A No-Reference Edge-Preservation Assessment Index for SAR Image Filters under a Bayesian Framework Based on the Ratio Gradient

Abstract

Denoising is an essential preprocessing step for most applications using synthetic aperture radar (SAR) images at different processing levels. Besides suppressing the noise, a good filter should also effectively preserve the image edge information. To quantitatively assess the edge-preservation performance of SAR filters, a number of indices have been investigated in the literature; however, most of them do not fully employ the statistical traits of the SAR image. In this paper, we review some of the typical edge-preservation assessment indices. A new referenceless index is then proposed. The ratio gradient is utilized to characterize the difference between two non-overlapping neighborhoods on opposite sides of each pixel in both the speckled and despeckled images. Based on these gradients and the statistical traits of the speckle, the proposed indicator is derived under a Bayesian framework. A series of experiments conducted with both simulated and real SAR datasets reveal that the proposed index shows good performances, in both robustness and consistency. For reproducibility, the source codes of the index and the testing datasets are provided.

1. Introduction

It is well known that, as with most of the active imaging systems, speckle noise is a serious image quality degradation factor for SAR data, which hinders the subsequent applications [1]. As reviewed by Foucher et al. [2] and Ma et al. [3], to deal with the despeckling issue many algorithms have been studied in the last few decades. The wavelet based filters [4,5], variational methods [6,7,8], nonlocal means based filters [9,10], and deep learning based filters [11,12,13] are hot topics in the area of SAR image despeckling. In addition to a good performance in reducing the speckle in the homogeneous areas, a good SAR filter should also effectively preserve the image information. Normally, for SAR data, the image information includes three main parts [14]: (1) the image edge information; (2) the polarimetric or radiometric information; and (3) the high returns from strong scatterers. It is necessary to objectively evaluate the image information preservation capability when developing a filter. To this end, a number of quantitative assessment indices have been developed in an attempt to reveal the performance of a filter in one or more of the three aforementioned aspects.

Retaining image edges or textures when applying an SAR filter is important for many applications. In this paper, we focus on the topic of evaluating the edge-preservation capability (EPC) of single-polarization SAR filters. Normally, when we talk about the “single-polarization SAR filter” we often refer to a filter that only handles the despeckling problem of SAR amplitude or intensity (the square of amplitude) data in the cases that the SAR images with only one polarization channel (HH, VV, HV or VH) are available. This also means that, for single look complex (SLC) single-polarization SAR data, the noise in the phase part is not concerned and suppressed by these filters, which, instead, is concerned by an interferometric SAR filter [15,16]. Generally speaking, SAR image EPC assessment indicators can be divided into two types: (1) full-reference indices; and (2) no-reference indices. Compared with the no-reference ones, the full-reference indices are much easier to obtain. The full-reference approaches assume that, if the noise-free image is known, the evaluation can be executed by directly comparing the structural information of the reference with that of the despeckled image. However, the problem is that no noise-free real-world SAR image exists. One solution is to average multitemporal SAR images of a same area, which, however, needs a large number of images and has to ensure that there are no major changes between different time sequences. A more common solution is to resort to a clean digital image and exert the speckle noise to generate a simulated SAR image. Then, some of the commonly used indicators in the digital image processing area, such as the peak signal-to-noise ratio (PSNR) [17] and the structural similarity (SSIM) index [18], can be directly applied to assess SAR filters. However, the shortcoming is apparent: the traits of the optical image simulated data are different from real-world SAR data in many aspects, making the evaluation results unreliable.

Recently, differing from the traditional full-reference approaches, Di Martino et al. [14] proposed a framework for the simulation of speckled and clean SAR images relevant to canonical scenes, based on a SAR raw signal simulator. The simulator employs complete physical models for the radar sensed targets by considering the scattering traits and the radar operational mode. As a result, the generated data are more in line with real SAR data. However, it needs to be pointed out that this simulation framework can only generate images with simple scenes. The real-world SAR images often have complicated scenes and high data dynamics. Therefore, the full-reference indices developed in the aforementioned simulation framework may lose their significance when evaluating the despeckling performance on complicated real-world SAR data.

All in all, a robust no-reference index needs to be developed to more objectively assess the EPC of SAR filters, which is the main goal of this paper. In this study, we developed a ratio gradient based EPC assessment index derived under a Bayesian framework. Specifically, the ratio gradient is first utilized to characterize the difference (i.e., the edge information) between two non-overlapping neighborhoods on opposite sides of each pixel in both the speckled and despeckled images. Then, based on the gradient values and the statistical traits of the speckle, the indicator is derived by employing Bayesian theory.

The rest of this paper is organized as follows. A brief review of the existing SAR EPC evaluation indices is provided and the proposal is introduced in Section 2. Then, in Section 3, the quantitative evaluation results obtained on several datasets are shown to reveal the good performance of the proposed index. Finally, our conclusions are drawn in Section 4.

2. Method

2.1. Review of the Classical SAR EPC Assessment Indices

In this section, we review some of the classical SAR EPC assessment indices, which are listed in

Table 1. A list of some classical EDGE-preservation indices.

	Need Ratio Image	Need Reference	Need Edge Detector	Range of Values
FOM	NO	YES	YES	$(0,1]$; 1 is best
EC	NO	YES	YES	$[0,1]$; 1 is best
EPD-ROA	NO	NO	NO	$(0,1]$; 1 is best
DEI	NO	NO	NO	$(0,1]$; 0 is best
DSL	YES	YES	YES	$[0,1]$; 0 is best
${\mathbf{\beta }}_{{r}{a}{t}{i}{o}}$	YES	NO	YES	$[0,1]$; 0 is best

. We categorize the indices into two classes, based on the type of image the indices operate on, namely, despeckled image based ones and ratio image (the point-to-point ratio between the speckled and the filtered image) based ones.

2.1.1. Despeckled Image Based Indices

The main idea of many despeckled image based indices is to compare the edges detected in the despeckled image with those in the original (clean or speckled) image, to evaluate the loss of edges. Two typical indices are the figure of merit (FOM) [19] and the edge correlation (EC) [20].

(1) FOM: The FOM [19] is formulated as:

$$\mathrm{FOM}=\frac{1}{\max (n_d,n_r)}{\displaystyle \sum _{i=1}^{n_d}\frac{1}{1+\gamma l_i^2}}$$

(1)

where $n_d$ and $n_r$ are, respectively, the number of edge pixels detected in the despeckled image and the reference; $l_i$ denotes the Euclidean distance between the ith edge pixel in the despeckled image and its nearest edge pixel in the reference; and $\mathsf{\gamma }$ is a constant value, which is usually fixed as 1/9 in practice. A higher FOM value indicates a better edge-preservation result.

(2) EC: the EC is also known as the $\beta$ index [20]. It evaluates a filter by inspecting the gradient correlations between the noise-free image and the filtered image, which is defined as:

$$\beta =\frac{{\Psi }({\Delta }u-\overline{{\Delta }u},{\Delta }\widehat{u}-\overline{{\Delta }\widehat{u}})}{\sqrt{{\Psi }({\Delta }u-\overline{{\Delta }u},{\Delta }u-\overline{{\Delta }u})\cdot {\Psi }({\Delta }\widehat{u}-\overline{{\Delta }\widehat{u}},{\Delta }\widehat{u}-\overline{{\Delta }\widehat{u}})}}$$

(2)

with the correlation function

$${\Psi }(A,B)={\displaystyle \sum _{i=1}^{N}A(i)\cdot }B(i)$$

(3)

where $\Delta$ denotes the gradient image obtained by the Canny detector [21]; N is the total pixel number of the image; u and $\widehat{u}$ are, respectively, the reference image and the filtered image; and $\overline{\Delta \mathit{u}}$ and $\overline{\Delta \hat{u}}$ are the average values of images $\Delta \mathit{u}$ and $\Delta \hat{u}$. The EC index ranges between 0 and 1, with unity for ideal edge preservation.

A key point of the indices, such as FOM and EC, is that an edge detector needs to be deployed. That is to say, the evaluation accuracy is highly dependent on the effectiveness of the edge detector. However, in many cases, distinguishing the fine edges from the noise is a challenging task. To cope with this issue, indices that do not require an edge detector have been proposed. Two representatives are the edge-preservation degree based on the ratio of average (EPD-ROA) [22] and the despeckling evaluation index (DEI) [23].

(3) EPD-ROA: The EPD-ROA [22] is one of the most widely used EPC assessment indices. It is a refined version of the edge-preservation index introduced in [24] that considers the multiplicative trait of the speckle, which is defined as:

$$\mathrm{EPD-ROA}={\displaystyle \sum _{i=1}^N|{\widehat{u}}_1(i)/{\widehat{u}}_2(i)|/|I_1(i)/I_2(i)|}$$

(4)

where I₁(i) and I₂(i) are two adjacent pixel values of the speckled image along a certain direction. The better the filter retains the edges, the higher the EPD-ROA value is.

(4) DEI: The DEI [23] is defined as the ratio between the standard deviation over a small neighborhood and that over a larger neighborhood in the filtered image, which is defined as:

$$\mathrm{DEI}=\frac{1}{N}{\displaystyle \sum _{i,j}\frac{\underset{|p-i|<s,|q-j|<s}{{\displaystyle \min }}(\mathrm{std}({\mathrm{W}}_{p,q}^m))}{\mathrm{std}({\mathrm{W}}_{i,j}^s)}},m<s$$

(5)

where ${\mathrm{W}}_{\mathit{p},\mathit{q}}^{\mathit{m}}$denotes the window centered at pixel (p, q) roaming in a neighborhood no greater than $\mathit{s}\times \mathit{s}$, which has the size of $\mathit{m}\times \mathit{m}$ pixels. The assumption is that, for a homogeneous image patch without structures, the numerator and the denominator will be very close. In theory, the DEI lies between 0 and 1, and a high value indicates a serious over-smoothing problem.

2.1.2. Ratio Image Based Indices

The speckle model in SAR images can be described as:

$$I=u\cdot n$$

(6)

where n is the noise component. For a filter that perfectly retains the edges, the ratio image between the speckled and filtered image should be the noise component. Hence, it has the appearance of pure speckle without any profiles of edges. This is the main idea behind the ratio image based indices. The despeckling structure loss (DSL) [25] and the ${\beta }_{\mathrm{ratio}}$[26] are two representatives.

(1) DSL: By considering the local correlation (LC) between the noise-free image and the ratio image, the DSL [25] is defined as:

$$\mathrm{DSL}=\frac{\mathrm{LC}(u,r)-\mathrm{LC}(u,r_{\mathrm{best}})}{\mathrm{LC}(u,r_{\mathrm{worst}})-\mathrm{LC}(u,r_{\mathrm{best}})}$$

(7)

with

$$\mathrm{LC}(u,r)=\frac{mean\{(u_i-{\overline{u}}_i\cdot (r_i-{\overline{r}}_i)\text{}}}{\sqrt{mean\{{(u_i-{\overline{u}}_i}^2\text{}}\cdot mean\{{(r_i-{\overline{r}}_i)}^2\text{}}}}$$

(8)

where i denotes any edge pixel detected in the reference image u; r is the ratio image; ${\mathit{r}}_{\mathrm{best}}$ is the best ratio image, i.e., the ratio image of an ideal filter; and ${\mathit{r}}_{\mathrm{worst}}$ is the worst ratio image obtained by multiplying the speckled data with a constant value, indicating that all the structures are filtered out. The range of the DSL value is between 0 and 1. Clearly, for the worst case, the value will be close to 1.

(2)${\beta }_{\mathit{ratio}}$: In [26], Gomez et al. proposed to refine the $\beta$ index [20] by replacing the reference image with the speckled image, the despeckled image with the ratio image, and the Canny edge detector by the ratio edge detector [27]:

$${\beta }_{\mathrm{ratio}}=\frac{\mathsf{\Psi }(\mathsf{\Delta }I-\overline{\mathsf{\Delta }I},\mathsf{\Delta }r-\overline{\mathsf{\Delta }r})}{\sqrt{\mathsf{\Psi }(\mathsf{\Delta }I-\overline{\mathsf{\Delta }I},\mathsf{\Delta }I-\overline{\mathsf{\Delta }I})\cdot \mathsf{\Psi }(\mathsf{\Delta }r-\overline{\mathsf{\Delta }r},\mathsf{\Delta }r-\overline{\mathsf{\Delta }r})}}$$

(9)

Clearly, differing from the $\beta$ index, ${\beta }_{\mathrm{ratio}}$ is a no-reference index that takes into account the multiplicative trait of the speckle. The more geometric content that appears in the ratio image, the higher the ${\beta }_{\mathrm{ratio}}$ value is.

2.2. Ratio Gradient Preservation Index

2.2.1. Motivation

It is well known that SAR intensity data can be modeled by a gamma distribution. However, from the review in Section 2, it was shown that most of the existing EPC assessment indices do not fully exploit the statistical traits of SAR data, which can make the evaluation results not robust or objective enough. In the following, we summarize some of the other drawbacks of the existing indices.

(1) Full-reference based indices are not the optimal choice for the evaluation of SAR filters since optical image simulated data are quite different from real SAR data.

(2) The evaluation results obtained by the edge detector based indices (no matter the detection is undertaken on the speckled or despeckled image) are often not robust, since a detection threshold often needs to be determined, which is nontrivial in most cases.

(3) The ratio image based indices were developed using the multiplicative trait of speckle. However, they are also not objective enough. The reason is that, unlike the noise in the raw speckled image, the noise in the ratio image cannot be simply regarded as following a certain statistical distribution. In such a case, distinguishing the fine texture from the noise or precisely measuring the gradient information is a difficult task.

In this paper, based on the above analyses, we present a new SAR EPC assessment indicator named the ratio gradient preservation index (RGPI). The proposed indicator does not require a clean reference image or an edge detector and, more importantly, can fully employ the statistical traits of SAR data.

2.2.2. The Ratio Gradient Preservation Index (RGPI)

The main idea behind the RGPI is to evaluate the gradient retention in the filtered image, given the gradient information obtained from the speckled image. The most commonly used gradient descriptor is the difference gradient, which is defined as the arithmetic difference between the two pixels contiguous to a given pixel along a certain direction. In the proposed index, we use the ratio gradient rather than the traditional difference gradient. This is because, as pointed out in [27], the radiometric distortion problem caused by a filter can significantly influence the values of the difference gradient; moreover, the ratio gradient is more suitable for the multiplicative nature of speckle noise.

Given the gradient information obtained from the speckled image, evaluation of the gradient retention in the filtered image can be regarded as solving the following conditional probability problem:

$$\mathrm{RGPI}=p(K=q|Q),Q\in \left[0,+\infty \right]$$

(10)

where Q and q are, respectively, the ratio gradient images obtained from the speckled and the filtered data. K is the unknown parameter, i.e., the ratio gradient for the unobserved clean data. A larger RGPI value indicates a better edge-preservation result. In a Bayesian framework, without prior knowledge of p (K = q) and p(Q), the probability $\mathit{p}(\mathit{K}=\mathit{q}|\mathit{Q})$ is proportional to $p\left(Q\right|K=q)$. In addition, the event Q can be considered as being uniform in its definition domain. Therefore, the RGPI is deduced as:

$$\begin{array}{l}\mathrm{RGPI}\propto p(Q|K=q)\\ \propto f(Q|K=q)\end{array}$$

(11)

where $f(\cdot )$ denotes the conditional probability density function (PDF).

For an image with N pixels and under an independence assumption for each pixel, the RGPI can be decomposed into the following product:

$$\mathrm{RGPI}={\displaystyle \prod _{i=1}^{N}f(Q(i)|K(i)=q(i))}$$

(12)

Naturally, taking the logarithm of (12) and considering the pixel number of the image, the RGPI is rewritten as:

$$\mathrm{RGPI}=\frac{1}{N}{\displaystyle \sum _{i=1}^N\ln \left[f(Q(i)|K(i)=q(i))\right]}$$

(13)

It can be seen from (13) that the only question for the RGPI is how to calculate $f(Q(i)|K(i)=q(i))$. Thanks to the study by Touzi et al. [27], we know that, for SAR intensity data, the PDF of the ratio gradient between two pixels is given by:

$$f(Q(i)|K(i)=q(i))=\frac{\mathsf{\Gamma }(2\mathrm{L})}{\mathsf{\Gamma }{(L)}^2}\left[\frac{q{(i)}^L}{{(Q(i)+q(i))}^{2L}}\right]Q{(i)}^{L-1}$$

(14)

where $\mathsf{\Gamma }(\cdot )$ denotes the gamma function, and L is the nominal number of looks (NNL) of the speckled image. It should be pointed out that, in theory, for the above derivation, equivalent number of looks (ENL) may be a better choice than NNL, since the noise level in the speckled image varies from pixel to pixel to a little degree. However, it is still an open and challenging issue to precisely estimate ENL, although many methods have been studied [28]; besides, the computational speed of the proposed index could be significantly slowed down if we calculate the ENL values.

2.2.3. Refinement of the RGPI

There are three issues that need to be further considered when applying the RGPI in practice.

(1) The first issue is to calculate the ratio gradient using the mean values of two square patches contiguous to a given pixel, instead of using the values of two single pixels. As revealed in [27], as the size of the patch increases, the ratio gradient becomes less sensitive to the speckle noise. In contrast, the small patch or pixel based ratio gradient has the advantage of being able to detect micro-edges. To achieve a balance between weakening the influence of speckle and reducing the speckle correlations, we set the size of the patch as $3\times 3$ in this study. Accordingly, as per the derivation in [27], the PDF in (14) becomes:

$$f(Q(i)|K(i)=q(i))=\frac{\mathsf{\Gamma }(2M\mathrm{L})}{\mathsf{\Gamma }{(ML)}^2}\left[\frac{q{(i)}^{ML}}{{(Q(i)+q(i))}^{2ML}}\right]Q{(i)}^{ML-1}$$

(15)

where M = 9 is the pixel number of a patch.

(2) The second issue is the direction when calculating the ratio gradient. To ensure the robustness of the proposed RGPI, the gradient information along several different directions should be considered. In this study, we chose the usual four directions (i.e., the vertical direction, the horizontal direction, and two different oblique directions (Figure 1)). The proposed index then becomes:

$$\mathrm{RGPI}=\frac{1}{4N}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j\in \Re }\ln \left[f\left(Q_j(i)|K_j(i)=q_j(i)\right)\right]}}$$

(16)

where $\Re$ denotes the set of four directions. Since the index involves division operation, to avoid the infinity problem, we use a simple strategy: for a certain pixel, if any pixel with zero value is found in the two opposite patches along a certain direction, the corresponding ratio gradient will not be considered.

(3) Last but not least, taking into account the homogeneity of the local area is also necessary. The formulation in (16) uses all the pixels to assess the EPC of a filter. However, pixels from homogeneous areas often outnumber the ones located around edges, and tend to dominate this quantity. In particular, as reported in some studies [3], some SAR filters can introduce artifacts (i.e., “false edges”) in homogeneous areas, although most of the edges of the original image are effectively retained. In such a circumstance, the performance of the RGPI could be degraded. To solve this problem, we integrate a weight for each pixel into the RGPI, leading to:

$$\mathrm{RGPI}=\frac{1}{4N}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j\in \Re }W(i)\cdot \ln \left[f\left(Q_j(i)|K_j(i)=q_j(i)\right)\right]}}$$

(17)

where the weight $W(i)$ is calculated based on the homogeneity of the local area centered at pixel i. In the proposed index, we utilize the local homogeneity descriptor employed in the classical Lee filter [29]:

$$W(i)=\frac{\mathrm{var}(I_{sub}(i))-{\overline{I}}_{sub}{(i)}^2/L}{(1+1/L)\mathrm{var}(I_{sub}(i))}$$

(18)

where $\mathrm{var}\left(I_{sub}\left(i\right)\right)$ and ${\overline{I}}_{sub}\left(i\right)$ are, respectively, the variance and mean of a local area in the speckled image. The weight is between 0 and 1, and the more homogeneous the local area is, the closer the value is to 0. W plays the role of weakening the impact of the pixels located in homogeneous areas. As shown in Figure 1, since the size of a patch in the ratio gradient operator is $3\text{×}3$, the size of the local area used to calculate W(i) is set as $7\text{×}7$.

2.2.4. Spatial Correlations of Speckle

An important issue should be pointed out here that the derivation of the proposed index is based on the assumption that the speckle component at each pixel of the SAR image is independently and identically distributed (IID assumption). However, during the SAR image formation, the raw data are usually oversampled and spectrally weighted to avoid Gibbs effects around point targets. In particular, during the SAR focusing operation, the matched filters are used. The above processes can introduce a spatial correlation of the noise, which may impact the validity of IID assumption [30]. Therefore, in theory, an initial decorrelation step is better to be taken on the speckled image, before applying the proposed IID assumption based index. Recently, Dalsasso et al. [31] reviewed the existing approaches that handle spatial correlations of speckle. These decorrelation approaches can be mainly divided into two categories, namely, the pre-processing approaches and post-processing approaches. The pre-processing approaches are performed by the data providers, which require the knowledge of sensor’s parameter and are sensor-specific. Clearly, the pre-processing approaches are often not feasible before using the proposed index. Instead, the post-processing approaches are better choices. Among them, the downsampling strategy and the method presented by Arienzo et al. [30] are two typical approaches. The downsampling strategy can directly reduce the spatial correlations of speckle by scarifying the resolution of SAR images. In [30], a blind decorrelation method is proposed, of which the main idea is to estimate and invert the point spread function by performing least square optimization. However, this blind decorrelation method can only handle the decorrelation issue of SLC SAR data, which limits its applicability before calculating the RGPI values

As aforementioned, to weaken the influence of speckle, the ratio gradient in the RGPI is calculated using the mean values of two square patches contiguous to a given pixel, instead of using the values of two single pixels. This strategy can be regarded as a downsampling or multi-look approach before calculating the ratio gradients, which is, therefore, a simple and fast means to reduce the speckle correlations before calculating the RGPI. To validate the robustness of RGPI in the case of correlated speckle, or to say, the reliability of the PDFs given by (14) and (15), we used the single-look simulated homogeneous image provided in [14], in which the speckle is spatial correlated (the link to get the image is provided in Appendix A). Then, we calculated the ratio gradients using single pixel values and mean values of $3\times 3$ image patch, respectively, and obtained the corresponding empirical Cumulative Distribution Function (CDF). Besides, by (14) and (15), it is easy to derive the corresponding theoretical CDF when L = 1. Figure 2 displays the theoretical and empirical CDF. Apparently, in the case of pixel based ratio gradient, the two curves generally show good agreement. Moreover, in the case of patch based ratio gradient, the two curves become very close to each other, which means that taking the mean value of patch can help to compensate the inaccuracy of the proposed index to a large degree when the speckle is spatial correlated. The issue of spatial correlations of speckle is further discussed in the experiment section.

3. Result

In this section, to inspect the performance of the proposed index, we describe the quantitative experiments undertaken on both simulated and real SAR datasets. The EPD-ROA [22] and DEI [23] were chosen to be compared with the RGPI, since they are all no-reference indices and do not need an edge detector. For reproducibility, the link to obtain the source codes of the RGPI which were implemented by MATLAB and the testing datasets are provided in the Appendix A. The computational cost of RGPI consist of two main parts: the calculation of the ratio gradient PDFs (see [12]) and the calculation of weights (see [18]). For an image with N pixels, assuming the computational burden of one PDF calculation step to be O₁(1) and one weight calculation step to be O₂ (1), the total computational complexity of RGPI is O₁(4N) + O₂ (N). For a computer with an i7-6500U CPU and an 8 GB RAM, the processing time on an image with $500\text{×}500$ pixels is about 23.5 s.

3.1. Experiments with Simulated SAR Data

In this part, to inspect the feasibility of RGPI on the evaluation of preserving the edge profiles, we use the simulated “building” image (Figure 3) provided by Di Martino et al. [14], with an isolated building placed on a homogeneous background. The results of three state-of-the-art filters, namely, the speckle reduction anisotropic diffusion (SRAD) filter [32], the probabilistic patch-based (PPB) filter [9] and the 3-D block matching-based SAR (SAR-BM3D) filter [10], were chosen to be evaluated by the indices (the links to get the simulated image and the source codes of the PPB and SAR-BM3D filter are provided in the Appendix A). From Figure 3, we can see that, compared with the SRAD filter, the PPB and SAR-BM3D filter perform much better in retaining the edges for the single-look speckled building images. To further inspect the EPC of the different filter, the range profile of the building of each filtered image compared with the clean reference is displayed in Figure 4, which clearly shows that the SAR-BM3D filter better retains the profile of the building than the PPB filter and the SRAD filter. The RGPI and DEI values listed in

Table 2. EPC assessment results for the simulated data.

		L = 1	L = 2	L = 4
EPD-ROA	SRAD	0.021	0.034	0.077
	PPB	0.042	0.075	0.159
	SAR-BM3D	0.037	0.053	0.144
DEI	SRAD	0.770	0.695	0.543
	PPB	0.672	0.553	0.399
	SAR-BM3D	0.611	0.522	0.371
RGPI	SRAD	−0.823	−0.442	−0.300
	PPB	−0.390	−0.230	−0.110
	SAR-BM3D	−0.308	−0.218	−0.102

are in agreement with Figure 3 and Figure 4. However, the EPD-ROA values disobey the observation that the SAR-BM3D filter performs better than the PPB filter; in addition, the values are very close to zero, due to the fact that the EPD-ROA is computed over all pixels and the pixels from homogeneous areas significantly outnumber the ones located around edges.

In theory, for a certain filter, the edges of an image can be better retained as the amount of noise decreases. In Table 2, we also list the assessment values when L = 2 and L = 4. We can see that the evaluation results obtained by the three indices all obey the above principle. However, as observed for the case of L = 1, the EPD-ROA values in the cases of L = 2 and L = 4 still do not precisely reflect the real state, which is that the SAR-BM3D filter better retains the edges than the PPB filter.

Since the RGPI is derived based on IID assumption, it is necessary to further inspect its validity on more images with correlated speckle. To this end, we used the clean “building” image provided in [14] and generated the speckle correlated single-look images by the approach presented in [30]. In this simulation process, the choice of parameter $\alpha$ which determines the degree of spectral smoothness is crucial. The lower the $\alpha$ is, the higher the level of spatial correlation that the SAR image has. For most commercial SAR sensors, the $\alpha$ value is between 0.6 and 0.7. The RGPI values for the filtered images with different $\alpha$ values are listed in

Table 3. RGPI values for the simulated data with different degree of speckle correlation.

		Pixel Based RGPI	Patch Based RGPI
$\alpha =0.7$	SRAD	−3.166	−0.711
	PPB	−2.212	−0.339
	SAR-BM3D	−2.001	−0.282
$\alpha =0.65$	SRAD	−3.435	−0.784
	PPB	−2.410	−0.350
	SAR-BM3D	−2.300	−0.332
$\alpha =0.6$	SRAD	−3.568	−0.801
	PPB	−2.566	−0.423
	SAR-BM3D	−2.603	−0.411

; in addition, the pixel based and patch based RGPI values are also listed to investigate the necessity of using patch based ratio gradient. (1) In theory, since all of the three filters are developed based on IID assumption, the performances of these filters will be degraded when the image has a higher level of noise correlation. The RGPI values in Table 3 well reflect this trend. (2) In most cases, for a given $\alpha$ value, the RGPI values well reflect the facts that the SAR-BM3D filter better preserves the edges than the PPB filter and the PPB filter is better than the SRAD filter. The only exception is observed in the case of the highest correlation, i.e., $\alpha$ = 0.6, in which the pixel based RGPI value of SAR-BM3D is a little lower than that of PPB. The reason for this is that the performances of these two filters on this image are close and, meanwhile, the strong noise level and high noise correlation impact the judgment of pixel based RGPI. This group of experiments reveals that the RGPI are generally robust for correlated speckle, especially when the patch based ratio gradients are used.

3.2. Experiments with Real SAR Datasets

The quantitative assessment values of the EPD-ROA, DEI and RGPI indices on three real-world SAR datasets are reported in this part. The first data was acquired by the L-band ALOS system (Figure 5a), which is single-look HH polarization data. For an image quality measurement, the need for consistency is one of the most important requirements. The usual way to test the consistency is to compare the assessment values obtained for a given filter using different filtering parameters. In theory, for an over-smoothed result, the EPD-ROA and RGPI will provide low values and the DEI will provide a high value, if they follow the principle of consistency. When applying the PPB filter, the iteration times (T) is one of the key parameters. In

Table 4. EPC assessment results for the ALOS image.

	SRAD	PPB (T = 4)	PPB (T = 3)	PPB (T = 2)	SAR-BM3D
EPD-ROA	0.676	0.889	0.890	0.894	0.896
DEI	0.762	0.725	0.610	0.303	0.287
RGPI	−0.217	−0.116	−0.096	−0.065	−0.066

, we list the EPC evaluation results for the images filtered by the PPB filter when T = 4, T = 3, and T = 2. To help the reader to better judge the EPC of PPB and SAR-BM3D, we display the corresponding ratio images. Clearly, from the filtered images and ratio images, we can see that the over-smoothing problem of the PPB filter is gradually relieved as the iteration times decrease. The values of the three indices are all consistent with this conclusion.

For the purpose of comparison, the evaluation values for the SRAD and SAR-BM3D filtered image are also listed. The profile of the river in the SRAD ratio image is quite notable, meaning that the SRAD filter performs poorly in preserving the edges of this image, and all the indices are in agreement with the observation. We can also see that the profile of the river is more distinguishable in the ratio image of the SAR-BM3D filtered data, indicating the better EPC of the PPB filter with T = 2 on this image. The values of the RGPI are in line with the above observation. Meanwhile, the EPD-ROA and DEI indices fail to reflect the real state.

The second real SAR image was acquired by the TerraSAR-X satellite in Ruhr, Germany (Figure 6a), which is single-look VV polarization data. The PPB filter and SAR-BM3D filter were both developed employing the idea of nonlocal means, and hence the size of the image patch (S) is an important parameter when applying these two filters. To confirm the consistency of the indices, we tuned the size of the patch in each filter to be $7\times 7$ and $5\times 5$, respectively, and we list the quantitative assessment values for the four filtered images in

Table 5. EPC assessment results for the TerraSAR-X image.

	PPB (S = 7 × 7)	SAR-BM3D (S = 7 × 7)	PPB (S = 5 × 5)	SAR-BM3D (S = 5 × 5)
EPD-ROA	0.617	0.679	0.682	0.692
DEI	0.731	0.700	0.606	0.524
RGPI	−0.532	−0.276	−0.417	−0.267

.

As pointed out in some studies [33,34], for a certain nonlocal means based filter, the image edges will be better preserved if small patches are used. This conclusion can be easily drawn by comparing Figure 6b,d. The values obtained by the three indices also allow this conclusion. In the meantime, it can be also seen from Figure 6 that, given the size of the patch, the SAR-BM3D filter always outperforms the PPB filter on edge preservation. The values obtained by the three indices once again follow the above observation. Furthermore, if we inspect Figure 6c,d, we can draw the conclusion that the SAR-BM3D filter with S = 7 × 7 better retains the edges than the PPB filter with S = 5 × 5. However, the values of the EPD-ROA and DEI violate the real state. From the aforementioned observation, we can again confirm the robustness and consistency of the proposed index.

To further support the above conclusions, a line was taken across a lake of the image (marked out by a red line in Figure 6a), and the intensity values of this line after despeckling by the different filters are plotted in Figure 7. The signals returned from the outside areas of the lake, which are dominated by double scattering and volume scattering, are much higher than those from the water. This means that the pixels with high value in Figure 7 are generally from the outside areas of the lake. Especially, the edges and strong point targets are more likely in the location of those pixels with notable higher value than the nearby pixels. From Figure 7, we can observe that, in the locations of the pixels with very high value, the original values are closer to the despeckled results by the SAR-BM3D filter (no matter S = 7 × 7, or S = 5 × 5) than to the PPB filter in most cases, indicating the better EPC of the SAR-BM3D filter. The above conclusion has an agreement with the RGPI values listed in Table 5.

To more comprehensively evaluate the performance of RGPI, the assessment result on an AirSAR dataset acquired in Kyoto, Japan, are reported. This four-look dataset have two different polarization channels (HH and HV) and two different frequencies (C band and L band). From Figure 8, we can observe that the SAR-BM3D filter outperforms the SRAD and PPB filter in retaining image details (especially for the forest area) for all the cases. The RGPI values listed in

Table 6. RGPI values for the AirSAR dataset.

	SRAD	PPB	SAR-BM3D
HH polarization (C band)	−0.088	−0.055	0.270
HV polarization (C band)	−0.078	−0.048	0.260
HH polarization (L band)	−0.143	−0.017	0.307
HV polarization (L band)	−0.144	0.029	0.303

are again in good agreement with the visual inspection. In addition, we found an interesting phenomenon that, for a certain filter, the values of the four filtered images are generally close to each other, indicating the satisfactory robustness of RGPI.

4. Conclusions

The edges in SAR images should be effectively retained when applying a filter, which is important for the target detection task. To help the user to better judge the performance of the different SAR filters in edge preservation, developing a quantitative assessment index is a meaningful task. The common drawback of most existing EPC assessment indices is that they are often simply extended from the indices developed for digital images, and they do not fully exploit the statistical traits of SAR data. This brings in the problems that the results are often not robust and consistent enough, when the existing indices are used to assess filters on different images (i.e., images with simple or complicated scenes, images with different level of speckle).

In this paper, we have presented a no-reference edge-preservation assessment index for single-polarization SAR filters. In the proposed index, the ratio gradient of two patches contiguous to each pixel in the speckled and despeckled images is utilized to reflect the edge information. Then, considering the conditional probability density function of the ratio gradient and under a Bayesian framework, the edge retention in the filtered image is evaluated. The results obtained with both simulated and real SAR images support the conclusion that, the proposed index is robust and reliable for images acquired in different frequencies and different polarization states, and for images with simple scenes or complicated scenes. For reproducibility, the source codes of the index and the testing datasets are provided.

There are also two issues that should be noticed when applying the proposed index. First, the original derivation of RGPI is based on the IID assumption, while the speckle in the real SAR data is spatially correlated. Although the patched based ratio gradients are used and the experiments in this paper show its feasibility in addressing the correlation problem, it would be better to care for the assessment result if they use RGPI when the original image is highly spatially correlated and with a high level of noise. Second, due to the use of patch based gradient ratios, some very narrow structures may hard to detect. In most cases, the above problem can be neglected, since the final purpose of RGPI is to rank the filters’ general performances by inspecting the preservation of ratio gradient for all pixels in an image. That is to say, the general performance of RGPI will not be significantly degraded if a limited number of fine structures are not well detected by the ratio gradients. However, in the extreme case that too many very narrow structures exhibit in the original image, the situation might be worse.