An optimal wavelet thresholding method for speckle noise reduction

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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

Speckle noise in the image formed by the replay of a laser hologram was removed by thresholding the wavelet coecients. The optimal threshold value was obtained by iteratively approaching the minimum of the dierence between the estimated noise standard deviation and the removed noise standard deviation. Results show that this method can eectively reduce speckle noise, while maintaining edge sharpness.

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 coecients for speckle reduction. Our approach minimizes the dierence 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 p coecients, t a threshold chosen as t =  2log(n)=(n), where n is the number of data points,  the noise level, and a constant, for reconstructing the wavelet coecients of the wavelet transform. Guo et al [6] used Donoho's soft thresholding on the speckle image

for noise reduction, and compared soft thresholding and hard thresholding, i.e. direct truncating the wavelet coecients, 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 dierence between the estimated noise standard deviation and the removed noise standard deviation.

2. SPECKLE NOISE MODEL Speckle has been identi ed as multiplicative noise. Images with multiplicative noise have the characteristic that the brighter the area, the noisier it is. It can be expressed by

z =x v (1) where x is the signal and v is the noise. The ij

ij

ij

ij

ij

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 v

z = vx + x(v ? v) (2) where v is the mean of v. As the mean of v is ij

ij

ij

equal to 1, (2) can be rewritten as

z =x +u (3) where u = x(v ? v). u has zero mean, and standard deviation  = x . We obtain an apij

ij

ij

ij

ij

ij

u

v

proximated additive noise model of the speckle noise image. It is then possible to use a proper

3 wavelet shrinkage method to reduce the noise from the observed image.

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 coef cients which represent the noise will not be the main part of the coecients. A hard thresholding scheme can be used to eliminate them. It can be expressed as  jwj > t w^ = w0 :: ifotherwise (4) where w are the wavelet coecients, t is the threshold. Performing the inverse transform of w^ will recover the noise reduced image. On the other hand, the thresholded coecients w, expressed as  jwj  t (5) w = w0 :: ifotherwise 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. u

u

u

u

4. THE ALGORITHM The algorithm we used is as follows: 1. Estimate the standard deviation ^ of the speckle noise u by calculating u

ij

r X (7) ^ = N1 (x ? x)2 where x is calculated by a local 77 winu

dow;

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2. Perform wavelet transform to obtain the wavelet coecients w; 3. Set initial threshold value t = t0 ; 4. Threshold wavelet coecients 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. u

u

u

where K is a tolerance of (6), k is the step of threshold t.

5. RESULTS Images used here were taken from the replay of a laser hologram at the Laser Laboratory, the Department of Engineering, University of Aberdeen, UK. The hologram was recorded using a ruby laser with wavelength 694 nm, and replayed using a Krypton gas laser with wavelength 647 nm. Our image capture system consists of a Photolux II CCD camera, a Matrox Meteor monochrome image grabber board with a resolution of 768582 (pixels)8 (bits), and a Pentium 90Mhz PC. The camera was mounted in front of the hologram. The reconstructed image was detected by the CCD image detector without a lens. The video signal was then digitized. The PC was running under the Linux system. In our algorithm, the wavelet transform was implemented by using the UBC Imager Wavelet Package V2.1 provided by Bob Lewis at the University of British Columbia, Canada. The Daubechies 4 wavelet base was used as the wavelet lter. As mentioned in [6], long base wavelet lter might oversmooth the details of the image and increase the computational cost. The results were compared with those of several existing speckle reduction lters. A set of

4 criteria was used. They are: (a) the standard deviation to mean ratio, considered as a measure of image speckle in homogeneous regions; (b) edge sharpness by using the three-pixel wide strips on both sides of an edge (introduced in [2]): this criterion shows the preservation of edges in the ltered image. The Newport USAF-1951 standard resolution target image was used for testing (Figure 1). The results are listed in Tables 1 and 2. Table 1 shows the standard deviation to mean ratio of 3 regions marked by big boxes, and the edge sharpness of 3 regions marked in small boxes in Figure 1. The results show that our method can reduce speckle noise like the median lter, but preserve edge sharpness like the Lee lter. Figures 1-4 could also be used for visual comparisons. Figures 5 and 6 show the images of a screw. The regions in the big boxes in Figure 5 were used for measuring the speckle removing in the homogeneous regions, the regions in the small boxes were used for measuring the edge sharpness. The results are listed in Table 2. These results demonstrate good speckle reduction, while maintaining edge sharpness.

REFERENCES 1. J.S.Lee, Digital image enhancement and noise ltering by use of local statistics, IEEE Trans. PAMI PAMI-2, No.2, 1980, 165-168. 2. J.S.Lee and I.Jurkevich, Speckle ltering of synthetic Aperture Radar images: a review, Remote Sensing Reviews vol.8, 1994, 313-340. 3. J.M.Durand, B.J.Gimonet and J.R.Perbos, SAR data ltering for classi cation, IEEE Trans. on Geoscience and Remote Sensing , Vol.GE-25, No.5, 1987, 629-637 4. J.W.Goodman, Statistical properties of laser speckle patterns, in Laser Speckle and Related Phenomena , J.C.Dainty, Ed., SpringerVerlag, Berlin, 1975, 9-75. 5. D.L.Donoho, De-noising by soft-thresholding, IEEE Trans on Info. Theory , Vol.41, No.3, 1995, 613-627. 6. H.Guo,J.E.Odegard, M.Lang, R.A.Gopinath, I.W.Selesnick and C.S.Burrus, Wavelet based speckle reduction with application to SAR based ATD/R, IEEE International Conf. on

Image Processing , Vol.1, 1994, 75-79.

Table 1 Numerical results of Figures 1-4 s/m ratio region 1 region 2 region 3 Original image 0.108 0.314 0.090 Median lter 0.042 0.104 0.050 Lee lter 0.072 0.271 0.066 Our method 0.043 0.133 0.050 edge sharpness region 1 region 2 region 3 Original image 121 135 151 Median lter 126 113 132 Lee lter 131 116 139 Our method 143 128 137

Table 2 Numerical results of Figures 5 and 6 s/m ratio region 1 region 2 region 3 Original image 0.103 0.078 0.107 Median lter 0.063 0.038 0.046 Lee lter 0.099 0.075 0.086 Our method 0.089 0.038 0.065 edge sharpness region 1 region 2 region 3 Original image 142 102 49 Median lter 107 72 37 Lee lter 113 79 46 Our method 122 83 51

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Figure 1. The original image.

Figure 3. The Lee lter result.

Figure 2. The median lter result.

Figure 4. Our method result.

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Figure 5. The original image.

Figure 6. Our method result.