边缘连续重叠组稀疏性在图像恢复中的应用

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"文章探讨了一种图像恢复模型，利用$l_2$型边缘连续重叠组稀疏性（EC-OGS）对图像的非零模式进行优化。此模型旨在捕捉图像梯度域中沿边缘分布的非零项特性。文章通过ADMM算法求解这一模型，并提出了一种新的快速近端算子算法来处理与EC-OGS正则化器相关的子问题。该方法可应用于图像去噪、去模糊和边缘检测等多个领域，数值实验显示了其在PSNR、视觉效果和边缘保留方面的优越性能。" 在图像处理和恢复的逆问题中，非零模式的识别和利用对于稀疏表示至关重要。传统的稀疏表示方法可能无法充分考虑到图像特征的连续性和结构信息，尤其是边缘信息。本文提出的$l_2$型边缘连续重叠组稀疏性（EC-OGS）正则化器是一种创新的方法，它基于一个观察：图像梯度域中的非零元素往往沿着图像的边缘集中。这种正则化器能够更好地保持图像的边缘特性，从而提高恢复质量。利用乘数的交替方向方法（ADMM）求解EC-OGS模型，这是一种广泛用于优化问题的有效工具，尤其适合处理包含非光滑正则化项的问题。在ADMM框架下，作者设计了一种快速的新算法，以计算涉及EC-OGS正则化的子问题的近端算子。近端算子的计算效率直接影响整个优化过程的速度，因此新算法的提出显著提升了模型求解的效率。在实际应用中，这个EC-OGS模型可以广泛应用于多种图像恢复任务。例如，它可以用于图像去噪，通过识别并保留图像的关键结构（如边缘），从而减少噪声的干扰；在去模糊任务中，模型有助于恢复被模糊遮挡的细节，特别是边缘区域；此外，模型还能用于边缘检测，因为它能有效地捕获边缘的连续性。实验结果表明，与传统方法相比，采用EC-OGS模型进行图像恢复可以获得更高的峰值信噪比（PSNR），同时在视觉上提供更清晰、更连贯的恢复图像。此外，这种方法在边缘保留方面表现出色，这对于保持图像的原始信息至关重要。这些优势证明了EC-OGS模型在图像恢复领域的实用价值和潜在影响力。 "l_2$型边缘连续重叠组稀疏性恢复"的研究为图像恢复带来了新的视角，通过考虑图像边缘的连续性，提高了稀疏表示的精确性和恢复效果。结合ADMM优化策略和快速近端算子算法，使得该方法不仅理论上具有吸引力，而且在实际应用中也具有高效性和实用性。

Pattern Recognition Letters 68 (2015) 211–216

Contents lists available at ScienceDirect

Pattern Recognition Letters

journal homepage: www.elsevier.com/locate/patrec

Image restoration with l

-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.

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

by the following minimization problem:

min

Hu − f 

+ 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

norm usually acts as the relaxed convex regularizer for l

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

Hφα − f 

+ λα

and min

Hu − f 

+ λDu

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

-type norm named Tikhonov regular-

izer (

φ(u) = ∇u

,∇ 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

+ ∇u

, 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



+ ∇u



. 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

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