一种以低秩矩阵重建的图像混合噪声去除算法

doi:10.3724/SP.J.1047.2015.00344

地球信息科学学报 ›› 2015, Vol. 17 ›› Issue (3): 344-352.doi: 10.3724/SP.J.1047.2015.00344

一种以低秩矩阵重建的图像混合噪声去除算法

孟樊^1,²(), 杨晓梅^1,^*(), 周成虎¹

1. 中国科学院地理科学与资源研究所,北京 100101
2. 中国科学院大学,北京 100049

收稿日期:2014-05-26 修回日期:2014-08-23 出版日期:2015-03-10 发布日期:2015-03-10
作者简介:
作者简介：孟樊(1984-),男,湖北襄阳人,博士生,研究方向为遥感图像处理与信息提取,智能计算,压缩感知与低秩矩阵重建。E-mail:mengf@lreis.ac.cn
基金资助:
国家“863”计划项目(2013AA122901、2012AA121201);国家自然科学基金项目(40971224)

A Novel Approach on Mixed Noise Removal Based on Low-rank Matrix Reconstruction

MENG Fan^1,²(), YANG Xiaomei^1,^*(), ZHOU Chenghu¹

1. Institute of Geographic Sciences and Natural Resources Research, CAS, Beijing 100101, China
2. University of Chinese Academy of Sciences, Beijing 100049, China

Received:2014-05-26 Revised:2014-08-23 Online:2015-03-10 Published:2015-03-10
Contact: YANG Xiaomei
About author:
*The author: SHEN Jingwei, E-mail:jingweigis@163.com

摘要/Abstract

摘要：

近年来,低秩矩阵重建在机器学习、图像处理、计算机视觉与生物信息学等众多科学与工程应用领域中,迅速发展为一个新的研究热点,其主要涉及矩阵填充与稳健主成分分析2大问题,即分别从精确且不完全的采样矩阵元与从大误差矩阵元的分布较为稀疏的观测矩阵中恢复出原始低秩矩阵。鉴此,本文定义了稳健矩阵填充,即从非完全且存在稀疏误差的采样矩阵元中精确恢复出原始低秩矩阵,通过最小化核范数与l₁-范数的组合构建了相应的凸优化模型,并提出了一种新颖的增广分部拉格朗日乘数法来求解此类最优化问题。通过将其应用于混合高斯与椒盐噪声去除的问题中表明,此算法对具有规则纹理及相似结构内容等低秩特征的影像中混合噪声的去除效果较好,其能同时去除影像中的椒盐噪声与高斯噪声,且有效保留影像中的纹理细节等信息;当影像中椒盐噪声密度较高而高斯噪声相对较小时,其去噪性能更佳。

关键词: 混合噪声去除, 矩阵填充, 稳健主成分分析, 低秩矩阵重建, 增广拉格朗日乘数法

Abstract:

This paper studies the problem of the restoration of images corrupted by mixed Gaussian-impulse noise. In recent years, low-rank matrix reconstruction has become a research hotspot in many scientific and engineering domains such as machine learning, image processing, computer vision and bioinformatics, which mainly involves the problems of matrix completion and robust principal component analysis. The two problems namely focus on recovering a low-rank matrix from an incomplete but accurate sampling subset of its entries, or from an observed data matrix with an unknown fraction of its entries being arbitrarily corrupted, respectively. Inspired by these ideas, the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors was considered, which was defined as a problem of matrix completion from corrupted samplings and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the l₁-norm in this paper. Meanwhile, a novel and effective algorithm called augmented subsection Lagrange multipliers was put forward to exactly solve the problem. For the mixed Gaussian-impulse noise removal, we regard it as the problem of matrix completion from corrupted samplings, and restore the noisy images following by an impulse-detecting procedure. Compared with some existing methods for mixed noise removal, the recovery quality of our method is dominant when the images possess low-rank features such as geometrically regular textures and similar structural contents. Especially when the density of impulse noise is relatively high and the variance of Gaussian noise is small, our method can outperform the traditional methods significantly not only in the simultaneous removal of Gaussian noise and impulse noise, and in the restoration of low-rank image matrix, but also in the preservation of textures and details of the image.

Key words: mixed noise removal, matrix completion, robust principal component analysis, low-rank matrix reconstruction, augmented Lagrange multipliers

孟樊, 杨晓梅, 周成虎. 一种以低秩矩阵重建的图像混合噪声去除算法[J]. 地球信息科学学报, 2015, 17(3): 344-352.DOI:10.3724/SP.J.1047.2015.00344

MENG Fan,YANG Xiaomei,ZHOU Chenghu. A Novel Approach on Mixed Noise Removal Based on Low-rank Matrix Reconstruction[J]. Journal of Geo-information Science, 2015, 17(3): 344-352.DOI:10.3724/SP.J.1047.2015.00344

图/表 8

图1

表1

混合噪声去除统计指标-Barbara

(p,σ)	统计指标	含噪影像	TVL1法去噪	Xiong and Yin法去噪	三边滤波法去噪	MCCS-ASLM法
$0.1, 5255$	PSNR(dB)	15.132	20.077	26.472	21.528	27.163
$0.1, 5255$	SSIM	0.956	0.940	0.984	0.962	0.986
$0.2, 5255$	PSNR(dB)	12.184	19.702	25.713	19.712	26.386
$0.2, 5255$	SSIM	0.913	0.934	0.981	0.949	0.982
$0.3, 5255$	PSNR(dB)	10.407	18.214	24.688	17.652	25.287
$0.3, 5255$	SSIM	0.870	0.915	0.976	0.933	0.977
$0.1, 15255$	PSNR(dB)	14.910	20.070	25.892	20.726	25.385
$0.1, 15255$	SSIM	0.938	0.941	0.978	0.954	0.975
$0.2, 15255$	PSNR(dB)	12.044	19.603	25.280	19.432	25.067
$0.2, 15255$	SSIM	0.896	0.934	0.976	0.945	0.973
$0.3, 15255$	PSNR(dB)	10.349	18.722	24.256	16.838	24.182
$0.3, 15255$	SSIM	0.855	0.923	0.971	0.928	0.970

表1

图2

图3

表2

混合噪声去除统计指标-RS image

(p,σ)	统计指标	含噪影像	TVL1法去噪	Xiong and Yin法去噪	三边滤波法去噪	MCCS-ASLM法
$0.1, 5255$	PSNR(dB)	15.620	26.444	36.079	33.976	36.246
$0.1, 5255$	SSIM	0.954	0.972	0.996	0.994	0.996
$0.2, 5255$	PSNR(dB)	12.509	24.635	34.918	30.145	35.170
$0.2, 5255$	SSIM	0.910	0.957	0.995	0.989	0.995
$0.3, 5255$	PSNR(dB)	10.691	23.340	33.037	27.797	33.691
$0.3, 5255$	SSIM	0.861	0.945	0.992	0.983	0.993
$0.1, 15255$	PSNR(dB)	15.144	25.980	31.581	29.618	28.385
$0.1, 15255$	SSIM	0.932	0.966	0.989	0.985	0.981
$0.2, 15255$	PSNR(dB)	12.393	24.405	30.978	27.657	27.943
$0.2, 15255$	SSIM	0.892	0.955	0.988	0.981	0.980
$0.3, 15255$	PSNR(dB)	10.739	23.039	30.168	26.382	27.481
$0.3, 15255$	SSIM	0.851	0.941	0.987	0.976	0.979

表2

图4

图5

图6

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一种以低秩矩阵重建的图像混合噪声去除算法

A Novel Approach on Mixed Noise Removal Based on Low-rank Matrix Reconstruction

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