## 3 Performance Analysis  
Quantitative and qualitative evaluation are available for analyzing the performance of different methods. Various performance metrics are calculated and compared with existing methods to evaluate the improvement in performance. A few of the computed parameters are explained in detail which include Peak Signal to Noise Ratio (_PSNR_), Mean Square Error (_MSE_), Feature Similarity Index (_FSIM_) and Structural Similarity Index (_SSIM_). _PSNR_ finds the quality of the final image. Equation (1) computes _PSNR_ as:  
\[PSNR\ =\ \log_{10}\left(\frac{255*255}{MSE}\right), \tag{1}\]  
where _MSE_ is the Mean Square Error. Equation (2) calculates the _MSE_ as:  
\[MSE\ =\ \frac{1}{MN}\sum_{i\ =\ 1}^{M}\sum_{j\ =\ 1}^{N}\ [I(i,j)-I^{\prime}(i,j)]^{2}, \tag{2}\]  
where _l_(_i,j_) is the original image and _I'_(_i,j_) represents the thresholded image. _PSNR_ and _MSE_ are mainly used to analyze the quality of compressed and reconstructed image. _SSIM_ measures the structural similarity between the source and final image while FSIM measures the feature similarity between the source and final image. There are two features that can be viewed in _FSIM_. They are Phase congruency and Gradient magnitude. Phase congruency (_PC_) is a dimensionless quantity which is significant in local structure map and is a primary feature in FSIM. _PC_ is contrast invariant with no effect of contrast information on human visual system. Gradient magnitude (GM) is another important feature in _FSIM_. _PC_ and GM are complementary to one another in describing the image local quality. It is computed using Equation (3) as  
\[FSIM\ =\ \frac{\sum\limits_{x\in X}S_{L}(x)PC_{m}(x)}{\sum\limits_{x\in X} PC_{m}(x)}, \tag{3}\]  
where \(x\) represents the whole image, \(S_{L}(x)\) denotes the similarity in the two images and \(PC_{m}\) is the phase congruency map. _SSIM_ is calculated using Equation (4):  
\[SSIM\ =\ \frac{(2\mu_{x}\mu_{y}+C_{1})(2\sigma_{xy}+C_{2})}{(\mu_{x}^{2}+ \mu_{y}^{2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})}, \tag{4}\]  
where \(\mu_{x}\) and \(\mu_{y}\) represents the sample means of \(x\) and \(y\), respectively; \(\sigma_{x}\) and \(\sigma_{y}\) give the sample variances of \(x\) and \(y\), respectively; and \(\sigma_{xy}\) describes the sample correlation coefficient between \(x\) and \(y\) and \(x\) and \(y\) are local windows in the input images. _SSIM_ and FSIM values vary between 0 and 1 with a value 1 representing best similarity. These values can analyze the extend by which the image has recovered after the image is distorted. Following are the performance measures for classification problem with number of correct and incorrect predictions tabulated in terms of count values divided for each classes. A matrix based representation called the confusion matrix is developed and it can be used to analyze the performance of the classifier. Confusion matrix gives the performance of classifier with respect to a test data. It is a matrix which is indexed in a dimension by the true class of the object and in the other by the class assigned by classifier [86]. It gives the relation between observed and predicted values in classification. It is using this matrix it can be found whether mislabeling of classes has occurred and the predictions are correct or not [87]. On arranging in matrix form, correct predictions are given along the main diagonal while the incorrect predictions are along the non-diagonals. This makes it simpler to check where the predictions are wrong. The four main error metrics computed from the matrix are True Positive (_TP_), True Negative (_TN_), False Positive (_FP_) and False Negative (_FN_). True Positive is when the sample is correctly classified as positive, _TN_ is when the sample is correctly classified as negative, _FP_ or False Positive is when the sample is incorrectly classified as positive and _FN_ or False Negative is when the sample is incorrectly classified as negative [88]. Consider an example of confusion matrix for a classification problem with multiple classes.