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IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 3, MARCH 1997
Smoothing Speckled Images Using an Adaptive Rational Operator G. Ramponi and C. Moloney Abstract— This letter presents an adaptive rational filter for use in smoothing correlated speckle noise in images while preserving edges. The filter adapts to local conditions via point estimates of the standard deviation to mean ratio. The filter’s performance is demonstrated by quantitative tests on synthetically corrupted speckled images.
TABLE I QUANTITATIVE COMPARISON NOISE-SMOOTHING RESULTS
OF ON
SELECTED FIG. 1(B)
I. INTRODUCTION
S
PECKLE can be observed as a granular appearance in images produced by coherent imaging systems, such as synthetic aperture radar (SAR), laser, and ultrasound. Since speckle may hinder the detection of image detail, it is typically regarded as noise. Considerable research has been conducted in the past into the nature and properties of speckle (e.g., [1], [2]) and the design of speckle smoothing filters (e.g., [3], [4]). Yet, since speckle noise may be modeled as a multiplicatively signal-dependent and correlated random process corrupting the underlying image, speckle removal continues to provide a challenge to smoothing filters. In this letter, the rational filter is investigated as a speckle smoothing operator. Rational filters have already proved to be able to outperform many conventional operators in reducing noise in images while preserving fine details. Different noise distributions have been considered, ranging from very short-tailed ones (e.g., uniform) [5] to medium- and long-tailed ones (e.g., contaminated Gaussian) [6].
The operator that is introduced in this letter has been specifically devised in order to cope with the characteristics of speckle noise. We first define a two-dimensional (2-D) rational filter expressed as
II. THE RATIONAL FILTER
(1)
The input/output relation of the filters in this family is expressed as the combination of some rational functions of the input variables. Very roughly speaking, the numerator terms have the role of performing noise smoothing, while the denominator terms may be interpreted as a control of the bandwidth of an equivalent linear filter: In the presence of strong image detail, the response of the filter is modulated so that lowpass effects are avoided. The filters can be applied more than once on the input data, in order to obtain a stronger smoothing action [5]. Manuscript received July 23, 1996. This research was supported in part by the European Esprit Project 20 229-Noblesse, and by the Natural Sciences and Engineering Research Council of Canada (NSERC). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. R. Mersereau. G. Ramponi is with the Department of Electronics, University of Trieste, Trieste, Italy (e-mail:
[email protected]). C. Moloney is with Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Nfld., Canada A1B 3X5 (e-mail:
[email protected]). Publisher Item Identifier S 1070-9908(97)02190-1.
where is the input pixel of interest, at the center of a 3 3 mask, surrounded by its eight neighbors denoted by 8. The output value is , and and are the filter’s parameters. As with other members of the same family, we can describe the operation of this rational filter by first considering the degenerate case that results when 0. In this case, (1) becomes a linear filter with coefficients for all , while the coefficient of is ; if 9, a simple averaging is performed. This filter becomes nonlinear and is able to maintain the sharpness of image detail 0. In this case, indeed, each contribution in if we choose the summation of (1) is modulated by the squared difference between the two pixels involved. Consequently, the output value depends more strongly on those neighboring pixels having similar gray levels, while those of differing gray levels have reduced contributions. This mechanism ensures edgepreserving smoothing; the larger the value of , the more sensitive the filter becomes to signal variations.
1070–9908/97$10.00 1997 IEEE
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(a)
(b) Fig. 1. Selected noise smoothing results. (a) Original image. (b) Original image corrupted with synthetic speckle.
It is interesting to observe that this realization of the rational filter is a form of the anisotropic diffusion operator proposed in [7]. The latter was conceived for creating a scale space in which fine details would not be blurred, in order to extract image edges. A modified version was proposed later to perform noise smoothing while enhancing corners and triple points, as a preprocessor stage prior to segmentation [8]. However, from our experience it appears that the simple operator in (1) can profitably be used also for smoothing noisy images. The performance of the above filter can be significantly improved with respect to speckle noise reduction by the modification we now introduce. We alter the denominator in (1) in order to make it a function of the local standard deviation
. It is known, indeed, that this ratio can be to mean ratio used to characterize speckle noise over homogeneous regions of speckled images [3]. To this purpose, we propose the filter (2) This filter performs a smoothing action that is more delicate in low luminance areas (where is small), and becomes stronger in bright image zones. In this way, it complies with the nature of speckle noise, which is multiplicative and, hence, has amplitude proportional to the local mean value of the signal.
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IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 3, MARCH 1997
(c)
(d) Fig. 1. (Continued.) Selected noise smoothing results. (c) Smoothing of (b) by adaptive rational filter. (d) Smoothing of (b) by Lee multiplicative filter. (See Table I for parameters.)
Each term in (2) can have a twofold interpretation. On the one hand, as in the other rational filters, it ensures edge preservation because it modulates the contribution of the couple of pixels to the smoothing filter. On the other hand, we may consider two such pixels to be elements of a stationary random process characterized by mean , and correlation coefficient between standard deviation neighboring pixels; then, provides a very local estimate of . Indeed
Now, if we assume that may be estimated locally by , we see that may be approximated using the couple and by (3) The right-hand term of this expression, squared, is proportional to the quadratic component of the denominator in (2). It should be emphasized that we are not claiming that such an expression is a good estimator of the ratio, which of course would not be true; we only state that we can
RAMPONI AND MOLONEY: ADAPTIVE RATIONAL OPERATOR
produce effective results by building this local information into the filter adaptation mechanism. Moreover, the above consideration allows the incorporation of possible knowledge of the statistical properties of the speckle in determining appropriate values for the parameter . III. SPECKLED IMAGES Synthetically corrupted speckled images were generated for use in quantitative tests of the rational filter. A speckle simulation method from [4] was used to produce images which exhibit the visual and statistical characteristics of multilook speckled SAR images. For example, Fig. 1(a) shows an original image consisting of a bi-gray-level image combined with a photographic image. A corresponding speckle corrupted version, shown in Fig. 1(b), was generated from the original as a simulated fourlook amplitude SAR image. Note the characteristic granular appearance of the speckle noise in Fig. 1(b), with the noise of greater intensity in brighter regions of the image than in darker regions. The noise may be described by the mean squared error (MSE) between the corrupted and original images as well as by relevant statistical properties. The MSE for Fig. 1(b) is 761, while the ratio estimated over several homogeneous regions of differing average intensity is approximately 0.23. Over the same homogeneous regions the correlation coefficient between horizontal and vertical neighboring pixels is approximately 0.49, and between diagonal neighbors approximately 0.25. These statistical properties of the speckle in Fig. 1(b) are consistent with those of four-look amplitude SAR speckle [4], [9]. IV. EXPERIMENTS
AND
DISCUSSION
The rational filter given by (2) has been applied to speckled images such as that of Fig. 1(b). In our experiments, the filter was passed several times over the image, in order to extend the effective filter action beyond the 3 3 size of its mask. Noise smoothing was measured quantitatively by the reduction of the MSE after filtering, as well as by the reduction in the ratio estimated over homogeneous regions. While the latter is, by definition, purely a measure of noise reduction, the MSE remaining after filtering can represent both the residual noise and any errors due to edge degradation. Edge preservation by a filter was also judged qualitatively, with a view to the retention of both dominant structural edges and of weak or thin edges. Fig. 1(c) shows the result of filtering Fig. 1(b) using filter parameters 4.6 and 870 and with eight passes of the filter over the input image. The remaining MSE between Fig. 1(a) and (c) is 181, while the ratio over homogeneous regions is reduced from 0.23 to 0.053. Visually, the noise is substantially reduced, while large structures are well preserved. The definition of most of the image edges is very good, even if some of the thin edges (such as the stripes on the balloon) are diminished by the filter’s action. Nevertheless, for the degree of noise smoothing attained by the filter, the detail preservation is good.
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For comparison, the result of filtering Fig. 1(b) by the Lee multiplicative filter [3] with a 9 9 mask is shown in Fig. 1(d), with the corresponding noise reduction reported in Table I. The visual edge preservation is comparable in this image to that of Fig. 1(c), but without the same degree of noise smoothing. Although the rational filter of (2) is adaptive, the performance of the filter depends on appropriate choices for the fixed parameters and , as well as for the number of passes of the filter. Experimental results with the image of Fig. 1(b), as well as with other synthetically speckled images, indicate that the filter is fairly robust in relation to the setting of its parameters. That is, if the resulting MSE is plotted against and , with the number of filter passes fixed, then the best parameter values can be seen to lie within a reasonably wide and smooth valley. For speckle noise such as that in Fig. 1(b), useful ranges for parameters and have been found to be [4.0, 5.0] and [700, 1100], respectively. Similarly, experimental results indicate that the number of filter passes best used with the above values of and is between six and ten. The large equivalent filter mask allows substantial noise smoothing in homogeneous areas, while the filter’s sensitivity to abrupt luminance transitions can prevent undue degradation of edges. Other representative filtering results are summarized in Table I, showing the robustness of our technique with respect to the choice of its parameters. Some preliminary experiments on real SAR images, using settings similar to those for Fig. 1(c), substantiate these results. V. CONCLUSIONS An adaptive version of the rational filter has been proposed for smoothing speckle noise, with the adaptation based on local estimates of the standard deviation to mean ratio. Quantitative results on synthetically corrupted images are encouraging, and suggest that the filter may be applied to naturally speckled images. Useful ranges for the three parameters of the filter are suggested by these quantitative tests. REFERENCES [1] J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Amer., vol. 66, pp. 1145–1150, 1976. [2] M. Tur, K. C. Chin, and J. W. Goodman, “When is speckle noise multiplicative?,” Appl. Opt., vol. 21, pp. 1157–1159, 1982. [3] J. S. Lee, “Speckle suppression and analysis for synthetic aperture radar images,” Comput. Graph. Image Processing, vol. 17, pp. 24–32, 1981. [4] M. R. Zaman and C. R. Moloney, “A comparison of adaptive filters for edge-preserving smoothing of speckle noise,” Proc. ICASSP-93, Minneapolis, MN, vol. 5, pp. 77–80, Apr. 1993. [5] G. Ramponi, “Detail-preserving filter for noisy images,” Electron. Lett., vol. 31, pp. 865–866, May 1995. [6] , “The rational filter for image smoothing,” IEEE Signal Processing Lett., vol. 3, pp. 63–65, Mar. 1996. [7] P. Perona and J. Malik, “Scale space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 629–639, July 1990. [8] M. Nitzberg and T. Shiota, “Nonlinear image filtering with edge and corner enhancement,” IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 826–832, Aug. 1992. [9] C. R. Moloney and S. Ward, “Adaptive filters for edge-preserving smoothing of airborne SAR image speckle noise,” Proc. Eusipco 94, Edinburgh, U.K., Sept. 1994, vol. III, pp. 1776–1779.
