Pattern Recognition Letters 68 (2015) 211–216
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Pattern Recognition Letters
journal homepage: www.elsevier.com/locate/patrec
Image restoration with l
2
-type edge-continuous overlapping
group sparsity
✩
Xiaowei He
∗
, Junli Fan, Zhonglong Zheng
College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, PR China
article info
Article history:
Received 10 January 2015
Availableonline23October2015
Keywords:
Inverse problem
Image restoration
Overlapping group sparsity
Edge-continuous
abstract
It is important and necessary to take account of the non-zero pattern in image sparsity representation. In this
paper, we present an image restoration model by introducing a novel edge-continuous overlapping group
sparsity regularizer (EC-OGS), based on our observation that the non-zero entries in an image gradient do-
main often distribute along its edges. The model is solved by the ADMM (alternating direction method of
multipliers), where a fast novel algorithm is proposed for computing the proximal operator in solving the
subproblem with EC-OGS regularizer. The proposed model can be applied to various image restoration tasks
including denoising, deblurring, and edge-detecting. The numerical experiments demonstrate the effective-
ness of our method in terms of PSNR, visual effect and edge preserving.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
Image restoration is an important task in the field of image
processing and usually formulated as a linear inverse problem. The
objective of image restoration is to estimate an image u from an
observed image f ∈ R
d
by the following minimization problem:
min
u
1
2
Hu − f
2
2
+ λφ(u) (1)
where
φ(u) is a regularization term, H is a linear operator which
is typically an identity operator in image denoising, a projection
operator in image inpainting or a blurring operator in image de-
blurring. The most two important subjects of this model are about
how to design a regularizer
φ(u) effectively and how to find good
computation methods for the model. The regularizer is usually with
some kinds of assumptions, such as the solution having the feature
of sparsity or group sparsity. In the field of sparse representation, l
1
norm usually acts as the relaxed convex regularizer for l
0
which is
exactly the norm for sparse coding. In recent years, sparsity-based
approaches have led to promising results for various image restora-
tion problems. The sparsity-based regularization problems can be
classified into the following two kinds: the first one assumes that the
unknown image u has the nature of sparse representation and can be
synthesized by a few atoms in a given dictionary
φ(synthesis-based
sparsity problems), while the second one assumes that the analysis
coefficients Du(D is the analysis operator) in the analysis domains
✩
This paper has been recommended for acceptance by C. Luengo.
∗
Corresponding author. Tel.: +86 137 5098 3697; fax: +86 579 8229 8188.
E-mail address: jhhxw@zjnu.cn, jhhxw@zjnu.edu.cn (X. He).
are sparse (analysis-based sparsity problems). They can be modeled
as min
u
1
2
Hφα − f
2
2
+ λα
1
and min
u
1
2
Hu − f
2
2
+ λDu
1
respectively. Elad and Aharon [17] proposed K-SVD algorithm for
learning a dictionary for image sparse representations. Nonlocal-
patches based sparse representation approaches [1,25,29,30,43] have
made a great success in the field of image restoration. [1,29,30] pro-
posed BM3D algorithm for image denoising based on an enhanced
sparse representation in transform domain. Dong and Zhang [43]
proposed a nonlocally centralized sparse representation (NCSR)
model to improve the performance of sparse representation-based
image restoration by suppressing the sparse coding noise. Mairal
et al., [25] present simultaneous sparse coding as a framework by
two approaches, namely exploiting self-similarities and learning
dictionary. Dong et al., [42] incorporate the image nonlocal self-
similarity into sparse representation for image interpolation. These
methods are mostly classified into the synthesis-based sparsity
problems. In this paper, we pay more attention to the analysis-based
sparsity problems in image gradient domain. We give a review about
analysis-based approaches as following.
A definition of
φ(u)asal
2
-type norm named Tikhonov regular-
izer (
φ(u) = ∇u
2
,∇ is the gradient operator) was proposed by
Tikhonov [40]. Although it has the virtue of simple computation,
the regularizer overly smoothes the edges which are very important
features in the natural images. To overcome smearing edges, a regu-
larizer based on total variation (TV) was proposed in [38] (in which
φ(u) =
∇u
2
x
+ ∇u
2
y
, the regularizer often called the isotropical TV
regularizer). The well-known model by this regularizer is called the
ROF model. Esedoglu and Osher [18] also proposed the anisotropical
ROF model where
φ(u) = ∇u
x
1
+ ∇u
y
1
. A remarkable advan-
tage of TV regularizer is good edge-protecting. Due to this, it is
http://dx.doi.org/10.1016/j.patrec.2015.10.001
0167-8655/© 2015 Elsevier B.V. All rights reserved.