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Procedia Engineering
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 2788 – 2792 www.elsevier.com/locate/procedia
Advanced in Control Engineering and Information Science
Directionlet-Based Bayesian Filter for SAR Image Despeckling Yi-Xiang Lua, Qing-Wei Gao a*, De-Xiang Zhanga, Dong Suna a
School of Electrical Engineering and Automation of Anhui University, Hefei 230039, China
Abstract Synthetic aperture radar (SAR) images are inherently affected by multiplicative speckle noise, which is due to the radar coherent wave. In this paper, we introduce a new SAR despeckling method based on directionlet transform using Bayesian MAP estimator. The directionlet coefficients of the logarithmically transformed reflectance image and the speckle image are modeled Cauchy PDF and additive Gaussian distribution, respectively. Then, we exploit this as a priori information to design a maximum a posterior (MAP) estimator. A new method for estimating the dispersion parameter of Cauchy distribution is also presented. Experiment results, carried out on both synthetic speckled image and real SAR image, show that the proposed scheme removes noise from SAR images more efficiently. Keywods: Synthetic aperture radar (SAR) image, directionlet transform, Bayesian MAP, speckle.
1.
Introduction
Speckle noise occurs inherently in SAR imaging system due to employing coherent radiation. The presence of speckle noise in SAR images reduces the detectability of targets and makes scene analysis and understanding very difficult. Thus, the removal of the speckle is a critical preprocessing step in tasks such as segmentation, detection and classification in the processing of the SAR images. The goal of despeckling algorithm is to remove noise while preserving all the image’s textural features.
* Corresponding author. Tel.: +8613856922457. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2011.08.525
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However, the fact that speckle noise in SAR images is multiplicative introduces significant difficulties in designing effective noise filtering algorithm. Many different spatial-domain despeckling filters using posteriori information of the SAR image have been proposed in the past few years. Among the more widely used filters are the Lee [1], Frost [2] and Gamma [3] filters. All these filters usually perform efficiently on most SAR images but with some limitations regarding resolution degradation and smoothing of uniform areas. With the advent of the multiresolution analysis, a significant breakthrough was made in the filed of despeckling. Since its introduction, wavelets have been universally regarded as extremely powerful tool for analysis of nonstationary signals and images. Basically, all wavelet methods involve as a first step the use of a logarithmic transform to separate the reflectance and speckle noise from the noisy image. Then, different wavelet despeckling approaches are adopted, including thresholding schemes which are based on Donoho’s pioneering work [4] and MAP estimator [5] which takes into consideration the statistics of the signal and speckle components. Although the discrete wavelet transform (DWT) can provides a better reduction of the speckle noise as compared with that of the spatial-domain filters, two drawbacks relating to wavelet-based methods must be noted, one is the DWT itself and the other is the statistical model. The DWT can’t capture edges and contours properly due to its isotropic property. Most models used in the literatures encounter some difficulties in practice, such as PDF’s expression [6] and parameters estimation [7]. To overcome these difficulties, in this paper, despeckling is tackled as a MAP estimation problem based on directionlet. This anisotropic transform captures the edges and contours of the images more effectively comparing with the general DWT. In order to recover the noise-free image using MAP estimator from the observation, Cauchy PDF was utilized to model the directionlet coefficients of the logtransformed reflectance. We employ the less computational complexity method called logarithmic moments to estimate the parameter of Cauchy distribution. Simulations are carried out to study the performance of the proposed filter and to compare it with wavelet-based soft-thresholding techniques and the spatial domain filters using typical image corrupted with synthetic speckle noise as well as real SAR image. 2.
Directionlet Transform and Statistical Modeling of SAR Images
2.1. Directionlet Transform The directionlet transform based on integer lattices was proposed by V. Velisavljevic [8]. Here, we briefly review the directionlet transform. Directionlets are constructed as basis functions of the so called skewed anisotropic wavelet transforms (S-AWT). These transforms make use of the two concepts: anisotropy and directionality. Anisotropy is achieved by an unequal iteration of 1-D transform steps along two directions, that is, the transform is applied more along one than along the other direction. Directionality is obtained by implementation of the sampling using lattice. 2.2. Statistics of Speckle Noise and Reflectance Image If we assume that the speckle noise is fully developed, the corresponding model of SAR image can be expressed as y (t , p ) = s (t , p )η (t , p ) (1) In this paper, we choose the log-normal distribution as the speckle noise model, because the directionlet coefficients of speckle noise are treated as Gaussian distribution in the denoising filter. It can be generated using η log − normal = exp(ε normal 2 log(M m ) + ln m) (2)
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where M and m are the mean and the median value of the distribution, respectively, and ε normal is a standard zero-mean, unit-variance Gaussian random variable. The equivalence between L (Number of Looks) in a speckle image and m in the expression (3) have been given in [9]. To transform the multiplicative speckle into additive noise, logarithmic operation is applied on both sides of (1) Y (t , p ) = S (t , p ) + ε (t , p ) (3) Since the directionlet transform is a linear operator, the coefficient at scale
j
can be written as
d ij , k = x ij , k + ξ ij , k
where
i
refers to the orientations at each scale, and
k
(4)
refers to the cosets.
Mallat [10] pointed out that the distribution of the wavelet coefficients is non-Gaussian, symmetric, and sharply peaked around zero with heavy tails. Thus, in this paper, we adopt the Cauchy PDF with zero location as the model of the directionlet coefficients of the log-transformed reflectance of SAR image. The Cauchy PDF is written as p x (x ) = γ π γ 2 + x 2 (5)
)
((
3.
Bayesian MAP Estimator and Parameter Estimation
MAP estimator is a powerful estimation strategy for random processes affected by noise. By applying the Bayes rule to (4), the estimator can be written as p x|d (x | d ) = pd | x (d | x ) p x (x ) pd (x ) (6) Minimizing the conditional risk and selecting the uniform cost function, the MAP estimator of the noisy observation d is ∧
x
x (d ) = arg max p x|d (x | d ) = arg max pd | x (d | x ) p s (s ) = arg max pξ (d − x ) p s (s ) = arg max pξ (ξ ) ps (s )
given (7)
In order to implement the MAP estimator in practice, we must estimate the parameters of the estimator. The standard deviation is obtained in the finest decomposition level by the measured directionlet coefficients as (8) σ ξ = Median( HHH1 ) 0.6745 We use a fairly simple method called logarithmic moments [11] to estimate the parameter γ of Cauchy PDF. If Y
X
is a Cauchy random variable, then we define a new random variable Y = log X . The moments of
of any order satisfy
we have
4.
( )
[
]
E Y k = d k dp k C1 ( p, α )γ p α | p = 0 ,
simplifying the equation and selecting the first order,
E (Y ) = log γ
(9)
Simulation Results
In this section, we first describe our despeckling scheme, and then present simulation results obtained by processing both synthetic SAR images despeckled by synthetic speckle noise and true SAR images. Finally, the performance of the proposed method are compared with those of Gamma-MAP (GMAP) filter and wavelet-based soft thresholding (WST) scheme. In order to reduce the computational complexity, we only selected two anisotropic transforms to decompose the images, and computed the
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mean of the despeckled results as the output. The corresponding generator matrices were M1 = [1,1;−1,1]T
M 2 = [− 1,1;1,1]T
(10)
Daubechies’ symmlet 8 wavelet was used for a 3-level decomposition of the images, while a 3× 3 mask was used in the Gamma-MAP filter. In the test experiment, two quantitative measures were computed to assess the performance of these methods, that is, signal-to-mean squared error (S/MSE) ratio and edge preservation [12] denoted as β . As to the evaluation of the performance of the true SAR images, equivalent number of looks (ENL) was used to evaluate the despeckling effectiveness. A homogeneous region used to computed the ENL has been highlighted in noisy SAR image. Table 1. S/MSE and β values of image Lena by the three schemes L=3
Method
S SME
L = 11
L=7
β
S SME
β
β
S SME
WST
16.7237
0.0773
17.7007
0.1017
19.3747
0.1636
GMAP
16.6138
0.1861
18.8620
0.2322
21.8159
0.3452
Proposed
16.8728
0.2640
19.0100
0.3422
22.7748
0.5707
Table 1. shows the values of S/MSE and β obtained by speckling the test images “Lena” shown in Fig.1(a) with 3-look, 7-look and 11-look simulated speckle noise, respectively. Note that the proposed method provides a substantial improvement in terms of β over the other methods. Also note that larger values of S/MSE are achieved by the proposed algorithm, that is, it despeckles the noise more efficiently. The despeckled images obtained by the three methods are also shown in Fig.1 (b)-(e).
Fig. 1. (a) Original image; (b)Speckled image with 11-look speckle noise; (c) Despeckled image using WST; (d) Despeckled image using GMAP; (e) Despeckled image using the proposed method.
Fig. 2. (a) Noisy SAR image; (b) Despeckled using WST; (c) Despeckled using GMAP; (d) Despeckled using the proposed method. Table 2. ENL values of image Dock by the three schemes
ENL
WST
GMAP
Proposed
18.0006
16.9222
17.2338
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Yi-Xiang Lu et al. / Procedia Engineering 15 (2011) 2788 – 2792 Author name / Procedia Engineering 00 (2011) 000–000
Fig. 2 illustrates the real noisy SAR image “Dock” and the results obtained by the all methods. The ENL values obtained by the different methods are listed in Table 2. Although the soft thresholding method achieves larger ENL values than our scheme, it over-smoothes the image and blurs many features. In terms of visual quality, our scheme achieves best result, especially in preserving the edges. 5.
Conclusion
In this paper, a SAR image despeckling method based on directionlet transform using Bayesian MAP has been proposed. The reflectance and speckle noise components of the SAR image in directionlet domain were modeled as zero location Cauchy PDF and Gaussian distribution. Under the assumption, Bayesian MAP filter was construted to despeckle the SAR noise. The experimental results confirm that the proposed algorithm allows efficient removal of speckle noise, outperforming the previously proposed filters. Acknowledgements This research is supported by National Nature Science Fund of China (60872163), and is also supported by Nature Science Fund of Anhui Province (KJ2011A013). References [1] Lee JS. Digital image enhancement and noise filtering by use of local statistics. IEEE Trans. Pattern Anal. Machine Intell, 1980; 2:165-168. [2] Frost VS, Stiles JA, Shanmugan KS and Holtzman JC. A model for radar images and its application to adaptive digital filtering of multiplicative noise. IEEE Trans. Pattern Anal. Machine Intell, 1982; 4:157-166. [3] Baraldi A and Parmiggiani F. A refined Gamma MAP speckle filter with improved geometrical adaptivity. IEEE Trans. On Geosci. And Remote Sensing, 1995; 33:1245-1257. [4] Donoho DL. Denoising by soft-thresholding. IEEE Trans. Inform. Theory. 1995; 41:613–627. [5] Foucher S, Benie GB and Boucher J. Multiscale MAP filtering of SAR images. IEEE Trans. mage Processing, 2001;10:49–60. [6] Achim A, Tsakalides P and Bezerianos A. SAR image denoising via Bayesian Wavelet shrinkage based on heavy-tailed modeling. IEEE Trans.Geosci.Remote Sens, 2003; 41:1089-1099. [7] Gleich D, Datcu M. Wavelet-based despeckling of SAR images using gauss-markov random fields. IEEE Trans. Geosci. Remote Sens, 2007; 45:1089-1099. [8] Velisavljevic V, Beferull-Lozano B, Vetterli M and Dragotti PL. Directionlets: Anisotropic multi-directional representation with separable filtering. IEEE Trans. Image Processing, 2006; 15:1916-1933. [9] Gagnon L and Jouan A. Speckle filtering of SAR images─a comparative study between complex wavelet-based and standard filters. Proc. SPIE, 1997; 3169:80-91. [10] Mallat S. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell, 1989; 11:674-693. [11] Ma XY and Nikias CL. Parameter estimation and blind channel identification in impulsive signal environments. IEEE Trans. Signal Processing, 1995; 43:2884-2897. [12] Sattar F, Floreby L, Salomonsson G and Lövström B. Image enhancement based on a nonlinear multiscale method. IEEE Trans. Image Processing, 1997; 6:888-895.
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