文章快速检索 高级检索

1. 湖南大学 电气与信息工程学院, 湖南 长沙 410082;
2. 中国公路工程咨询集团有限公司, 北京 100097

A method of image denoising based on two-dimensional GARCH model
LI Junze1, YUAN Xiaofang1, ZHANG Zhenjun1, WANG Yaonan1, WANG Guofeng2
1. School of Electrical and Information Engineering, Hunan University, Changsha 410082, China;
2. China Highway Engineering Consulting Corporation, Beijing 100097, China
Abstract: An image denoising method based on the statistical model for wavelet coefficients is proposed. It uses a two-dimensional Generalized Autoregressive Conditional Heteroscedasticity (2D-GARCH) model for modeling the wavelet coefficients. A novel wavelet coefficients model is also used to make better use of the important characteristics of wavelet coefficients such as "sharp peak and heavy tailed" marginal distribution and the dependencies between the coefficients. It utilizes maximum likelihood estimation based on fruit fly optimization algorithm (ML Estimation based on FOA) to estimate the model parameters instead of using traditional linear programming in order to improve the accuracy of the modeling. The minimum mean square error estimation (MMSE Estimation) is applied to estimating the parameters of the wavelet coefficients of the original image that is not affected by noise. Experimental results showed that compared to the present widely-used denoising methods the proposed method is more effective in image denoising, and it may achieve higher peak signal-to-noise ratio (PSNR) and good visuality.
Key words: wavelet transform     statistical modeling     two-dimensional GARCH model     FOA     image denoising

1 2D-GARCH模型的小波系数建模 1.1 2D-GARCH模型

hijyij的条件方差，εij是独立同分布的二维标准正态分布，即εij~N(0,1)，(p1,p2,q1,q2)为模型的阶数。从式(2)可以看出，每一个确定的空间位置上的变量值yij的条件方差hij是由与之邻近的变量值yi-k,j-l和其条件方差hi-k,j-l决定的。现定义随机变量yij邻近位置上的变量值和条件方差的集合为

1.2 2D-GARCH模型参数估计

1) 在满足约束条件的前提下初始果蝇群体位置InitΓ_axis.其中Γ={{α0,α01,α10,α11,β01,β10,β11},b}。

2) 赋予果蝇利用嗅觉搜索食物的随机方向与距离Γi=Γ_axis+RandomValue。

3) 将Γi代入式(10)求出果蝇个体的味道浓度

4) 找出果蝇群体中味道浓度最高的果蝇

[bestSmellbestIndex]=max(Smelli)

5) 保留最佳味道浓度与各参数的坐标，此时果蝇群体利用视觉往该方向飞去：

6) 进入迭代寻优，重复执行步骤2)~4)，并判断味道浓度是否优于前一迭代味道浓度，若是则执行步骤5)。

 参数 估计值 参数 估计值 α0 1.3e-4 β10 0.115 α01 0.195 β11 0.039 α10 0.277 b1 0.041 α11 0.111 b2 0.048 β01 0.153 b3 0.068
 图 1 第2层水平子带的小波系数直方图Fig. 1 Histogram of the wavelet coefficients in horizontal subband at the second level of decomposition

2 图像去噪算法

 图 2 本文去噪算法流程图Fig. 2 Flow chart of the proposed agorithm for image denoising

1) 含噪图像小波分解：对被噪声污染的图像进行N层离散小波分解，得到不同尺度和方向上的子带。

2) 2D-GARCH(1,1,1,1)模型参数求解：利用果蝇优化算法求解式(10)，得到各个尺度上的细节子带的模型参数。

3) 原始图像小波系数估计：在上一步估计出各尺度上的细节子带的模型参数的基础上，利用最小均方误差估计原始图像的小波系数。

4) 图像重构：通过离散小波逆变换重构去噪后的图像。令xn分别表示原始图像和加性噪声，y=x+n表示观测到的含噪图像。首先，对含噪图像进行任意尺度的离散小波变换，从而得到了不同尺度和方向上的子带。令XijNijyij分别代表xny的任意子带上的小波系数，那么有

σYij2表示Yij的条件方差即i>σYij2=hijσN2表示噪声方差，可利用鲁棒性中值估计求得[13]

3 实验结果分析

 dB 噪声标准差 噪声 Wiener BayesShrink LAWML 本文方法 10 28.13 31.87 31.73 32.01 33.12 15 24.60 29.96 30.01 30.33 31.17 20 22.11 28.16 28.38 28.59 29.43 25 20.16 26.66 26.95 27.29 27.84 30 18.61 25.33 26.01 26.32 26.75
 图 3 σ=20时Cars图像4种算法的去噪比较Fig. 3 Comparison of four denoising methods for Cars image of σ=20

4 结束语

 [1] SIMONCELLI E P.Modeling the joint statistics of image in the wavelet domain[C]//SPIE's International Symposium on Optical Science,Engineering,and Instrumentation.International Society for Optics and Photonics.Denver,USA,1999:188-195. [2] ANTONIADIS A,BIGOT J,SAPATINAS T.Wavelet estimators in nonparametric regretssion:A comparative simulation study[J].Journal of Statistical Software,2001,6(6):1-83. [3] DOHONO D L.Denoisingby soft-thresholding[J].IEEE Transactions on Information Theory,1995,41(3):613-627. [4] CHANG S,YU B,VATTERELI M.Wavelet thresholding for multiple noisy image[J].IEEE Transactions on Image Processing,2000,9(9):1631-1635. [5] CHANG S,YU B,VATTERELI M.Spatially adaptive wavelet thresholding with context modeling for imaged noising[J].IEEE Transactions on Image Processing,2000,9(9):1522-1531. [6] CHANG S,YU B,VATTERELI M.Adaptive wavelet thresholding for image denoising and compression[J].IEEE Transactions on Image Processing,2000,9(9):1532-1546. [7] ACHIM A,BEZERIANOS A,TSAKALIES P.SAR image denoising via Bayesian wavelet shrinkage based on heavy tailed modeling[J].IEEE Transactions on Geoscience and Remote Sensing,2003,41(8):1773-1784. [8] BOLLERSLEV T.Generalized autoregressive conditional heteroscedasticity[J].Journal of Econometrics,1986,31(3):307-327. [9] NOIBOAR A,COHEN I.Two-dimensional GARCH model with application to anomaly detection[C]//13th European Signal Processing Conference.Istanbul,Turkey,2005:1594-1597. [10] AMIRMAZLAGHANI M,AMIRNDAVAR H.Speckle suppression in SAR image using the 2D GARCH model[J].IEEE Transactions on Image Processing,2009,18(2):250-259. [11] AMIRMAZLAGHANI M,AMIRNDAVAR H.Two novel Bayesian multiscale approaches for speckle suppression in SAR images[J].IEEE Transactions on Geoscience and Remote Sensing,2010,40(7):2980-2993. [12] PAN W T.A new fruit fly optimization algorithm:taking the financial distress model as an example[J].Knowledge Based Systems,2012,26(2):69-74. [13] DOHONO D L,JOHNSTONE I M.Ideal spatial adaptati on via wavelet shrinkage[J].Biometrika,1994,81(3):425-455. [14] LEE J S.Digital image enhancement and noise filtering by use of local statistics[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,1980,2(2):165-168. [15] 胡海平,莫玉龙.基于贝叶斯估计的小波阈值图像降噪方法[J].红外与毫米波学报,2002,21(1):74-76.HU Haiping,MO Yulong.Method of wavelet threshold denoising based on bayesian esitimation[J].Journal of Infrared and Millimeter Waves,2002,42(9):74-76. [16] 谢杰成,张大力,徐文立.一种小波去噪方法的几点改进[J].清华大学学报:自然科学版,2002,42(9):1269-1272.XIE Jiecheng,ZHANG Dali,XU Wenli.Several improvements for a wavelet denoising method[J].Journal of Tsinghua University:Science and Technology,2002,42(9):1269-1272.
DOI: 10.3969/j.issn.1673-4785.201403066

0

#### 文章信息

LI Junze, YUAN Xiaofang, ZHANG Zhenjun, WANG Yaonan, WANG Guofeng

A method of image denoising based on two-dimensional GARCH model

CAAI Transactions on Intelligent Systems, 2015, 10(01): 62-67.
DOI: 10.3969/j.issn.1673-4785.201403066