Speckle noise filtering in SAR images using fuzzy logic and particle swarm optimization

Abstract

Speckle noises inherently exist in Synthetic Aperture Radar (SAR) image. It reduces the quality of the image and must be removed. Recently, several fuzzy based noise filters have been proposed and it is claimed that they are more superior to the existing state of the art classical filters. However, most of these filters are designed for removing impulse noise. This paper presents a new fuzzy weighted mean filter for SAR images. The weights are computed using fuzzy rules. These rules can differentiate variation in pixel values due to noise and edges. In edge region, the neighboring pixels are assigned with less weight, and thereby preserving the edge pixels. The value of parameters used in defining fuzzy membership functions is determined using Particle Swarm Optimization (PSO) technique. The proposed fuzzy filter is comparatively assessed using Equivalent Number of Looks (ENL), Mean of Ratio (MoR), Signal-to-Noise Ratio (SNR), and Edge-Preservation Factor (EPF) with some of the existing noise removal techniques. It is found that the proposed filter can suppress speckle noise present in SAR images and simultaneously preserve the image details.

Keywords

Fuzzy logic image denoising noise cancellation particle swarm optimization synthetic aperture radar

1. Introduction

SAR images are formed by coherent interaction of transmitted electromagnetic signals with the target. The reflected echoes are captured and processed to construct images of the target. Echoes from target may be out of phase even though they are coherent in frequency. One of the reasons is the difference in the distance of waves travelling back from the target, caused by multiple bounce due to variations in the surface roughness. The motion of the antenna also changes phase, the phase of each elementary signal changes according to the distance between the target and the antenna [5]. Out of phase waves interfere constructively or destructively to produce stronger or weaker signals. Interference of these out of phase, but coherent wavelets results in the granular pattern known as speckle noise [13, 9]. Consequently the speckle noise consists of bright spots and dark spots. Speckle has the characteristics of a random multiplicative noise [17]. A fully developed speckled image NI can be modeled as $\textit{NI}=\textit{CI}+\eta\times\textit{CI}$ , where CI is a clean image and $\eta$ is uniformly distributed random noise with mean zero and variance $v$ [25, 3]. The speckle noises alter the intensity value of pixels in the SAR images. Highly corrupted (noisy) pixel has different intensity value from the neighboring pixels. Speckle noise degrades the quality of the image and causes serious difficulty in analyzing the images. It must be removed before using the image for high level processing like segmentation, classification, image understanding etc.

Reducing noise and preserving fine edges present in the images are two important noise suppression considerations, as edges contain most of the important information [15]. It is the balance between these two considerations that determines the success of a filter. Nonlinear filtering techniques are preferred because they provide better results in suppressing noise and preserving edges present in the image than linear methods [18]. Some of the classical nonlinear speckle noise filtering techniques includes median filter, non-local mean filter, Lee filter, Frost filter and Weiner filter. Median filtering is based on a window which slides over each pixel in the image and replaces the central pixel by the median of neighboring pixel value [11]. Non-local mean filter computes the mean of all pixels in the image, weighted by how similar these pixels are to the target pixel. Non-local mean [1] filter improves the results by giving a greater post-filtering clarity, and less loss of detail in the image compared with mean filter. The Lee filter [14] computes mean and variance within each window. In the areas of low signal activity estimated pixel approaches local mean, whereas in the areas of high signal activity (like edge) estimated pixel favors noisy pixel, thus retaining the edge information. The Frost filter [26] updates the central pixel with a weighted sum calculated based on the distance from filter center, damping factor and local variance. Wiener Filter [2] is based on least squared principle; it minimizes the Mean Square Error (MSE) between the actual output and the desired output. Both global statistics and local statistics are used in Wiener filtering. Soft-computing techniques like artificial neural network [28], support vector machine [22], particle swarm optimization [27] and genetic algorithm [30] are gaining popularity in reducing noise from images as they are intelligent (can be trained) and robust in filtering noise.

The image formed by SAR is a 2-D projection of a 3-D natural surface and some amount of imprecision is inherent in the images. The amount of noise and distribution of noise in SAR images is unpredictable. Due to the presence of noise, it is also confusing to determine an edge in SAR image [10]. Therefore, a noise filtering system needs to be capable of resolving vague and uncertain information. Fuzzy sets are capable of handling uncertainty associated with vagueness and imprecision in information. In classical set theory, the membership of elements in a set is 1 if the element belongs or 0 if it does not belong to the set. Fuzzy set theory permits partial membership of elements (value can be real) in the interval [0, 1]. A membership function maps each point in the input space to a membership value. Fuzzy logic uses IF-THEN rule for taking decisions. Generally fuzzy IF-THEN rules are written as “IF p is Q THEN r is S”, where Q and S are membership functions. Fuzzy rule allows dealing complex problem in a simpler way by using linguistic terms.

Recently, various fuzzy based nonlinear noise filters have been proposed. However, most of the existing fuzzy filters are developed for removing impulse noise. FIRE (Fuzzy Inference Ruled by Else-action) filter proposed by Russo [7] is based on IF-THEN-ELSE reasoning. FIRE filter uses the pixel intensity difference between the central pixel (to be filtered) and its neighbors denoted by $\Delta x$ , and fuzzy rules are used to evaluate correction parameter $\Delta y$ . The filtered pixel is defined by $y(i,j)=x(i,j)+\Delta y$ . The fuzzy rules in FIRE filter uses two triangular shaped membership function named as “positive” and “negative”. This filter has good detail preserving behaviour. Fuzzy weighted moving averaging (MAV) and Fuzzy weighted median (MED) are also a popular class of noise filtering technique in which the weights are computed using window function. Based on the different window function several different fuzzy filters can be obtained [18], it includes the asymmetrical triangular fuzzy filter with median center (ATMED), symmetrical triangular fuzzy filter with moving average center (TMAV) and asymmetrical triangular fuzzy filter with moving average center (ATMAV). Schulte et al. [23] proposed a two stage fuzzy filtering technique. In the first stage the presence of noise is detected by using gradient value (difference between the central pixel and its neighbors). If most of the gradient values are large then the central pixel is considered to be noisy. In the second phase if the pixel is detected as noisy then it is filtered iteratively based on membership function termed as more or less impulsive noise. Dimitri Van De Ville [4] proposed a fuzzy based filter which is capable of distinguishing local variation due to noise and variation due to image structure like sharp edges in the image. This technique consists of two phases; in the first phase it computes fuzzy derivative, i.e. the difference between the central pixel and its neighbor and the second phase performs fuzzy smoothing by considering the contribution of the neighboring pixels. A robust Fuzzy Median Filter is presented in [16], it detects noise using fuzzy rules and the filter parameters are controlled based on these rules. Suppose $x(i,j)$ denotes the central pixel of two dimensional filter window and $m(i,j)$ is the median. Output of the fuzzy median filter [16] can be expressed as $y(i,j)=m(i,j)+\mu[x(i,j)]\times(x(i,j)-m(i,j))$ where $\mu[]$ denotes the membership function. If $\mu[x(i,j)]=0$ an impulsive noise is considered to be located at $x(i,j)$ and if $\mu[x(i,j)]=1$ impulsive noise is not located at $x(i,j)$ . If the difference between $x(i,j)$ and $m(i,j)$ is large and the average difference of the closest two values in the neighborhood is large then impulsive noise in assumed to be present. The membership function is approximated by a step like function such that the height of each step reduces the MSE for training data.

The output of fuzzy filters depends on the fuzzy rules, the notion of fuzzy sets (membership function), parameter used in defining the membership function and the defuzzification process. Generally, one of the main issues in designing fuzzy filter is determining the parameters used in defining the fuzzy membership function. These parameters are either determined by using trial and error method or by using an optimization algorithm. The proposed fuzzy based speckle noise filter uses PSO to determine these parameters. PSO is a computational method that optimizes a solution by iteratively improving the candidate solution [6, 29]. PSO is robust and fast in solving nonlinear, non-differentiable and multi-modal problems [12].

2. Fuzzy speckle noise filter

The proposed fuzzy filter differentiates pixel variation due to noise and edges. In an edge region the neighboring pixels are given less weight in computing the weighted mean, with an intention to preserve the image structure such as fine edges. The proposed filter consists of fuzzy rules, which are based on the observation that the speckle noise corrupted pixels are different from its neighbors. Consider the moving window shown in Fig. 1.

Figure 1.

Relationship of the pixel to be filtered $P_{0}$ and its neighbors.

If the difference between the pixel intensity to be filtered ( $P_{0}$ ) and its immediate eight neighbors are large, then there is a very high probability that the central pixel ( $P_{0}$ ) is noisy [16]. This can be express as the following fuzzy rule:

Rule 1: If $D_{k}$ is LARGE then $P_{0}$ is NOISY

Where $D_{k}=|P_{k}-P_{0}|$ and $k\epsilon{\{}1,2,3,4,5,6,7,8{\}}$ , LARGE is a fuzzy membership function defined in Eq. (1). Eight such rules are required for all the eight immediate neighbors. If $P_{0}$ is part of fine details like edges in the image, then it may also give large $D_{k}$ value [23]. As an example, consider Fig. 2. Let us assume that in a homogeneous region (without noise) there is an edge of size one pixel along the direction $P_{18}$ , $P_{5}$ , $P_{0}$ , $P_{1}$ and $P_{10}$ (lightly shaded region). Then $D_{2},D_{3},D_{4},D_{6},D_{7}$ and $D_{8}$ will be large and the difference of relative neighbors will be similar. Relative neighbors are the two pairs of pixels which are in the direction of the immediate neighbor from $P_{0}$ . In Fig. 2, the two dotted lines ${\{}(P_{1},P_{12})$ and $(P_{5},P_{16}){\}}$ represents the relative neighbors with respect to the solid line ( $P_{0}$ , $P_{3}$ ). The difference of relative neighbor are computed as $D_{31}=|P_{1}-P_{12}|$ , $D_{32}=|P_{5}-P_{16}|$ (lightly shaded pixels minus corresponding dark shaded pixels). Similarly, we can have eight $D_{k}$ for all the immediate neighbors and corresponding $D_{k1}$ and $D_{k2}$ values for the relative neighbors.

Figure 2.

Pixels involved in determining edges.

From the preceding discussions, we can develop fuzzy rules that separate noisy pixels and edges.

Rule 2(a): If $D_{k1}$ is SMALL and $D_{k}$ is SMALL and $D_{k2}$ is SMALL then $P_{0}$ is EDGE. Rule 2(b): If $D_{k1}$ is LARGE and $D_{k}$ is LARGE and $D_{k2}$ is LARGE then $P_{0}$ is EDGE.

SMALL is a fuzzy membership function defined in Eq. (2). We can formulate eight such rules for all $D_{k}$ . Rule 1 can be used for detecting high frequency component (noise and edges) present in the images and Rule 2 can be used to separate noise from edges. These two rules are combined in Rule 3.

Rule 3(a): If $D_{k}$ is LARGE and $D_{k1}$ is SMALL and $D_{k2}$ is SMALL then $P_{0}$ is NOISY. Rule 3(b): If $D_{k}$ is SMALL and $D_{k1}$ is LARGE and $D_{k2}$ is LARGE then $P_{0}$ is NOISY.

Rule 3 is based on the perception that if the difference between the immediate neighbor and the relative neighbor is large, then there is a very high probability of the pixel $P_{0}$ being corrupted with speckle noise, and the chance of pixel $P_{0}$ being part of the edge is low. In noise detection phase, sixteen fuzzy rules defined in Rule 3 are used, two for each $D_{k}$ values. The LARGE membership function is defined as:

$\displaystyle f(x)=\left\{{\begin{array}[]{ll}0,&x\leqslant a_{l}\\ \frac{x-a_{l}}{b_{l}-a_{l}},&a_{l}<x<b_{l}\\ 1,&x\geqslant b_{l}\\ \end{array}}\right.$ (1)

Where $a_{l}$ and $b_{l}$ are parameters computed using PSO, such that $a_{l}\leqslant b_{l}$ . Membership function SMALL is defined as:

$\displaystyle f(x)=\left\{{\begin{array}[]{ll}1,&x\leqslant a_{s}\\ \frac{b_{s}-x}{b_{s}-a_{s}},&a_{s}<x<b_{s}\\ 0,&x\geqslant b_{s}\\ \end{array}}\right.$ (2)

Where $a_{s}$ and $b_{s}$ are parameters, such that $a_{s}\leqslant b_{s}$ and the values are determined using PSO. The membership functions are graphically represented in Fig. 3.

Figure 3.

Graphical representation of SMALL and LARGE.

The result ( $n l$ ) obtained after evaluating the fuzzy rules is the noise level of $P_{0}$ with respect to the neighbours, highly corrupted pixels have larger value, less noisy and edge pixels has smaller value. Each pixel $x(i,j)$ in the image is replaced by $y(i,j)$ which is computed using Eq. (3).

$\displaystyle y(i,j)=\frac{\sum\limits_{r=-2,s=-2}^{r=2,s=2}{W(i+r,j+s)\times x% (i+r,j+s)}}{\sum\limits_{r=-2,s=-2}^{r=2,s=2}{W(i+r,j+s)}}$ (3)

Where the weight $W(i+r,j+s)$ is computed using Eq. (4). $l m$ is the mean of the current window (local mean). The noise level $n l$ with respect to itself is set to 1 (assume it is noisy).

$\displaystyle W(p,q)=\frac{nl}{(lm-x(p,q))}$ (4)

Where $p=i+r$ and $q=j+s$ . Smaller weight is given to a neighbouring pixel if $n l$ is small, as the pixel to be filtered may be an edge pixel or not noise. If $n l$ is large (close to 1) then weight approaches $1/(lm-x(p,q))$ , which intends to bring the filtered pixel, closer to the local mean.

One of the most difficult tasks while working with fuzzy system is determining the appropriate membership functions and the associated parameters. Result of fuzzy system depend largely on these problem oriented parameters and usually determined using trial and error or optimization algorithm. Membership functions defined in Eqs (1) and (2) are suitable for LARGE and SMALL. The degree of membership for a value in the set LARGE (refer Fig. 3) increases with the increase in input values of $D_{k},D_{k1}$ and $D_{k2}$ . Input values less than $a_{l}$ are not included in the set LARGE and degree of membership will be zero, input values greater than $b_{l}$ are members of the set LARGE with degree one, input value in the range of $a_{l}$ to $b_{l}$ will have partial membership. Determining the value of $a_{l}$ and $b_{l}$ is crucial as these are the points that will decide the membership of set LARGE. Similarly for fuzzy set SMALL determining the parameters $a_{s}$ and $b_{s}$ is crucial. These parameters can be accurately determined by using the PSO algorithm.

PSO was developed by James Kennedy and Russell Eberhat after being inspired by the bird flocking behavior [20]. $N$ particles (birds) are randomly set in motion in the solution space (looking for a food). In every iteration, each particle $P[i]$ checks the fitness (distance from the food) of themselves and the neighbors and moves towards the particle having better solution, velocity $V[i]$ controls the movement of $i^{th}$ particles. The fitness value is evaluated by fitness function $FV()$ . Each particle updates the best solution, it has achieved so far Pbest[i] and the best solution achieved by any particle in the population Gbest [31].

[h] Steps involved in PSO[1] $i\leftarrow 1$

while $i\leqslant N$ do

$P[i]\leftarrow\textit{ rand()}$

$F[i]\leftarrow FV(P[i])$

$\textit{Pbest}[i]\leftarrow F[i]$

$\textit{Gbest}\leftarrow\infty$

$V[i]\leftarrow 0$

$i\leftarrow i+1$

end while

$i\leftarrow 1,j\leftarrow 1$

while MAXIT $\geqslant$ j do

while i $\leqslant$ N do

$F[i]\leftarrow FV(P[i])$

$\textit{if Pbest}[i]>F[i]\textit{ then}$

$\textit{Pbest}[i]\leftarrow F[i]$

end if

if Pbest[i] $<$ Gbest then

Gbest $\leftarrow$ Pbest[i]

end if

$i\leftarrow i+1$

end while

while i $=$ 1 to N do

$V[i]\leftarrow V[i]+C1\times\textit{rand}()\times(\textit{Pbest}[i]-P[i])+C2% \times\textit{rand}()\times(\textit{Gbest}-P[i])$

$P[i]\leftarrow P[i]+V[i]$

$i\leftarrow i+1$

end while

$j\leftarrow j+1$

end while

In Algorithm 1, $\textit{rand}()$ is a random number, procedure $FV()$ is defined in Algorithm 2, MAXIT is the maximum number of iterations, $C1$ and $C2$ are constant. $V[i]$ calculates the particles new velocity by using its previous velocity $V[i]$ , current position $P[i]$ , its own best experience $\textit{Pbest}[i]$ and the groups best experience Gbest. Using $V[i]$ the particle moves toward a new position $P[i]$ [21].

[h] Fitness function[1] $\textit{procedure }FV(f)$

CI $\leftarrow$ Clean Reference Image

NI $\leftarrow$ Noisy version of CI

FI $\leftarrow$ Fuzzy Filter(NI, f)

$\textit{err}\leftarrow\textit{sum}(CI-FI)^{2}/(R\times C)$

return err

end procedure

Where $C I$ is a clean SAR image that we want to achieve, $N I$ is a noisy image. $R\times C$ is the size of SAR image. The procedure $FV()$ takes the position of the particle $P[i]$ and returns the MSE.

PSO is based on supervised learning approach. The objective is to find the best possible parameter which will give the highest fitness value. Once the parameters associated with membership functions LARGE and SMALL are determined, the proposed fuzzy speckle noise filter can be used for filtering speckle noises from similar SAR images.

3. Experimentation and results

The dataset that we have considered in comparing the SAR despecking filters are downloaded from http://www.grip.unina.it/. The image includes the most common elementary scenes encountered in typical SAR images, as follows [3]:

i)
Homogeneous: a flat region with constant electromagnetic parameters shown in Fig. 4a
ii)
Digital elevation model (DEM): a non flat orography region with constant electromagnetic parameters shown in Fig. 4b
iii)
Squares: Several regions on flat terrain, separated by straight contours having different electromagnetic parameters shown in Fig. 4c
iv)
Corner: a reflector placed on a homogeneous background shown in Fig. 4d
v)
Building: an isolated building placed on a homogeneous background shown in Fig. 4e

Figure 4.
Elementary scenes encountered in typical SAR images. (a) Homogeneous. (b) DEM. (c) Square. (d) Corner. (e) Building.

The SAR images available in http://www.grip.unina.it/ are generated using a complete SAR simulator considering proper physical models for the sensed surface, scattering and the radar operation mode. The sensor parameters for the first four images are those of typical European remote sensing (ERS) or environmental satellite (ENVISAT) sensors, it generates SAR images with a ground or range resolution and azimuth resolution of 19.9 meters and 4 meters, respectively. For building, high-resolution sensor parameters of Cosmo/SkyMed are used to ensure resolution of 3.6 meter in range and 2.6 meter in azimuth adequate for identifying a building. The clean reference images shown in Fig. 5 are generated by averaging 512-independent intensity SAR images; it provides a stronger reduction of speckle than the best known techniques in the literature. The size of the square and DEM images is 512 $\times$ 512 pixels and 256 $\times$ 256 pixels for the other cases.

Figure 5.
Clean reference images. (a) Homogeneous. (b) DEM. (c) Square. (d) Corner. (e) Building.

The proposed fuzzy filter is implemented in two phases; training and testing. The training phase adopts a supervised learning approach, and the objective is to determine the value of parameters $a_{l}$ , $b_{l}$ , $a_{s}$ and $b_{s}$ presented in proposed fuzzy speckle noise filter using PSO. The parameters associated with PSO are initialized as follows; the number of particle $N$ is set to 20, maximum number of iteration MAXIT is set to 20, personal learning coefficient ( $C1=$ 0.015) and global learning coefficient ( $C2=$ 0.02). Resized (64 $\times$ 64) noise free square image shown in Fig. 5c is considered as a reference image, and resized (64 $\times$ 64) noisy version in Fig. 4c is filtered using the proposed fuzzy filter (presented in Section 2) to match the reference image as close as possible by adjusting the fuzzy parameters using PSO. All the four parameters are trained at once; the particles are placed in four dimension random positions and the PSO determines the best possible position of the particle. The square image consists of both homogeneous and heterogeneous region, which the filter should be able to differentiate. In the testing phase noisy images are filtered by setting the fuzzy parameter determined in training phase.

The performance of the proposed fuzzy filter is comparatively evaluated with some of the existing classical and fuzzy based filters. We have considered classical mean, non-local mean, median, Frost, fuzzy FIRE, TMAV, ATMAV and ATMED filter for comparisons. All implemented filters are based on 3 $\times$ 3 window except the proposed filter, as it requires 5 $\times$ 5 window in differentiating noise and edge pixels. A good speckle noise filter must have the characteristic of reducing speckle in homogeneous areas, preserving details and radiometric, and absence of artifacts in the filtered image [3, 19]. Accordingly, we have adopted ENL, MoR, SNR and EPF as quantitative measures for evaluating the performance of the propose speckle noise filter. ENL is the number of single-look images that must be averaged to obtain the equivalent despeckled image in a perfectly uniform region. The ENL is defined as

$\displaystyle\textit{ENL}=\mu_{v}^{2}/\sigma_{v}^{2}$ (5)

Where $\mu_{v}$ and $\sigma_{v}$ denote the estimated mean and standard deviation of the filtered image. MoR is the ratio of the mean intensity value of speckled image and the despeckled image. Significant variation of MoR from one indicates radiometric distortion [3]. A popular approach to evaluate the despeckling filter when the clean reference images are available is SNR. It provides meaningful information about the closeness of the clean reference images CI and the filtered image FI. The SNR [8] is defined as

$\displaystyle\textit{SNR}=20\times\text{log}_{10}\left[{\frac{\sum\limits_{i=0% }^{r-1}\sum\limits_{j=0}^{c-1}{FI(i,j)^{2}}}{\sum\limits_{i=0}^{r-1}\sum% \limits_{j=0}^{c-1}[CI(i,j)-FI(i,j)]^{2}}}\right]$ (6)

The closer value of CI and FI will give larger SNR. EPF measures the ability to preserve the details after filtering the image [12]. EPF is computed using the following formula:

$\displaystyle\textit{EPF}=\frac{\sum(\Delta CI-\Delta\overline{CI})\times(% \Delta FI-\Delta\overline{FI})}{\sqrt{\sum(\Delta CI-\Delta\overline{CI})^{2}% \times(\Delta FI-\Delta\overline{FI})^{2}}}$ (7)

Where $\Delta CI$ and $\Delta FI$ are the high pass filtered version of desired clean image CI and denoised (filtered) image FI, $\Delta\overline{CI}$ and $\Delta\overline{FI}$ are the mean value of $\Delta CI$ and $\Delta FI$ respectively. The high pass filtering is obtained by using Laplacian operator [8, 24]. Noisy image used for evaluating the performances are shown in Fig. 4.

Table 1 presents the result of filtering homogeneous area in Fig. 4a by different filters. The randomness in the homogeneous regions is caused by speckle. By determining the ENL, MoR and SNR in a homogeneous region one can analyze the speckle suppression capability, radiometric preservation and closeness of the filter image to the ideal clean image.

Table 1
Result for homogeneous region

ENL MoR SNR

Clean 436.9726 1.0077 –

Noisy 0.9996 – 6.0422

Mean 6.9563 0.9977 17.9626

Non-local mean 74.8274 1.0020 36.6982

Median 4.0274 1.3270 11.0358

Frost 9.6317 1.0003 20.4880

FIRE 2.7145 1.0133 4.7026

TMAV 5.0535 0.7355 12.9027

ATMAV 6.4210 1.0073 17.3605

ATMED 5.6731 1.1141 15.9057

Proposed 73.8200 1.0903 32.8100

From the experimental result presented in Table 1, it is observed that the proposed filter is capable of the suppressing speckle noise as it has high ENL in homogeneous region. It improves the signal quality of image since the SNR is increased after filtering. And it also preserves the radiometric as there is no significant deviation from one in MoR. Figure 6a shows the result of filtering Fig. 4a using the proposed filter. The Non-local mean filter performs slightly better than the proposed filter for all the measures in a perfectly homogeneous area. In Table 1, the proposed fuzzy filter clearly outperforms the classical fuzzy filters, frost, mean and median in filtering speckle noise. In DEM, signal variations occur at different scales. Differentiating speckle from edges of different size and orientation is challenging. Table 2 shows the performance of the filtering DEM Fig. 4b.

Figure 6.
Despeckle images using proposed filter. (a) Homogeneous. (b) DEM. (c) Square. (d) Corner. (e) Building.

Table 2
Result for DEM

MoR SNR EPF

Clean 1.0079 – 6.1835

Noisy – 6.2152 3.2871

Mean 1.0019 3.9815 0.6357

Non-local mean 1.2009 1.7356 2.6968

Median 2.0402 2.5037 0.5112

Frost 1.0109 7.4395 2.8383

FIRE 0.9956 5.4446 3.6065

TMAV 0.5345 4.9280 1.9158

ATMAV 1.0283 3.6671 1.4921

ATMED 1.3745 0.1958 0.2964

Proposed 1.0902 4.7010 3.6896

In Table 2, looking at the EPF value the proposed filter preserve the edges present in the noisy image, it even improves the edges with a small amount. There is no significant deviation in MoR. However, the SNR has reduced after filtering. The fuzzy FIRE filter too preserves the edges, but reduces the SNR. The Frost filter improves the quality (SNR) of the images but distort edges. Other filters do not perform well in filtering DEM image containing complex signal variations. Filtered image of DEM using the proposed filter is shown in Fig. 6b.

Table 3
Result for square

MoR SNR EPF

Clean 0.9990 – 12.1978

Noisy – 6.0298 2.1167

Mean 1.0025 17.9241 1.5986

Non-local mean 1.0030 36.3661 9.9594

Median 1.3203 11.0196 0.3581

Frost 1.0003 20.3473 7.1417

FIRE 1.0096 4.1955 5.8784

TMAV 0.7339 12.8690 3.7278

ATMAV 1.0088 17.3182 4.5570

ATMED 1.1150 15.8322 4.2091

Proposed 1.0907 32.5538 10.1256

Table 3 presents the result of filtering square image. It can be observed that all the filters except mean and median preserve strong edges. All the filters except FIRE filter improve the SNR after filtering. There is a significant deviation of MoR in median and TMAV filtered image. The proposed filter reconstructs strong edges and improves the quality of images with an insignificant deviation of MoR. The filtered image shown in Fig. 6c has smooth reconstructed edges.

Table 4
Result for corner

MoR SNR EPF

Clean 0.9944 – 1.0056

Noisy – 48.7431 1.0055

Mean 1.0047 12.9022 0.4405

Non-local mean 1.0040 61.8231 1.0054

Median 1.4249 36.4657 0.2453

Frost 1.0011 26.5594 1.0038

FIRE 1.0100 41.6012 1.0059

TMAV 0.6412 1.3713 1.6025

ATMAV 1.0428 21.4332 0.4410

ATMED 1.1742 27.0726 0.0774

Proposed 1.0064 48.7482 1.0674

Table 5
Result for building

MoR EPF

Clean 1.000029 3.653114

Noisy – 3.653225

Mean 1.000003 0.564290

Non-local mean 1.002660 3.652114

Median 1.209589 3.180514

Frost 1.056504 3.501994

FIRE 1.012211 3.657228

TMAV 0.886048 0.842154

ATMAV 0.638985 0.575838

ATMED 1.086523 3.344115

Proposed 0.9999 3.5075

Figure 7.
Pentagon SAR image. (a) Clean. (b) Noisy. (c) Mean. (d) Median. (e) NL mean. (f) ATMAV. (g) ATMED. (h) FIRE. (i) Frost. (j) TMAV. (k) Proposed.

Table 6
Result for pentagon

MoR SNR EPF

Clean 0.9953 – 10.3578

Noisy – 30.5139 7.7587

Mean 1.0046 38.6277 7.8199

Non-local mean 1.0075 32.9874 8.2050

Median 1.0105 37.7078 8.9114

Frost 1.0043 31.7732 9.0032

FIRE 0.9983 31.4714 7.9628

TMAV 0.9850 38.2820 8.2782

ATMAV 1.9038 37.7732 9.0397

ATMED 1.0058 39.1975 9.2581

Proposed 1.0043 40.0298 9.6263

The performance of filtering corner image shown in Fig. 4d is presented in Table 4. The TMAV reconstruct edges, but at the cost of radiometric distortion. Mean, median, ATMAV and ATMED does not preserve edges in corner image. The local mean filter improves the SNR significantly. The image shown in Fig. 5d is the filtered image using the proposed approach. It can be seen that the speckle noise are suppressed, but the side lobe could not be reconstructed as the filter is not able to differentiate the thin and fragmented lines from noise.

Table 5 presents the performance of filtering building image shown Fig. 4e, it is a very high resolution image simulated considering the reflection from nearby building walls and rough terrain with double and triple reflections. The reflection lines can be used for estimating the height of the building; therefore it is important to preserve the radiometric after filtering. The proposed filter and non-local mean preserve the radiometric after despeckling as there is a very small deviation of MoR from one. The reflection lines in filtered image Fig. 6e is still visible. It is also worth mentioning that the Mean, median, TMAV and ATMAV does not preserve edges.

It is a general observation that the non-local filter perform well in images containing large homogeneous region, but it is not suitable for using in complex image like DEM with many edges of different shape and size. The proposed fuzzy filter can suppress speckle noise and preserve edges with a negligible deviation of MoR from one in all the images shown in Fig. 4. It performs much better than the classical fuzzy filters in suppressing speckle noise from SAR images.

Figure 7a shows a clean pentagon SAR image, Fig. 7b is the speckle noise corrupted image and Fig. 7c–k are the images filtered by the filters considered in this work. It can be observed in Fig. 7k that the proposed filter can suppress speckle noise after filtering. The filtered image is slightly blurred due to distortion of very small details, but the strong edges are preserved, differentiating noise form details of size one or two pixels is extremely difficult. Table 6 shows the result of filtering pentagon image. It is clear from the result that the proposed filter improves the quality of images and preserved the strong edges in the filtered image.
4. Conclusion

	ENL	MoR	SNR
Noisy	0.9996	–	6.0422
Mean	6.9563	0.9977	17.9626
Non-local mean	74.8274	1.0020	36.6982
Median	4.0274	1.3270	11.0358
Frost	9.6317	1.0003	20.4880
FIRE	2.7145	1.0133	4.7026
TMAV	5.0535	0.7355	12.9027
ATMAV	6.4210	1.0073	17.3605
ATMED	5.6731	1.1141	15.9057
Proposed	73.8200	1.0903	32.8100

	MoR	SNR	EPF
Clean	1.0079	–	6.1835
Noisy	–	6.2152	3.2871
Mean	1.0019	3.9815	0.6357
Non-local mean	1.2009	1.7356	2.6968
Median	2.0402	2.5037	0.5112
Frost	1.0109	7.4395	2.8383
FIRE	0.9956	5.4446	3.6065
TMAV	0.5345	4.9280	1.9158
ATMAV	1.0283	3.6671	1.4921
ATMED	1.3745	0.1958	0.2964
Proposed	1.0902	4.7010	3.6896

	MoR	SNR	EPF
Clean	0.9990	–	12.1978
Noisy	–	6.0298	2.1167
Mean	1.0025	17.9241	1.5986
Non-local mean	1.0030	36.3661	9.9594
Median	1.3203	11.0196	0.3581
Frost	1.0003	20.3473	7.1417
FIRE	1.0096	4.1955	5.8784
TMAV	0.7339	12.8690	3.7278
ATMAV	1.0088	17.3182	4.5570
ATMED	1.1150	15.8322	4.2091
Proposed	1.0907	32.5538	10.1256

	MoR	SNR	EPF
Clean	0.9944	–	1.0056
Noisy	–	48.7431	1.0055
Mean	1.0047	12.9022	0.4405
Non-local mean	1.0040	61.8231	1.0054
Median	1.4249	36.4657	0.2453
Frost	1.0011	26.5594	1.0038
FIRE	1.0100	41.6012	1.0059
TMAV	0.6412	1.3713	1.6025
ATMAV	1.0428	21.4332	0.4410
ATMED	1.1742	27.0726	0.0774
Proposed	1.0064	48.7482	1.0674

	MoR	EPF
Clean	1.000029	3.653114
Noisy	–	3.653225
Mean	1.000003	0.564290
Non-local mean	1.002660	3.652114
Median	1.209589	3.180514
Frost	1.056504	3.501994
FIRE	1.012211	3.657228
TMAV	0.886048	0.842154
ATMAV	0.638985	0.575838
ATMED	1.086523	3.344115
Proposed	0.9999	3.5075

	MoR	SNR	EPF
Clean	0.9953	–	10.3578
Noisy	–	30.5139	7.7587
Mean	1.0046	38.6277	7.8199
Non-local mean	1.0075	32.9874	8.2050
Median	1.0105	37.7078	8.9114
Frost	1.0043	31.7732	9.0032
FIRE	0.9983	31.4714	7.9628
TMAV	0.9850	38.2820	8.2782
ATMAV	1.9038	37.7732	9.0397
ATMED	1.0058	39.1975	9.2581
Proposed	1.0043	40.0298	9.6263

A fuzzy based weighted mean filter is presented in this article. The weights are computed using fuzzy rules and the membership function parameters used for defining the rules are determined using PSO. Proposed filter is comparatively evaluated with some of the existing noise filters. We have adopted ENL, MoR, SNR and EPF for quantitative analysis of the results and found convincing results on common elementary scenes encountered in typical SAR images. The proposed filter suppresses speckle noise and also preserves edges in the filtered image.

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	ENL	MoR	SNR
Clean	436.9726	1.0077	–
Noisy	0.9996	–	6.0422
Mean	6.9563	0.9977	17.9626
Non-local mean	74.8274	1.0020	36.6982
Median	4.0274	1.3270	11.0358
Frost	9.6317	1.0003	20.4880
FIRE	2.7145	1.0133	4.7026
TMAV	5.0535	0.7355	12.9027
ATMAV	6.4210	1.0073	17.3605
ATMED	5.6731	1.1141	15.9057
Proposed	73.8200	1.0903	32.8100