A Recursive Filter for Despeckling SAR Images...A Recursive Filter for Despeckling SAR Images G. R. K. S. Subrahmanyam, A. N. Rajagopalan, and R. Aravind Abstract—This correspondence

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 10, OCTOBER 2008 1969

A Recursive Filter for Despeckling SAR Images

G. R. K. S. Subrahmanyam, A. N. Rajagopalan, and R. Aravind

Abstract—This correspondence proposes a recursive algorithm for noisereduction in synthetic aperture radar imagery. Excellent despecklingin conjunction with feature preservation is achieved by incorporating adiscontinuity-adaptive Markov random field prior within the unscentedKalman filter framework through importance sampling. The performanceof this method is demonstrated on both synthetic and real examples.

Index Terms—Importance sampling, Markov random field (MRF),speckle, synthetic aperture radar (SAR), unscented Kalman filter (UKF).

I. INTRODUCTION

The synthetic aperture radar (SAR) imaging technique has becomepopular because of its usability under varied weather conditions, itsability to penetrate through clouds and soil, and due to the indepen-dence of SAR image resolution with regard to sensor height. Since SARsystems rely upon coherence properties of the scattered signals, theyare highly susceptible to interference effects. A SAR image is a meanintensity estimate of the radar reflectivity of the region being imaged.The difference between a particular measurement and the true meanvalue is referred to as speckle noise. Fully-developed speckle can bemodeled as random multiplicative noise. If � represents the originalimage and � is speckle noise, then the degraded observation � is givenby the relation

�� (1)

where �� indicates the pixel location. Noise � is assumed to be in-dependent of � with unit mean and variance �� . Speckle severely im-pedes automatic scene segmentation and interpretation, and limits theresolution of the SAR image as well as its utility. The multiplicativenature of speckle complicates the noise reduction process. A specklesuppression filter should effectively filter homogeneous areas, retainimage texture and edges (both straight and curved), and preserve fea-tures (linear as well as point-type).

There exist several methods for SAR speckle reduction. In the Leefilter [1], the multiplicative model is first approximated by a linearcombination of the local mean and the observed pixel. The minimummean-square error criterion (MMSE) is then applied to determine theweighting constant. The method by Kuan et al. [2] uses a nonstationaryimage model and is an extension of Lee’s local statistics algorithm. TheFrost filter [3] is an adaptive and exponentially weighted averagingfilter. The weights are based on the coefficient of variation which isthe ratio of the local standard deviation to the local mean of the de-graded image. The enhanced Lee and enhanced Frost filters proposedby Lopes et al. [4] divide an image into homogeneous areas, hetero-geneous areas and isolated point targets based on the value of the co-efficient of variation (low, intermediate, and high, respectively). Theseadaptive filters approach the local mean at homogeneous regions; at

Manuscript received July 25, 2007; revised May 19, 2008. Current versionpublished September 10, 2008. A. N. Rajagopalan was supported by the A. vonHumboldt Foundation, Germany.The associate editor coordinating the reviewof this manuscript and approving it for publication was Prof. Mark R. Bell.

The authors are with the Department of Electrical Engineering, IIT Madras,Chennai-600 036, India (e-mail: [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TIP.2008.2002160

points of high activity they tend to retain the original observation pixel.The disadvantages are over-smoothing of image texture or ineffectivedenoising around edges. An adaptive block-Kalman filter (ABKF) hasbeen proposed in [5] which uses a block-wise varying AR model. How-ever, the autocorrelation coefficients of the noise-free image must beidentified accurately. Wavelet despeckling approaches also exist andare based on modifying the (log-transformed) speckle noisy waveletcoefficients according to some rule (shrinkage) and reconstructing thefiltered image from them. A wavelet method based on soft-thresholdinghas been proposed in [6]. Xie et al. [7] have proposed a despecklingalgorithm that fuses Bayesian wavelet denoising with a regularizingprior. A MAP estimator with an alpha-stable prior within the waveletframework is proposed in [8]. Bayesian shrinkage which relies on edgeinformation is discussed in [9]. Argenti et al. [10] propose despecklingin the undecimated wavelet domain using a space-varying generalizedGaussian distribution for the wavelet coefficients. An adaptive MAP es-timator with a heavy-tailed Rayleigh signal model has been suggestedin [11].

In this paper, we propose a novel recursive scheme based on the un-scented Kalman filter (UKF) [12], [13] to suppress speckle noise inSAR images while effectively preserving the features. We show that theUKF can be formulated to incorporate a discontinuity-adaptive Markovconditional PDF for the prior and to account for multiplicative noise inthe estimation procedure. The first two moments of the prior are es-timated using importance sampling. A small set of sigma points cap-ture the prior and the speckle noise statistics, and these are propagatedthrough the multiplicative measurement equation of the UKF to arriveat the final image estimates. Our approach is spatially adaptive (as in[10]) but is computationally less intensive. As compared to [9], thediscontinuity preserving prior incorporated in our filter accounts foredges without the need for explicit edge detection. Unlike the ABKF[5], our method does not require inference of the AR parameters of theoriginal image. In contrast to the filters in [9]–[11], the proposed filterdoes not require parameter estimation or optimization to incorporatenon-Gaussian prior.

In Section II, we review the principle of unscented transformationwhich forms the basis for the UKF. In Section III, we formulate a dis-continuity-adaptive prior and discuss a Monte Carlo procedure for itsmoment estimation. In Section IV, we propose a speckle suppressionfilter which is a judicious combination of the UKF and the edge-pre-serving prior through importance sampling. Experimental results andcomparisons are given in Section V. We conclude with Section VI.

II. UNSCENTED TRANSFORMATION

The unscented transformation (UT) calculates the moments ofa random variable that has undergone a nonlinear transformation.It is founded on the intuition that it is easier to approximatea probability distribution than it is to approximate an arbitrarynonlinear function or transformation [12]. The principle in UT isto 1) capture the Gaussian approximation of the prior distributionwith a very small set of carefully chosen deterministic samplesknown as sigma points, 2) propagate these samples through thenonlinearity, and 3) determine the moments of the posterior fromthe transformed samples.

Consider propagating an��-dimensional random variable � throughan arbitrary nonlinear function � � �� to generate � � ��.Assume � has mean � and covariance ��. To estimate the first twomoments of � using UT, we proceed as follows: A set of �� sigma points �� with weights ��, � �� are deter-ministically chosen so that they completely capture the true mean andcovariance of the (prior) random variable �. A selection scheme that

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satisfies this requirement [13], [14] is

��

�

��

��

��

��

� � ��

��

� � ��

��

��

�

��

� �� (2)

Here, � �� represents the �th row or column [12] ofthe matrix square root of �� . Superscripts and � on theweight �� refer to its use in the calculation of mean and covariance,respectively. The weights �

�� satisfy ��

�� . Parameter

�� controls the spread of the sigma point distribution around �and is usually set to a value between 0 and 1. The term �� canbe used to incorporate prior knowledge about �. Each sigma pointis instantiated through the nonlinear function as �� ,� � � �� . The estimated mean and covariance of � com-puted as� � ��

�� and��

��

are accurate to the second-order (third-order for symmetric priors)[12] of the Taylor series expansion of �� for any nonlinear function.When UT is used in the Kalman filter formulation, it leads to theunscented Kalman filter (UKF) [12], [13].

III. DISCONTINUITY-ADAPTIVE PRIOR AND MOMENT ESTIMATION

In recursive filtering, the (current) state is usually predicted fromits past through an auto-regressive (AR) model. The drawback lies inthe identification of AR parameters which ideally requires the originalimage. In addition, strong local linear dependence as dictated by theAR model renders it difficult to achieve both effective noise suppres-sion and feature preservation simultaneously. In the past, attempts havebeen made to circumvent this limitation by introducing space-varyingAR models [15] or by using analytically tractable non-Gaussian resid-uals [16]. However, these methods assume an additive noise model. Wepropose to handle this problem by incorporating an edge-preservingprior [17], [18].

Bayesian methods enable effective incorporation of prior knowl-edge. Statistical dependence incorporated using a Markov randomfield (MRF) model provides great flexibility in enforcing contextualconstraints. An MRF possesses Markovian property, i.e., the value ofa pixel depends only on the values of its neighboring pixels and onno other pixel [17]. The prior is usually modeled as a homogeneousGaussian MRF (for mathematical convenience and simplicity). AGMRF prior with a MAP estimate has been used for SAR despecklingin [19]. However, the issue with GMRF is that it imposes smoothnessconstraint ubiquitously, which inevitably leads to over-smoothing ofedges. A better way to handle the situation is to use an appropriatenon-Gaussian conditional density function to model the originalimage.

We model the original image � by a non-Gaussian MRF with condi-tional probability density function

��

��

��

��

� (3)

Here, � is a normalizing constant, the neighborhood �� ,where �� is the order of the nonsymmetric half plane (NSHP)

support. The term �� . The quantity �� is a measure of the local de-pendencies. For this MRF model [17], the smoothing strength ofthe regularizer increases monotonically as � increases within a band�� . Outside, the smoothing decreases and becomeszero as �. Since this feature enables preservation of imagediscontinuities, it is called a discontinuity adaptive MRF (DAMRF)model [17], [20].

In a recursive formulation, the predicted mean and variance of theprior are required for the update stage of the filter. Since it is analyti-cally not possible to estimate the variance of the non-Gaussian prior in(3), we resort to a Monte Carlo approach known as importance sam-pling (IS) [21]. This technique can be used to determine the estimatesof the moments of any (non-Gaussian) target PDF (say ��) from thesamples of another roughly approximate PDF �� (termed as sam-pler PDF) that includes the (nonzero) support of target PDF in it and iseasy to sample. In importance sampling, we draw� samples, �� from the sampler PDF � which concentrates on those points where thefunction � is large. If these were under �, we can determine the mo-ments of � directly from them. When we use samples from � to de-termine any estimates under �, in the regions where � is greater than�, the estimates are over-represented. In the regions where � is lessthan �, they are under-represented. To account for this, we use cor-rection weights �� in determining the estimatesunder �. For example, to find the mean of the distribution � we use� �

��

�� . As � the estimate � tends

to the actual mean of �. In our problem, the DAMRF distribution cor-responds to �. We choose the heavy-tailed Cauchy distribution as theimportance function �.

IV. SPECKLE SUPPRESSION USING UKF

When the observation is linearly related to the state, and the mod-eling errors are additive white Gaussian, then the Kalman filter (KF)provides an optimal estimate of the state by propagating the first twomoments recursively. In speckle reduction, the multiplicative model ofspeckle noise complicates recursive propagation in the KF formulation.

The unscented Kalman filter (UKF) provides a mathematicallytractable way to propagate the first two moments even in the presenceof multiplicative noise. We present an algorithm that uses importancesampling to incorporate the DAMRF prior within the recursive UKFframework for speckle noise reduction. We refer to this method asISUKF. The conditional PDF based on the DAMRF model effectivelyuses the already estimated pixels in the NSHP support. We employ im-portance sampling to estimate the first two moments of the conditionalPDF. The UKF uses these predicted mean and variances to determinethe sigma points and propagates them through the multiplicative modelto yield the final estimate of the original pixel intensities. The steps inthe proposed algorithm are as follows.

1) Using the DAMRF model (Section III) at each pixel, we constructthe state conditional PDF using the past pixels in the NSHP sup-port and the values of �� and � as

��

� ��

�(4)

where �� . For a first-order NSHP support,�� . To effectively smooththe speckle over the entire image, we allow �� to be space-variantas described in Section V.

2) Importance sampling is employed to estimate the mean andcovariance of the above PDF. We draw samples ��,



Fig. 1. (a) Original image. (b) Noisy version. Output of (c) Enhanced Lee (FOM � ��, S/MSE � �� dB), (d) Frost (FOM � ��, S/MSE � �� dB),and (e) the proposed method (FOM � ��, S/MSE � �� dB).

� � �� from a Cauchy sampler. To facilitate effec-tive sampling, the location of the sampler density � is varieddepending on the mean of the estimated first-order NSHP pixels,and the scale is chosen sufficiently high so as to include the sup-port of � in �. Thus, the importance function has a space-varyingmean and a heavy-tailed distribution. The samples are weightedby �� (Section III). The mean � andvariance �� of � are computed as

� ��

��

��

��

��

��

��

��

3) The above statistics are directly used to obtain the one-step aheadpredicted sigma points for the UKF as �� and �� . To incorporate the observation noisemodel, we augment the predicted state statistics with the noisestatistics as

��

��

��

� ��

��

� ��

We adopt the sigma point selection scheme [(2) in Section II]on the augmented statistics to determine the corresponding sigmapoints as per the equation shown at the bottom of the page, where�� is the dimension of �� and �� is the augmented sigma pointmatrix of dimension�� . Note that only 5 sigma pointsare needed to propagate the state dynamics.

4) Next, we apply the measurement model independently on each ofthe sigma points to obtain the sigma points corresponding to themeasurement as

��

�

��

This is in accordance with the multiplicative noise model of(1). Here, ��

�� and �� are formed from the

first �� rows of �� and the subsequent � rows,

respectively. Operation “��” represents element-by-elementmultiplication. Using these sigma points the following statis-tics can be estimated as per the equation shown at the bottomof the next page. Here, �

�� and �

�� are weights (2),

while �� (or � ��) represents the

th sigma point which is the th column of �� (or

��).

The Kalman gain�� . Positive semi-definiteness

of �� is guaranteed [14] by choosing � � � in the expressionfor � in (2).

5) We update the predicted mean using the most recent observation�� as

��

The estimated intensity is given by �� .In the proposed framework, covariance prediction is based on the

variance of the DAMRF prior which explicitly uses neighboring esti-mated pixels. This allows us to model the local uncertainty better.

V. EXPERIMENTAL RESULTS

We present results using the ISUKF method and also compare theperformance of our approach with both standard as well as recentmethods using SAR metrics. It must, however, be noted that the visualquality of the output is very important in eventually determining theperformance of any method.

We observed that a low value of �� is required in the DAMRF modelfor low-intensity regions in comparison to high intensity regions to per-form equivalent amounts of smoothing. To achieve good overall perfor-mance, we vary �� monotonically as a function of the mean value �(of the first-order NSHP pixels) as �� where � controls theamount of smoothing depending on image texture. It typically takes avalue between 0.01 and 0.03. In the following experiments, the numberof samples� � �� in the importance sampling step. We set�� ,�� and � � � for UKF [14].

Quantitative comparisons are made using the following.1) Signal-to-mean square error ratio �S/MSE� �

�� .Higher the S/MSE value the closer will be the filtered image tothe original.

2) Edge correlation factor (ECF) [22] should be close to unity foroptimal edge preservation.

3) Equivalent number of looks (ENL) measures the speckle contentin a homogeneous region and is given by ��

�

where � and � are the mean and standard deviation, respec-tively, over a chosen uniform region in the image. Theoretically,the square of the reciprocal of the speckle contrast ratio (com-puted as above) is a measure of the number of independent im-ages that must be averaged to obtain equivalent despeckling in per-

��

��



Fig. 2. (a) Original aerial image. (b) Degraded with 3-look Nakagami-distributed speckle [11]. Image estimated using (c) Rayleigh prior MAP estimator [11], and(d) the proposed ISUKF filter.

TABLE IQUANTITATIVE COMPARISON WITH OTHER FILTERS

fectly uniform regions [7], [10], [23]. A large �� correspondsto better speckle suppression.

4) Figure of merit (FOM) [6] gives a quantitative evaluation of de-tection of true edges and suppression of false edges and rangesbetween 0 and 1. A high value indicates superior edge rendition.

The edge image shown in Fig. 1(a) is degraded with simulatedfully-developed 1-look exponential speckle [Fig. 1(b)]. The outputof the Enhanced Lee [4], Frost [3], and ISUKF method are shown inFig. 1(c)–(e), respectively. Fig. 1(c) and (d) has a grainy appearancein the white uniform region due to residual speckle. Even though thenoise level is quite high, the proposed method is not only effective indespeckling but also preserves the curved edges between the dark andbright regions. This is also reflected in the FOM and S/MSE valueswhich are higher than those of the standard filters.

We also compared our method with the adaptive MAP technique in[11] which is based on a heavy-tailed Rayleigh model. In Fig. 2(a)–(c),

we reproduce the original, the 3-look degraded image, and the outputimage from [11], respectively. Even though the method proposed in[11] reduces speckle, the output is blurred. The image estimated usingthe proposed approach [Fig. 2(d)] is sharp and even the fine details arerecovered well. The output of our method has higher S/MSE and��values as given in Table I(a). Because the output of [11] is blurred, its�� value is marginally higher. The ISUKF filter is able to preserveeven curved edges (as can be inferred from its high ECF value) and itsoutput is much closer to the original image (texture-wise also).

Next, we considered the case of a 1-look “Horse track” image that isaffected by real speckle noise [Fig. 3(a)]. The output of the Frost filter,shown in Fig. 3(b), contains noticeable residual speckle in uniform re-gions. Also, the edge boundaries are noisy. The image estimated bythe ISUKF method is shown in Fig. 3(c). Our method is significantlymore effective in smoothing speckle over uniform regions (such as thetop-left and bottom-right gray regions); yet the edges are sharp and

��

��

��

��

��

��

��

��

��

��

��

��



Fig. 3. (a) 1-look “Horse track” image. Estimated output using (b) the Frost filter and (c) our method.

Fig. 4. (a) 2-look “Bedfordshire” image [9]. Result using (b) edge-based Bayesian wavelet filter [9] and (c) the ISUKF method.

clear. Even the small white blobs on the top-right corner are recoveredwell. Table I(b) gives a quantitative comparison with other filters. Thecropped 60� 60 region that was used to calculate the�� is centeredat (115, 55) as given in Table I. The performance of our method is evi-dently superior even in the real case.

Fig. 4(a) shows a 2-look “Bedfordshire” image in Southeast Eng-land with real speckle. The image estimated using a Bayesian waveletfilter which relies on edge information is reproduced from [9] and isgiven in Fig. 4(b). The output obtained by using the ISUKF filter onthe degraded image of Fig. 4(a) is shown in Fig. 4(c). We note thatthe edges are recovered without any blurring effects while the homo-geneous regions are almost free from noise. The isolated point targetswhich are two white spots in the dark homogeneous region on the leftbecome strikingly visible in Fig. 4(c). A quantitative comparison of thetwo methods is given in Table I(c).

Finally, we compared our method with a very recent despecklingtechnique [10] developed in the undecimated wavelet (UDWL) do-main. Fig. 5(a) shows a NASA/JPL AIRSAR 4-look image of an airport[10]. This is a very difficult example as it contains lots of weak edges.The outputs of the method in [10] and the proposed method are shownin Fig. 5(b) and (c), respectively. It is quite evident that the ability ofthe ISUKF method in capturing soft edges is superior to that of [10].Even the �� for our method is higher [Table I(d)].

These examples clearly demonstrate the effectiveness of the ISUKFfilter in suppressing speckle while simultaneously preserving edges and

fine features. It leads to better visual quality while comparing favor-ably in quantitative performance with existing filters. The method in[9] requires only �� operations for wavelet transformation andedge detection. However, the expectation-maximization step used forparameter estimation in [9] is computationally quite involved and de-pends on the number of iterations. The method in [11] incurs a highcomputational cost in formulating the Rayleigh prior and takes about15 min to execute in MATLAB on a Pentium-IV 1.8-GHz PC for a1100� 1100 image. The UDWL reportedly requires a few minutes [10]to filter a 512� 512 image on a 1.4-GHz PC running MATLAB. Thecomputational requirement for our method is as follows: at each pixel,it requires �� exponential functional evaluations, �� additions andmultiplications in prediction, ��

��

�� multiplications and

additions for matrix square root computation in the sigma point calcu-lation, and �� operations for updation. Since �� , theoverall complexity is approximately ��. On a Pentium-IV PCwith 256-MB RAM and for a 200� 200 image, our method executesin less than 20 s in MATLAB.

VI. CONCLUSION

We proposed a recursive spatial domain unscented Kalmanfilter-based despeckling technique for SAR imagery. A small set ofsigma points were used to capture and propagate the first two momentsthrough the SAR filter equation for estimating the original image. Themethod explored the advantage of incorporating non-Gaussian prior in



Fig. 5. (a) 4-look “Airport” image [10]. Result using (b) MAP estimator in UDWL domain [10], and (c) the proposed method.

recursive estimation for speckle reduction. When tested on syntheticas well as real examples, the proposed method was found to be veryeffective.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their useful sug-gestions.

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A Recursive Filter for Despeckling SAR Images...A Recursive Filter for Despeckling SAR Images G. R. K. S. Subrahmanyam, A. N. Rajagopalan, and R. Aravind Abstract—This correspondence