An optimal wavelet thresholding method for speckle noise reduction

z =x v (1) where x is the signal and v is the noise. The
ij ij ij ij ij v
speckle noise has a negative exponent distribution 4] with mean 1 and variance 2 . Lee 1] has shown that an optimal linear approximation of (1) can be obtained by
1. INTRODUCTION
Speckle phenomena are caused by the interference of coherent light or waves which are scattered from the rough surface of an object or nonuniform transmission medium. When high resolution images are needed from the replay of laser holograms, speckle noise is the main noise source which causes the degradation of the images. In our applications, a hologram is recorded of a 3D mechanical structure. To check for any failures in the structure, especially nding features as small as 40 m, such as a crack on a pipe, we can not directly use the images captured from the replay of the hologram. In this situation, the noise ltering techniques is a very important issue. There are various speckle reduction ltering approaches to process SAR images 1{3]. Some appear to have good visual interpretation, some have good speckle smoothing capability. The use of these lters relies on the requirements of a particular application. This paper proposes a method using optimal thresholding of wavelet coe cients for speckle reduction. Our approach minimizes the di erence between the standard deviations of the estimated and removed noise. Donoho 5] has proposed a wavelet thresholding procedure for optimally recovering functions from data with additive Gaussian noise: so called soft thresholding. Donoho soft thresholding algorithm uses a nonlinear function: w = sgn(w)(jwj ? t)+ , ~ where w is wavelet coe cients, t a threshold chop 2log(n)=(n), where n is the numsen as t = ber of data points, the noise level, and a constant, for reconstructing the wavelet coe cients of the wavelet transform. Guo et al 6] used Donoho's soft thresholding on the speckle image
Speckle noise in the image formed by the replay of a laser hologram was removed by thresholding the wavelet coe cients. The optimal threshold value was obtained by iteratively approaching the minimum of the di erence between the estimated noise standard deviation and the removed noise standard deviation. Results show that this method can e ectively reduce speckle noise, while maintaining edge sharpness.
z = vx + x(v ? v) (2) where v is the mean of v. As the mean of v is
can be rewritten as
ij ij ij ij ij ij
z =x +u (3) where u = x(v ? v). u has zero mean, and standard deviation = x . We obtain an approximated additive noise model of the speckle noise image. It is then possible to use a proper
u v
3 wavelet shrinkage method to reduce the noise from the observed image. 2. Perform wavelet transform to obtain the wavelet coe cients w; 3. Set initial threshold value t = t0 ; 4. Threshold wavelet coe cients w to get w by using equation (5); 5. Perform inverse transform of w to obtain noise image; 6. Calculate ^ of the noise image; 7. Calculate = ^ ? ^ . If K , go to 9; 8. Renew threshold t = t + k . Go to 4; 9. Subtract noise image from the original image to obtain speckle reduced image; 10. End.
for noise reduction, and compared soft thresholding and hard thresholding, i.e. direct truncating the wavelet coe cients, results. From results obtained by Guo, the hard thresholding appears to give a signi cant improvement in reducing speckle noise. The question is how to choose a proper threshold for hard thresholding. Guo used the estimated noise variance to determine a range of thresholds. Here, we present a novel thresholding approach which is to minimize the di erence between the estimated noise standard deviation and the removed noise standard deviation.
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An optimal wavelet thresholding method for speckle noise reduction
R.Yu, A.R.Allen, and J.Watson Department of Engineering, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
u u u
3. AN OPTIMAL THRESHOLDING APPROACH
Speckle noise appears as a form of chaotic and unordered high frequency noise. In the wavelet transform of the speckle noise image, those coefcients which represent the noise will not be the main part of the coe cients. A hard thresholding scheme can be used to eliminate them. It can be expressed as w = w :: if jwj > t ^ (4) 0 otherwise where w are the wavelet coe cients, t is the threshold. Performing the inverse transform of w will recover the noise reduced image. On the ^ other hand, the thresholded coe cients w, expressed as (5) w = w :: if jwj t 0 otherwise represent the noise component. If we perform an inverse transform of w, we will obtain the noise image. The measure of this noise image can be used to adjust the threshold t to obtain an optimal value. To do this, we set a performance function: f = minf ^ ? ^ g (6) where ^ is the estimated standard deviation of speckle noise, ^ is the standard deviation of the removed noise obtained from inverse transform of w. The algorithm is to nd the best t so that (6) approaches a minimum.