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\begin{document} 

\title[Article Title]{Optimal Condition Evaluation for WNN Based Speckle Filtering Algorithms in SAR Image Denoising}
\author*[1,2]{\fnm{G.,} \sur{Preethisree}}\email{preethisreeg@lpsc.gov.in} 
\author[2,3] {\fnm{V.,} \sur{Vivekanand}}\email{v\_vivekanand@vssc.gov.in}
\author[2]{\fnm{Deepak} \sur{Mishra}}\email{deepak.mishra@iist.ac.in}
\author[3]{\fnm{R.,} \sur{Sethunadh}}%\email{r\_sethunadh@vssc.gov.in}  

%----------- 
 \affil[1]{
\orgdiv{Liquid Propulsion System Centre}, 
%\orgname{Indian Space Research Organisation}
\orgaddress{ 
 	%\street{Street}, 
	%\city{Thiruvananthapuram}, 
	%\postcode{100190}, 
	%\state{State}, 
	%\country{INDIA}
	}
}
%-----------  
\affil[2]{
\orgdiv{Indian Institute of Space Science and Technology}, 
%\orgname{Department of Space}.
\orgaddress{
 	%\street{Street}, 
	%\city{Thiruvananthapuram}, 
	%\postcode{10587}, 
	%\state{State}, 
	%\country{INDIA}
	}
}
%----------- 
\affil[3]{
\orgdiv{Vikram Sarabhai Space Centre}, 
%\orgname{Indian Space Research Organisation} 
\orgaddress{ 
 	%\street{Street}, 
	 \city{Thiruvananthapuram}, 
	%\postcode{100190}, 
	%\state{State}, 
	 \country{INDIA}
	}
} 
%%==================================%%
%%  
%%==================================%%

 \abstract
{
This paper investigates the application of the weighted nuclear norm minimization method for despeckling synthetic aperture radar images, for various input configurations and acquisition scenarios. 
%
The  approach, originally designed for additive white Gaussian noise, was adapted to handle SAR-specific multiplicative noise using a homomorphic transformation. Speckle filtering was performed in the logarithmic domain, followed by exponential reconstruction to extract the image. 
Various similarity measures were used to evaluate the performance. 
%
Results reveal that despeckling  log-transformed images showed superior performance compared to other evaluated methods, achieving better visual quality and structural preservation. 
%
While different similarity measures applied to log-transformed images provided comparable results, the WNNM method consistently demonstrated its ability to enhance image quality and objective metrics, and   its effectiveness strongly dependent on identifying similar segments in noisy images.
% 
The result of the evaluation  are presented with extensive simulations results on various test images and SAR data.
}
 %-----------------
\keywords{Speckle Noise, Denoising, Image Segmentation, Norm Minimization, Weighted Nuclear Norm}
 %-----------------

\maketitle
%-----------------

%-----------------
\section{Introduction}
%-----------------
% First Edit 2023  October  4
% Viv Edit   2025  January 20
The random interference   in Synthetic Aperture Radar (SAR) imaging  process  
result in speckle noise. 
%
This is mainly due to   multiplicative noise perturbation, 
consequently degrades the visual quality of the SAR images and
undermines the   of the image processing methods used for data interpretation \cite{SAR-RFI}. 
%
The speckle noise is to be removed  before to the image recovery to minimize the information loss \cite{SARdenoise}
The  methods for additive noise removal are not effective in this scenario.
%  
However, the multiplicative noise can be transformed into additive form of noise by transforming the image matrix with    logarithm;
but the   transformation  changes the normal  distribution of the image too.
The  despecking approaches based on   non-stationary or non-spacial statistics of the image are used in such cases. These approaches, however are not effective and are computationally intensive. 
%
The image despeckling is essential preprocessing step to avoid artifacts, during extraction of the information from the SAR images.
%
Various methods are available for despeckling such as spatial filtering, 
transform domain filtering  \cite{choi-1} and non local filtering. 
%
The spatial filtering is  simple, but result  in over smoothing,  loss of  resolution and textual information content.
%
The transform domain methods   suppress speckle noise, 
but lead to pixel distortions and artifacts \cite{denoise_review}.
%
Total variation methods preserve the edges well but produces staircase artifacts  in the filtered images. 
%-------------
%-------------

\par
The non local filtering methods uses the similarity of non local segments and uses segment based denoising techniques. These methods achieved speckle suppression with good texture preservation.
 %
The Non-Local Means (NLM) filter is a segment based denoising technique that exploits  the self similarity of the image for noise removal and produced good despeckling results.
%
This approach use the  weighted filters. The weights were calculated based on similarity between the intensity values of the segments surrounding the pixels. The $\ell_2$  distance is used as metric for finding similarity  between segments.
% 
%
The probabilistic patch-based (PPB) \cite{ppb} algorithm is also a non-local filtering method, where the similarity between  segments are determined by the noise distribution rather than the $\ell_2$  distance. 
This   achieves better speckle suppression by iteratively updating the weights \cite{patch_review}. 
%
%
The Block matching 3D algorithm \cite{bm3d} uses the self similarity and the sparsity of images.  It combines similar image segments with three dimensional (3D) transformation to enhance the sparsity level. 3D wavelet transformation is applied on the segments and the wavelet  coefficients of the   image is estimated through Weiner filtering.  
%
SAR-Block-Matching 3D (SAR-BM3D)  algorithm \cite{ImprovedSAR} extends the BM3D algorithm to SAR images containing speckle noise. It combines the non  local methods and the transform methods.  The Wiener filtering is replaced with a local Least Mean Squares Estimator (LMSE) \cite{LMSE} that can estimates the wavelet coefficients of the image more precisely.
%
Guided Non Local Means (GNLM)  \cite{GNLM1} uses the structure information of the co-registered optical images to provide beneficial image filtering\cite{GNLM2}, as it is easier to find the best predictor in a noise free  image.  
%  
The low rank representation methods use  the low rank property of the similar segment group formed by the image segments for despeckling \cite{Lowrank1}.
%
The machine learning based low rank denoising is described in \cite{LEANFIL}.
The multiplicative noise model is converted to additive noise model before using low-rank methods \cite{Lowrank2}. 
These methods uses the soft thresholding of the singular values of the similar image segments for denoising \cite{Lowrank3}.
%
The weighted nuclear norm (WNN) \cite{wnnm} is used as the measure of self similarity for grouping the similar segments and the WNN minimization problem is solved using the soft thresholding operator \cite{wnnm2}. 
%
The WNN minimization  method applies logarithmic transformation to the noisy images to convert the speckle noise is into additive noise \cite{WNN-Grey}.
%
Then by matching the local blocks of images based on similarity measure, approximates the low rank matrix grouped by the similar blocks of the reference segment.
%
WNN minimization method is applied on these segment groups to denoise the image segment group. Finally exponential transformation is applied to revert back the image. 
%
The non homomorphic methods directly works on multiplicative noise \cite{HOMOmorph1}.
These methods are also effective but are complex in terms of computation \cite{HOMOmorph2}. 
%
A parametric analysis of the WNN minimization  based speckle filtering algorithms is presented here. 
The WNN minimization algorithm uses homomorphic filtering with $\ell_2$ based similarity measure for denoising. 
%
The performance of WNN minimization method under homomorphic and non homomorphic approaches with different similarity measure such as euclidean ($\ell_2$) and logarithmic ratio of pixel intensities are discussed.
%
The paper is organized as follows. 
The section 2 discusses about the speckle noise affects on SAR images,    
the noise filtering method used,
different configurations of the noise removal methods and 
the parameter measures used in the performance evaluation;
%
Section 3 presents the experimental setup for the simulation of various test cases and the denoising of test images. This is followed by a comprehensive evaluation of the proposed denoising configurations on SAR image scan data, aimed at experimentally determining the optimal SAR data acquisition scenario for these methods to achieve maximum efficiency.
%
%
%The final section gives the parametric $psnr$ model, followed by the  discussion on the results and comparison with the benchmarking algorithms.
%------------------------------------------
%------------------------------------------
%------------------------------------------
%------------------------------------------
\section{The Speckle Noise}
%------------------------------------------
%Write something about this... 
%Brief detailing about how the speckle noise affects SAR images 
%How the speckle noise are introduced 
%Write about the number of looks in SAR imaging 
%------------------------------------------
%------------------------------------------
The synthetic aperture radar (SAR) imaging is sensitive to speckle noise due to tis coherent nature of image data acquisition.
%
The SAR system induce random interference patterns into the image due to back scattering of the radiations.
%
The back scattered signals contain both in-phase and quadrature-phase components, which add constructively or destructively to introduce  interference pattern at random locations, generally indicated as the    speckle noise in the SAR image.
%
The variation in the  pixel intensity   due to this noise speckle  spoils the  scene contrast and alters the spatial statistics of the pixels in the scene. 
 %
The degradation of this visual quality of the SAR image  prevents the image data interpretation.
This noise needs to be eliminated before applying any data interpretation   methods.
%
The multi-look scene scanning method is used to reduce this kind of noise speckle, where several independent samples of  image are averaged to estimate the good quality scene image, however, the averaging process degrades the  resolution.
%
The speckle noise is modeled  as     independent and identically distributed  (iid)  multiplicative noise (\ref{mul}).
%--------------------
\begin{equation}\label{mul}
	I = X \times U
\end{equation}
%--------------------
where $I$ is the observed or measured scene reformatted as pixel intensity matrix,  
$X$ is the underlying target scene,  the noise free ground truth 
and $U  \sim \Gamma(L)$ is the  multiplicative  speckle noise,
with single parameter Gamma distribution.
Here  $L$ is number of independent measurements or number of looks and
the probability distribution   is modeled  as $P(U=u) = \frac{L^L}{\Gamma(L)}u^{L-1}e^{-Lu}$, 
with mean = $1$ and variance = $\frac{1}{L}$. 
%----------------------------------------
%----------------------------------------
%\begin{equation}\label{gamma}
%P(U=u) = \frac{L^L}{\Gamma(L)}u^{L-1}e^{-Lu}
%\end{equation}
%----------------------------------------

%----------------------------------------
%----------------------------------------
%----------------------------------------
%----------------------------------------
\subsection{The Speckle Filtering  }
%---------------------------------------- 
%
The image is transformed into $log$ image  to change  the   multiplicative noise component into additive component (\ref{EQlog}).  
%---------------------------
\begin{equation}\label{EQlog}
\ln (I) = \ln (X) + \ln (U)
\end{equation}
%---------------------------
Over this log transformed image, the  weighted nuclear norm minimization is applied,
which is a segment based denoising technique using the rank minimization method.
% 
%---------------------------
In this method, the reference segment and the other   segments for similarity grouping are vectorized as column vector and stacked to form the sub images.  
%
The similar segments for this segment segregation are determined  using on the   distance metric (\ref{block})  
%---------------------------
\begin{equation}\label{block}
BSM_{(i,j)} = \sum_{k=1}^{n} 
log \left[\frac {\sqrt{X_i(k)}} {\sqrt{X_j(k)}}  + \frac {\sqrt{X_j(k)} }{\sqrt{X_i(k)} } \right] 
\end{equation}
%--------------------------- 
The denoising  is  reformulated as   matrix rank minimization problem and   the matrix model is represented as (\ref{wnnm_noise_model}),
where $Y_j$ is the sub image formed from the similar segment group, $X_j$ and $N_j$ are ground truth image and the noise respectively.
%---------------------------
\begin{equation}\label{wnnm_noise_model}
	Y_j = X_j + N_j
\end{equation}
%---------------------------
 The rank minimization  is defined as (\ref{wnnm}),
where $\|.\|_F$  is Frobenius norm and  $\sigma^2$ is the assumed noise variance. 
%---------------------------
\begin{equation}\label{wnnm}
\hat X_j =\min \limits_{X_j} {\frac{1}{\sigma^2} \|Y_j-X_j\|^2_F}+ \|X_j\|_{(w,*)}
 \end{equation}
%--------------------------- 
%
The $\|X_j\|_{(w,*)}$   is  weighted nuclear norm  (\ref{nuclear norm}) of $X_j$.
%---------------------------
%---------------------------
 \begin{equation}\label{nuclear norm}
\|X_j\|(w,*) = \sum_k w_k \sigma_k (X_j)
 \end{equation}
%---------------------------
After arranging the singular values of $X_j$ in non ascending order 
$(\sigma_{1}(X_j)\geq \sigma_{2}(X_j) \geq.....\geq \sigma_{k}(X_j))$, 
the closed form solution to the minimization problem with thresholding  is obtained from the singular value decomposition of   $Y_j = U\Sigma V^T$    and thresholding  (\ref{wsvt}). 
%---------------------------
%---------------------------
\begin{equation}\label{wsvt}
\hat{X_{j}} = US_{w}(\Sigma)V^T
\end{equation}
%---------------------------
%---------------------------
where  $S_{w}$ is the soft thresholding operator (\ref{soft}) and $w_k$ the threshold     (\ref{wnnm_weight}).
%---------------------------
\begin{equation}\label{soft}
S_{w}(\Sigma)_{kk}= \max( {\Sigma_{ii}- w_{k},  0} ), \quad \Sigma_{kk} \in \mathbb{R}
\end{equation}
%---------------------------
%---------------------------
%---------------------------
\begin{equation}\label{wnnm_weight}
w_{k}=\frac {c\sqrt {n}}{ (\sigma_{k}(X_{j})+\epsilon)}
\end{equation}
%---------------------------
where $c$ is a positive constant, $\epsilon$ is a finite value and $n$ is the number of similar segments.
%-----------------
%-----------------
The threshold of the singular values $\sigma_{k}(X_{j})$ is set to inversely proportional to its magnitude, the larger singular values with the significant information are retained. 
The $\sigma_k(X_j)$ for weight calculation is initialized using (\ref{init_est}).
%----------------- 
\begin{equation}\label{init_est}
\hat{\sigma}_{k}(X_j) = \sqrt {\max( 0,   \sigma_k^2(Y_j)-n\sigma^2 )}
\end{equation}
%----------------- 
%-----------------
A comparison of WNN minimization based despeckling method with different approaches are presented. 
Four   cases of image analysis are identified.   The benchmarking algorithms use are PPB and SAR BM3D.
%-----------------
\par
Method 1: WNN-L2,  The original image is set as input  and the euclidean  ($\ell_2$) distance  is the metric for similar segment identification. 
\par
Method 2: WNN-Log, The original image is set as input   and the $log$ ratio of the pixel intensities is  the metric for similar segment identification. 
\par
Method 3 : Log-WNN-L2, The $log$ transformation of the image is set as   input  and the euclidean ($\ell_2$)  distance between the pixels is use as metric for similar segment identification.
\par
Method 4 : Log-WNN-Log, The $log$ transformation of the image is set as the input and the $log$ ratio of the pixel intensities is used as metric for similar segment identification
%----------------- 
%-----------------
%----------------- 
%-----------------
%--------------------------- 
%---------------------------
\subsection{The Parameter Measures }
%---------------------------
%---------------------------  
The   quality  of the denoised images are measured in terms of the mean squared error (MSE)  (\ref{MSE})  and the peak signal to noise ratio (PSNR)  (\ref{psnr}).
%-------------------
\begin{equation}\label{MSE}
	MSE = \frac{\|X-\hat{X}\|_2^2} {M N}
\end{equation}
%-------------------
%-------------------
\begin{equation}\label{psnr}
	PSNR = 10 \log ( { 255 ^2} / {MSE})
\end{equation}
%-------------------
%-------------------
where $X$ is  unperturbed image     and $ \hat{X} \in \mathbb{R}^{M\times N}$  the recovered image.
%-------------------
\color{black} 
The pixel vise  comparison of the image  is  done  by determining the signal to noise ratio (SNR) (\ref{SNR}), where $X_{i,j}$ is the individual pixel value.
%-------------------
%-------------------
\begin{equation}\label{SNR}
	SNR =  10 \log \frac{1}{MN} \sum_{i=1}^{M} \sum_{j=1}^{N} 
	\frac{ X_{i,j} ^2}{(X_{i,j}-\hat {X}_{i,j})^2} 
\end{equation}
%-------------------
%-------------------
The pixel information  entropy   $H(X_{i,j})$  is estimated  from the probability $P(X_{i,j} = x_{i,j} )$ of that  pixel value (\ref{LogInfo}).
%-------------------
%-------------------
\begin{equation}\label{LogInfo}
	H(X_{i,j}) = P(X_{i,j}) \log P(X_{i,j}) 
\end{equation}
%-------------------  
%------------------- 
The pixel entropy is taken as the weight  for computing  
iWSNR the information weighted mean squared signal to noise ratio  (\ref{iWSNR}).
%------------------- 
\begin{equation}\label{iWSNR}
iWSNR = 
	10 \log \frac{1}{MN} \sum_{i=1}^{M} \sum_{j=1}^{N} 
	\frac{ H(X_{i,j})  X_{i,j} ^2}
	{(X_{i,j}-\hat {X}_{i,j})^2} 
\end{equation}
%-------------------
%-------------------
The structural similarity index (SSIM) (\ref{ssim})   and  
the multiscale structural similarity index (MS-SSIM) \cite{MS-SSIM} are also used in the evaluation of the algorithm.
%------------------- 
\begin{equation}\label{ssim}
	SSIM(X, \hat X) = 
	\frac{(2\mu_{X}\mu_{\hat {X}}  + c_1)   (2\sigma_{X, \hat X}+c_2)}
		 {(\mu_X^2 + \mu_{\hat X}^2 + c_1) (\sigma_X^2 + \sigma_{\hat X}^2 + c_2)}
\end{equation}
%-------------------
where $c_1$, $c_2$ are constants  and
$\mu_X$,  $\sigma_X$  and   $\mu_{\hat{X}}$,  $\sigma_{\hat X}$  are the mean and variances of the 
reference image $X$ and  the recovered signal $\hat X$. The covariance between $X$ and $\hat X$ is presented as $\sigma _{X\hat X}$
%-------------------
The edge preservation index (EPI)   determined as (\ref{epi}) is also used in the comparison.
%-------------------
\begin{equation}\label{epi}
EPI = 
\frac
{\sum_{i=1}^{M} \sum_{j=1}^{N}  (X_{i,j} -  \mu_X )    (\hat X_{i,j} -  \mu_{\hat{X}} )   }
{\sum_{i=1}^{M} \sum_{j=1}^{N}  (X_{i,j} -  \mu_X)^2 (\hat X_{i,j} -  \mu_{\hat{X}})^2 }
\end{equation}
%---------------------------------------------------
%---------------------------------------------------
%---------------------------------------------------
%------------------------------------------------------
\begin{figure}
\centering 	 
%---------------- 
	\begin{minipage}[b]{1\textwidth}
	 \includegraphics[width=0.48\textwidth]{Block_diagram.png}  
	\end{minipage}
 %---------------- 
\caption{The algorithm evaluation setup. The reference image is perturbed with multiplicative noise.
The image segments with similarity are arranged in groups. The SVD and   value thresholding are  performed to eliminate speckle noise. The matrix is reverted back to the image format  }  
\label{fig:Block_diagram}
\end{figure}
%-------------------------------------------------------
%---------------------------------------------------
\begin{figure*}
\centering 	 
%---------------- 
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{PSNR-Barbara.png}   
	 \\ \centering\tiny (a)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{PSNR-Camera.png} 
	\\ \centering\tiny (b)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{PSNR-House.png}
		\\ \centering\tiny (c)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{PSNR-Leafs.png}
		\\ \centering\tiny (d)
	\end{minipage}
%---------------- 
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Barbara.png}   
	 \\ \centering\tiny (e)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Camera.png} 
	\\ \centering\tiny (f)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-House.png}
		\\ \centering\tiny (g)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Leafs.png}
		\\ \centering\tiny (h)
	\end{minipage}
%---------------------------	
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Barbara.png}   
	 \\ \centering\tiny (i)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Camera.png} 
	\\ \centering\tiny (j)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-House.png}
		\\ \centering\tiny (k)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Leafs.png}
		\\ \centering\tiny (l)
	\end{minipage}
%---------------------------	
\iffalse
	\begin{minipage}[b]{.22\textwidth}
	 \includegraphics[width=1.2\textwidth]{PSNR-Monarc.png}
		\\ \centering\tiny (f)
	\end{minipage}
	\begin{minipage}[b]{.22\textwidth}
	 \includegraphics[width=1.2\textwidth]{PSNR-Parrot.png}
		\\ \centering\tiny (g)
	\end{minipage}
	\begin{minipage}[b]{.22\textwidth}
	 \includegraphics[width=1.2\textwidth]{PSNR-Starfish.png}
		\\ \centering\tiny (h)
	\end{minipage}
	\begin{minipage}[b]{.22\textwidth}
	 \includegraphics[width=1.2\textwidth]{PSNR-Lena.png}
		\\ \centering\tiny (i)
	\end{minipage}
        %---------------- 
\fi
\caption{
PSNR, SSIM  and EPI  of the  images denoised using various algorithms on   test images.
%  (i) Barbara,  (ii) Camera, (iii) House, (iv) Leafs.  % (e) Monarc, (f) Parrot, (g) Starfish.
The window size used is $31\time31$ and segment size is $7 \time 7$ and number of looks is varied from $1$ to $16$.
The  reference used are the BM3D-SAR and PPB methods.
The cases $2,3,4$ are used in the simulation. }  
\label{fig:PSNR}
\end{figure*} 
%---------------------------------------------------
%---------------------------------------------------
%---------------------------------------------------
%---------------------------------------------------
%------------------------------ 
\section{Simulation and Evaluation}
%------------------------------ 
The cases described in the previous section are simulated using the  setup shown in Figure \ref{fig:Block_diagram}.
%
The experimental results obtained from 
applying  the bench mark methods and  the proposed methods  on  
the speckle induced images for  different test cases  are presented.
%
Eight  simulated images are taken. 
Speckle noise with different number of  looks $L=1,2,4,8, 6 $ are induced into the image.
%
The image $Y$ is divided into  overlapping set of segments  ($ S = \{Y_j : j=1,2,.......(N-d+1)^2\}$,  $ \|S\|_0 =  (N- d+1)^2$ ) each with  size $d \times d$  and the  similarity between them with in the search window  size  is $n \times n$ is found,  using the logarithmic  ratio of pixel intensities or the euclidean ($\ell_2$) distance. 
%    
The segment $Y_j \in S$ is taken as the indicator segment for the pixel at its center.  
%   
The  segments are then converted to a column vector and  are   arranged according to their similarity with the adjacent one.   This transforms the image matrix into low rank representation.  %
The method is applies on the  given noise induced  images and also on its 
$log$ transformed image,  to evaluate the effect of  initial transformation. 
%---------------- 
%----------------
The best estimate of each segment $\hat{Y}_j$  obtained by the singular value thresholding and the $exponential$ transformation is applied to recover the despeckled image. 
%
The simulations are done on the test images for  the test cases identified  and compared with benchmark methods PPB, SARBM3D.  
%
%---------------------------------------------------
%---------------------------------------------------
The variation of peak signal to noise ratio (PSNR)  for various reconstructed image for different  number of looks are shown in   
 Figure \ref{fig:PSNR}(a-d) and the corresponding plots for  the structural similarity index (SSIM) and the
edge preservation index (EPI)  are shown in Figures \ref{fig:PSNR}(e-h) and (i-l) respectively.
%
The   images of size $256 \times256$  are taken from $Matlab$ image processing repository.  
The evaluation metrics are calculated for the original   images too, for reference.  The number of looks used in the evaluation are $L= 1, 2, 4, 8$  and $16$ and the noise variance is  changed from $0.01$ to $ 3.5$.
%
%---------------------------------------------------
%--------------------------------------------------- 
%
%This shows the variation in the PSNR of the images with WNNM algorithm with different approaches. 
%
\par
No significant improvement is observed in case-1 simulation,  
when WNN minimization and $\ell_2$ similarity measure is applied directly on the images.
%
In case-2 simulation with  $log$ distance, the PSNR recovery is inferior to the bench mark algorithms, 
%
while case-3 and case-4 simulations  with homomorphic filtering and with $\ell_2$ and $log$ distance measurements respectively, the PSNR performance are comparable for different noise variance from $0.01$ to $0.14$ and is better than the bench mark algorithms. 
%
In case 4, when noise variance is increased beyond $0.2$, the PSNR recovery degrades for all test images simulated.
%
For higher values of noise variance the PSNR recovery shows no improvement, when number of looks is increased beyond 8. 
%
The  BM3D and PPB methods with both iterative and non iterative method  have better PSNR values and have limited influence on number of looks. 
%
From Figure \ref{fig:PSNR} it can be observed that   for noise variance $\sigma\ge 0.2$  the performance degrades especially when $L \ge 8$. 
%
%---------------------------------------------------
\par 
%---------------- 
%---------------- 
The variation of SSIM for various methods and test images are shown in Figure \ref{fig:PSNR}(e-h)
For a given noise variance, the SSIM increases as the number of looks $L$ increases, 
then saturates to a value near 1. 
However if noise variance increased beyond 0.2, the performance degrades.
%
The SSIM measurements are better for $Log$-WNN minimization  compared to direct WNN minimization. 
As $\sigma$ increases above $0.2$ the performance deteriorates and for higher $\sigma$ the increase in number of looks do not have significant effects. 
%
The variation of edge preservation shown in Figure \ref{fig:PSNR}(i-l)  is better in WNN minimization method compared to BM3D  and PPB methods.  The performance of $log$-WNN  minimization  is better than direct WNN minimization.
The  $Log$-WNN minimization with $\ell_2$ and $log$  distance measure provide same edge preservation measure for variation in $\sigma$ from $0.14 $ to $3.5$.  
%
The maximum performance is observed for $\sigma = 0.2$, beyond this value that performance degrades. 
The EPI metric also shows an increases with number of looks for  given noise variance.
%
In the illustrated figures,  the legend  WN-Log-WxSx-$\sigma x$  indicate   case-2 simulation and
LogWN-Eu-WxSx-$\sigma x$  indicates case-3 simulation corresponds to WNN minimization on $Log$ transformed image with $\ell_2$ norm as pixel similarity measure. 
%
WNN minimization on $Log$ transformed image with $Log$ of pixel ratio as distance measure is indicated with legend  LogWN-Log-WxSx-$\sigma x$, corresponding to the case-4. The noise variance is indicated in $\sigma$, the window size and segment size are indicated as $W[x]S[x]$. 
%---------------------------------------------------
%---------------------------------------------------
%----------------  
%----------------    
 %---------------
\begin{figure*}
\centering
	%---------------- 
	\begin{minipage}[b]{1\textwidth}
	 \includegraphics[width=0.16\textwidth]{Average-PSNR-LogWL2.png}   
	\includegraphics[width=0.16\textwidth]{Average-iWSNR-LogWL2.png}   
%
	 \includegraphics[width=0.16\textwidth]{Average-SSIM-LogWL2.png}   
	 \includegraphics[width=0.16\textwidth]{Average-MS-SSIM-LogWL2.png}   
	 \includegraphics[width=0.16\textwidth]{Average-EPI-LogWL2.png}   
	 \includegraphics[width=0.16\textwidth]{Average-INFO-LogWL2.png}   
	 %\\ \centering\tiny (a) avg. Structural Similarity Index
	\end{minipage}
      %---------------- 
	%---------------- 
	\begin{minipage}[b]{1\textwidth}
	\includegraphics[width=0.16\textwidth]{Average-PSNR.png}  
	\includegraphics[width=0.16\textwidth]{Average-iWSNR.png}   	
%
	\includegraphics[width=0.16\textwidth]{Average-SSIM.png}  
	\includegraphics[width=0.16\textwidth]{Average-MS-SSIM.png}   
	\includegraphics[width=0.16\textwidth]{Average-EPI.png}  
	\includegraphics[width=0.16\textwidth]{Average-INFO.png}  
	% \\ \centering\tiny (b) avg. Multi Scale Structural Similarity Index
	\end{minipage}
      %---------------- 
\caption{ 
Averaged performance metric for   Log-WNN-L2 method and Log-WNN-Log method.
PSNR,   iWSNR SSIM,  MS-SSIM,  EPI, and information content; averaged over nine different images recovered from speckle noise perturbed image matrix. The multiplicative noise perturbation is varied from $0.01 \le \sigma \le 2.0$. }
\label{fig:AvgSSIM}
\end{figure*}
%----------------------------------
%----------------------------------  
%---------------------------------------------------
%---------------------------------------------------
\subsection{Average Variations}
%---------------------------------------------------
\par
The average  variations of the evaluation metric  corresponding to the test case-3  and case-4  is shown in Figure \ref{fig:AvgSSIM}.  
 %
The average values of PSNR,  iWSNR  SSIM and EPI increases with increase in the noise level from $ 0.01 \le \sigma \le  0.2 $. 
%
For number of looks up to $4$  here is no large difference in the performance; but for $L \ge 4 $ significant improvement is noticed. 
%
The average PSNR increases with increase in the noise level from $  0.01 \le \sigma \le 0.2 $ and from $0.4$ onwards the performance degrades.   
%%
The average SSIM increases with increase in the noise level from $  0.01 \le \sigma \le  0.2$ and from $0.4$ onwards the performance degrades and no improvement  observed with respect to number of looks. 
%%
The average EPI increases with increase in the noise level from $  0.01 \le \sigma \le  0.2$ and from $0.4$ onwards the performance degrades.
%
%
In general it is noted that for   $0.01 <  \sigma < 0.2$  the average values of the metric are better compared to other range. 
%
 The average PSNR varies from 12dB to 33dB corresponds to the number of looks $2 $ and $16$. 
%
With respect to noise variance, when $L = 16 $ the PSNR variation is from 23dB to 33dB, with maximum value for $\sigma = 0.14$ and for  $\sigma \ge 0.4$  the performance  degrades.
%
The   average  change of   SSIM,  MS-SSIM,  EPI  and  the Information contents for 
various values of number of looks and noise variance is studied and 
the results for the test  case-3 and case-4  are shown in Figure \ref{fig:AvgSSIM}.
%
The average value of SSIM and EPI shows similar trend as that of PSNR.
%
The average value of SSIM varies from $0.2$ to $0.93$  when the noise variance $\sigma$ increased from $0.01$ to $0.2$ after that it degrades. 
%
In the case of EPI measure, the corresponding values are $0.6$ to $0.98$  for different number of looks and reduces with further increase in variance. 
%
%----------------   
%----------------------------------
%----------------------------------  
\subsection{Comprehensive Evaluation}
%-------------------------------------  
The noisy data acquisition is simulated for different number of looks 
($1$ to $16$) and the recovery  is performed over  noise levels $[0.01 \le \sigma \le 4.0]$.
%
The reconstruction performance is evaluated using Peak SNR,  SNR, Information Loss,  information weighted SNR (iWSNR),   SSIM, MS-SSIM,  EPI  and  the information content  values.
%
The methods evaluated are WNN-L2 ,  WNN-Log, Log-WNN-L2 and Log-WMM-Log algorithms.
%
The combined variation ($L, \sigma$ vs $metric$) corresponding to  WNN-L2  and WNN-Log methods are shown in  
Figure \ref{TEST-LW2-PSNR}(1),(2) respectively,  followed by the   graphs of Log-WWN-L2 and Log-WNN-Log in Figure  \ref{fig:AvgMESH}.  
%
In all the evaluations, the multiplicative noise  is changed from $0.01 \le \sigma \le 2.0$ and
 the number looks in the data acquisition process is varied from $1$ to $18$. 
%
The optimal parameter are selected from the maximum of  the performance  graphs.
The   level set  curves of  Log-WWN-L2 and  Log-WNN-Log methods for  optimal parameter  identification are shown in the Figure \ref{fig:OptomalSigma}.
%
The optimal  noise variance for best possible recovery   is identified as  $\sigma = 0.14$.
%---------------------------------- 
%----------------------------------  
 %---------------
\begin{figure*}
\centering 	 
%---------------- 
\iffalse
	\begin{minipage}[b]{.22\textwidth}		 \includegraphics[width=1\textwidth]{Average-PSNR-WNNL2.png}   	  
	\\ \centering\tiny (a1) 	\end{minipage} 
	\begin{minipage}[b]{.22\textwidth}	 	\includegraphics[width=1\textwidth]{Average-INFO-WNNL2.png}   	  
	\\ \centering\tiny (b1) 	\end{minipage}
	\begin{minipage}[b]{.22\textwidth}	       \includegraphics[width=1\textwidth]{Average-MS-SSIM-WNNL2.png}  	  
	\\ \centering\tiny (c1) 	\end{minipage}
	\begin{minipage}[b]{.22\textwidth}	       \includegraphics[width=1\textwidth]{Average-EPI-WNNL2.png}  	  
	\\ \centering\tiny (d1) 	\end{minipage} 
\fi
%---------------- 
%---------------- 
	\begin{minipage}[b]{.20\textwidth}\includegraphics[width=1\textwidth]{Average-PSNR-mesh-WNNL2.png}   
	\\ \centering\tiny (a1) 	\end{minipage} 
	\begin{minipage}[b]{.20\textwidth}\includegraphics[width=1\textwidth]{Average-IWSNR-mesh-WNNL2.png}    
	\\ \centering\tiny (b1) 	\end{minipage}
	\begin{minipage}[b]{.20\textwidth} \includegraphics[width=1\textwidth]{Average-MS-SSIM-mesh-WNNL2.png}
	\\ \centering\tiny (c1) 	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}\includegraphics[width=1\textwidth]{Average-EPI-mesh-WNNL2.png}  	  
	\\ \centering\tiny (d1) 	\end{minipage} 
%---------------- 
%---------------- 
%---------------- 
	\begin{minipage}[b]{.20\textwidth}\includegraphics[width=1\textwidth]{Average-PSNR-mesh-WNNLog.png}    
	\\ \centering\tiny (a2) 	\end{minipage} 
	\begin{minipage}[b]{.20\textwidth}\includegraphics[width=1\textwidth]{Average-IWSNR-mesh-WNNLog.png}  
	\\ \centering\tiny (b2) 	\end{minipage}
	\begin{minipage}[b]{.20\textwidth} \includegraphics[width=1\textwidth]{Average-MS-SSIM-mesh-WNNLog.png}
	\\ \centering\tiny (c2) 	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}\includegraphics[width=1\textwidth]{Average-EPI-mesh-WNNLog.png}  	  
	\\ \centering\tiny (d2) 	\end{minipage} 
%----------------  
\caption{
Method 1, 2: WNN-L2 algorithm  vs WNN-Log algorithm: The   test images are perturbed with  speckle noise and  recovered using  WNN minimization after $\ell_2$ distance based block sorting. 
%The acquisition  simulated for  number of looks  $1 \le L \le 16$  and  noise levels $0.01 \le \sigma \le 4.0$. 
The evaluation metric:  (a) avg. Peak SNR,  (b) avg. iWSNR, (c) avg. MS-SSIM and (d) avg. EPI.
%and the corresponding ($L, \sigma; metric $) mesh plots. 
}  
\label{TEST-LW2-PSNR}
\end{figure*} 
%----------------------------------
%----------------------------------  
%----------------------------------
%----------------------------------
%----------------------------------
%---------------------------------- 
%---------------------------------- METHOD 3, 4
 %---------------
\begin{figure*}
\centering 	 
%---------------- 
	\begin{minipage}[b]{.20\textwidth}   
	\includegraphics[width=1.0\textwidth]{Average-PSNR-mesh-LogWL2.png}   	  
	\\ \centering  \tiny(a1)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	\includegraphics[width=1.0\textwidth]{Average-SSIM-mesh-LogWL2.png}     
	\\ \centering \tiny (b1)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	\includegraphics[width=1.0\textwidth]{Average-MS-SSIM-mesh-LogWL2.png}     
	\\ \centering \tiny (c1)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	\includegraphics[width=1.0\textwidth]{Average-EPI-mesh-LogWL2.png}    
	\\ \centering  \tiny (d1)
	\end{minipage}
%---------------- 
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.0\textwidth]{Average-PSNR-mesh.png}   
	\\ \centering \tiny (a2)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	 \includegraphics[width=1.0\textwidth]{Average-SSIM-mesh.png} 
	\\ \centering \tiny  (b2)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	\includegraphics[width=1.0\textwidth]{Average-MS-SSIM-mesh.png}   	 
	\\ \centering  \tiny (c2)
	\end{minipage}
	\begin{minipage}[b]{.20\textwidth}
	\includegraphics[width=1.0\textwidth]{Average-EPI-mesh.png}     
	\\ \centering \tiny (d2)
	\end{minipage} 
%---------------- 
%---------------- 
\caption{ 
Method 3, 4: Log-WNN-L2 vs Log-WNN-Log: 
The $log$ transformed  image is used for block sorting and WNN minimization.
The evaluation metric:  (a) avg. PSNR, (b) avg. SSIM, (c) agv. MS-SSIM and (d) agv. EPI.
%
%(e-h) the corresponding variations of Log-WNN-Log. The multiplicative noise  is varied from $0.01 \le \sigma \le 2.0$ and the number looks varied from $1$ to $18$. 
}  
\label{fig:AvgMESH}
\end{figure*} 
%----------------------------------
%------------------------------------------------------------------------------  
%------------------------------------------------------------------------------  
                     \section{SAR Image Despeckling}
%------------------------------------------------------------------------------  
%------------------------------------------------------------------------------  
%------------------------------------------------------------------------------  
%
The real SAR data used in the despecking algorithm evaluation is collected from USA Air Force Research Laboratory's  Moving and Stationary Target Acquisition and Recognition (MSTAR) in X-band SAR sensor in one foot resolution spotlight mode; sample images are shown in the Figure \ref{SARimg} \cite{MSTAR}.
%
The images are perturbed with multiplicative speckle noise and the images are recovered using Log-WNN-Log and Log-WNN-L2 methods, described.   The acquisition is simulated for different number of looks  $(L = 1 $ $ to $ $6)$ and noise levels $[0.01 \le \sigma \le 3.5]$
%
%----------------------------------
%----------------------------------
\par
{Despeckling using Log-WNN-L2}: 
%----------------------------------
%----------------------------------
%
The noise is induced into the SAR images of size $256 \times 256$ and 
$log$ transformation is applied before using the image as input for $\ell_2$ distance based segment regrouping.
%
The variation of PSNR, SSIM, MS-SSIM  and EPI metric  
with number of looks and noise variance is studied and shown in 
the Figure \ref{SAR-LW2-PSNR}.
%
The optimal number of looks for a given noise level is obtained from the level set corresponding to the maximum parametric value.
%
As in the case of test images, the average PSNR value increases with  number of  looks. 
Significant improvement is observed for noise variance is between $ 0.01 \le \sigma  \le 0.2$.
%
When the noise variance  $\sigma = 0.2$, the PSNR value improves  till $L= 4$
and further increase in $L$ do not have any effect on PSNR.
%
For higher values of noise variance from $ 0.4 \le \sigma \le 3.5 $ and  $ 1 < L \le 5$ finite improvement in PSNR value is observed, but beyond that no change in PSNR is observed.
%
The average SSIM value in this case increases with increase in number of looks and saturate to a 
value near $1$ and  improvement observed for noise variance from $ 0.01  \le \sigma \le 0.2$.
%
When  noise variance is $0.2$  the SSIM value improves as $L$ increased to  $4$ and further increase deteriorate the  performance.
%----------------------------------
%----------------------------------
%----------------------------------
%----------------------------------METHOD 4
 %---------------
\begin{figure}
\centering 	 
%---------------- 
%---------------- 
\iffalse
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.0\textwidth]{High-PSNR-parameters-LogWL2.png}   	
		\\ \centering   \tiny(a)
	\end{minipage}
      %---------------- 
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.0\textwidth]{High-SSIM-parameters-LogWL2.png}   	
		\\ \centering  \tiny (b)
	\end{minipage}
      %---------------- 
      %---------------- 
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.0\textwidth]{High-MS-SSIM-parameters-LogWL2.png}   	
		\\ \centering  \tiny (c)
	\end{minipage}
      %---------------- 
      %---------------- 
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.0\textwidth]{High-EPI-parameters-LogWL2.png}   	
		\\ \centering  \tiny (d)
	\end{minipage}
      %---------------- 
\fi
	\begin{minipage}[b]{.17\textwidth}
	 \includegraphics[width=1.0\textwidth]{High-PSNR-parameters.png}   	
	\\ \centering  \tiny (a)
	\end{minipage}
      %---------------- 
	\begin{minipage}[b]{.17\textwidth}
	 \includegraphics[width=1.0\textwidth]{High-SSIM-parameters.png}   	
	\\ \centering  \tiny (b)
	\end{minipage}
      %---------------- 
	\begin{minipage}[b]{.17\textwidth}
	 \includegraphics[width=1.0\textwidth]{High-MS-SSIM-parameters.png}   	
	\\ \centering  \tiny (c)
	\end{minipage}
      %---------------- 
      %---------------- 
	\begin{minipage}[b]{.17\textwidth}
	 \includegraphics[width=1.0\textwidth]{High-EPI-parameters.png}   	
	\\ \centering  \tiny (d)
	\end{minipage}
      %----------------  
\caption{  
Method 4:  Optimal parameter estimation for Log-WNN-Log from test images:  
The noise  is increased from $0.01 \le \sigma \le 2.0$ and the number of looks varied from $1$ to $16$.
The optimal values are indicated in \color{red}$*$\color{black}.
%
%(a) to (d) The optimal parameter for determined from various evaluation metric.
%(e) to (h) maxima determined from the level set.
}   
\label{fig:OptomalSigma}
\end{figure}
%----------------------------------
%----------------------------------  	
%---------------------------------- SAR IMAGE SCAN
%----------------------------------    
\par
{Despeckling using Log-WNN-Log}: 
%----------------------------------
%----------------------------------
%
The input image is $log$ transformed before applying logarithmic ratio for similar segment  identification and grouping; WNN minimization is applied over this matrix.
%
The SAR images of size $256 \times 256$ are perturbed with speckle noise and used as input.
%
The variation of PSNR, SSIM, MS-SSIM  and EPI with different number of looks and noise variance  are given in 
Figures \ref{SAR-LWL-PSNR}. 
% 
The optimal number of looks for a given noise level is obtained from the level set corresponding to the maximum parametric value.
%
The average PSNR value increases with number  of looks $L$. 
%
Significant improvement in PSNR value observed when $L = 1, 2$ and 
noise variance  $ 0.01 \le \sigma \le 0.2$ and finite improvement when noise variance $0.4 \le \sigma \le 3.5$; 
but for higher values of  $L$ no change in PSNR observed.
%
The average SSIM value increases with increase in $L$ and saturate to a value near $1$ and significant improvement observed for noise variance from $ 0.01 \le \sigma \le  0.2$; further increase in $L$ reduces the SSIM performance.
%----------------------------------
%---------------------------------- 
%----------------------------------
\par
The optimal choice of number of looks $L$ and value of   noise variance $\sigma$  are found from the maxima of the metrics contour plots.
The optimal value of  $(L, \sigma )$ combination for the Log-WNN-Log method  is $L \ge 2$ and $\sigma < 1$, in the case of Matlab repository test images; and  $L \ge 4$ and $\sigma < 1$ in the case of MSTAR SAR images  \cite{MSTAR}; however, the optimal values are $L \ge 2$ and $\sigma < 1$ if the method is Log-WNN-L2.  This indicates that the similar segment grouping through $\ell_2$ method gives better recovery performance in the case of SAR image matrix even in multiplicative  the speckle noise cases.
% 
%----------------------------------
%----------------------------------
%----------------------------------
%----------------------------------
%---------------------------------- 
%----------------------------------
%\subsection{Timing Analysis}
%
\par
The average computation time for the  Log-WNN-L2  reconstruction method on the  test images and SAR  data are  shown in  Figure \ref{ALG-TIME}.
%
The computation time for both cases are in the range of $170$ to $185$ seconds, 
on an  $Intel$  $i5$  processor system with $8$ $GB$ RAM.
%on a Intel Pentium $i7$ processor system with $16GB$ memory running iOS.
%
it is seen that as the number of looks $L$ increases, the computation time reduces initially and then it is almost maintain the same.
But, when the noise variance is less, the algorithm takes more time to converge  than when the noise variance is more.
%
In the case of  Log-WNN-Log method  for the same noise variance  the computation time is relatively less for the standard test images, in the range of $110$ to $150$ seconds.  But for SAR images, the average computation time is in the range of $165$ to $185$ seconds.
%----------------------------------
%---------------
\begin{figure}
\centering 	 
%---------------- 
	\begin{minipage}[b]{0.5\textwidth}
	 \includegraphics[width=1\textwidth]{Img79131725.png}   
	\end{minipage}
%---------------- 
%---------------- 
\caption{ 
The set of SAR scan images used in the algorithm evaluation, taken from SDMS http://www.sdms.afrl.af.mil}  
\label{SARimg}
\end{figure} 
%----------------------------------
%----------------------------------
%----------------------------------
%----------------------------------
\begin{figure*}
\centering 	 
%---------------- 
	\begin{minipage}[b]{1\textwidth}
\centering
%---------------- 
 	 \includegraphics[width=.15\textwidth]{Average-PSNR-mesh-LogWL2-SAR.png}   
  	 \includegraphics[width=.15\textwidth]{Average-SSIM-mesh-LogWL2-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-MS-SSIM-mesh-LogWL2-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-EPI-mesh-LogWL2-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-IWSNR-mesh-LogWL2-SAR.png}  
	 \includegraphics[width=.15\textwidth]{Average-INFO-mesh-LogWL2-SAR.png}  
%----------------  
	 \includegraphics[width=.15\textwidth]{Average-PSNR-LogWL2-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-SSIM-LogWL2-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-MS-SSIM-LogWL2-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-EPI-LogWL2-SAR.png}  
 	 \includegraphics[width=.15\textwidth]{Average-iWSNR-LogWL2-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-INFO-LogWL2-SAR}  
%----------------  
	 \includegraphics[width=.18\textwidth]{High-PSNR-parameters-LogWL2-SAR.png}  
	 \includegraphics[width=.18\textwidth]{High-SSIM-parameters-LogWL2-SAR.png}  
	 \includegraphics[width=.18\textwidth]{High-MS-SSIM-parameters-LogWL2-SAR.png}  
	 \includegraphics[width=.18\textwidth]{High-EPI-parameters-LogWL2-SAR.png}  
	\end{minipage}
%---------------- 
%----------------  
%----------------  
%----------------    
%---------------- 
\caption{
Log-WNN-L2 method on SAR data:
The MSTAR SAR images are perturbed with  speckle noise, 
the acquisition is simulated   number of looks  ($1 \le L \le 16$) and   noise levels $[0.01 \le \sigma \le 4.0]$.
The reconstruction is evaluated using PSNR, SSIM, MS-SSIM,  EPI, iWSNR and InfoLoss metric values averaged over 10 SAR scenes.
%The Peak SNR of the reconstructed image averaged over 10 SAR scenes.
%(b) The structural similarity index.
%(c) The multi scale structural similarity index.
%(d) The edge preservation index.
The optimal values  indicated in \color{red}$*$\color{black}.
}  
\label{SAR-LW2-PSNR}
\end{figure*} 
%----------------------------------
%----------------------------------
%----------------------------------
%----------------------------------
\begin{figure*}
\centering 	 
%---------------- 
	\begin{minipage}[b]{1\textwidth}
\centering
 	 \includegraphics[width=.15\textwidth]{Average-PSNR-mesh-SAR.png}   
 	 \includegraphics[width=.15\textwidth]{Average-SSIM-mesh-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-MS-SSIM-mesh-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-EPI-mesh-SAR.png} 
 	 \includegraphics[width=.15\textwidth]{Average-IWSNR-mesh-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-INFO-mesh-SAR.png}  
	%---------------- 
	 \includegraphics[width=.15\textwidth]{Average-PSNR-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-SSIM-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-MS-SSIM-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-EPI-SAR.png}    
 	 \includegraphics[width=.15\textwidth]{Average-iWSNR-SAR.png}   
	 \includegraphics[width=.15\textwidth]{Average-INFO-SAR}  
	%---------------- 
	 \includegraphics[width=.18\textwidth]{High-PSNR-parameters-SAR.png}  
	 \includegraphics[width=.18\textwidth]{High-SSIM-parameters-SAR.png}  
	 \includegraphics[width=.18\textwidth]{High-MS-SSIM-parameters-SAR.png}  
	 \includegraphics[width=.18\textwidth]{High-EPI-parameters-SAR.png}  
		%\\ \tiny  \centering (a) avg. PSNR
	\end{minipage}
%----------------  
%---------------- 
\caption{
Log-WNN-Log method on SAR data:
%The speckle induced images are recovered  using  Log-WNN-Log  method. 
The acquisition is simulated for different number of looks ($1\le L \le 16$) and  noise levels $[0.01 \le \sigma \le 4.0]$.
The reconstruction is evaluated using PSNR, SSIM, MS-SSIM,  EPI, iWSNR, InfoLoss metric values.
%(a) The Peak SNR  
%(b) The structural similarity index.
%(c) The multi scale structural similarity index.
%(d) The edge preservation index.
The optimal values are indicated in \color{red}$*$\color{black}.
}  
\label{SAR-LWL-PSNR}
\end{figure*} 
%----------------------------------
%----------------------------------
%---------------------------------- 
%----------------------------------
%----------------------------------

%---------------------------------- 
%----------------------------------
%----------------------------------
%----------------------------------
%----------------------------------
%----------------------------------   
%---------------------------
%---------------------------  
%---------------------------
\section{Conclusion}
%--------------------------- 
The study of the weighted nuclear norm minimization (WNNM) method for despeckling across various test cases is presented. Details of the implementation are discussed, and the PSNR, SSIM, and EPI metrics were calculated for eight simulated images. The WNNM method, originally developed for additive white Gaussian noise, was adapted for SAR images.
A homomorphic transformation was first applied, with speckle filtering implemented in the logarithmic domain, followed by an exponential transformation to reconstruct the image. Various similarity measures were applied to evaluate performance improvement. The results of despeckling direct images and log-transformed images were also analyzed. For direct images, significant improvement was not observed for either similarity measure. However, for log-transformed images, the performance, in terms of PSNR, SSIM, and EPI, was superior compared to other methods evaluated.
\par
The application of different similarity measures on log-transformed images did not yield significant advantages, as the results were comparable.
The findings indicate that the WNNM method effectively enhances the visual quality of denoised images, preserves the local structure of the image, and improves objective index values. Furthermore, the WNNM method demonstrates better performance than state of the art denoising methods. However, its performance depends on the ability to identify similar segments within the noisy image. Despeckling methods that use log-transformed images for multiplicative noise filtering are found to be highly effective.
%----------------------------------  
%----------------------------------  
%----------------------------------  
%---------------
\begin{figure}
\centering 	 
%---------------- 
%----------------   
%	\begin{minipage}[b]{.42\textwidth}
% 	 \includegraphics[width=1\textwidth]{Average-TIME-LogWL2.png}    
%		\\ \centering\tiny (a) Test Image
%	\end{minipage} 
%	\begin{minipage}[b]{.23\textwidth}
% 	 \includegraphics[width=1\textwidth]{Average-TIME-LogWL2-SAR.png}   
%		\\ \centering\tiny (b) SAR Image
%	\end{minipage} 
	\begin{minipage}[b]{.42\textwidth}
 	 \includegraphics[width=1\textwidth]{Average-TIME.png}    
	\\ \centering\tiny (d) SAR Image
	\end{minipage} 
%	\begin{minipage}[b]{0.23\textwidth}
% 	 \includegraphics[width=1\textwidth]{Average-TIME-SAR.png}    
%		\\ \centering\tiny (e) SAR Image
%	\end{minipage} 
%---------------- 
%---------------- 
\caption{
Computation Time: 
The average computation time for the  Log-WNN-L2   method on SAR  data. 
%(c), (d) The   computations times for Log-WNN-Log  method.
%The despeckling is carried out for   noise levels $[0.01 \le \sigma \le 4.0]$
}  
\label{ALG-TIME}
\end{figure} 
%----------------------------------
%----------------------------------  
\section*{Acknowledgement}
The authors acknowledge the authors of different algorithms for providing implementation code of their algorithms.  
%The research is made possible using the facility available at Indian Institute of Space Science and Technology,  and  Indian Space Research Organisation.
%----------------------
\section*{Declaration}
Funding: The  research on denoising algorithms is supported by Indian Space Research Organisation.%, Government of India.
%----------------------
Conflict of Interest: None of the authors has conflict of interest with any of the matter presented in the document. 
%----------------------
Data Availability: The data and simulation code   are available up on request.
%----------------------
Code Availability: The algorithm is developed using MATLAB  is available on request.
%----------------    
%----------------    
%----------------    

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\fi


\iffalse
\begin{figure*}
\centering 	 
%---------------- 
	\begin{minipage}[b]{.24\textwidth}		 \includegraphics[width=1\textwidth]{Average-PSNR-WNNLog.png} 	  
	\\ \centering\tiny (a1) 	\end{minipage} 
	\begin{minipage}[b]{.24\textwidth}	 	\includegraphics[width=1\textwidth]{Average-INFO-WNNLog.png}     
	\\ \centering\tiny (b1) 	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}	       \includegraphics[width=1\textwidth]{Average-MS-SSIM-WNNLog.png}	  
	\\ \centering\tiny (c1) 	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}	       \includegraphics[width=1\textwidth]{Average-EPI-WNNLog.png} 	  
	\\ \centering\tiny (d1) 	\end{minipage} 
%---------------- 
%---------------- 
	\begin{minipage}[b]{.24\textwidth}\includegraphics[width=1\textwidth]{Average-PSNR-mesh-WNNLog.png}    
	\\ \centering\tiny (a2) 	\end{minipage} 
	\begin{minipage}[b]{.24\textwidth}\includegraphics[width=1\textwidth]{Average-IWSNR-mesh-WNNLog.png}  
	\\ \centering\tiny (b2) 	\end{minipage}
	\begin{minipage}[b]{.24\textwidth} \includegraphics[width=1\textwidth]{Average-MS-SSIM-mesh-WNNLog.png}
	\\ \centering\tiny (c2) 	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}\includegraphics[width=1\textwidth]{Average-EPI-mesh-WNNLog.png}  	  
	\\ \centering\tiny (d2) 	\end{minipage} 
%----------------  
%---------------- 
\caption{
Method 2: WNN-Log algorithm: The  test images are perturbed with multiplicative speckle noise and the images are recovered using  WNN minimization after $log$ distance based block sorting.  
%The acquisition  simulated for  number of looks  $1 \le L \le 16$  and  noise levels $0.01 \le \sigma \le 4.0$.  
The evaluation metric:   (a) Peak SNR,  (b) Information Loss, (c) MS-SSIM and (d) EPI  and  and the coresponding ($L, \sigma; metric $) mesh plots.
}  
\label{TEST-LWL-PSNR}
\end{figure*} 
\fi
%----------------------------------

%---------------------------------------------------
\iffalse
\begin{figure*}
\centering 	 
%---------------- 
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Barbara.png}   
	 \\ \centering\tiny (a)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Camera.png} 
	\\ \centering\tiny (b)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-House.png}
		\\ \centering\tiny (c)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Leafs.png}
		\\ \centering\tiny (d)
	\end{minipage}
%---------------------------	
%---------------------------	
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Monarc.png}
		\\ \centering\tiny (e)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Parrot.png}
		\\ \centering\tiny (f)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Starfish.png}
		\\ \centering\tiny (g)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{SSIM-Lena.png}
		\\ \centering\tiny (h)
	\end{minipage}
        %---------------- 
\caption{ 
SSIM: The structural similarity index of the  images denoised using various algorithms.
(a) Barbara,  (b) Camera, (c) House, (d) Leafs,   
(e) Monarc, (f) Parrot, (g) Starfish.
%
The window size used is $31\time31$ and segment size is $7 \time 7$ and number of looks is varied from $1$ to $16$.
The performance comparison reference used are the BM3D-SAR and PPB methods.
The cases $2,3,4$ are used in the simulation.
%
}  
\label{fig:SSIM}
\end{figure*} 
%---------------------------------------------------
%---------------------------------------------------
\begin{figure*}
\centering 	 
%---------------- 
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Barbara.png}   
	 \\ \centering\tiny (a)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Camera.png} 
	\\ \centering\tiny (b)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-House.png}
		\\ \centering\tiny (c)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Leafs.png}
		\\ \centering\tiny (d)
	\end{minipage}
%---------------------------	
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Monarc.png}
		\\ \centering\tiny (e)
	\end{minipage} 
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Parrot.png}
		\\ \centering\tiny (f)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Starfish.png}
		\\ \centering\tiny (g)
	\end{minipage}
	\begin{minipage}[b]{.24\textwidth}
	 \includegraphics[width=1.2\textwidth]{EPI-Lena.png}
		\\ \centering\tiny (h)
	\end{minipage}
        %---------------- 
\caption{
EPI: The edge preservation index of the  images denoised using various algorithms.
(a) Barbara,  (b) Camera, (c) House, (d) Leafs,   (e) Monarc, (f) Parrot, (g) Starfish.
%
The window size used is $31\time31$ and segment size is $7 \time 7$ and number of looks is varied from $1$ to $16$. The performance comparison reference used are the BM3D-SAR and PPB methods.
The cases $2,3,4$ are used in the simulation.
%
}  
\label{fig:EPI}
\end{figure*}
\fi
%---------------------------------------------------
%---------------------------------------------------
%                 average trend	 
%---------------- 
%----------------  
\iffalse
\begin{figure*}
\centering 	 
	%---------------- 
	\begin{minipage}[b]{1\textwidth}
	 \includegraphics[width=0.31\textwidth]{Average-PSNR-LogWL2.png}   
	 \includegraphics[width=0.31\textwidth]{Average-SNR-LogWL2.png}   
	 \includegraphics[width=0.31\textwidth]{Average-iWSNR-LogWL2.png}   
	\end{minipage}
      %---------------- 
	%---------------- 
	\begin{minipage}[b]{1\textwidth}	
	\includegraphics[width=0.31\textwidth]{Average-PSNR.png}   	
	\includegraphics[width=0.31\textwidth]{Average-SNR.png}  
	\includegraphics[width=0.31\textwidth]{Average-iWSNR.png}  
	 %\\ \centering\tiny (b) avg. Signal to Noise Ratio
	\end{minipage}
      %---------------- 
      %---------------- 
\caption{ 
Averaged performance metric for Log-WNN-L2 method and Log-WNN-Log method.
The PSNR,   SNR and iWSNR  the information weighted SNR all averaged over different test images  
recovered from speckle noise perturbed reference test images. 
The multiplicative noise perturbation is varied from $0.01 \le \sigma \le 2.0$. 
}  
\label{fig:AvgPSNR}  
\end{figure*}
\fi
%---------------- 
%----------------------------------  
\iffalse
%----------------------------------
 \section{Parametric Modeling }
%----------------------------------
%----------------------------------
%---------------------------------- 
                 \iffalse
\begin{equation}\nonumber
psnr = ( 0.5 + e^{-\frac{(1+\sigma)}{2}}) . A_{m} 
 \end{equation}

\begin{equation} 
\left(   
%
1 -  \frac{1- ( 0.5 + e^{-\frac{(1+\sigma)}{2}})(1+\sigma)  \frac{ A_0 }{A_m}}{1+\sigma^{2}(L+L_{0})^{2}}
%
 \right)
\end{equation}

\begin{equation*}
 = ( 0.5 + e^{-\frac{(1+\sigma)}{2}}) . A_{m} 
\left(   
%
1 -  \frac{1- ( 0.5 + e^{-\frac{(1+\sigma)}{2}})(1+\sigma)  A}{1+\sigma^{2}(L+L_{0})^{2}}
%
 \right)
\end{equation*}

where $A =\frac{ A_0 }{A_m} $

\begin{equation*}
 = ( 0.5 + e^{-\frac{ \sigma_{1} }{2}}) . A_{m} 
\left(   
%
1 -  \frac{1- ( 0.5 + e^{-\frac{ \sigma_{1}}{2}})\sigma_{1}  A}{1+\sigma^{2}(L+L_{0})^{2}}
%
 \right)
\end{equation*}

where 
$\sigma_{1} = \sigma + 1$ 
\begin{equation*}
 = ( 0.5 + e^{-\frac{ \sigma_{1} }{2}}) .
\left(   
%
1 -  \frac{1-   \frac{\sigma_{1}  A}{2} + e^{-\frac{ \sigma_{1}}{2}} \sigma_{1}  \alpha}{1+\sigma^{2}(L+L_{0})^{2}}
%
 \right)  A_{m} 
\end{equation*} 
where $ \alpha =\frac{ A_0 }{A_m} $ 
                    \fi
%--------------------------
%--------------------------
%--------------------------
The effect of parameter variation on the performance metric of the denoising methods are shown the the mesh plot Figure \ref{fig:AvgMESH}.   The observed PSNR variation is modeled using the empirical relation  (\ref{EQmodel}).
The   formula is proposed by identifying a function which generates the similar characteristic curve as obtained from the simulation. 
The exponential curve  $e^{-\frac{(1+\sigma)}{2}}$ is the starting point of the   modeling and modified to generate the required curve meeting the actual observation.
%--------------------------
%--------------------------
\begin{equation}
\label{EQmodel}
PSNR = q . A_{m} 
\left(   
%
1 -  \frac{1- q(1+\sigma)  \frac{ A_0 }{A_m}}{1+\sigma^{2}(L+L_{0})^{2}}
%
 \right)
\end{equation}
%--------------------------
where $ 0.01 < \sigma < 2$,  ${A_m}$ is the maximum estimated PSNR and
${A_0}$ minimum SNR of the image.  $L_{0}$ is the initial number of looks.
And $q$ is defined empirically as (\ref{EQq}).
%--------------------------
\begin{equation}
\label{EQq}
q  = 0.5 + e^{-\frac{(1+\sigma)}{2}}
\end{equation}
%------------------------------------
The model simulated PSNR is given in Figure \ref{PSNR-model}.
The PSNR variation is observed to be following the actual variation obtained from the simulation, when number looks  $L$ is varied. 
%----------------------------------
%----------------------------------
%----------------------------------
%----------------------------------
\begin{figure}
\centering 	 
	\begin{minipage}[b]{.45\textwidth}
	\includegraphics[width=1.0\textwidth]{PSNR-model.png}   	
      \end{minipage}
\caption{ 
PSNR recovery model of Logarithmic input processed weighted nuclear norm minimization with
logarithmic ratio as metric for similar segment grouping.  The noise estimate is varied from $0.01 \le \sigma \le 2.0$. 
and the number of looks varied from $1$ to $16$.}  
\label{PSNR-model}
\end{figure} 
%----------------------------------
%----------------------------------
%----------------------------------
%----------------------------------  
\fi  
%\subsection 
%---------------------------
%-------------------------  ALGORITHM II
%--------------------------------
%------------------------
\iffalse
\begin{algorithm} 
\providecommand{\SetAlgoLined}{\SetLine}
\textbf{Task:} 	 \quad denoise $\mathbf{Y} \in \mathbb{R}^{M \times N}$   \\ 
\textbf{Initialization:}    
		\text{ } $threshold$:  $\theta$,   $norm$:  $\ell_p$\\ 
		\text{ } \qquad \qquad $sub-Img$: $n \times n$,    $segment$:  $d \times d$\\
\textbf{Do}(1)    $ \forall$   sub-Img   $Y_p$   $\subset  \mathbf{Y}$\\
{	
\text{ }\text{ }  $select$:     subImg  $Y_p$  $ \in \mathbb{R} ^{n \times n} $      \\  
\text{ }\text{ } \textbf{Do}(2) $ \forall$  segment $[d\times d]$   $\subset   Y_p$ \\
{
	\text{ } \qquad (vectorize:)  \\
	\text{ } \qquad $collect$:    pixels $P_{i,j}$  \\
	\text{ } \qquad  $\{  P_{i,j}  \vert   i \dots  i \pm d/2, j \dots  j  \pm d/2  \} \rightarrow D_{ij} \in \mathbb{R}^{d^2}$\\
	\text{ } \qquad (sort)    \\
	\text{ } \qquad \quad $measure$: $\ell_p$  distance  $\forall$  $D_{ij}$  \\
	\text{ } \qquad \quad $arrange$:  $ \mathbf{D} = $    $[D_{1} \dots  D_{d^{2}} ]  \in  {R}^{ {d^2} \times {d^2} }$ \\
	\text{ } \qquad (minimize:)    \\
	\text{ } \qquad \quad $sv-decompose$:       $\mathbf{D} = \mathbf{U \Sigma  V}^{T}$\\
	\text{ } \qquad \quad $threshold$:            $\theta (\mathbf{\Sigma}) =  \mathbf{\Sigma}_{\theta}  $\\
	\text{ } \qquad \quad $rank- min$:            $\mathbf{D}_{\theta} = \mathbf{U \Sigma_{\theta}  V}^{T}$\\
	\text{ } \qquad (denoise:)\\
	\text{ } \qquad \quad $arrange$:  $ \mathbf{D}_{\theta} \rightarrow$ 
		 $[ \bar{D}_{1} \dots   \bar{D}_{d^{2}} ]  $  \\
	\text{ } \qquad \quad  $  \bar{D}_{1}  \rightarrow \{ \bar{P}_{i,j}  \} $  $i \dots i \pm  d/2$, $j \dots  j  \pm d/2 $\\
	\text{ } \qquad (replace:)    \\
	\text{ } \qquad \quad ${P}_{i,j} = \bar{P}_{i,j}$ \\
 }
\text{ }\text{ } \textbf{End}(2)\\
 }
\text{ }\text{ } $ Y_{p}  \leftarrow  Y_{p}^{*} = \{ {P}_{i,j}  \}$\\	
\textbf{End}(1)\\ 
\textbf{Output:}  $\mathbf{Y}^{*} \in \mathbb{R}^{M \times N}$
\caption{OLD ALGORITHM FOR FORMAT REFERENCE}
\label{Algo1}
\end{algorithm} 
%-----------------------------
%---------------------------
%--------------------------- 
%---------------- 
%---------------- 
%---------------------------
% PSNR
\begin{table*}
\begin{center}
%\tiny
\caption{\textbf{The OLD TABLE SET TO REFER FORMAT :  do not change format or additional formatting}} 
 %\tabcolsep=0.35cm
\begin{tabular}{ |c| c| c| c| c| c| c| c| c| c|} 
% \toprule
\hline
 &\multicolumn{9}{c|}{$\sigma = 30$} \\
\hline
Image &NLM &BM3D &\multicolumn{5}{c|}{WNNM ($l_p norm$)} &GSR-WNNM &WSNM \\
\hline
&&&p=0.2&p=0.5&p=0.8&p=1&p=2 &  &\\
\hline
Monarch &27.16 &28.36 & 28.80 &28.85 &28.87 &28.88 &28.89 & 28.83 &28.92 \\
\hline
Cameraman &27.33 &28.64 &28.71 &28.73 &28.74 &28.75 &28.75 &28.51 &28.78\\
\hline
Barbara &26.53 &29.08 &29.41 &29.45 &29.47 &29.47&29.49 &29.37 & 29.43\\
\hline
House & 29.44 &32.09 &32.40 & 32.42 &32.43 &32.43 &32.44 &32.55 & 32.43 \\
\hline
Leaves&26.73 &27.81 &28.43 & 28.49 &28.52 & 28.52 &28.52 &28.57 &28.52\\
\hline
Lena &27.65 &29.46 &29.64 &29.66 &29.67 &29.68 &29.68 &29.65 &29.67\\
\hline
Parrot &28.76 &30.33 &30.61 &30.63 &30.63 &30.63 &30.63 &30.47 &30.61\\
\hline
Starfish &26.19 &27.65 &27.99 & 28.03 &28.06 &28.07 & 28.07 &28.02 &28.03\\
\hline
\hline
&\multicolumn{9}{c|}{$\sigma = 100$}\\
\hline
Image &NLM &BM3D &\multicolumn{5}{c|}{WNNM ($l_p norm$)} &GSR-WNNM &WSNM\\
\hline
&&&p=0.2&p=0.5&p=0.8&p=1&p=2 & &\\
\hline
Monarch &18.94 &22.52 &22.82 &22.89 &22.92 &22.94 &22.95 &22.80 &23.00\\
\hline
Cameraman &20.18 & 23.08 &23.32 &23.34 &23.34 &23.35 &23.36 &23.32 &23.38\\
\hline
Barbara &20.30 & 23.20 &23.46 &23.48 &23.49&23.49 &23.49 & 23.04 &23.40\\
\hline
House & 21.43 & 25.87 &26.63 &26.65 &26.65 &26.66 & 26.66 &26.61 &26.69\\
\hline
Leaves &17.25 &20.09 &21.43 &21.5 &21.54 &21.55 &21.56 &21.64 &21.71\\
\hline
Lena &20.59 &23.87 &24.34 &24.34 &24.35 &24.35 &24.35 &24.22 &24.33\\
\hline
Parrot &21.30 &24.60 &24.69 &24.71 &24.71 &24.71&24.72 &24.65 &24.72 \\
\hline
Starfish &19.66 &22.1 & 22.21 &22.22 &22.23 &22.23 &22.23 &21.92 &22.22\\
\hline
\end{tabular} 
\end{center}
\end{table*}  
\label{table:PSNR}
\fi
%--------------------------- 