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SPECKLE DENOISING BASED ON BIVARIATE SHRINKAGE FUNCTIONS AND 
DUAL-TREE COMPLEX WAVELET TRANSFORM 
Shuai Xing a *, Qing Xu a , Dongyang Ma a 
a Zhengzhou Institute of Surveying and Mapping, 450052 Zhengzhou, China - xing972403@163.com 
Commission I, WG 1/2 
KEY WORDS: Transformation, Algorithms, SAR, Radiometric, Processing, Image 
ABSTRACT: 
Bivariate shrinkage functions (bsf) statistically denoted as joint probability density functions (pdf) and noise pdf, can be united by 
MAP to denoise image. Because the intensity of speckle in synthetic aperture radar (SAR) image is hypothesized to be distributed 
according to Rayleigh distribution, SAR image denoising modal based on bsf and dual-tree complex wavelet transform (DT-CWT) 
is constructed and reduced. Local variance estimation and wiener filter are used to estimate noise variance and noisy wavelet 
coefficients variance respectively, and they are used to choose an appreciated threshold to denoise SAR image. Experiment results 
demonstrate that PSNR and ENL values of denoised images are extremely larger than the speckle denoising algorithms based on 
discrete wavelet transform (DWT) and edge features have been perfectly preserved. 
1. INTRODUCTION 
The SAR image is produced by coherently receiving echo. Echo 
overlapping inevitably produced speckle noise. Speckle is a 
serious obstacle of SAR image object recognition and even 
makes some ground features disappear. (Xiao Guochao et. al, 
2001) So speckle has to be removed before any interpretations. 
Prof. Donoho (David Donoho L., 1995) in 1995 has proposed the 
soft-thresholding algorithm, and proved that the filtered image 
/(*) could be computed by nonlinear threshold of wavelet 
coefficients. 
But the soft-thresholding has two problems. One is that the real 
biorthogonal wavelet transform (RBWT) has a disadvantage, 
lack of shift invariance. It means that a shift of the input image 
can produce aliasing in the reconstructed image. (Nick 
Kingsbury et al., 1997) RBWT without sub-sample can produce 
shift invariance with huge redundancy. Prof. Nick Kingsbury 
(Nick Kingsbury, 1998a; Nick Kingsbury, 1998b; Peter de Rivaz 
et al., 2001) has developed a dual-tree algorithm with a real 
biorthogonal wavelet basis, and an approximate shift invariance 
can be obtained with limited redundancy by doubling the 
sampling rate at each scale, which is achieved by computing two 
parallel sub-sampled wavelet trees respectively. (Yi Xiang et al., 
2004; Yang Mengzhao et al., 2005; Yi Xiang et al., 2005; Wang 
Hongxia et al., 2005) Zhang Chunhua et al. (Zhang Chunhua et 
al., 2005) have used soft-thresholding and hard-thresholding 
based on DT-CWT to despeckle SAR images, and proved 
DT-CWT was better than RBWT in speckle denoising. 
The other problem of the soft-thresholding is that the 
dependences between the coefficients of two adjacent scales 
have been neglected. In fact they are significantly dependent, 
since the wavelet coefficients of child scale are derived from the 
parent scale. Yi Xiang et al. (Yi Xiang et al., 2005) used an 
interscale model to classify the coefficients into two classes: 
significant coefficients and insignificant coefficients. Then the 
former was denoised with the MAP estimator based on an 
intramodel, and the later was denoted as noise and set zero. But 
their interscale model couldn’t exactly describe the relationship 
of the wavelet coefficients of two adjacent scales. Wang 
Hongxia et al. (Wang Hongxia et al., 2005) used only one 
threshold to judge the dependency, which was only effective on 
some particular conditions. Levent S.endur and Ivan W. 
Selesnick (Levent S.endur et al., 2002a; Levent S.endur et al., 
2002b; Levent S.endur et al., 2002c) have analyzed the 
dependencies between the child and parent coefficients in detail 
and proposed 4 models of bivariate shrinkage functions (bsf). 
Bsf statistically was denoted as joint probability density 
functions (pdf) between the wavelet coefficients of two adjacent 
scales, it could be united with noise pdf by MAP estimator to 
denoise image. And bsf have been successfully used in denoise 
optical images with Gaussian noise. 
In this paper, speckle is hypothesized multiplicative noise 
according to Rayleigh distribution, and a speckle denoising 
modal based on bsf and DT-CWT is constructed. Section II 
introduces the model based on bsf in detail. The speckle 
denoising algorithm is described in section III. Section IV shows 
experiment results of 8 real SAR images and section V is 
conclusions. 
2. THE SPECKLE DENOISING MODAL BASED ON 
BSF 
Speckle is usually hypothesized a multiplicative noise 
Where % represents a real SAR image gray value, x 
represents an un-noised gray value, n represents speckle noise. 
n can be approximately described as a Rayleigh probability 
density function (Marc Simard et al., 1998).