2724 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 7, JULY 2011

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1 2724 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 7, JULY 2011 SAR Image Despeckling Based on Local Homogeneous-Region Segmentation by Using Pixel-Relativity Measurement Hongxiao Feng, Student Member, IEEE, Biao Hou, Member, IEEE, and Maoguo Gong, Member, IEEE Abstract This paper provides a novel pointwise-adaptive speckle filter based on local homogeneous-region segmentation with pixel-relativity measurement. A ratio distance is proposed to measure the distance between two speckled-image patches. The theoretical proofs indicate that the ratio distance is valid for multiplicative speckle, while the traditional Euclidean distance failed in this case. The probability density function of the ratio distance is deduced to map the distance into a relativity value. This new relativity-measurement method is free of parameter setting and more functional compared with the Gaussian kernelprojection-based ones. The new measurement method is successfully applied to segment a local shape-adaptive homogeneous region for each pixel, and a simplified strategy for the segmentation implementation is given in this paper. After segmentation, the maximum likelihood rule is introduced to estimate the true signal within every homogeneous region. A novel evaluation metric of edge-preservation degree based on ratio of average is also provided for more precise quantitative assessment. The visual and numerical experimental results show that the proposed filter outperforms the existing state-of-the-art despeckling filters. Index Terms Homogeneous-region speckle-product model, local homogeneous-region segmentation, pixel-relativity measurement, synthetic aperture radar (SAR) image despeckling. I. INTRODUCTION SYNTHETIC aperture radar (SAR) images contain abundant terrain information for extensive applications in ground and sea monitoring, archeology, disaster assessment, and so on. Speckle is an inherent phenomenon accompanying SAR image for the coherent imaging property of SAR. Sometimes, speckle is useful, such as in SAR-based glacier-flow monitoring. However, in many cases, for example, image clas- Manuscript received July 28, 2009; revised June 25, 2010 and November 23, 2010; accepted January 9, Date of publication March 10, 2011; date of current version June 24, This work was supported in part by the National Natural Science Foundation of China ( , , , , and ), by the China Postdoctoral Science Foundation funded Project ( ), by the National Research Foundation for the Doctoral Program of Higher Education of China ( ), by the Program for Cheung Kong Scholars and Innovative Research Team in the University of China under Grant IRT0645, by the China Postdoctoral Science Foundation Special funded Project ( ), and by the National High Technology Research and Development Program (863 Program) of China (2009AA12Z210). The authors are with the Key Laboratory of Intelligent Perception and Image Understanding of the Ministry of Education, Xidian University, Xi an , China ( hxfeng@mail.xidian.edu.cn; avcodec@163.com; gong@ieee.org). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TGRS sification and object detection, the existence of speckle leads to the degradation of image quality and makes an undesired effect on the SAR images visual and automatic interpretation. Therefore, speckle removal is a key and indispensable step in SAR image preprocessing. In this paper, we treat the speckle as a noise and consider that the SAR image consists of true terrain backscatterer and speckle from the viewpoint of despeckling. The purpose of despeckling is to suppress speckle and restore the true backscatterer. Generally, the most popular speckle-suppression methods could be classified into two groups: multilook processing and filtering methods. Multilook processing reduces the variance of speckle while degrading the image spatial resolution. Compared with multilook processing, filtering methods could avoid this shortage to a certain degree. During the past several decades, plenty of filtering methods have been presented to suppress speckle. Among these filters, many classical filters have been widely applied, including the following: Lee filter [1], Kuan filter [2], Frost filter [3], Gamma-MAP filter [4], enhanced Lee or Frost filter [5], improved Sigma filter [6], and so on. These classical filters assume that the true backscatterer value is a local stationary random process. In fact, the assumption does not completely hold true for the case of a local sliding square window, including strong edge or texture, and thus, this causes oversmoothing in the texture or edge regions. Although the series of enhanced filters could effectively preserve texture and edge, it is difficult to distinguish homogeneous regions from texture or edge accurately. Wavelet has been theoretically proven effective in nonstationary signal analysis and is also applied in SAR image despeckling [7] [10]. The wavelet-based methods can preserve edge and texture but are prone to bring noticeable artifacts, such as Gibbs-like ringing in uniform and edge in nearby areas [11], [12] due to the own drawbacks of the wavelet and the similar frequency characteristics of noise and edge. As a modified version, undecimated wavelet transform holds a translation-invariant property. The despeckling approaches based on it obtain remarkable results [13] [15] and obviously restrain the artifacts in uniform regions. Under certain conditions, the nonstationary issue can be transformed into stationary issue. Within a local homogeneous region, the true backscatter value is a constant [5], and then the local scene can be treated as a stationary scene. As a curve is piecewise approximated by short lines, a SAR image consists of many small homogeneous regions which vary in sizes and shapes, and a single pixel is the minimum size among /$ IEEE

2 FENG et al.: SAR IMAGE DESPECKLING BASED ON LOCAL HOMOGENEOUS-REGION SEGMENTATION 2725 these regions. If the homogeneous regions could be determined successfully, the nonstationarity issue will be transformed into be a stationary one. Accordingly, SAR image despeckling can be treated as a simple linear-image restoration within these regions. This framework can be summarized in two steps: local homogeneous-region segmentation and estimation of the true signal with the segmented results. Homogeneous-region segmentation is a key issue for SAR image despeckling. Lopes et al. [5] addressed that SAR image scene is segmented based on the heterogeneity of scene and suggested that the scene is divided into three classes: homogeneous, heterogeneous, and extremely heterogeneous scene, by using variation coefficient. This approach has been widely applied in despeckling methods, such as the enhanced Lee and Frost filters [5]. Applying the assumption of gammadistribution to homogeneous scenes, a similar method of scene segmentation is operated using a chi-square test instead of variation coefficient [16]. Argenti et al. proposed a scene-segmentation method [17] in the wavelet domain with the clean wavelet coefficients. An improved segmentation method of [17] is presented in [14], which is implemented in undecimated wavelet domain and segments the coefficients into homogeneous and different-level heterogeneous classes with their texture energy. These aforementioned approaches consider the scene segmentation globally without exception and only make a coarse segmentation for homogeneous regions. A refined method of homogeneous-region segmentation is proposed in [18], which determines a homogeneous region for every pixel by using region variance. The local equivalent number of looks (ENL) estimation [19] is applied in the improved version of [18]. The methods in [18] and [19] segment image into many overlapping local homogeneous regions for every pixel. They are consistent with the ideas of classical filters, such as the Lee filter, and apply a sliding square window with varying size based on local-scene properties, instead of a fixed one. Through the use of varying sliding window, the local stationary assumption for classical speckle filters can be satisfied better within the window. However, the homogeneous regions segmented by these methods are usually rectangular and not accurate, particularly for texture or edge areas, which make these filters only effective on expanded homogeneous regions, such as expanded agricultural areas. It is important to describe the relativity between two noisy pixels robustly in image denoising, segmentation, or classification. Buades et al. [20] presented an approach of relativity measurement between two pixels utilizing the Euclidean distance between two noisy image patches and designed non local mean (NL-means in [20]) filter based on this effective and robust measurement method. Unfortunately, the method proposed in [20] is effective only in handling additive noise but is invalid in the case of multiplicative noise [21]. In order to solve this problem, Coupe et al. [22] proposed a kind of relativity-measurement method for speckled image. It is assumed that the noise is a mixture of additive and multiplicative noise, and the multiplicative noise obeys Gaussian distribution. Based on these assumptions, the Pearson distance is defined to measure the relativity of two image patches. Another distance is defined for the multiplicative model in [21], which describes the relativity through a ratio of Euclidean distance to image patch product. Deledalle et al. [23] developed an iterative approach of relativity measurement, and the relativity is designed with a combination of two distances: the distance between current restored-image patches and the distance between noisy patches which is similar to the method in [21]. By utilizing prior knowledge, this method improves the precision of relativity measurement compared with [21] and [22] and is shown to be an absorbing relativity-measurement framework. This paper proposed a segmentation-based despeckling approach for SAR images. The authors focus on segmenting a local homogeneous region for every pixel, which is inspired by the works in [18] and [19]. Ratio distance is developed to represent the distance between two speckled-image patches. The work in [21] gives us the idea to research the probability density function (PDF) of ratio distance. By utilizing this PDF, the ratio distance is mapped into a relativity value which is utilized to segment a local homogeneous region for the interested pixel. Maximum likelihood (ML) rule is applied to estimate the true signal within the obtained homogeneous region. The whole approach consists of local region segmentation and true-signal estimation, which is well consistent with the aforementioned two-step framework. Edge-preservation ability plays an important role in the performance evaluation of despeckling filters. Edge-preservation index (EPI) was introduced in [24] and has been applied in [25] [27], which calculates the spatial edges by absolute differences of two adjacent pixels along some direction. It is demonstrated in this paper that EPI is not an effective tool to evaluate the edge preservation for real SAR image and can only be employed for simulated speckle image (i.e., the real image should beforehand be obtained). Inspired by the idea of the ratio distance, a new metric for the evaluation of edge preservation is presented, and its rationality is shown in theory and experiments. This paper is mainly composed of three parts: pixel-relativity measurement, local homogeneous-regions segmentation, and ML estimation. The remainder of this paper is organized as follows. In Section II, the basic speckle statistical characteristics and multiplicative model in homogeneous region are introduced for multilook amplitude SAR image. A novel definition of pixel relativity for SAR image under multiplicative model is developed in Section III. The segmentation of homogeneous region and the ML estimation for true signal are exhibited in Sections IV and V. Section VI explains the proposed edgepreservation evaluation metric and provides the experimental results for real and simulated SAR images, and the conclusion is made in Section VII. II. SPECKLE STATISTICAL CHARACTERISTICS Under the assumption of fully developed speckle [28], an L-look amplitude SAR image obeys the generalized gamma distribution (GGD) [29, p. 108] (also named Nakagami Rayleigh distribution [23]) p A (A) =2 ( ) L ) L 1 ( R Γ(L) exp LA2 A (2L 1), A 0. R (1)

3 2726 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 7, JULY 2011 Here, L represents the number of looks, A indicates the observed SAR image. R represents the underlying radar reflectivity, i.e., true backscattering terrain factor. The m-order moment of GGD is equal to R m/2 Γ(L + m/2)/(l m/2 Γ(L)). Inthis paper, we will only discuss the amplitude image. Multilook amplitude SAR speckle is multiplicative [31], which can be described as Y = X Z (2) where Y represents the observed signal, X is the underlying reflectivity for the amplitude SAR image and is equal to R, and Z indicates an uncorrelated multiplicative speckle. X depends on the properties of the ground and is considered to be a nonstationary random process, while Z is stationary. Z can be treaded as white noise [3], [29, p. 153] with a unit mean. Generally, an image could be divided into many uniform regions that vary in shapes and sizes. In this paper, it is assumed that the SAR image consists of many homogeneous regions, and a single pixel is the minimum case of these regions. Since X is a constant within a homogeneous region [5], the product model is rewritten as Y Ω = C Ω Z Ω (3) where Ω represents a homogeneous region and C Ω represents a constant that is dependent on Ω. It is worth noticing that two interesting conclusions can be obtained from the simplified product model E[Y Ω ]=C Ω E[Z Ω ]=C Ω E[Z] =C Ω = R Ω σ 2 Z Ω = σ2 Y Ω C 2 Ω = σ2 Y Ω E 2 [Y Ω ] = LΓ2 (L) Γ 2 1. (4) (L +0.5) These conclusions show that, within a homogeneous region, the randomness of the observed signal is only caused by a speckle, and the speckle variance is an invariant value that is dependent on the number of looks. III. PIXEL-RELATIVITY MEASUREMENT Image is 2-D signal, and the pixel value represents the digital observed value of this 2-D signal in spatial domain. For the whole image, these pixels can be marked with different classes, i.e., image segmentation, while the relativity measurement between these pixels plays an essential role in image segmentation. The determination of homogeneous regions could be treated as a special issue of image segmentation. It goes without saying that relativity measurement is crucial to homogeneousregion segmentation. A. Traditional Measurement of Pixel Relativity and Problem Plenty of distance measurements are developed to represent the relativity of two samples in the field of pattern classification. Between these distances, Euclidean distance is an essential and widely used measurement in pattern recognition and classification. In practice, Euclidean distance is usually mapped by a Gaussian kernel radial basis function ( ) R i =exp β P 0 P i 2 2 (5) where P 0 is the central pixel value and P i represents the pixel value on location i. R i represents the relativity between the two pixels. The parameter β acts as a degree of kernel function, which controls the decay of the function. The definition of (5) is used in many neighborhood-averaging filters, for example, the Similar Univalue Segment Assimilating Nucleus (SUSAN) filter [32] and the Bilateral filter [33]. Definition (5) can supply a reliable relativity estimation of two pixels for noise-free image. However, SAR images are contaminated by speckle, and thus, (5) becomes unreliable. A possible solution is to expand the sample capacity. In practice, the Euclidean distance of P 0 and P i is usually replaced by the Euclidean distance between two square-image patches N 0 and N i, which contain P 0 and P i in their centers, respectively, i.e., the distance is developed for NL-means filter [20] (we only consider the case of SAR image here) ( ) R i =exp β Y N0 Y Ni 2 2,G (6) where Y N0 and Y Ni denote two patches which have the same size and shape in the observed SAR image Y, and G represents the standard Gaussian kernel function. Unfortunately, (6) is only robust for additive noise and no more effective for multiplicative noise, which was mentioned in [21]. However, a theoretical proof of the invalidity is absent from the thesis. Here, we will make a detailed proof as follows. Considering convenience of demonstration, the product model can be transformed into an additive model with a signaldependent noise [9], [13], [14] Y = X Z = X + X(Z 1) = X + S. (7) Let M ΔY = Y N0 Y Ni 2 2 = Y N0 (k) Y Ni (k) 2 where M denotes the number of pixels of image patches Y N0 and Y Ni. The expectation is calculated as E[ΔY ]= = = = E ( Y N0 (k) Y Ni (k) 2) ( ( E XN0 (k) X Ni (k) ) + ( S N0 (k) S Ni (k) ) 2) E ( ΔX(k)+ΔS(k) 2) { [ΔX(k)] 2 +2ΔX(k)E [ΔS(k)] + E ΔS(k) 2} E[ΔS(k)]=0 = ΔX + E ΔS(k) 2. (8)

4 FENG et al.: SAR IMAGE DESPECKLING BASED ON LOCAL HOMOGENEOUS-REGION SEGMENTATION 2727 Here E ΔS(k) 2 = E S N0 (k) S Ni (k) 2 = [ E S N0 (k) 2 2E [S N0 (k) S Ni (k)] + E S Ni (k) 2]. Considering the white-noise assumption for speckle, Z N0 (k) is independent of Z Ni (k)(i 0) E [S N0 (k) S Ni (k)] and thus = E [X N0 (k)(z N0 (k) 1) X Ni (k)(z Ni (k) 1)] = X N0 (k) X Ni (k)e [Z N0 (k) 1] E [Z Ni (k) 1] = 0 E ΔS(k) 2 = [E S N0 (k) 2 + E S Ni (k) 2] Thus we get = =σz 2 { [X N0 (k)] 2 +[X Ni (k)] 2} E(Z 1) 2 E[ΔY ]= X N0 X Ni σ2 Z { [X N0 (k)] 2 +[X Ni (k)] 2}. ( ) X N X N i 2 2. (9) Equation (9) shows that the Euclidean distance is robust when the true signal X is a constant, i.e., it is only robust for ideal uniform regions of SAR image. However, the second term of (9) will vary according to local scenes in edge and texture regions, and therefore, (6) will no longer be valid. B. Ratio Distance for Multiplicative Noise Model The distinction between two image patches could not only be denoted as the difference of these patches but could also be denoted as the ratio of the patches. In this paper, a universal ratio distance is developed the idea of which arises from the ratio of average (ROA) operator [34]. Let symbol./ denote the dot quotient of two vectors; the ratio distance is then defined as M d i = Y Ni./Y N0 2 2,G = G(k) Y Ni (k) 2 Y N0 (k), Y N0 (k) 0. (10) Although the aforementioned definition is simple, it is very robust for multiplicative noise. The following formula will show the validity proof of the ratio distance in theory. Due to the assumption of white noise for speckle, we obtain E[ Z Ni (k)/z No (k) 2 ]=E Z Ni (k) 2 /E Z No (k) 2, and so E[d i ]= G(k)E Y Ni (k) 2 Y N0 (k) = G(k) X Ni (k) 2 E Z Ni (k) 2 X N0 (k) E Z No (k) 2 = G(k) X Ni (k) 2 ( σz 2 +1) X N0 (k) (σ 2 Z +1) = X Ni./X N0 2 2,G. (11) Equation (11) shows the validity and robustness of the ratio distance since, as expected, it conserves the true relativity between pixels. C. Relativity Measured by the PDF of Ratio Distance The ratio distance cannot be directly applied as a relativity after a mapping of Gaussian kernel function because of its property of relativity representation: A ratio distance that is closer to one involves a bigger relativity, while farther from one involves a smaller relativity. Therefore, the ratio distance should be mapped as a rational normalized relativity. Azzabou mapped the Euclidean distance into relativity with the PDF of the distance [21], which gives us the enlightenment to resolve the previous problem. Here, we set r i,k = Y N i (k) Y N0 (k). (12) Expression (12) contains true signal ratio, which make it difficult to handle. Here, it is assumed that Y Ni (k) and Y N0 (k) are two different observed values of real signal X N0 (k), i.e., Y Ni (k) and Y N0 (k) are located within the same homogeneous region. Thus, (12) can be transformed into r i,k = Z N i (k) Z N0 (k). (13) Let p(r i,k ) represent the PDF of r i,k and set X =1;after that, (2) is substituted into (1). We get the PDF of speckle Z as p Z (Z) = 2LL Γ(L) exp( LZ2 )Z (2L 1), Z 0. (14) For convenience of representation, we set t = Z Ni (k) and s = Z N0 (k). Utilizing (14), we get p(r i,k )= = s p T (sr i,k ) p S (s)ds s 2LL (sr i,k ) 2L 1 Γ(L) 2LL (s) 2L 1 Γ(L) exp ( L(sr i,k ) 2) exp( Ls 2 )ds

5 2728 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 7, JULY 2011 Fig. 1. Variation of p(r i,k ) following the number of looks. ratio distance. The multiplicative noise is assumed as Gaussian distribution in [22], while it is absent in this paper. From the aspects of distance definition, the distance for noisy-image patches in [23] is similar to the definition in [21]; the major difference of [23] and [21] is that the prior knowledge of the middle estimated result is introduced to the final relativity measurement in [23]. Compared with this paper, [23] developed a new kind of framework for relativity calculation, while this paper focuses on the relativity measurement for speckled-image patches. Under certain conditions, our method can be combined into the framework proposed in [23]. The relativity calculation in [23] requires that it manually set the degree parameter of the Gaussian kernel function, while our approach is free of this parameter setting. = 4L2L (r i,k ) 2L 1 } Γ 2 (L) {{ } =α + 0 = α 2L 1 β s 4L 1 exp ( (r i,k ) 2 +1 ) }{{} =β + 0 (2L 1)! = α β 2L 1 = 2(2L 1)! Γ 2 (L) s 2 s 4L 3 exp( βs 2 )ds = + 0 s exp( βs 2 )ds = α ds (2L 1)! 2β 2L (r i,k ) 2L 1. (15) [(r i,k ) 2 2L +1] The study of p(r i,k ) indicates that the maximum value of p(r i,k ) locates at r = (2L 1)/(2L +1), and r approaches one when the number of looks increases gradually. Fig. 1 shows the variation of p(r i,k ) following L. Utilizing the definition of (15), we define the relativity between pixel P 0 and P i as follows: R i = p(r i,k ) 2 2,G = M [ ( )] 2 YNi (k) G(k) p. (16) Y N0 (k) An important progress of the new relativity measurement in (16) is that it is free of parameter control and only depends on the statistical property of the multiplicative noise. Relativity measurement between two speckled images patches has been studied widely [21] [23]. The major contributions of this paper on relativity measurement are the following: 1) A robust and universal ratio distance for two speckled image patches is developed; 2) a new relativity measurement is produced by the PDF of the ratio distance instead of traditional Gaussian kernel projection; and 3) the novel measurement method proposed by us is almost nonparametric, which only requires the setting of the size of the comparison window. The distance proposed in [21] can be seen as a special case of IV. LOCAL HOMOGENEOUS-REGION ESTIMATION WITH PIXEL-RELATIVITY MEASUREMENT As mentioned in Section III, the measurement of pixel relativity for a SAR image provides a tool for image segmentation. In this section, a method of homogeneous-region segmentation will be developed by applying the relativity-measurement approach proposed in the previous section. Let X 0 (n) be the nth true pixel value and Ω H (n) denote a local homogeneous region surrounding X 0 (n). N is the neighborhood of X 0 (n) and is larger than Ω H (n), i.e., Ω H (n) ={X 0 (i) R i >T R,X 0 (i) N} (17) where T R is a relativity threshold and R i is calculated using (16). The neighborhood N is square and is similar to the searching window in the NL-means algorithm [20]. Generally, a complete pixel traverse in neighborhood N could determine a homogeneous region. However, this complete traverse is time consuming. In this paper, we approximate the local homogeneous region defined in (17) with a polygonal region and give a kind of simplified traversal method which determines the polygon. A set of directions is defined as {d k (n) d k (n) =2kπ/D, k =0,...,D 1} for the nth pixel. Starting with the central pixel Y 0 (n) and traversing along a direction d k (n), an optimal scale Sk (n) can be found. The traverse can be seen in Fig. 2. The optimal-scale set {Sk (n),k =0,...,D 1} is determined after traversing along all directions. Based on these optimal scales, a homogeneous region Ω H (n) is determined. Fig. 3 shows an example of a homogeneous region constructed by eight directions. It is worth noticing that the shape of homogeneous region Ω H (n) will be consistent with the case in [18] and [19] when the direction number D is equal to four, and all of them are rectangular. Nevertheless, two novelties of our method are addressed compared with [18] and [19]: 1) The relativity between two noisy pixels is utilized to segment the local homogeneous region, while local statistical characters (local variance or ENL) are applied in [18] and [19], and 2) the homogeneous region constructed by pixel relativity is shape adaptive, and a higher segmentation precision is made compared with a shape-fixed window in [18] and [19].

6 FENG et al.: SAR IMAGE DESPECKLING BASED ON LOCAL HOMOGENEOUS-REGION SEGMENTATION 2729 Fig. 2. Implementing process of determining the optimal-scale set. where K R is a normalized scale factor. T R max represents the maximum value of T R, which can be obtained once the looks of the SAR image is available. We substitute r = (2L 1)/(2L +1) [which corresponds to the maximum value of (15)] into (15) and then get T R max = G(k) 4[(2L 1)!]2 (4L 2 1)(2L 1) 2L 2 Γ 4 (L)(4L) 2L. (19) V. D ESPECKLING WITH ML The multiplicative model is rewritten in homogeneous region Ω H (n) as follows: Y ΩH (n)(k)=c ΩH (n) Z ΩH (n)(k), 1 k M ΩH (n) (20) Fig. 3. One homogeneous region constructed by eight directions. The parameter T R is the most important factor for the segmentation result. Here, it was developed as follows: T R = K R T R max (18) where the total number of pixels within Ω H (n) is M ΩH (n). Considering the GGD of observed signal Y ΩH (n), themlrule is applied to estimate the parameter R of GGD ˆR ΩH (n) =maxl [ Y ΩH (n)(1),y ΩH (n)(2),..., Y ΩH (n) ( MΩH (n)) ; RΩH (n)]. (21)

7 2730 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 7, JULY 2011 The likelihood function is defined as L ( [ R ΩH (n)) = L YΩH (n)(1),y ΩH (n)(2),..., ( ] Y ΩH (n) MΩH (n)) ; RΩH (n) M ΩH (n) ( ) L L 1 = 2 R ΩH (n) Γ(L) exp { L [ Y ΩH (n)(k) ] } 2 R ΩH (n) [ Y ΩH (n)(k) ] (2L 1). (22) By letting f(r ΩH (n)) =lnl(r ΩH (n)), we obtain f ( M ΩH { (n) ) R ΩH (n) = L ln R ΩH(n) L[ Y ΩH (n)(k) ] 2 R ΩH (n) } +ln 2 Γ(L) + L ln L +(2L 1)Y Ω H (n)(k). (23) By setting df (R ΩH (n))/dr ΩH (n)) =0, we get the estimation ˆR ΩH (n) = 1 M ΩH (n) M ΩH (n) [ YΩH (n)(k) ] 2. (24) Equation (24) is an unbiased estimation of ˆR ΩH (n). Since the true reflectivity value is constant within Ω H (n), we get the despeckled result of Y (n) as ˆX(n) = ˆRΩH (n) = 1 M ΩH (n) M ΩH (n) [ YΩH (n)(k) ] 2. (25) VI. EXPERIMENTAL RESULTS AND DISCUSSION In this section, we present the experimental results by applying the proposed approach and compare our method with some classical and recent filters. The performance evaluation of filters is an important and basic issue on SAR image despeckling. As pointed out in [30, pp ], a good despeckling method demonstrates the following properties: 1) speckle reduction in extended uniform regions; 2) edge and texture preservation; 3) absence of artifacts in uniform regions; and 4) radiometric preservation. The performance of despeckling methods can be evaluated based on the aforementioned properties with quantitative indicators and visual impression. A. Novel Evaluation Approach of Edge-Preservation Degree The traditional EPI [24] [27] can be represented as m I D1 (i) I D2 (i) i=1 EPI = m (26) I O1 (i) I O2 (i) i=1 where m is the pixel number of the selected area. I D1 (i) and I D2 (i) represent the adjacent pixel values of the despeckled Fig. 4. Artificial edge images. image along a certain direction. Similarly, I O1 (i) and I O2 (i) represent the adjacent pixel values of the speckled image. As shown in (8) and (9), m i=1 I D1(i) I D2 (i) cannot robustly indicate the edges for the multiplicative-noise model. In this paper, we revised the traditional EPI and present a novel measurement method of edge-preservation degree based on ROA (EPD-ROA) m I D1 (i)/i D2 (i) i=1 EPD-ROA = m. (27) I O1 (i)/i O2 (i) i=1 The validity of EPD-ROA was theoretically revealed in (11). It is similar to EPI; when the EPD-ROA is closer to one, it means better ability of edge preservation. Three artificial edge images (Fig. 4) are contaminated with different-level synthetic speckle, and the edge-preservation degree of these images are calculated with EPI and EPD-ROA. In the experiments, the despeckled images are represented by clean artificial images and by calculating the EPI and the EPD-ROA for the whole image along the horizontal (HD) and vertical (VD) directions. The experimental results are shown in Table I. Since the despeckled image is replaced with clean image, the ideal value of EPI or EPD-ROA should be equal to one. Table I exhibits that EPD-ROA is very robust and close to ideal value of one when the looks is larger than two, while all of the EPI values are far from the ideal value. Although the values of EPD- ROA are not very close to one when L =1or 2, the degree of adjacency is far higher than EPI. It is worth pointing out that EPI and EPD-ROA are all gradually close to the ideal value when the looks increase, but EPD-ROA is stable for different types of edge images and is only affected by the speckle level, whereas EPI varies for all the images. B. Evaluating Index-of-Despeckling Performance In this paper, EPD-ROA and two other different types of quantitative evaluation indicators are applied to evaluate the performance of the different filters. 1) ENL: ENL measures the degree of speckle reduction in a homogeneous region, which is given by [35] ENL = S C ( μhr S C = σ HR ) 2 { 1, intensity image 4/π 1, amplitude image (28) where μ HR and σ HR are the mean and the standard deviation of a chosen homogeneous region, respectively. In practice, we

8 FENG et al.: SAR IMAGE DESPECKLING BASED ON LOCAL HOMOGENEOUS-REGION SEGMENTATION 2731 TABLE I COMPARISON OF MEASUREMENT OF THE EDGE-PRESERVATION DEGREE should artificially choose the homogeneous region to be as large as possible. A large ENL value corresponds to better speckle suppression. 2) Ratio Image: Since the speckle is multiplicative, the ratio of the original SAR image to the despeckled one is a speckle. Therefore, the statistical characteristics of the ratio image could indicate the performance of the despeckled algorithms. It usually evaluates the filters performance by the following two aspects. 1) Mean of ratio image: It measures the degree of radiometric preservation, the ideal value of which is a unit. The mean is closer to one, which means a better performance of radiometric preservation. 2) Variance of ratio image: Under ideal conditions, it is calculated by the following formula [30, pp ]: σ 2 Z = L eγ 2 (L e ) Γ 2 (L e +0.5) 1 (29) where L e represents the effective number of looks [30, pp ]. In this paper, we estimated the effective looks of a real SAR image with the method in [36]. In fact, the aforementioned ideal conditions correspond to a uniform scene. Generally, the real image contains edges and textures, and thus, the genuine variance of the ratio image should be smaller than this ideal value. However, if the value is bigger than the ideal one, (29) can reveal the fact that the despeckling result makes a deviation from the underlying result. C. Discussion of the Parameters for the Proposed Method Several parameters are needed to be set in the proposed method. Among these parameters, the scale factor K R in (18) is the most important parameter, which decides the value of relativity threshold T R and has a serious influence on the despeckled results. The other parameters are the size of image patch N 0 or N i in (16) (we set the size equal to (2f +1) (2f +1)) and the maximal value h k max of scale S k (n) in Section IV. Several groups of experiments are devised to obtain the effect of these parameters on the results. In order to acquire a better shapeadaptive homogeneous region, we set the direction number D as eight and choose the same h max for every scale S k (n). The peak signal-to-noise ratio (PSNR) is used to evaluate the despeckled results. Fig. 5. Test of despeckled result variation following K R. First, we test the parameter K R of three classical natural noise-free images: Peppers, Lena, and Cameraman are corrupted by different-level synthetic speckle. In the experiments, the speckle level is set as L =2, 4, and 8, and the image patch size f and the maximal value h max are empirically set as three and seven. The results are shown as curves in Fig. 5. It is easy to acquire the information from these curves that the threshold K R affects the results seriously, and all the curves contain a peak value. After observing these peak values, an exciting result is obtained that shows that almost all of the maximal values appear as K R within the interval [0.35, 0.4]. This conclusion shows the nicer practicability and operability of our method. In practice, a smaller K R value results in a smoother despeckled result, and a high degree of smoothness corresponds to high level of speckle. Therefore, we suggest that K R be set as for relatively slight speckle, such as the case of L larger than two and as 0.35 for L =1or 2. Second, we set K R =0.375 and f =3. It considers that Lena contains typical uniform and edge regions, which are used to test the influences of parameter h max on the results. In the experiments, noise-free Lena is corrupted by synthetic speckle with L =2, 4, and 8, and we choose h max =4, 5,...,11. Fig. 6(a) shows the relationship between h max and PSNR. The curves in Fig. 6(a) shows that the despeckling results are slightly affected by the value of h max, and the degree of influence decreases when the speckle level decreases. A relatively stable result can be obtained if h max is larger than five. However, the PSNR will decrease clearly in case h max is over eight, because a bigger h max value produces a smoother result. Consequently, we suggest that h max be set as six or seven in practice.

9 2732 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 7, JULY 2011 Fig. 6. Lena. Influence of (a) h k max and (b) f on the despeckling results for speckled Fig. 8. Experimental results for a two-look synthetic curving-edge image. (a) A two-look speckled image ( ). (b) Gamma-MAP. (c) Frost. (d) Wu Maitre. (e)map-uwd-s. (f) LHRS-PRM. Fig. 7. Experimental results for a two-look synthetic regular-edge image. (a) A two-look speckled image ( ). (b) Gamma-MAP. (c) Frost. (d) Wu Maitre. (e) MAP-UWD-S. (f) LHRS-PRM. Finally, the parameter f is tested. It is similar to the test of h max ; Lena is also used in this test and contaminated with L =2, 4, and 8 synthetic speckle. We set K R =0.375 and h max =7, and choose f =1, 2,...,6. The results are shown as curves in Fig. 6(b). These curves reveal that size f has very slight influence on the despeckling results, and a stable result can be obtained once f is more than two. Here, it is notable that the time complexity will increase seriously when f increases. Considering the stability and time cost, we suggest that f should be set as three in practice. Based on aforementioned experiments, it can be known that f is relatively stable, and the degree of smoothness increases as the value of K R or h max decreases or increases, respectively. In contrast, K R has a greater impact on the degree of smoothness than h max. D. Experimental Results for Synthetic and Real SAR Images In this section, three synthetic and four real SAR images are tested; visual and numerical results are obtained for these images. Two of the synthetic images are synthetic regular-edge image [Fig. 4(a)] and synthetic curving-edges image [Fig. 4(b)], which are all corrupted with a two-look speckle and shown in Figs. 7(a) and 8(a), respectively. The third synthetic image is shown in Fig. 9(a) and contaminated with a four-look speckle. The results of our approach (local homogeneous-region segmentation with pixel-relativity measurement, named LHRS- PRM) for these synthetic speckle images have been com- Fig. 9. Experimental results for a four-look synthetic image. (a) A four-look synthetic image ( ). (b) Gamma-MAP. (c) Frost. (d) Wu Maitre. (e) MAP-UWD-S. (f) LHRS-PRM. pared with those of Frost filter [3], Gamma-MAP filter [4], Wu Maitre filter [19], and MAP filter based on undecimated wavelet decomposition and image segmentation (MAP-UWD-S) [14]. In all experiments, the relativity threshold K R was set as 0.375, and the window sizes f of N 0 and N i were all set to three for the relativity calculation. The parameter h max was set as seven for the local homogeneousregion estimation, and the direction number D has been set as eight for all experiments. The sliding window size was 5 5for the Frost and Gamma-MAP filters. The despeckled results for the different algorithms are shown in Figs Table II gives all the numerical evaluation results for these images, which contains the evaluations of PSNR, ratio image, and EPD-ROA along HD and VD for the whole image.

10 FENG et al.: SAR IMAGE DESPECKLING BASED ON LOCAL HOMOGENEOUS-REGION SEGMENTATION 2733 TABLE II NUMERICAL EVALUATION FOR SYNTHETIC SPECKLE IMAGES Fig. 10. Experimental results for Horsetrack. (a) Original real SAR image ( ). (b) Frost. (c) Wu Maitre. (d) MAP-UWD-S. (e) LHRS-PRM. (f) LHRS-PRM-C. Fig. 11. Experimental results for Field. (a) Original real SAR image ( ). (b) Frost. (c) Wu Maitre. (d) MAP-UWD-S. (e) LHRS-PRM. (f) LHRS-PRM-C. The visual comparison of Figs. 7 and 8 indicates that LHRS- PRM is free of artifacts in uniform regions, and the performance smoothness markedly precedes the Gamma-MAP, Frost, and Wu Maitre filter. A very clean uniform region is obtained with our approach, while speckle residue appears in the results obtained with the other filters except with that of MAP-UWD-S. In terms of edge preservation, LHRS-PRM produces clear and sharp edges, while blur phenomenon appears in the results obtained with the Gamma-MAP, Frost, and Wu Maitre filters. Although MAP-UWD-S possesses nicer ability of smoothing, it produces Gibbs-like ringing and edge blurring, particularly in Fig. 7(e). The results shown in Fig. 9 show that LHRS- PRM possesses comparative preserving performance of texture and small objects compared with the Gamma-MAP, Frost, and Wu Maitre filters. MAP-UWD-S could hold a similar preserving performance, whereas edge blurring is unavoidable. Therefore, the finer performance of edge preservation is one of the virtues of our method compared with the other filters. After the presentation of the results for synthetic speckle images, four real SAR images are tested: 1) Ku-band 1-m

11 2734 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 7, JULY 2011 Fig. 12. Experimental results for Yellow River. (a) Original real SAR image ( ). (b) Frost. (c) Wu Maitre. (d) MAP-UWD-S. (e) LHRS-PRM. (f) LHRS-PRM-C. resolution Horse Track near Albuquerque, New Mexico (Fig. 10(a) and named Horsetrack the effective number of looks is about 4); 2) British Defense Research Agency (DRA) SAR X-band 3-m resolution, a rural scene in Bedfordshire, England (Fig. 11(a) and named Field the effective number of looks is about 3.2); 3) Radarsat-2 C-band 8-m resolution, near Yellow River estuary, China (Fig. 12(a) and named Yellow River the effective number of looks is 4); and 4) X-band 3-m resolution in a town near Xi an, China, produced by a Chinese institute (Fig. 13(a) and named Town the effective number of looks is about 4). The comparisons were performed with Frost filter [3], Wu Maitre filter [19], MAP-UWD-S [14], and LHRS-PRM. In the experiments, the setting for all the parameters is completely identical to the setting for the earlier experiments of synthetic images. In Section IV, the authors give a simplified traversal method when one determines a homogeneous region, i.e., the aforementioned approach: LHRS-PRM. Here, we will also show the results obtained by applying complete traversal in the whole spatial neighborhood, named LHRS-PRM-C. In the experiments, its parameter setting is entirely consistent with LHRS-PRM. The readers can download a C-code implementation of LHRS-PRM and LHRS-PRM-C from the website: Figs show the visual results for the four real SAR images. The contrasts for these results indicate that the speckle suppression ability of LHRS-PRM is excellent. The Frost filter produces distinct speckle residue within uniform regions; Fig. 13. Experimental results for Town. (a) Original real SAR image ( ). (b) Frost. (c) Wu Maitre. (d) MAP-UWD-S. (e) LHRS-PRM. (f) LHRS-PRM-C. pointillist-like small regions appear in the results obtained with the Wu Maitre filter. MAP-UWD-S is always accompanied with ringing effect despite its excellent ability of speckle reduction, and the phenomenon is more evident for the Field [Fig. 11(d)]. In contrast, LHRS-PRM effectively removes the speckle in uniform regions, for example, the left area of Horsetrack [Fig. 10(e)] and the middle area of Field [Fig. 11(e)]. Compared with the other filters, LHRS-PRM possesses the best smoothing ability in uniform regions, which is validated by the ENL comparisons in Table III [note: four uniform regions are chosen in Horsetrack and Field, respectively, which are marked in Figs. 10(a) and 11(a)]. In terms of edge and texture preservation, Frost filter can retain the edge and texture features to a certain extent, but the results for uniform regions are not satisfactory. MAP-UWD-S keeps the texture features while resulting in edge blurring. The Wu Maitre filter obtains a tradeoff between smoothing and edge preservation to a certain degree. In contrast, our algorithm preserves the texture well and produces clear edges and without artifacts. Table IV reveals that LHRS-PRM obtained the optimal EPD- ROA in most cases compared with the other algorithms. The radiometric-preservation ability of our approach is comparable with that of the Frost or Wu Maitre filter, but MAP-UWD-S usually produces a relatively larger bias. The visual comparisons indicate that the performance of LHRS-PRM is very close to its completely traverse version: LHRS-PRM-C. It is worth pointing out that LHRS-PRM actually produces sharper edges than LHRS-PRM-C does, which

12 FENG et al.: SAR IMAGE DESPECKLING BASED ON LOCAL HOMOGENEOUS-REGION SEGMENTATION 2735 TABLE III ENL COMPARISON TABLE IV NUMERICAL EVALUATION FOR REAL SAR IMAGES TABLE V IMPLEMENTING-TIME COMPARISON also can be proved through the EPD-ROA comparisons in Table IV. Table V exhibits the contrast of implementing time. It shows that the simplified traverse version possesses noticeably low time cost and is close to one-fifth of the cost of the completely traverse one. Local homogeneous-region segmentation is a critical step for the whole algorithm. A similar idea is also applied in the Wu Maitre filter. In order to compare the segmentation performance between the Wu Maitre filter and LHRS-PRM, a comparative experiment was devised. Because the shape of homogeneous region Ω H (n) is rectangular when the direction number D is equal to four, which is consistent with the case in the Wu Maitre filter, we designed a new method: the segmentation result of the Wu Maitre filter is companied with the ML estimation in this paper. Fig. 14 shows the results for the Wu Maitre-ML filter and the four-directional LHRS- PRM (note: our method can be seen as four-directional LHRS- PRM+ML). It is obvious that an entirely blurring result is produced by the Wu Maitre ML filter, while a very satisfactory result is obtained with the four-directional LHRS-PRM. This fact reveals that the segmentation performance of LHRS-PRM is far higher than the Wu Maitre filter. A drawback of the proposed approach is that it is not very good at processing single-look image since the mapping function is not precise for very high level speckle. This would be solved by making an effective modification on the mapping function when one deals with a single-look image. For computational purpose of the proposed algorithm, it assumes that the maximum size of a homogeneous region is (2S max +1) (2S max +1), the size of window N 0 is M M, the total pixel number of image is N, and the average complexity of the algorithm is [4(S max +1)M 2 +(1+ (2S max +1) 2 )/2]N. Here, 4(S max +1)M 2 N represents the complexity of the homogeneous-region determination, while [1 + (2S max +1) 2 ]N/2 represents the homogeneous-region estimation with ML rule. Since S max and M are commonly

13 2736 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 7, JULY 2011 PASCO China Corporation for providing the Radarsat-2 data, and the anonymous reviewers and editors for their valuable comments and helpful suggestions which greatly improved this paper s quality. Fig. 14. Visual comparison of the results obtained with (a) Wu Maitre+ML and (b) four-directional LHRS-PRM. far less than N, the complexity level of our algorithm is only O(N). VII. CONCLUSION A novel despeckling methodology has been proposed for SAR image based on local homogeneous-region segmentation with pixel-relativity measurement and ML estimation. The simplified speckle model in homogeneous regions enables the unmanageable multiplicative model to become a simple linear model. The new method of relativity measurement between two pixels only considered the distribution of speckle. It is very robust for speckled SAR image and is almost free of parameter setting and very functional in practice. A homogeneous region is segmented for each pixel by utilizing the proposed pixelrelativity measurement. In particular, the segmentation is accurate for uniform and textured or edged areas. The satisfied estimated result of a true signal is obtained by the ML rule based on the segmented result. Considering the whole algorithm, our method consists of two steps: homogeneous-region segmentation and ML estimation. Classical and recent speckle filters are compared with the proposed approach, and visual impression shows that very clean uniform regions and clear sharp edges are obtained by the proposed method. The numerical results indicate that our method possesses remarkable speckle-suppression ability and good ability of edge and radiometric preservation. It is noticeable that the proposed approach is pointwise; thus, it could easily do parallel processing for software or hardware implementation. ACKNOWLEDGMENT The authors would like to thank T. 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