ and $75<\sigma\leq 100$, respectively.
The searching matched patches $k$ is set to be 60, 80, 90 for $\sigma\leq 50$,
$50<\sigma\leq 75$ and $75<\sigma\leq 100$, respectively. The detailed setting
of the involved parameters $c$, $\delta$, $\gamma$ and $\tau$ are shown in
Table II. We run denoising experiments for a large range of noise standard
deviations ($\sigma$= 20, 30, 40, 50, 75 and 100).

TABLE II: The detailed involved parameters setting of $c,\delta,\gamma,\tau$. Noise level | GSRC-BM3D | GSRC-EPLL
---|---|---
$\sigma$ | c | $\delta$ | $\gamma$ | $\tau$ | c | $\delta$ | $\gamma$ | $\tau$
$\sigma\leq 20$ | 0.2 | 0.2 | 0.7 | 1e-4 | 0.3 | 0.1 | 0.5 | 5e-4
$20<\sigma\leq 30$ | 0.4 | 0.1 | 0.5 | 7e-4 | 0.3 | 0.1 | 0.5 | 5e-4
$30<\sigma\leq 40$ | 0.2 | 0.2 | 0.7 | 6e-5 | 0.3 | 0.1 | 0.5 | 6e-4
$40<\sigma\leq 50$ | 0.5 | 0.1 | 0.4 | 6e-5 | 0.5 | 0.1 | 0.4 | 4e-4
$50<\sigma\leq 75$ | 0.9 | 0.1 | 0.3 | 6e-5 | 0.9 | 0.1 | 0.3 | 1e-4
$75<\sigma\leq 100$ | 1 | 0.1 | 0.3 | 2e-4 | 0.9 | 0.1 | 0.3 | 2e-4
TABLE III: PSNR (dB) values of denoising results for four competing state-of-the-art image denosing methods. Top Left: BM3D [9]; Top Right: EPLL [44]; Bottom Left: GSRC-BM3D; Bottom Right: GSRC-EPLL. $\sigma$ | 20 | 30 | 40 | 50 | 75 | 100
---|---|---|---|---|---|---
Barbara | 31.24 | 29.85 | 29.08 | 27.58 | 27.26 | 25.99 | 26.42 | 24.86 | 24.53 | 23.00 | 23.20 | 21.89
31.65 | 31.56 | 29.50 | 29.42 | 27.92 | 27.85 | 26.80 | 26.54 | 24.80 | 24.52 | 23.57 | 23.42
Elaine | 32.51 | 32.16 | 30.52 | 30.15 | 28.95 | 28.73 | 27.96 | 27.63 | 25.93 | 25.60 | 24.48 | 24.16
32.63 | 32.65 | 30.61 | 30.66 | 29.09 | 29.14 | 28.07 | 28.02 | 26.14 | 26.11 | 24.74 | 24.67
flower | 30.01 | 30.01 | 27.97 | 27.95 | 26.48 | 26.55 | 25.49 | 25.51 | 23.82 | 23.59 | 22.66 | 22.39
30.47 | 30.41 | 28.28 | 28.32 | 26.92 | 26.93 | 25.92 | 25.89 | 24.17 | 24.16 | 22.94 | 22.89
foreman | 34.54 | 33.67 | 32.75 | 31.70 | 31.29 | 30.28 | 30.36 | 29.20 | 28.07 | 27.24 | 26.51 | 25.91
34.81 | 34.82 | 33.34 | 33.34 | 32.00 | 32.13 | 31.06 | 31.04 | 29.12 | 29.12 | 27.75 | 27.77
Hill | 30.20 | 30.12 | 28.41 | 28.28 | 27.11 | 27.04 | 26.28 | 26.10 | 24.71 | 24.50 | 23.62 | 23.47
30.29 | 30.28 | 28.45 | 28.47 | 27.24 | 27.23 | 26.35 | 26.29 | 24.82 | 24.78 | 23.82 | 23.78
House | 33.77 | 32.99 | 32.09 | 31.24 | 30.65 | 29.89 | 29.69 | 28.79 | 27.51 | 26.70 | 25.87 | 25.21
34.08 | 34.03 | 32.63 | 32.54 | 31.47 | 31.45 | 30.43 | 30.45 | 28.48 | 28.53 | 26.95 | 26.97
lena | 31.52 | 31.25 | 29.46 | 29.18 | 27.82 | 27.78 | 26.90 | 26.68 | 25.17 | 24.75 | 23.87 | 23.46
31.87 | 31.83 | 29.76 | 29.78 | 28.31 | 28.30 | 27.31 | 27.22 | 25.56 | 25.52 | 24.39 | 24.41
lin | 32.83 | 32.62 | 30.95 | 30.67 | 29.52 | 29.32 | 28.71 | 28.26 | 26.96 | 26.36 | 26.00 | 25.05
33.06 | 33.03 | 31.18 | 31.13 | 29.85 | 29.86 | 28.93 | 28.90 | 27.19 | 27.17 | 25.99 | 25.97
Monarch | 30.35 | 30.49 | 28.36 | 28.36 | 26.72 | 26.89 | 25.82 | 25.78 | 23.91 | 23.73 | 22.52 | 22.24
31.07 | 31.10 | 28.83 | 28.88 | 27.43 | 27.45 | 26.38 | 26.41 | 24.43 | 24.49 | 23.06 | 23.09
Parrot | 32.32 | 32.00 | 30.33 | 30.00 | 28.64 | 28.60 | 27.88 | 27.53 | 25.94 | 25.56 | 24.60 | 24.08
32.64 | 32.58 | 30.75 | 30.68 | 29.38 | 29.33 | 28.35 | 28.29 | 26.33 | 26.31 | 24.94 | 24.96
pentagon | 28.23 | 27.96 | 26.41 | 26.06 | 25.10 | 24.79 | 24.21 | 23.83 | 22.59 | 22.18 | 21.45 | 21.12
28.61 | 28.57 | 26.63 | 26.60 | 25.35 | 25.30 | 24.42 | 24.39 | 22.89 | 22.83 | 21.81 | 21.73
peppers | 30.49 | 30.46 | 28.66 | 28.66 | 27.26 | 27.35 | 26.17 | 26.35 | 24.43 | 24.49 | 23.17 | 23.25
30.72 | 30.72 | 28.81 | 28.84 | 27.51 | 27.56 | 26.53 | 26.56 | 24.67 | 24.70 | 23.44 | 23.46
Average | 31.50 | 31.13 | 29.58 | 29.15 | 28.07 | 27.77 | 27.16 | 26.71 | 25.30 | 24.81 | 23.96 | 23.52
31.83 | 31.80 | 29.90 | 29.89 | 28.54 | 28.54 | 27.55 | 27.50 | 25.72 | 25.69 | 24.45 | 24.43

Figure 7: Denoising results of BM3D, EPLL, GSRC-BM3D and GSRC-EPLL on test image $\emph{Monarch},\emph{pentagon}$ and peppers with $\sigma=20,40$ and 50, respectively. (a) Original image; (b) Noisy Image; (c) Pre-filtering BM3D [9] results. (d) Pre-filtering EPLL [44] results; (e) GSRC-BM3D results; (f) GSRC-EPLL results. TABLE IV: PSNR (dB) Comparison of NCSR [19], GID [52], LINC [53], MS-EPLL [54], AST-NLS [55], WNNM [27], GSRC-BM3D and GSRC-EPLL. | $\sigma=20$ | $\sigma=30$
---|---|---
| NCSR | GID | LINC | MS- | AST- | WNNM | GSRC- | GSRC- | NCSR | GID | LINC | MS- | AST- | WNNM | GSRC- | GSRC-
EPLL | NLS | BM3D | EPLL | EPLL | NLS | BM3D | EPLL
Barbara | 31.10 | 29.81 | 31.70 | 30.07 | 31.43 | 31.60 | 31.65 | 31.56 | 28.68 | 27.35 | 29.53 | 27.72 | 29.13 | 29.68 | 29.50 | 29.42
Elaine | 32.39 | 31.23 | 32.59 | 32.50 | 32.45 | 32.55 | 32.63 | 32.65 | 30.25 | 28.98 | 30.39 | 30.53 | 30.32 | 30.75 | 30.61 | 30.66
flower | 30.05 | 29.12 | 30.30 | 30.10 | 30.28 | 30.34 | 30.47 | 30.41 | 27.86 | 27.01 | 28.13 | 28.05 | 28.20 | 28.26 | 28.29 | 28.32
foreman | 34.42 | 33.08 | 34.76 | 34.09 | 34.55 | 34.73 | 34.81 | 34.82 | 32.61 | 30.92 | 32.93 | 32.34 | 32.79 | 33.00 | 33.34 | 33.34
Hill | 30.03 | 29.06 | 30.14 | 30.20 | 30.22 | 30.27 | 30.29 | 30.29 | 28.15 | 27.05 | 28.29 | 28.41 | 28.37 | 28.41 | 28.45 | 28.47
House | 33.81 | 32.68 | 33.82 | 33.27 | 33.87 | 34.01 | 34.08 | 34.03 | 32.01 | 30.50 | 32.26 | 31.71 | 32.26 | 32.52 | 32.63 | 32.54
lena | 31.48 | 30.33 | 31.80 | 31.48 | 31.63 | 31.72 | 31.87 | 31.83 | 29.32 | 28.36 | 29.82 | 29.46 | 29.50 | 29.45 | 29.76 | 29.78
lin | 32.66 | 31.74 | 33.04 | 32.80 | 32.84 | 33.00 | 33.06 | 33.03 | 30.65 | 29.63 | 31.03 | 30.96 | 30.83 | 31.07 | 31.18 | 31.13
Monarch | 30.52 | 29.75 | 30.64 | 30.59 | 30.84 | 31.10 | 31.07 | 31.11 | 28.38 | 27.68 | 28.74 | 28.49 | 28.70 | 28.91 | 28.83 | 28.88
Parrot | 32.25 | 31.24 | 32.50 | 32.21 | 32.42 | 32.66 | 32.64 | 32.58 | 30.20 | 29.33 | 30.64 | 30.29 | 30.37 | 30.78 | 30.75 | 30.68
pentagon | 28.27 | 27.39 | 28.43 | 27.99 | 28.49 | 28.49 | 28.61 | 28.57 | 26.27 | 25.32 | 26.42 | 26.05 | 26.57 | 26.67 | 26.63 | 26.60
peppers | 30.35 | 29.78 | 30.50 | 30.60 | 30.61 | 30.70 | 30.72 | 30.72 | 28.41 | 27.86 | 28.79 | 28.85 | 28.75 | 28.84 | 28.81 | 28.84
Average | 31.45 | 30.43 | 31.69 | 31.32 | 31.64 | 31.76 | 31.83 | 31.80 | 29.40 | 28.33 | 29.75 | 29.41 | 29.65 | 29.86 | 29.90 | 29.89
| $\sigma=40$ | $\sigma=50$
| NCSR | GID | LINC | MS- | AST- | WNNM | GSRC- | GSRC- | NCSR | GID | LINC | MS- | AST- | WNNM | GSRC- | GSRC-
EPLL | NLS | BM3D | EPLL | EPLL | NLS | BM3D | EPLL
Barbara | 27.25 | 25.77 | 27.77 | 26.05 | 27.41 | 27.86 | 27.92 | 27.85 | 26.13 | 24.52 | 26.27 | 25.06 | 26.43 | 26.83 | 26.80 | 26.54
Elaine | 28.91 | 27.67 | 28.96 | 29.13 | 28.69 | 29.05 | 29.09 | 29.14 | 27.68 | 26.54 | 27.73 | 27.98 | 27.68 | 28.08 | 28.07 | 28.02
flower | 26.35 | 25.60 | 26.79 | 26.64 | 26.75 | 26.85 | 26.92 | 26.93 | 25.31 | 24.42 | 25.47 | 25.56 | 25.77 | 25.80 | 25.92 | 25.89
foreman | 31.52 | 29.61 | 31.31 | 31.05 | 31.29 | 31.54 | 32.00 | 32.13 | 30.41 | 28.64 | 30.33 | 30.04 | 30.46 | 30.75 | 31.06 | 31.04
Hill | 26.91 | 25.87 | 26.84 | 27.18 | 27.05 | 27.29 | 27.24 | 27.23 | 26.01 | 25.05 | 26.03 | 26.28 | 26.16 | 26.14 | 26.35 | 26.29
House | 30.79 | 29.02 | 31.00 | 30.47 | 30.91 | 31.31 | 31.47 | 31.46 | 29.61 | 27.76 | 29.87 | 29.47 | 30.13 | 30.32 | 30.43 | 30.45
lena | 28.00 | 26.98 | 28.13 | 28.05 | 28.00 | 28.43 | 28.31 | 28.30 | 26.94 | 25.82 | 26.94 | 26.97 | 27.08 | 27.26 | 27.31 | 27.22
lin | 29.27 | 28.44 | 29.94 | 29.68 | 29.39 | 29.78 | 29.85 | 29.86 | 28.23 | 27.50 | 28.85 | 28.69 | 28.50 | 28.83 | 28.93 | 28.90
Monarch | 26.81 | 26.32 | 27.14 | 27.06 | 27.20 | 27.47 | 27.43 | 27.45 | 25.73 | 25.28 | 25.88 | 25.93 | 26.12 | 26.18 | 26.38 | 26.41
Parrot | 28.77 | 28.01 | 29.26 | 28.94 | 28.87 | 29.33 | 29.38 | 29.33 | 27.67 | 26.79 | 28.23 | 27.90 | 27.92 | 28.15 | 28.35 | 28.29
pentagon | 24.93 | 23.95 | 24.96 | 24.75 | 25.22 | 25.41 | 25.35 | 25.30 | 23.94 | 22.81 | 23.85 | 23.81 | 24.31 | 24.46 | 24.42 | 24.39
peppers | 27.09 | 26.47 | 27.39 | 27.57 | 27.37 | 27.70 | 27.51 | 27.56 | 26.02 | 25.48 | 26.47 | 26.55 | 26.36 | 26.56 | 26.53 | 26.56
Average | 28.05 | 26.97 | 28.29 | 28.05 | 28.18 | 28.50 | 28.54 | 28.55 | 26.97 | 25.88 | 27.16 | 27.02 | 27.24 | 27.44 | 27.55 | 27.50
| $\sigma=75$ | $\sigma=100$
| NCSR | GID | LINC | MS- | AST- | WNNM | GSRC- | GSRC- | NCSR | GID | LINC | MS- | AST- | WNNM | GSRC- | GSRC-
EPLL | NLS | BM3D | EPLL | EPLL | NLS | BM3D | EPLL
Barbara | 24.06 | 22.43 | 24.04 | 23.19 | 24.40 | 24.79 | 24.80 | 24.52 | 22.70 | 21.40 | 22.39 | 22.11 | 23.19 | 23.26 | 23.57 | 23.42
Elaine | 25.34 | 24.54 | 25.42 | 25.99 | 25.51 | 25.94 | 26.14 | 26.11 | 23.77 | 23.21 | 23.92 | 24.52 | 23.94 | 24.54 | 24.74 | 24.67
flower | 23.50 | 22.72 | 23.30 | 23.68 | 23.87 | 23.88 | 24.17 | 24.16 | 22.22 | 20.69 | 21.96 | 22.50 | 22.50 | 22.70 | 22.94 | 22.89
foreman | 28.18 | 26.71 | 28.11 | 28.14 | 28.54 | 28.48 | 29.12 | 29.12 | 26.55 | 25.33 | 26.55 | 26.84 | 27.32 | 27.39 | 27.75 | 27.77
Hill | 24.43 | 23.62 | 24.13 | 24.72 | 24.42 | 24.70 | 24.82 | 24.79 | 23.27 | 22.75 | 23.21 | 23.74 | 23.33 | 23.63 | 23.82 | 23.78
House | 27.16 | 25.23 | 27.56 | 27.45 | 28.06 | 28.25 | 28.48 | 28.53 | 25.49 | 22.38 | 26.11 | 25.99 | 26.52 | 26.68 | 26.95 | 26.97
lena | 25.02 | 23.78 | 25.12 | 25.11 | 25.32 | 25.38 | 25.56 | 25.52 | 23.63 | 22.43 | 23.67 | 23.91 | 24.17 | 24.08 | 24.39 | 24.41
lin | 26.22 | 25.50 | 26.86 | 26.91 | 26.72 | 26.94 | 27.19 | 27.17 | 24.85 | 24.14 | 25.23 | 25.63 | 25.42 | 25.67 | 25.99 | 25.97
Monarch | 23.67 | 22.77 | 23.91 | 23.92 | 24.11 | 24.31 | 24.43 | 24.49 | 22.10 | 20.73 | 22.13 | 22.44 | 22.68 | 22.95 | 23.06 | 23.09
Parrot | 25.45 | 24.87 | 26.20 | 25.92 | 25.93 | 26.32 | 26.33 | 26.31 | 23.94 | 23.54 | 24.48 | 24.38 | 24.61 | 24.85 | 24.94 | 24.96
pentagon | 22.04 | 21.31 | 22.26 | 22.19 | 22.49 | 22.65 | 22.89 | 22.83 | 20.92 | 19.53 | 21.05 | 21.15 | 21.24 | 21.56 | 21.81 | 21.73
peppers | 24.07 | 23.44 | 24.44 | 24.74 | 24.47 | 24.61 | 24.67 | 24.70 | 22.68 | 22.09 | 22.83 | 23.51 | 23.18 | 23.19 | 23.44 | 23.46
Average | 24.93 | 23.91 | 25.11 | 25.16 | 25.32 | 25.52 | 25.72 | 25.69 | 23.51 | 22.35 | 23.63 | 23.89 | 24.01 | 24.21 | 24.45 | 24.43

Figure 8: Denoising images of _lin_ by different methods ($\sigma=30$). (a)
Original image; (b) Noisy image; (c) NCSR [19] (PSNR= 30.65dB, SSIM=0.8632);
(d) GID [52] (PSNR= 29.63dB, SSIM=0.8287); (e) LINC [53] (PSNR= 31.03dB,
SSIM=0.8670); (f) MS-EPLL [54] (PSNR= 30.96dB, SSIM=0.8688); (g) AST-NLS [55]
(PSNR= 30.83dB, SSIM=0.8465); (h) WNNM [27] (PSNR= 31.07dB, SSIM=0.8657); (i)
GSRC-BM3D (PSNR= 31.18dB, SSIM =0.8743); (j) GSRC-EPLL (PSNR = 31.13dB,
SSIM=0.8704).

###  V-B Performance Comparison with the State-of-the-Art Methods

In this subsection, we validate the performance of the proposed GSRC and
compare it with recently proposed state-of-the-art denoising methods,
including BM3D [9], EPLL [44], NCSR [19], GID [52], LINC [53], MS-EPLL [54],
AST-NLS [55] and WNNM [27]. For all the competing methods, the source codes
are obtained from the original authors. We used the default parameters in
their software packages.

First, we compare GSRC-BM3D, GSRC-EPLL with BM3D and EPLL method,
respectively. In Table III, we report the PSNR results for different noise
variances for the 12 test images in Fig. 6. It can be seen that GSRC-BM3D,
GSRC-EPLL are significantly better than BM3D and EPLL with an average gain of
about 0.40dB and 0.79dB, respectively. The visual quality comparisons in the
case of $\sigma=20,40$ and 50 for test images _Monarch_ , _pentagon_ and
_peppers_ are provided in Fig. 7, respectively. It can be found out that the
over-smooth phenomena and undesirable artifacts are generated by BM3D and EPLL
methods, respectively. In contrast, the proposed GSRC not only reduces most of
the artifacts, but also provides better denoising performance on both edges
and textures than BM3D and EPLL methods. Therefore, these results validate the
usefulness of the proposed GSRC model through the pre-filtering BM3D and EPLL.

Second, to further verify the performance of the proposed GSRC in image
denoising, we compare it with six representative algorithms: NCSR [19], GID
[52], LINC [53], MS-EPLL [54], AST-NLS [55] and WNNM [27]. Gaussian white
noise with standard deviation $\sigma$=20, 30, 40, 50, 75 and 100 is added to
the 12 test images. The PSNR results by the competing denoising methods are
shown in Table IV. It can be seen that the proposed GSRC has achieved highly
competitive denoising performance to other leading methods. Based on the pre-
filtering BM3D [9], the proposed GSRC achieves 0.61dB, 1.68dB, 0.39dB, 0.52dB,
0.32dB and 0.11dB improvement on average over NCSR, GID, LINC, MS-EPLL, AST-
NLS and WNNM, respectively. Meanwhile, based on the pre-filtering EPLL [44],
the proposed GSRC achieves 0.59dB, 1.66dB, 0.37dB, 0.50dB, 0.30dB and 0.09dB
improvement on average over NCSR, GID, LINC, MS-EPLL, AST-NLS and WNNM,
respectively. The visual comparisons of the competing methods at noise level
30 and 75 are shown in Fig. 8 and Fig. 9, respectively.

TABLE V: Average PSNR (dB) results of different denoising algorithms for Gaussian denoising with noise level 20, 30, 40, 50, 75 and 100 on BSD200 dataset [51]. $\sigma$ | 20 | 30 | 40 | 50 | 75 | 100
---|---|---|---|---|---|---
NCSR [19] | 29.89 | 27.92 | 26.58 | 25.65 | 24.04 | 23.00
GID [52] | 28.87 | 27.00 | 25.87 | 24.97 | 23.37 | 22.20
LINC [53] | 29.92 | 27.94 | 26.61 | 25.64 | 23.98 | 22.91
MS-EPLL [54] | 29.95 | 28.02 | 26.73 | 25.84 | 24.29 | 23.27
AST-NLS [55] | 29.98 | 28.02 | 26.68 | 25.80 | 24.20 | 23.17
WNNM [27] | 30.11 | 28.17 | 26.88 | 25.96 | 24.42 | 23.37
GSRC-BM3D | 30.12 | 28.15 | 26.89 | 26.01 | 24.49 | 23.49
GSRC-EPLL | 30.11 | 28.17 | 26.91 | 26.00 | 24.48 | 23.49
TABLE VI: Average PSNR (dB) results of APS and No-APS scheme on 12 test images. Pre-filtering | BM3D
---|---
$\sigma$ | 20 | 30 | 40 | 50 | 75 | 100
No-APS | 31.71 | 29.74 | 28.38 | 27.37 | 25.56 | 24.25
APS | 31.83 | 29.90 | 28.54 | 27.55 | 25.72 | 24.45
Pre-filtering | EPLL
$\sigma$ | 20 | 30 | 40 | 50 | 75 | 100
No-APS | 31.71 | 29.80 | 28.42 | 27.36 | 25.55 | 24.26
APS | 31.80 | 29.89 | 28.55 | 27.50 | 25.69 | 24.43

To further demonstrate our performance, we comprehensively evaluate the
proposed GSRC on 200 test images from the BSD dataset [51]. Table V lists the
average PSNR comparison results for a collection of 200 test images among
eight competing methods at six noise levels ($\sigma$=20, 30, 40, 50, 75 and
100). The visual comparisons of the denoising methods for test images 130066
and 223004 with $\sigma=50$ and 100 are shown in Fig. 10 and Fig. 11,
respectively. Obviously, one can observe that the proposed GSRC achieves very
competitive denoising performance compared to WNNM.

In addition, we apply the proposed GSRC to some real noisy images. Fig. 12
shows the denoised images yielded by BM3D and our approach. It can be seen
that the proposed GSRC can not only reduce the noise effectively, but also
preserve the finer details. The results indicate the feasibility of the
proposed GSRC for some practical image denoising tasks.

TABLE VII: Average run time (s) on the 12 test images (size: $256\times 256$) with different methods. Methods | NCSR [19] | GID [52] | LINC [53] | MS-EPLL [54] | AST-NLS [55] | WNNM [27] | GSRC-BM3D | GSRC-EPLL
---|---|---|---|---|---|---|---|---
Average Time (s) | 348 | 346 | 257 | 191 | 459 | 202 | 72 | 148

To sum up, it can be easily found that BM3D, EPLL, NCSR, GID, LINC, MS-EPLL,
AST-NLS and WNNM still generate some undesirable artifacts and some details
are lost. By contrast, the proposed GSRC is able to preserve the sharp edges
and suppress undesirable artifacts more effectively than other competing
methods. Such experimental findings clearly demonstrate that the GSRC model is
a stronger prior for the class of photographic images containing large
variations in edges/textures.

Figure 9: Denoising images of _House_ by different methods ($\sigma=75$). (a)
Original image; (b) Noisy image; (c) NCSR [19] (PSNR= 27.16dB, SSIM =0.7749);
(d) GID [52] (PSNR= 25.23dB, SSIM=0.7052); (e) LINC [53] (PSNR= 27.56dB,
SSIM=0.7850); (f) MS-EPLL [54] (PSNR= 27.45dB, SSIM=0.7738); (g) AST-NLS [55]
(PSNR= 28.06dB, SSIM=0.7720); (h) WNNM [27] (PSNR= 28.25dB, SSIM=0.7883); (i)
GSRC-BM3D (PSNR= 28.48dB, SSIM=0.7992); (j) GSRC-EPLL (PSNR = 28.53dB,
SSIM=0.7998).

Figure 10: Denoising images of _130066_ by different methods ($\sigma=50$).
(a) Original image; (b) Noisy image; (c) NCSR [19] (PSNR= 25.69dB,
SSIM=0.7800); (d) GID [52] (PSNR= 25.40dB, SSIM=0.7152); (e) LINC [53] (PSNR=
26.51dB, SSIM=0.7938); (f) MS-EPLL [54] (PSNR= 25.67dB, SSIM=0.7833); (g) AST-
NLS [55] (PSNR= 26.27dB, SSIM=0.7715); (h) WNNM [27] (PSNR= 26.43dB,
SSIM=0.7888); (i) GSRC-BM3D (PSNR= 26.69dB, SSIM=0.7937); (j) GSRC-EPLL (PSNR
= 26.62dB, SSIM=0.7908).

Figure 11: Denoising images of _223004_ by different methods ($\sigma=100$).
(a) Original image; (b) Noisy image; (c) NCSR [19] (PSNR= 23.73dB,
SSIM=0.6882); (d) GID [52] (PSNR= 22.66dB, SSIM=0.5997); (e) LINC [53] (PSNR=
23.68dB, SSIM=0.6756); (f) MS-EPLL [54] (PSNR= 24.21dB, SSIM=0.6869); (g) AST-
NLS [55] (PSNR= 24.24dB, SSIM=0.6861); (h) WNNM [27] (PSNR= 24.48dB,
SSIM=0.7025); (i) GSRC-BM3D (PSNR= 24.70dB, SSIM =0.7182); (j) GSRC-EPLL (PSNR
= 24.67dB, SSIM=0.7157).

Figure 12: Visual comparisons of denoising results on real noisy images with
unknown noise characteristics.

###  V-C Comparison between APS and No-APS Scheme

In this subsection, in order to demonstrate the designed adaptive patch
selection (APS) scheme effectively, we compare it with No-APS scheme. Table VI
shows the average PSNR results of APS and No-APS schemes on 12 test images. It
can be seen that the average PSNR results of APS scheme are better than No-
APS. Therefore, the proposed APS scheme can boost the accuracy of nonlocal
similar patch selection under the task of image denoising.

###  V-D Computational Cost

Efficiency is another key factor in evaluating an algorithm. We then compare
the speed of the proposed GSRC and six representative algorithms. All
experiments are conducted under the Matlab 2012b environment on a machine with
Intel (R) Core (TM) i3-4150 with 3.56Hz CPU and 4GB memory. The average run
time (s) of the competing methods on the 12 test images (size: 256$\times$256)
is shown in Table VII. Denoising 12 test images, NCSR, GID, LINC, MS-EPLL,
AST-NLS and WNNM take, on average, roughly 348s, 346s, 257s, 191s, 459s and
202s, respectively. For the test images, the proposed GSRC requires only 72s
and 148s on average based pre-filtering BM3D and EPLL, respectively.
Obviously, it can be seen that that the proposed GSRC used less computation
time than these representative methods. Note that the run time of the proposed
GSRC includes the pre-filtering process.

##  VI Conclusion

In this paper, we proposed a novel prior model named group sparsity residual
constraint (GSRC) that exploited two kinds of nonlocal self-similar (NSS)
prior and explored its application into image denoising. To boost the
performance of group sparse-based image denoising, the group sparsity residual
was proposed, which is defined as the difference between the group sparse code
of noisy image and the group sparse code of the original image. Therefore, the
problem of image denoising was translated into one that reduces the group
sparsity residual. Since the original image was unknown, to reduce the group
sparsity residual, we first obtained some good estimation of the group sparse
coefficients of the original image by pre-filtering and then the group sparse
coefficients of the noisy image were used to approximate the estimation. To
enhance the accuracy of nonlocal similar patches selection, an adaptive patch
search scheme was designed. In addition, to fuse these two NSS priors better,
an iterative shrinkage algorithm was adopted to solve the GSRC model.
Extensive experimental results have shown that the proposed GSRC can not only
leads to visible PSNR improvements over many state-of-the-art methods such as
BM3D and WNNM, but also preserves the image local structures, suppresses
undesirable artifacts and results in a competitive speed. In the future, we
will extend the proposed GSRC to other applications such as image deblurring,
image super-resolution and image deblocking.

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