Speckle Noise Reduction Technique for SAR Images Using Statistical Characteristics of Speckle Noise and Discrete Wavelet Transform

Synthetic aperture radar (SAR) images map Earth’s surface at high resolution, regardless of the weather conditions or sunshine phenomena. Therefore, SAR images have applications in various fields. Speckle noise, which has the characteristic of multiplicative noise, degrades the image quality of SAR images, which causes information loss. This study proposes a speckle noise reduction algorithm while using the speckle reducing anisotropic diffusion (SRAD) filter, discrete wavelet transform (DWT), soft threshold, improved guided filter (IGF), and guided filter (GF), with the aim of removing speckle noise. First, the SRAD filter is applied to the SAR images, and a logarithmic transform is used to convert multiplicative noise in the resulting SRAD image into additive noise. A two-level DWT is used to divide the resulting SRAD image into one low-frequency and six high-frequency sub-band images. To remove the additive noise and preserve edge information, horizontal and vertical sub-band images employ the soft threshold; the diagonal sub-band images employ the IGF; while, the lowfrequency sub-band image removes additive noise using the GF. The experiments used both standard and real SAR images. The experimental results reveal that the proposed method, in comparison to state-of-the art methods, obtains excellent speckle noise removal, while preserving the edges and maintaining low computational complexity.


Introduction
Synthetic Aperture Radar (SAR) images employ active sensors that detect microwave radiation, which has longer wavelength than visible light that is detected in passive sensors, such as the optical sensor. Therefore, the surface of the Earth can be observed at high resolution, regardless of weather conditions and sun phenomena [1]. The active sensor of SAR images is also used with satellites or unmanned aerial vehicles (UAVs), as the development of the active sensor technology applied to SAR images enabled high-resolution target detection and identification. SAR images are widely used in applications of a variety of fields, such as the military, agricultural, weather forecasting, and environmental analysis, etc. [2]. Due to the advantages of SAR images and their various applications, research on the technology behind SAR images is being actively conducted around the world (image enhancement [3][4][5], image classification [6,7], image segmentation [8,9], etc.).
In contrast with the optical sensor, the active sensor of the SAR is accompanied by speckle noise that arises from the coherent imaging mechanism. Speckle noise in SAR images is generated by the random interference of many elementary reflectors within one resolution cell [10]. This noise has different features from the noise observed in images that were obtained by passive sensors, such as the in the non-homomorphic framework. Subsequently, a two-sided generalized Gamma distribution was used in an earlier step to process the heavy-tailed nature of the wavelet coefficients of the noise-free reflectivity. Based on this, the maximum a-posteriori (MAP) method was used in an analytical wavelet shrinkage function. The MAP method employed a heterogeneity-adaptive threshold to select the best estimates of the noise-free wavelet coefficients. Controlling a tunable parameter is difficult; hence, this parameter affects the optimal heterogeneity-adaptive weight function. As a result, the algorithm exhibits a blurring phenomenon around the edge areas in the actual SAR images. Rajesh et al. [38] presented a combination of a spatial filter as a preprocessing step and adaptive threshold in the frequency domain. The algorithm employs a Wiener filter, among other spatial filters, and an adaptive soft threshold of wavelet coefficients in the wavelet domain. The Wiener filter shows excellent additive noise removal performance in the image [40]. The algorithm did not consider this ability of the Wiener filter, which results in a low speckle suppression performance. Despite extensive efforts, as mentioned above, conventional algorithms exhibit low performance in terms of speckle noise removal, edge information preservation, and computational complexity.
In this study, we employ a speckle reducing anisotropic diffusion (SRAD) filtering method as a preprocessing filter to reduce the speckle noise and preserve the edge information. The SRAD filtering result image is applied to a logarithmic transform for the conversion of multiplicative noise to an additive noise. The soft threshold, guided filter (GF), and improved guided filter (IGF) in the wavelet domain are employed to further reduce the additive noise in the SRAD filtering result image. A diagonal sub-band image in the wavelet domain has a lower energy than the vertical and horizontal sub-band images. Hence, the IGF applied as a new edge-aware weighting method is employed in the diagonal sub-band image to preserve weak edges and remove the noise. To the same end, the soft threshold is applied to the vertical and horizontal sub-band images. Moreover, the GF removed the noise in the approximate sub-band image. Finally, a noise-free image is obtained by an exponential transform and wavelet reconstruction. The proposed algorithm is implemented to remove the speckle noise, preserve the edges, and reduce computational complexity.
This paper is organized, as follows. Section 2 describes the evaluation metrics and the proposed algorithm in detail. In Section 3, simulated and real SAR images are used to analyze the experimental results of qualitative, quantitative, and computational complexity. Section 4 presents a discussion. Section 5 concludes the paper.

Proposed Algorithm
In this study, we propose an algorithm for the reduction of speckle noise and the preservation of the edges in the SAR image ( Figure 1). The proposed algorithm employs the SRAD filtering method as a preprocessing filter instead of directly applying the wavelet domain, as the SRAD can be directly applied to the SAR image, because it uses the image without log-compressed data [39]. However, the SRAD filtering result image still includes the speckle noise, which represents a form of multiplicative noise. Since most of the filtering methods are developed for reducing the AGWN, the logarithmic transform is applied to the resulting SRAD image to convert the multiplicative noise into additive noise [41], after which the resulting SRAD image contains additive noise. Subsequently, the two-dimensional (2D) DWT transforms the SRAD filtering result image, which represents the logarithmic transform, into four sub-band images (vertical sub-band image (LH), horizontal sub-band image (HL), diagonal sub-band image (HH), and approximate sub-band image (LL)). We employed the DWT performed until two-level decomposition. An effect of algorithm and analysis of results is tested at one to two decomposition level. The two-level decomposition of the DWT shows the best results [42]. Most of the speckle noise occurs in high-frequency sub-band images [43]. Therefore, the soft threshold of the wavelet coefficients is only applied to the horizontal and vertical sub-band images, which have similar energy, to preserve the original signal and remove the noise signal. However, the diagonal sub-band image has a low energy when compared to the vertical and horizontal sub-band images. For the diagonal sub-band image, we employ an IGF that is based on a new edge-aware weighting method Remote Sens. 2019, 11, 1184 5 of 27 to preserve a low original signal and suppress the noise signal using this new edge-aware weighting method. The approximate sub-band image contains significant components of the image and is less affected by the noise [44]; however, the noise exists in the approximate sub-band image. The GF [45] is employed to reduce the speckle noise and preserve the edges in the approximate sub-band image. Each sub-band image, once the noise is removed, is reconstructed by wavelet reconstruction, and the exponential transform is performed to reverse the logarithmic transform. Finally, we obtain the despeckled image. of the image and is less affected by the noise [44]; however, the noise exists in the approximate subband image. The GF [45] is employed to reduce the speckle noise and preserve the edges in the approximate sub-band image. Each sub-band image, once the noise is removed, is reconstructed by wavelet reconstruction, and the exponential transform is performed to reverse the logarithmic transform. Finally, we obtain the despeckled image.

Speckle Reducing Anisotropic Diffusion
As mentioned above, the AD [18] performs poorly in terms of edge preservation in the presence of speckle noise. It removes the additive noise from the image, thereby causing a loss of detailed information. The SRAD method modifies the AD filter to improve the edge detection accuracy in speckled images. The instantaneous coefficient of variation (ICOV) is merged into the edge detector. The method for removing speckle noise using ICOV is described below.
The output image ( , ; ) is obtained by a PDE model of SRAD when an intensity image ( , ; 0) has finite power and no zero values over the image of the 2D coordinate grid (Equation (1)).
where is the initial noisy image, is time, and is the divergence operator. ∇ is the gradient operator, is the boundary of , ⃗ is the unit vector, and ( , ; ) is the output image. ( ) is the diffusion coefficient.
The diffusion coefficient ( ) plays a crucial role in SRAD by determining the diffusion scale. It encourages diffusion in the homogeneous regions and restriction near the edges of the image. The formula of the diffusion coefficient can be alternatively expressed as: or Here, the ICOV ( , ; ) can detect edges of the images and speckle noise. The ICOV exhibits high values at edge regions and low values in the homogeneous regions. It can be estimated using the following equation:

Speckle Reducing Anisotropic Diffusion
As mentioned above, the AD [18] performs poorly in terms of edge preservation in the presence of speckle noise. It removes the additive noise from the image, thereby causing a loss of detailed information. The SRAD method modifies the AD filter to improve the edge detection accuracy in speckled images. The instantaneous coefficient of variation (ICOV) is merged into the edge detector. The method for removing speckle noise using ICOV is described below.
The output image I(x, y; t) is obtained by a PDE model of SRAD when an intensity image I(x, y; 0) has finite power and no zero values over the image of the 2D coordinate grid Ω (Equation (1)).
I(x, y; 0) = I 0 (x, y), where I 0 is the initial noisy image, t is time, and div is the divergence operator. ∇ is the gradient operator, ∂Ω is the boundary of Ω, → n is the unit vector, and I(x, y; t) is the output image. c(q) is the diffusion coefficient.
The diffusion coefficient c(q) plays a crucial role in SRAD by determining the diffusion scale. It encourages diffusion in the homogeneous regions and restriction near the edges of the image. The formula of the diffusion coefficient can be alternatively expressed as: or where ∇ 2 represents the Laplace operator, f 0 is the coefficient of variation at the time, and T is the threshold of the diffusion coefficient (Equations (2) and (3)). The value of c(q) tends to zero when f 2 (x, y; t) − f 2 0 (t) is greater than T, as the diffusion stops. In the opposite case, the value of c(q) approaches 1 when f 2 (x, y; t) − f 2 0 (t) is less than T, thus the diffusion is applied as the filter in the homogeneous regions. The threshold value of the diffusion coefficient has an effect on the reduction of speckle noise and in the preservation of edge information. where Here, z(t) and var[z(t)] are the intensity mean and variance, respectively, over a homogeneous region at t. The f 0 (t) of an automatic determination can be estimated, as follows: where ρ is a constant and f 0 is the coefficient of variation in the observed image. The SRAD filtering method can directly process the data and preserve important information in the image without performing log-compression [39]. Therefore, the SRAD filtering technique can be used as a preprocessing filter.

Logarithmic Transformation
Equation (8) shows the model degraded by the speckle noise in the SAR images [46]. The resultant image is the product of the speckle noise and original image.
where R(x, y) is degraded image of the SAR image. O(x, y) is the original image, M(x, y) is the speckle noise, and A(x, y) is the additive noise. Since the additive noise affects the SAR images less than multiplicative noise, it is ignored, and Equation (9) is obtained.
If a model of the multiplicative noise represents the speckle noise, as in Equation (9), it is difficult to separate the original image and the noise component. When a logarithmic transform is applied to SAR images containing the multiplicative noise (speckle noise), the speckle noise appears in the form of additive noise, as follows: F(x, y) = L(x, y) + S(x, y) where F(x, y), L(x, y) and S(x, y) are the logarithms of R(x, y), O(x, y), and M(x, y), respectively. F(x, y) represents the characteristics of an AWGN with an average of 0 and a variance of σ 2 . In this Remote Sens. 2019, 11, 1184 7 of 27 study, we use a logarithmic transform to convert the multiplicative noise into AWGN and attempt to additionally remove the noise in the wavelet domain.

Discrete Wavelet Transform
The DWT is employed to remove noise in the various high-and low-frequency coefficients of SAR images. It analyzes multiresolution sub-band images by adjusting the scaling and translation parameters; hence, as the scaling parameter increases, extending the signal lowers the spatial resolution. The extended scaling parameter can obtain a sub-band image representing low-frequency coefficients. In the opposite case, a high-frequency sub-band image can be obtained. The translation parameter moves along the time axis. As this parameter value increases, it moves to the right. With these two parameters, the DWT can obtain an approximate sub-band image and detailed sub-band images.
For the 2D image, the basic idea of the DWT is described, as follows. One-level DWT transforms the SAR images with speckle noise into four sub-band images: the approximate sub-band image (LL 1 ) and three detailed sub-band images (vertical coefficients (LH 1 ), horizontal coefficients (HL 1 ), and diagonal coefficients (HH 1 )) ( Figure 2b). Figure 2c shows the results of the two-level wavelet decomposition. The two-level DWT decomposes the LL 1 sub-band image that was obtained from the one-level wavelet decomposition in the same manner to obtain four sub-band images (LL 2 , LH 2 , HL 2 , and HH 2 ). The approximate sub-band image (LL 2 ) contains the low-frequency coefficients, and detailed sub-band images (LH 1 , HL 1 , HH 1 LH 2 , HL 2 , and HH 2 ) depict the high-frequency coefficients. The detailed sub-band images present most information regarding the image, including the noise and edge information. The approximate sub-band image includes important information about the SAR images, such as the texture.

images.
For the 2D image, the basic idea of the DWT is described, as follows. One-level DWT transforms the SAR images with speckle noise into four sub-band images: the approximate sub-band image ( ) and three detailed sub-band images (vertical coefficients ( ), horizontal coefficients ( ), and diagonal coefficients ( )) ( Figure 2b). Figure 2c shows the results of the two-level wavelet decomposition. The two-level DWT decomposes the sub-band image that was obtained from the one-level wavelet decomposition in the same manner to obtain four sub-band images ( , , , and ). The approximate sub-band image ( ) contains the low-frequency coefficients, and detailed sub-band images ( , , , , and ) depict the high-frequency coefficients. The detailed sub-band images present most information regarding the image, including the noise and edge information. The approximate sub-band image includes important information about the SAR images, such as the texture.

Soft Threshold
Various threshold methods exist ( [32][33][34]). The most commonly used wavelet functions are the soft and hard threshold. These threshold methods are used to reduce the speckle noise in the SAR images. Although both thresholds set to zero when the coefficients are smaller than the threshold, these thresholds have the main difference. The former function suppresses the coefficients that are larger than the threshold, while the latter function leaves them unchanged [43].
The hard threshold removes the coefficients that are below the threshold value , as determined by the noise variance. The hard threshold is depicted, as follows: where is the wavelet coefficient and is the threshold value. represents the wavelet coefficient after the hard threshold is applied. The hard threshold is known to have discontinuity in the noise-free image, since the wavelet coefficient at the threshold is suddenly zeroed. In the hard threshold method, the wavelet coefficients that do not exceed the given threshold value are zeroed. The other wavelet coefficients remain unchanged. Therefore, the hard threshold yields artifacts in the despeckled image [47]. The soft threshold applies the signum function in its model to overcome these issues of the hard threshold (Equation (13)).

Soft Threshold
Various threshold methods exist ( [32][33][34]). The most commonly used wavelet functions are the soft and hard threshold. These threshold methods are used to reduce the speckle noise in the SAR images. Although both thresholds set to zero when the coefficients are smaller than the threshold, these thresholds have the main difference. The former function suppresses the coefficients that are larger than the threshold, while the latter function leaves them unchanged [43].
The hard threshold removes the coefficients that are below the threshold value T, as determined by the noise variance. The hard threshold is depicted, as follows: where w is the wavelet coefficient and T is the threshold value. W hard represents the wavelet coefficient after the hard threshold is applied. The hard threshold is known to have discontinuity in the noise-free image, since the wavelet coefficient at the threshold is suddenly zeroed. In the hard threshold method, the wavelet coefficients that do not exceed the given threshold value are zeroed. The other wavelet coefficients remain unchanged. Therefore, the hard threshold yields artifacts in the despeckled image [47]. The soft threshold applies the signum function in its model to overcome these issues of the hard threshold (Equation (13)).
Here, sgn depicts the signum function. W so f t is the wavelet coefficient after the shrinkage of the soft threshold.
In the soft threshold method, the wavelet coefficients are zero if they are below the threshold. The wavelet coefficients above the threshold are shrunk by the threshold value. Hence, the soft threshold provides smooth results without artifacts. When compared to the hard threshold, the soft threshold generally exhibits excellent preservation of detail at the expense of computational complexity [43]. We apply the soft threshold to these sub-band images, since the horizontal and vertical sub-band images have a similar energy [48].

Guided Filter
Zhang and Gunturk [47] mentioned that noise may exist in the approximate sub-band image and detailed sub-band images in the wavelet domain. Gao et al. [43] have divided the 2D SAR images into low, medium, and high-frequency sub-band images using a 2D fast Fourier transform (FFT). The authors applied the approximate sub-band image through low-pass filtering to reduce the noise in the approximate sub-band image. This method shows low speckle noise removal and edge preservation ability. In [47,49], the authors applied the BF [13] to the approximate sub-band image to suppress speckle noise. BF is capable of preserving the edges and shows an excellent noise removal performance since it is difficult to distinguish between original signal and noise in the approximate sub-band image; however, it exhibits gradient distortion and high complexity [47]. We applied the GF [45] in the approximate sub-band image to overcome these problems. The process of removing the noise using the GF is as follows. GF models the output image q i for the guidance image I k of the window ω k region with the center pixel k in the image, as follows: where a k and b k are linear coefficients estimated form the window ω k . Equation (15) removes unwanted texture or noise to determine the linear coefficients.
Here, p i and n i denote the input image and noise component, respectively. The linear coefficients are obtained by Equation (16) to minimize the difference between the input image p i and the output image where ε is a normalization parameter that serves to prevent a k from becoming infinitely large. The minimization method of the liner coefficient in Equation (16) is as follows: Remote Sens. 2019, 11, 1184 Here, µ k and σ 2 k are the mean and variance of the guidance image in the window ω k . |ω| represents the number of pixels in the mask ω k , and p k = 1 |ω| p i . As the window size ω k and ε adjust, the noise is removed and the edge areas are preserved. Therefore, these parameters are adjusted according to the characteristics of the approximate sub-band image to remove the additive noise and to preserve edge information.

A New Edge-Aware Weighting
The horizontal and vertical sub-band images of DWT have the same energy, while the diagonal sub-band image has lower energy at the same scale [48]. We propose new edge-aware weighting for effectively detecting and preserving weak edge information (Equation (19)). The gradient operator is an effective operator for detecting sharp edge regions in the image and protecting against unnecessary blurring phenomena around the edges. However, the gradient operator produces wide and blurred edges when the edge regions are not sharp. In contrast, since a Laplacian operator uses a second-order derivative operator that has a zero crossing level, it can detect the weak information of the edges while using the zero crossing level. Therefore, we can detect weak edges in the diagonal sub-band image in the wavelet domain.
Here, ∆ and ∇ are the Laplacian and Gradient operators, respectively. The value of h is larger than 1 when h is located at weak edges; however, it is smaller than 1 if h is in the homogeneous regions.

The Proposed Filter
The new edge-aware weighting h of Equation (19) is incorporated into the cost function E(a k , b k ) of Equation (20). As mentioned above, an IGF obtains a solution that minimizes the input image p i and the output image q i , while maintaining the linear model of Equation (14). A cost function with applied new edge-ware weighting is expressed. as follows: The optimal values of a k and b k are computed as: The final value ofq i is given as follows:q Here, a k and b k are the mean values of a k and b k within the window, respectively.

Evaluation Metrics
We used peak signal-to-noise (PSNR), structural similarity (SSIM), and equivalent number of looks (ENL) to compare the performance of speckle noise reduction in the SAR images [41]. The PSNR depicts the maximum signal-to-noise ratio. The PSNR is an objective measurement method that is used to evaluate image quality, and it is defined as follows: where the mean square error (MSE) is given by: where M and N represent the number of pixels in the vertical and horizontal directions of the image, respectively. Y(x, y) is the pixel value at the position of the original image (x, y) and Z(x, y) is the pixel value at the coordinates of (x, y) in the filtered image. The filtered image Z(x, y) has a smaller MSE as the image approaches the original image Y(x, y). Larger PSNR values imply better noise reduction performances. The SSIM is an index that indicates the similarity between the original image Y(x, y) and the filtered image Z(x, y). The SSIM is given, as follows: Here, µ x and µ x are the mean values of x and y, respectively. σ 2 x and σ 2 y present the variance of x and y, respectively. cov xy is the covariance of x and y. c 1 and c 2 are the two variables used to stabilize the division that can occur with a weak denominator. When the value of SSIM is closer to 1, there is no difference between the original and the filtered image. The equivalent number of look (ENL) is used to evaluate the speckle noise reduction performance of the homogeneous regions in the image. It is a standard metric in the absence of reference images that is widely used to evaluate despeckling performance. The ENL is defined as: where µ z and σ z are the estimated mean and standard deviation of the filtered SAR image. Larger ENL values indicate excellent speckle noise removal ability.

Experiments on Standard Images
In this study, we selected 8-bit gray standard images (Airplane, Baboon, Barbara, Boat, Cameraman, Fruits, Hill, House, Lena, Man, Monarch, Napoli, Peppers, and Zelda) with 256 × 256, 512 × 512, and 748 × 512 pixels in order to evaluate the performance of speckle noise removal and edge preservation ( Figure 3). Speckle noise (σ = 0.04) was added to each image. The existing methods (NLM [14], Guided [45], Frost [50], Lee [51], Bitonic [52], weighted-least-squares (WLS) [53], non-local low-rank (NLLR) [54], anisotropic diffusion filter with memory based on speckle statistics (ADMSS) [55], SRAD [39], SRAD-Guided [56], SAR-BM3D [24]) and the proposed algorithm were used to compare the speckle noise suppression performance. Tables 1 and 2 present the simulation conditions for the standard images. The optimal parameters of the SRAD-guided method are the same as those of the proposed algorithm ( Table 2). The MATLAB 2018b software was used in a computer environment [Intel (R) Core (TM) i5-8500 CPU @ 3.0 GHz with 16 GB RAM].  Tables 3 and 4 exhibit the PSNR (dB) and SSIM values of the despeckled standard images while using the existing filtering methods and the proposed algorithm. The best and second-best values among all of the despeckling methods are denoted in red and blue color, respectively. Table 3 (Table 3).
In Table 4, the performances of the existing filtering methods and the proposed algorithm are compared in terms of SSIM. As aforementioned, the SRAD filtering technique provides the best edge preservation performance in two images (Baboon = 0.65 and Napoli = 0.77). In the Airplane, Barbara, Cameraman, Fruits, House, Lena, Monarch, and Zelda, SAR-BM3D provides the best edge preservation performance while the proposed algorithm exhibits the second-best performance (Airplane, Barbara, House, Lena, Monarch, and Zelda). The SAR-BM3D and the proposed method show the same edge preservation performance in Cameraman, Fruits, and Hill. The proposed algorithm shows the highest edge preservation performance in the following images: Boat = 0.73; Man = 0.77; and Peppers = 0.84. We confirm that the results obtained by the proposed algorithm, which is displayed in Tables 3 and  4, demonstrate an excellent performance and rank at least second among the existing filtering techniques.      (Table 3). In Table 4, the performances of the existing filtering methods and the proposed algorithm are compared in terms of SSIM. As aforementioned, the SRAD filtering technique provides the best edge preservation performance in two images (Baboon = 0.65 and Napoli = 0.77). In the Airplane, Barbara, Cameraman, Fruits, House, Lena, Monarch, and Zelda, SAR-BM3D provides the best edge preservation performance while the proposed algorithm exhibits the second-best performance (Airplane, Barbara, House, Lena, Monarch, and Zelda). The SAR-BM3D and the proposed method show the same edge preservation performance in Cameraman, Fruits, and Hill. The proposed algorithm shows the highest edge preservation performance in the following images: Boat = 0.73; Man = 0.77; and Peppers = 0.84. We confirm that the results obtained by the proposed algorithm, which is displayed in Tables 3 and 4, demonstrate an excellent performance and rank at least second among the existing filtering techniques.
From Tables 3 and 4, in the standard images, we analyze the performances of each method to evaluate the soft threshold, IGF, and GF in the wavelet domain (Table 5 Figure 4b-e,g,h. Figure 4e, which is compared to Figure 4b-d,g,h exhibits reduced speckle noise but not quite. As shown in Figure 4f,i,j, some edges are lost in the edge regions, whereas the homogeneous regions remain with the speckle noise. The speckle noise reduction and edge preservation performance is noticeable in SAR-BM3D and the proposed algorithm (Figure 4k,l). The SAR-BM3D and the proposed algorithm exhibit similar performance with respect to the edge preservation, and show the strongest speckle removal ability in the homogeneous regions. However, the proposed algorithm has the best speckle noise removal performance in the homogeneous areas, as SAR-BM3D exhibits artifacts in these regions.   homogeneous regions remain with the speckle noise. The speckle noise reduction and edge preservation performance is noticeable in SAR-BM3D and the proposed algorithm (Figure 4k,l). The SAR-BM3D and the proposed algorithm exhibit similar performance with respect to the edge preservation, and show the strongest speckle removal ability in the homogeneous regions. However, the proposed algorithm has the best speckle noise removal performance in the homogeneous areas, as SAR-BM3D exhibits artifacts in these regions. The GF, Frost filter, Lee filter, Bitonic filter, NLLR method, ADMSS method, and SRAD filter do not perform well for speckle noise removal in the homogeneous regions, and the speckle noise persists in the filtering result images (Figure 5b-e,g-i). The WLS filter and the SRAD-Guided algorithm perform better than the above filtering methods with regard to speckle noise reduction (Figure 5f,j). However, the WLS filter and the SRAD-Guided algorithm exhibit a blurring phenomenon in the image. The filtering result image that was obtained by the proposed algorithm had similar visual quality as SAR-BM3D. The SAR-BM3D achieves excellent edge preservation The GF, Frost filter, Lee filter, Bitonic filter, NLLR method, ADMSS method, and SRAD filter do not perform well for speckle noise removal in the homogeneous regions, and the speckle noise persists in the filtering result images (Figure 5b-e,g-i). The WLS filter and the SRAD-Guided algorithm perform better than the above filtering methods with regard to speckle noise reduction (Figure 5f,j). However, the WLS filter and the SRAD-Guided algorithm exhibit a blurring phenomenon in the image. The filtering result image that was obtained by the proposed algorithm had similar visual quality as SAR-BM3D. The SAR-BM3D achieves excellent edge preservation performance, however it exhibits artifacts in the homogeneous regions (Figure 5k). In Figure 5l, the proposed algorithm exhibits strong speckle noise removal ability while maintaining the edges. The qualitative result of Figure 5 represents the same result as in Figure 6. performance, however it exhibits artifacts in the homogeneous regions (Figure 5k). In Figure 5l, the proposed algorithm exhibits strong speckle noise removal ability while maintaining the edges. The qualitative result of Figure 5 represents the same result as in Figure 6.

Experiments on Real SAR Images
In this section, two real SAR images showing different scenes are used for the evaluation of the conventional filtering methods and the proposed algorithm on real SAR images (Figure 7). The real SAR image1 shows a scene from a photojournal [256 × 256, 8 bit, X-band] [57]. The real SAR image2 depicts a rural scene in Bedfordshire [512 × 512, 8 bit, X-band] [2,58]. As mentioned in Section 3.1, the optimal parameters of the conventional methods maintained the same values as those in the standard image (Table 1). Table 6 shows the optimal parameters of the proposed algorithm.

NLM Guided Frost
Lee Bitonic WLS NLLR ADMSS SRAD SRAD-Guided   Figure 8 shows the results of the simulated SAR images. Figure 8 shows that some filters, such as Guided, Frost, Lee, Bitonic, NLLR, and SRAD, do not exhibit strong speckle noise removal ability (Figure 8b-e,g,i). Tables 6 and 7 illustrate that the WLS filter represents the best ENL value; however, it exhibits a blurring phenomenon in the image (Figure 8f). The SAR-guided method when compared with the SAR-BM3D and the proposed methods show inferior speckle noise removal and edge preservation performances. The SAR-BM3D method has an excellent speckle noise reduction and edge preservation abilities; however, artifacts in the homogeneous regions are observable (Figure 8k). The proposed algorithm compared with the SAR-BM3D method has a strong speckle noise removal ability; however, it does exhibit limited low edge preservation performance in some edge areas (Figure 8l). Table 9. ENL results of each method in the proposed algorithm for each real SAR images.

SAR image1
Time step = 0.01 Exponential decay rate  Tables 7 and 8 depict that the WLS filter outperforms all of the filtering methods in terms of the ENL, while the proposed method ranks second in the speckle noise suppression performance. From Tables 7 and 8, the data from Table 9 are analyzed for evaluating the performance of each method in the proposed method for the real SAR images. In the real SAR image1 and image2, the soft threshold, the IGF, and the GF methods, as compared to the SRAD filtering result image, shows enhanced noise suppression ability. In the SAR image1, the soft threshold, the IGF, and the GF techniques have enhanced noise removal ability (ROI-1 (ENL = 114.62 (+0.52)) and -2 (ENL = 81.59 (+0.58), ROI-1 (ENL = 118.84 (+4.74)) and -2 (ENL = 84.09 (+2.29), ROI-1 (ENL = 136.52 (+21.90)) and -2 (ENL = 97.05 (+16.04)). The noise reduction performance of the soft threshold (ROI-1 (ENL = 147.76 (+0.85)) and ROI-2 (ENL = 118.50 (+1.33)), IGF (ROI-1 (ENL = 148.93 (+2.02)) and ROI-2 (ENL = 119.27 (+2.10)), and the GF (ROI-1 (ENL = 203.02 (+56.11)) and ROI-2 (ENL = 157.24 (+40.07)) methods in the SAR image2 exhibits improved ability (Table 9).  Figure 8 shows the results of the simulated SAR images. Figure 8 shows that some filters, such as Guided, Frost, Lee, Bitonic, NLLR, and SRAD, do not exhibit strong speckle noise removal ability (Figure 8b-e,g,i). Tables 6 and 7 illustrate that the WLS filter represents the best ENL value; however, it exhibits a blurring phenomenon in the image (Figure 8f). The SAR-guided method when compared with the SAR-BM3D and the proposed methods show inferior speckle noise removal and edge preservation performances. The SAR-BM3D method has an excellent speckle noise reduction and edge preservation abilities; however, artifacts in the homogeneous regions are observable (Figure 8k). The proposed algorithm compared with the SAR-BM3D method has a strong speckle noise removal ability; however, it does exhibit limited low edge preservation performance in some edge areas (Figure 8l).

Computational Complexity
Tables 10 and 11 in this section present the time costs of the existing methods and the proposed algorithm on 14 standard images and two real SAR images. The experimental environments are those that are referred to in Section 3.1. Table 10 shows that the proposed method is much faster than Lee, NLLR, ADMSS, and SAR-BM3D. The running time of the proposed algorithm with 14 standard images is approximately 5.06 s on average. Table 11 denotes that the proposed algorithm has less computational complexity than the NLLR, ADMSS, and SAR-BM3D methods. The average execution time in the real SAR images is approximately 4.50 s.  Tables 12 and 13 present the computing time of each step of the proposed method for the standard and real SAR images. The SRAD method has a high computing time, because the SRAD filter uses an iterative method to remove the speckle noise (standard images = 4.76 s (91.56%); SAR images = 4.12 s (91.56%)). In finding an optimal threshold value for classifying an original signal and a noise signal, the computing time of the soft threshold method is low, because, when compared to the computing time of the SRAD filter, it takes approximately 0.11 s (standard images) and 0.10 s (real SAR images). The IGF and the GF work very fast, because they only take approximately 6% (standard images) and 9% (real SAR images), respectively, of the total time. The main reason for this low time consumption is the use of a box filter in the GF [45]. The box filter can efficiently use a computational complexity in O(N) time by employing the integral image method [59]. The IGF is a method developed based on the GF; hence, it has a low computation time.

Discussion
This study used the statistical characteristics of speckle noise and the DWT to remove the speckle noise in SAR images. The proposed algorithm applies the SRAD filter, soft threshold, GF, and the IGF.
The speckle noise in SAR images is modelled as multiplicative noise. However, most of the filtering methods were developed for AWGN, as additive noise in imaging and sensing systems is most common. Therefore, conventional filtering methods are unable to remove speckle noise. The SRAD filtering method, in contrast, uses the ICOV to directly apply a diffusion process in all areas, except for the edge regions, by separating the edge areas and noise from SAR images with speckle noise. The SRAD filter exhibits excellent speckle noise removal and edge preservation. From the experimental results, the SRAD filtering scheme demonstrates the best speckle noise suppression and edge preservation performance among the single filtering methods. Based on this finding, the SRAD filtering technique was used as a preprocessing filter. In order to further remove the speckle noise remaining in the SRAD filtering result image, the logarithmic transform is used to convert the multiplicative noise (speckle noise) to additive noise. The SRAD filtering result image with the additive noise is decomposed into one low-frequency sub-band image (LL 2 ) and six high-frequency sub-band images (LH 1 , HL 1 , HH 1 LH 2 , HL 2 , and HH 2 ) while using a two-level DWT. In the wavelet domain, horizontal (HL 1 , HL 2 ) and vertical (LH 1 , LH 2 ) sub-band images have the same energy, while, in comparison, the diagonal (HH 1 , HH 2 ) sub-band images have lower energies on the same scale. The former sub-band images were applied to soft threshold to remove the additive noise. The IGF method with new edge-aware weighting based on the Gradient and the Laplacian operators is applied to the latter sub-band images in order to remove the additive noise and preserve low edge information. We applied the guide filter to remove the additive noise that is present in the approximate (LL 2 ) sub-band image. In some of the standard images, the proposed algorithm does not represent the best speckle noise removal and edge preservation performance in terms of PSNR and SSIM (Tables 3 and 4). When the soft threshold, IGF, and GF are employed in the wavelet domain after the SRAD filter application in the proposed algorithm, when compared with the SRAD filter, the proposed algorithm shows enhanced speckle noise and edge preservation performance in the Airplane (PSNR = 0.48 dB; SSIM = 0.  (Table 5). Among these methods, the soft threshold shows low speckle noise and edge preservation abilities in most standard images (average: PSNR = −0.19 dB; SSIM = −0.01). The IGF method exhibits a limited improvement in the speckle noise suppression performance (PSNR = +0.04 dB (avg.)). The GF technique contributes most of the speckle noise removal and edge preservation performances among each method in the wavelet domain (PSNR = +0.24 dB, SSIM = +0.02 (average)). When the same method that is applied in the standard images is applied to real SAR images, the proposed algorithm shows an improved speckle noise reduction performance in the ROI of the two real SAR images (SAR image1 (ROI-1: ENL = 27.68; ROI-2: ENL = 18.91) and SAR image2 (ROI-1: ENL = 58.98; ROI-2: ENL = 43.50)). From the ENL results in the real SAR images, we analyzed the contributions of the soft threshold, IGF, and GF to the noise suppression performance ( Table 9). The soft threshold shows improved speckle noise rejection performance for ROI-1 (ENL = +0.69) and -2 (ENL = +0.85) in SAR image1 and 2. The IGF method exhibits enhanced speckle noise removal ability over the soft threshold (ROI-1 (ENL = +3.38) and ROI-2 (ENL = +2.20)). The GF technique was confirmed to have improved speckle noise reduction performance in terms of ENL = +39.27 at ROI-1 and ENL = +68.56 at ROI-2. The GF method has been found to have the greatest contribution to speckle noise removal in the wavelet domain.
The proposed algorithm shows the performance within the second-best value in all of the standard images and real SAR images in Table 3, Table 4, Table 7, and Table 8. The proposed method performs better than any nonlinear filter and hybrid method in the different images that have characteristics that include low-frequency components. Although the SAR-BM3D method, when compared to the conventional algorithms, exhibits excellent speckle noise reduction and edge preservation abilities, it employs noise reduction based on the NLM filter. The computational complexity of the SAR-BM3D algorithm is high, since the NLM filter needs to search regions. Therefore, it is difficult to obtain real time observations using the SAR-BM3D. The proposed algorithm exhibits a 10-30 times lower computational complexity when compared with SAR-BM3D (Table 10). In the proposed algorithm, the SRAD filter has a high computational complexity, because it uses an iterative method to remove the speckle noise (standard images = 4.76 s (91.56%); SAR images = 4.12 s (91.56%)); however, the time that is consumed by the soft threshold in finding an optimal value to classify an original signal and a noise signal is low (0.11 s (standard images) and 0.10 s (SAR images)). Moreover, the IGF and the GF have a low computational complexity (standard image ≈ 6%; real SAR images ≈ 9%), because the box filter in the IGF and the GF can efficiently employ computing time (O(N)). Moreover, the proposed method exhibits speckle noise suppression and edge preservation performance similarly to SAR-BM3D (Tables 3 and 4). In the real SAR images, the WLS filter exhibits the best speckle noise removal performance in terms of ENL (Tables 7 and 8). However, the resulting WLS image exhibits a blurring phenomenon ( Figure 8f). As mentioned in [60], high ENL values do not always imply the best speckle noise suppression performance; further, blurring is observed in the image. Tables 7 and 8 indicate that SAR-BM3D and the proposed method provide satisfactory speckle noise removal (Figure 8k,l). As mentioned above, the proposed algorithm has low computational complexity, which is about 8-13 times that of the SAR-BM3D method ( Table 11). The experimental results demonstrate that the proposed method exhibits excellent speckle noise reduction, while preserving edge information and maintaining low computational complexity.

Conclusions
In summary, we proposed a novel algorithm that is based on statistical characteristics of speckle noise and the DWT to remove speckle noise from the SAR images. For this purpose, the SRAD filtering method, which can be directly applied to the SAR image, is used as a preprocessing filter. The logarithmic transform is employed to convert the multiplicative noise in the resulting SRAD image to additive noise. In order to further remove the additive noise from the SRAD filter result image, the two-level DWT converts the SRAD filter result image into one approximate sub-band image and six detailed sub-band images. The IGF is applied to diagonal sub-band images, which have lower energy within the same scale, to remove the additive noise and preserve edge information. Meanwhile, the horizontal and vertical sub-band images, which exhibit higher energy than the diagonal sub-band images, are treated with the soft threshold. The GF is applied to remove the additive noise that is present in the approximate sub-band image. The experiments in this study used both standard images and real SAR images. The experimental results demonstrate that the proposed method is able to obtain excellent speckle noise removal and edge preservation at low computational complexity when compared with the state-of-the art methods. In future research, we aim to study a novel filtering technique that can remove noise while preserving edge information in the approximate sub-band image.
Author Contributions: H.C. designed the methodology, implemented the simulation, and wrote this paper. J.J. wrote and edited this paper.
Funding: This research was not funded.