Chapter 3. Study and Analysis of Different Noise Reduction Filters

Transcription

1 Chapter 3 Study and Analysis of Different Noise Reduction Filters Noise is considered to be any measurement that is not part of the phenomena of interest. Departure of ideal signal is generally referred to as noise. Noise arises as a result of unmodelled or unmodellable processes going on in the production and capture of real signal. It is not part of the ideal signal and may be caused by a wide range of sources, e. g. variation in the detector sensitivity, environmental variations, the discrete nature of radiation, transmission or quantization errors etc. It is also possible to treat irrelevant scene details as if they are image noises e.g. surface reflectance textures. The characteristics of noise (Russo, 2005) depend on its source, as does the operator which reduces its effects. Many image processing packages contains operators to artificially add noise to an image. Deliberately corrupting an image with noise allows us to test the resistance of an image processing operator to noise and assess the performance of various noise filters. There are two ways of image corruption by noise: noise addition and noise multiplication. A model of an image degraded by additive random noise is given by (Gonzalez and Woods, 2002). g( x, y) f ( x, y) n( x, y) (3.1) Where n( x, y) represents the signal independent additive random noise. The level of noise is generally expressed by its variance. In order to compare the performance of the original, degraded and processed images, some measures of error are necessary. The Peak Signal-to-Noise Ratio (PSNR) is often used for the characterization of signal. 60

2 PSNR is the ratio between possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. PSNR = 10 log10 (255 2 /MSE) (3.2) Higher PSNR value provides higher image quality. Mean square error (MSE) is given by: MSE N i j1 f ( i, j) F( i, j) 2 N 2 (3.3) Where, f is the original image F is the filtered image and N is the size of image. MSE is an estimator in many ways to quantify the amount by which a filtered/noisy image differs from noiseless image. 3.1 Classification of Different Types of Noise Digital images acquired with an electronic camera are typically corrupted with noise due to the optical system, light sensor and associated electronics. The transmission of video images is often accompanied by noise depending on environmental conditions. Video images transmitted via satellite are very susceptible to the electronic interference due to sunspot activity. Noise is classified based upon the shape of its probability density function (pdf). The mean and variance are important parameters to characterize the noise. Mean value m, gives the average brightness of the noise and square root of variance gives the average peak-to-peak gray level deviation of the noise (Xu and Lai, 1998). The mean and variance are defined as: gmax m k p ( k) (3.4) k0 n 61

3 gmax 2 2 ( kpn ( k) m) k0 (3.5) Where, p n (k) is the frequency of occurrence of noise amplitude, k and g max is the maximam gray level of the image. Ideally, k varies from - to +, however, since the pixel levels are limited in the range [0,L-1], the noise amplitude level k also lies in [0, L- 1]. Now we want to describe different types of noise that we have studied for our work Gaussian Noise The most common type of noise that is found in an image is Gaussian noise, which is the result of many unknown noises from independent sources added together. Gaussian noise is expressed by the probability density function (pdf) as (Xu and Lai, 1998): 2 ( km) 1 2 pn ( k) e ; 2 k (3.6) Where k=grey level, m mean and standard deviation In the pdf, mean is located at the peak, having highest probability of occurrence and the width is determined by the standard deviation. Gaussian noise is defined over infinite range; however, the digitized image has finite range. Hence the noise values that exceed the gray level range are deposited at the 0 and g max points on the pdf. For 99.7% of the gray levels, the peak-to-peak gray level deviation is equal to 6. Figure 3: Gaussian Noise Distribution Function 62

4 3.1.2 Salt and Pepper Noise In the salt and pepper noise model only two possible values are possible, a and b, and probability of obtaining each of them is less than 0.1(otherwise, the noise would vastly dominate the image). For an 8 bit/pixel image, the typical intensity value g for pepper noise is close to 0 and for salt noise is close to 255. The probability density function (PDF) for salt and pepper noise (Meher, 2004) model is given by: A for g a (" pepper ") PDF salt and pepper= B for g b(" salt" ) (3.7) Figure 3.1: Salt and Pepper Noise Distribution Function The salt and pepper noise is generally caused by malfunctioning of camera s sensor cells, by memory cell failure or by synchronization errors in the image digitizing or transmission. 3.2 Image Denoising Image denoising is a delicate and difficult task. A trade-off between noise reduction and the preservation of actual image features occurs, in order to enhance the relevant image content. Reducing noise has always been one of the standard problems of image processing. A multitude of methods have been proposed to remove noise as it is well 63

5 known that every source of noise creates a different type of noise. The purpose of denoising is to suppress the noise from the observed signal, and help the recovery of functions of that signal. In statistical terms (Hazma and Krim, 2001), this corresponds to a non parametric regression, where an orthogonal basis expansion is used to estimate the unknown function using a time regression setting Conventional Filters Filters are mainly used to suppress either the high frequencies in the image, i.e., for smoothing the image, or the low frequencies, i.e., for enhancing or detecting edges in the image.suppose that an image-processing operator F acts on the two input images A and B and produces output images C and D respectively. If the operator F is linear (Gonzalez and Woods, 2002; Jain, 1989), then F (a A+ b B) = a C + b D (3.8) Where a and b are constants. This means that each pixel in the output of a linear operator is the weighted sum of a set of pixels in the input image. For example, the threshold operator is non-linear (Astola and Kuosmanen, 1997) because individually, corresponding pixels in the two images A and B may be below the threshold, whereas the pixel obtained by adding A and B may be above threshold. Similarly, the absolute value operation is non-linear: -1+1 ~ = (3.9) Simple Mean Filter The moving average or mean filter (MF) is a simple linear filter. The idea of mean filtering is simply to replace each pixel value in an image with the mean (`average') 64

6 value of its neighbors, including itself (Gonzalez and Woods, 2002; Jain, 1989). This has the effect of eliminating pixel values, which are unrepresentative of their surroundings. Often a 3 3 square kernel shown in Figure 3.2 is used. 1 1 X Figure 3.2: 3 3 averaging kernel used in mean filtering It is very simple to implement in hardware and software. The computational complexity is very low. It works fine for low power AWGN. As the noise power increases, its filtering performance degrades. If the noise power is high, then a larger window should be employed for spatial sampling to have better local statistical information. As the window size increases, MF produces a reasonably high blurring effect and thus thin edges and fine details in an image are lost. There are two main problems with mean filtering, which are: A single pixel with a very unrepresentative value can significantly affect the mean value of all the pixels in its neighborhood. When the filter neighborhood straddles an edge, the filter will interpolate new values for pixels on the edge thereby blurring the edge. This may be a problem if sharp edges are required in the output Disk Filter Disk filter (Gonzalez and Woods, 2002) uses a circular averaging filter (pillbox) within the square matrix of side 2*radius+1. The pillbox has circular top and straight sides. For example, if the lens of a camera is not properly focused, each point in the image will be projected to a circular spot on the image sensor. In other words, the pillbox is the point 65

7 spread function of an out-of-focus lens. The Disk filter convolved image will appear blurry and have less defined edges, but will be lower in random noise. These are called smoothing filters, for their action in the time domain, or low-pass filters, for how they treat the frequency domain Median filter The median filter is a nonlinear digital filtering technique, often used to remove noise. Such noise reduction is a typical pre-processing step to improve the results of later processing (for example, edge detection on an image). Median filtering is very widely used in digital image processing because, under certain conditions, it preserves edges while removing noise. Median filter (Eng and Ma, 2001) is a spatial filtering operation, so it uses a 2-D mask that is applied to each pixel in the input image. To apply the mask means to centre it in a pixel, evaluating the covered pixel brightnesses and determining which brightness value is the median value. Figure presents the concept of spatial filtering based on a 3x3 mask, where I is the input image and O is the output image. Figure 3.3: Shows concept of spatial filtering The median value is determined by sorting all the pixel values from the surrounding neighborhood into numerical order and then replacing the pixel being 66

8 considered with the middle pixel value (Chan et al., 2005; Wang and Hang, 1999). If the neighborhood under consideration contains an even number of pixels, the average of the two middle pixel values is used. Figure 3.3 illustrates calculation of median value. Figure 3.4: The median value of a pixel neighborhood As can be seen, the central pixel value of 150 is rather unrepresentative of the surrounding pixels and is replaced with the median value: 124. A 3 3 square neighborhood is used here; however, larger neighborhoods will produce more severe smoothing. By calculating the median value of a neighborhood rather than the mean filter, the median filter has two main advantages over the mean filter: The median is more robust than the mean and so a single very unrepresentative pixel in a neighborhood will not affect the median value significantly. Since the median value must actually be the value of one of the pixels in the neighborhood, the median filter does not create new unrealistic pixel values, when the filter straddles an edge. For this reason, the median filter is much better at preserving sharp edges than the mean filter. The main problem of the median filter is its high computational cost (for sorting N pixels, the temporal complexity is O (N log N), even with the most efficient sorting algorithms).in General, the median filter allows a great deal of high spatial frequency 67

9 detail to pass while remaining very effective at removing noise on images where less than half of the pixels in a smoothing neighborhood have been effected. (As a consequence of this, median filtering can be less effective at removing noise from images corrupted with Gaussian noise). Unlike the mean filter, the median filter is non-linear. This means that for two images A (x) and B (x), we have Median A (x) +B (x) Median A (x) + Median (x) (3.10) Gaussian Filter The Gaussian Filter (Krystek, 1996; Vanherck, 1994) is a 2-D convolution operator that is used to `blur' images and remove detail and noise. In this sense, it is similar to the mean filter, but it uses a different kernel that represents the shape of a Gaussian (`bell-shaped') hump. This kernel has some special properties as explained below. The Gaussian distribution in 1-D has the form: G x 2 x e (3.11) 2 where is the standard deviation of the distribution. We have also assumed that the distribution has a mean of zero (i.e. it is centered on the line x=0). The distribution is illustrated in Figure 3.5. Figure 3.5: 1-D Gaussian distribution with mean 0 and =1 68

10 In 2-D, an isotropic (i.e. circularly symmetric) Gaussian has the form: G x 2 2 x y 1 2 2, y e 2 (3.12) 2 This distribution is shown in Figure 3.6. Figure 3.6: 2-D Gaussian distribution with mean (0, 0) and =1 The idea of Gaussian smoothing is to use this 2-D distribution as a `point-spread' function, and this is achieved by convolution. Since the image is stored as a collection of discrete pixels we need to produce a discrete approximation to the Gaussian function before we can perform the convolution. In theory, the Gaussian distribution is non-zero everywhere, which would require an infinitely large convolution kernel, but in practice it is effectively zero more than about three standard deviations from the mean, and so we can truncate the kernel at this point. Figure 3.7 shows a suitable integer-valued convolution kernel that approximates a Gaussian filter with a = 1.0. Figure 3.7: Discrete approximation to Gaussian function with =1.0 69

11 Once a suitable kernel has been calculated, then the Gaussian smoothing can be performed using standard convolution methods. The convolution can in fact be performed fairly quickly since the equation for the 2-D isotropic Gaussian shown above is separable into x and y components. Thus the 2-D convolution can be performed by first convolving with a 1-D Gaussian in the x direction, and then convolving with another 1-D Gaussian in the y direction. (The Gaussian is in fact the only completely circularly symmetric operator which can be decomposed in such a way.) Figure 3.8 shows the 1-D x component kernel that would be used to produce the full kernel shown in Figure 3.7 (after scaling by 273, rounding and truncating one row of pixels around the boundary because they mostly have the value 0. This reduces the 7x7 matrix to the 5x5 shown above.). The y component is exactly the same but is oriented vertically. Figure 3.8: one of the pair of 1-D convolution kernels used to calculate the full kernel more quickly A further way to compute a Gaussian smoothing with a large standard deviation is to convolve an image several times with a smaller Gaussian. While this is computationally complex, it can have applicability if the processing is carried out using a hardware pipeline. The Gaussian filter not only has utility in engineering applications. It is also attracting attention from computational biologists because it has been attributed with some amount of biological plausibility, e.g. some cells in the visual pathways of the brain often have an approximately Gaussian response. 70

12 3.2.4 The Laplace Operator The Laplacian operator (Taubin, 1995; Zhang, 2003) is a 2-D isotropic measure of the 2nd spatial derivative of an image. It is particularly good at finding the fine detail in an image. Any feature with a sharp discontinuity (like noise) will be enhanced by a Laplacian operator. Thus, one application of a Laplacian operator is to restore fine detail to an image which has been smoothed to remove noise. The Laplacian L(x, y) of an image with pixel intensity values I(x, y) is given by: L 2 2 I I (3.13) x y x, y 2 2 The Laplacian operator is implemented as a convolution between an image and a kernel. The Laplacian kernel can be constructed in various ways but is generally used as a 3-by-3 kernel and shown in the figure below Figure 3.9: Laplacian 3x3 kernel In image convolution, the kernel is centered on each pixel in turn, and the pixel value is replaced by the sum of the kernel mutipled by the image values. In the particular kernel we are using here, we are counting the contributions of the diagonal pixels as well as the orthogonal pixels in the filter operation Laplacian of Gaussian Laplacian filters are derivative filters used to find areas of rapid change (edges) in images. Since derivative filters are very sensitive to noise, it is common to smooth the 71

13 mage (e.g., using a Gaussian filter) before applying the Laplacian. This two-step process is called the Laplacian of Gaussian (LoG) operation f ( x, y) f ( x, y) f ( x, y) 2 2 L( x, y) (3.14) x y There are different ways to find an approximate discrete convolution kernal that approximates the effect of the Laplacian. A possible kernel is Figure 3.10: Commonly used discrete approximations to the Laplacian filter. This is called a negative Laplacian because the central peak is negative. We have defined the Laplacian using a negative peak because this is more common; however, it is equally valid to use the opposite sign convention. To include a smoothing Gaussian filter, combine the Laplacian and Gaussian functions to obtain a single equation: x y x y LOG( x, y) (1 )exp( ) (3.15) and is shown in Figure 3.11 Figure 3.11: The 2-D Laplacian of Gaussian (LoG) function. The x and y axes are marked in standard deviations ( ). 72

14 A discrete kernel for the case of σ = 1.4 is given by Figure 3.12: Discrete approximation to LoG function with Gaussian = 1.4 The LoG operator takes the second derivative of the image. Where the image is basically uniform, the LoG will give zero. Wherever a change occurs, the LoG will give a positive response on the darker side and a negative response on the lighter side. At a sharp edge between two regions, the response will be zero at a long distance from the edge, positive just to one side of the edge, negative just to the other side of the edge, zero at some point in between, on the edge itself. Figure 3.13 illustrates the response of the LoG to a step edge. Figure 3.13: Response of 1-D LoG filter to a step edge. 73

15 The left hand graph shows a 1-D image, 200 pixels long, containing a step edge. The right hand graph shows the response of a 1-D LoG filter with Gaussian = 3 pixels Wiener Filter Wiener (1949) proposed the concept of Wiener filtering. There are two methods: (i) Fourier-transform based method (frequency-domain) and (ii) mean-squared based method (spatial-domain) for implementing Wiener filter. The former method is used only for complete restoration (denoising and deblurring) whereas the latter is used for denoising. In Fourier transform based method of Wiener filtering, normally a priori knowledge of the power spectra of noise and the original image is required. But in mean-squared method, no such a priori knowledge is required. Hence, it is easier to use the mean-squared method for image denoising. Wiener filter is based on the leastsquared principle, i.e. the filter minimizing the mean-squared error (MSE) between the actual output and the desired output (Shui, 2005). Image statistics vary too much from a region to another even within the same image. Thus, both global statistics (mean variance etc. of the whole image) and local statistics (mean, variance etc. of a small region or sub-image) are important. Wiener filtering is based on both the global statistics and local statistics and is given by: f 2 2 f x y g gx, y g, 2 f n (3.16) where, g is the local mean, 2 f is the local signal variance, 2 n is the noise variance and f x, y denotes the restored image. For (2a+1) (2b+1) window of noisy image g(x, y), the local mean g 2 and local Variance g are defined by: 74

16 a b 1 g gs, t L sa tb (3.17) where, L, is the total number of pixels in a window, i.e. L = (2a+1) (2b+1); and 2 a b 2 1 g gs, t g (3.18) L 1 s a tb The local signal variance 2 f used in (3.16) is calculated from 2 g with a priori knowledge of noise variance, 2 n simply by subtracting 2 2 n from g with the assumption that the signal and noise are not correlated with each other. From (3.16) it may be observed that the filter-output is equal to local mean, if the current pixel value equals local mean. Otherwise, it outputs a different value; the value being some what different from local mean. If the input current value is more (less) than the local mean, then the filter outputs a positive (negative) differential amount taking the noise variance and the signal variance into consideration. Thus, the filter output varies from the local mean depending upon the local variance and hence tries to catch the true original value as far as possible. In statistical theory, Wiener filtering is a great land mark. It estimates the original data with minimum mean-squared error and hence, the overall noise power in the filtered output is minimal. Thus, it is accepted as a benchmark in 1-D and 2-D signal processing Unsharp Filter Unsharp filterering technique is used commonly in the printing industry for crispening the edges. A signal proportional to the unsharp or low-pass filtered, version 75

17 of the image is subtracted from the image. This is equivalent to adding the gradient, or a high-pass signal, to the image. Steps for Unsharp Filtering are: Blur the image Subtract the blurred version from the original (this is called the mask) g mask ( x, y) f ( x, y) f ( x, y) (3.19) Where f ( x, y) an original is image and ( x, y) f is the blurred image Add the mask to the original g( x, y) f ( x, y) k. g ( x, y) mask where k is a weight. We can better understand the operation of the unsharp sharpening filter by examining its frequency response characteristics. If we have a signal as shown in Figure 3.14(a), subtracting away the lowpass component of that signal (as in Figure 3.14(b)), yields the highpass, or `edge', representation shown in Figure 3.14(c). Figure 3.14: Calculating an edge image for unsharp filtering This edge image can be used for sharpening if we add it back into the original signal, as shown in Figure

18 Figure 3.15: Sharpening the original signal by adding the edge image The unsharp filter is implemented as a window-based operator, i.e. it relies on a convolution kernel to perform spatial filtering. It can be implemented using an appropriately defined lowpass filter to produce the smoothed version of an image, which is then pixel subtracted from the original image in order to produce a description of image edges, i.e. a highpassed image Lee Filter The Lee filter (Lee, 1980) developed by Jong-Sen Lee is an adaptive filter which changes its characteristics according to the local statistics in the neighborhood of the current pixel. Lee filters utilize the statistical distribution of the DN values within the moving kernel to estimate the value of the pixel of interest. The filter assumes a Gaussian distribution for the noise in the image data. The Lee filter is based on the assumption that the mean and variance of the pixel of interest is equal to the local mean and variance of all pixels within the user-selected moving kernel. The Lee filter is able to smooth away noise in flat regions, but leaves the fine details (such as lines and textures) unchanged. It uses small window (3 3, 5 5, 7 7).The formula used for the Lee filter is: DN out Mean KDN Mean (3.20) in Where Mean=Average of pixels in a moving window 77

19 Var ( x) K= 2 2 Mean Var ( x ) (3.21) and Variance within window Mean within window 2 Sigma 1 2 Var ( x) Mean within window 2 (3.22) The distinct characteristic of the filter is that in the areas of low signal activity (flat regions) the estimated pixel approaches the local mean, whereas in the areas of high signal activity (edge areas) the estimated pixel favours the corrupted image pixel, thus retaining the edge information. It is generally claimed that human vision is more sensitive to noise in a flat area than in an edge area. The major drawback of the filter is that it leaves noise in the vicinity of edges and lines. However, it is still desirable to reduce noise in the edge area without sacrificing the edge sharpness. Some variants of Lee filter available in the literature handle multiplicative noise and yield edge sharpening Frost Filter The Frost filter (Frost et al., 1982) is an adaptive and exponentially-weighted averaging filter based on the coefficient of variation which is the ratio of the local standard deviation to the local mean of the degraded image. The Frost filter replaces the pixel of interest with a weighted sum of the values within the nxn moving kernel. The weighting factors decrease with distance from the pixel of interest. The weighting factors increase for the central pixels as variance within the kernel increases. This filter assumes multiplicative noise and stationary noise statistics and follows the following formula: 78

20 DN k e nxn t (3.23) Where 4 2 n 2 2 I K= normalization constant I = local mean σ = Local variance = image coefficient of variation value t X X Y 0 Y 0, and n= moving kernel size 3.3 Estimation of Statistical Parameters Over the past few years, various filters have been proposed based on numerous purposes. Consequently, criteria quantifying the performances of the filters are desired. Two general classes of criteria are used as the basis for such evaluations (Gonzalez and Woods, 2002; Wang et al., 2002): (a) Objective fidelity criteria A simple and convenient mechanism for quantitating the differences between two images by letting functions represents images (b) Subjective fidelity criteria Human observers evaluate different images and averaging their evaluations. 79

21 The parameters which are used in the filter performance evaluation are Mean Square Error (MSE) (Girod,1993), Peak Signal to Noise Ratio (PSNR) (Eskicioglu and Fisher, 1995), Correlation Coefficient (COC) (Rodgers and Nicewander, 1995), and Mean Structural Similarity Index Measure (MSSIM) (Eskicioglu and Fisher, 1995), UQI (Universal Quality Index (Wang and Bovik, 2002). These are already explained in chapter I. 3.4 Simulation Results All the filters like Mean, Circular Mean, Median, Gaussian, Laplacian, Laplacian of Gaussian, Wiener, Unsharp, Lee and Frost filters are simulated on MATLAB 7.0 platform. The experiments are performed on 40 gray scale images (taken from the database of images Berkeley Segmentation Dataset, Matlab test images and this database brings a set of 200 images of natural scenes and their ground truth produced manually) and results on a few images (natural gray scale images; Lena and Cameraman, and two synthetic sharp edge images; Test_corners image and Test_pattern) shown in figure 3.16 below (Table 3.1 to 3.32) have been presented. To analyse the performance of various filters in the noisy environment, first the image is corrupted with Gaussian noise of variance =.05, 0.1, 0.25 and 0.5 and salt and pepper noise of density=.05, 0.1, 0.25 and 0.5. The Mean Square Error (MSE), Peak-Signal-to- Noise Ratio (PSNR), Correlation Coefficient (COC), Mean Structural Similarity Index Measure (SSIM) and Universal Quality Index (UQI) are taken as performance measures. Results of performance evaluation of various filters have been shown qualitatively as well as quantitatively. The figure 3.16 shows a few test images used in study. 80

22 Test Corners Test Pattern Lena Cameraman Figure 3.16: Shows a few test images used for simulation along with their names 81

23 3.4.1 Visual Results of Various Filters on Images Corrupted With Salt and Pepper Noise Test_corners Image Results I. Filtering performance of various filters operated on Test_corners image corrupted with salt and pepper noise of density=0.05 Figure 3.17: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) II. Filtering performance of various filters operated on Test_corners image corrupted with salt and pepper noise of density=0.1 Figure 3.18: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 82

24 III. Filtering performance of various filters operated on Test_corners image corrupted with salt and pepper noise of density=0.25 Figure 3.19: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) IV. Filtering performance of various filters operated on Test_corners image corrupted with salt and pepper noise of density=0.5 Figure 3.20: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 83

25 Test_pattern Image Results I. Filtering performance of various filters operated on Test_pattern image corrupted with salt and pepper noise of density=0.05 Figure 3.21: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) II. Filtering performance of various filters operated on Test_pattern image corrupted with salt and pepper noise of density=0.1 Figure 3.22: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 84

26 III. Filtering performance of various filters operated on Test_pattern image corrupted with salt and pepper noise of density=0.25 Figure 3.23: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) IV. Filtering performance of various filters operated on Test_pattern image corrupted with salt and pepper noise of density=0.5 Figure 3.24: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 85

27 Lena Image Results I. Filtering performance of various filters operated on Lena image corrupted with salt and pepper noise of density=0.05 Figure 3.25: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) II. Filtering performance of various filters operated on Lena image corrupted with salt and pepper noise of density=0.1 Figure 3.26: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 86

28 III. Filtering performance of various filters operated on Lena image corrupted with salt and pepper noise of density=0.25 Figure 3.27: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) IV. Filtering performance of various filters operated on Lena image corrupted with salt and pepper noise of density=0.5 Figure 3.28: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 87

29 Cameraman Image Results I. Filtering performance of various filters operated on Cameraman Image corrupted with salt and pepper noise of density=0.05 Figure 3.29: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) II. Filtering performance of various filters operated on Cameraman image corrupted with salt and pepper noise of density=0.1 Figure 3.30: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 88

30 III. Filtering performance of various filters operated on Cameraman image corrupted with salt and pepper noise of density=0.25 Figure 3.31: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) IV. Filtering performance of various filters operated on Cameraman image corrupted with salt and pepper noise of density=0.5 Figure 3.32: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 89

31 3.4.2 Visual Results of Different Filters on Images Corrupted with Gaussian noise Test_corners Image Results I. Filtering performance of various filters operated on Test_corners image corrupted with Gaussian noise of variance =0.05 Figure 3.33: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) II. Filtering performance of various filters operated on Test_corners image corrupted with Gaussian noise of variance =0.1 Figure 3.34: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 90

32 III. Filtering performance of various filters operated on Test_corners image corrupted Gaussian noise of variance =0.25 Figure 3.35: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) IV. Filtering performance of various filters operated on Test_corners image corrupted with Gaussian noise of variance =0.5 Figure 3.36: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 91

33 Test_pattern Image Results I. Filtering performance of various filters operated on Test_pattern image corrupted with Gaussian noise of variance =0.05 Figure 3.37: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) II. Filtering performance of various filters operated on Test_pattern image corrupted with Gaussian noise of variance =0.1 Figure 3.38: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 92

34 III. Filtering performance of various filters operated on Test_pattern image corrupted with Gaussian noise of variance =0.25 Figure 3.39: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) IV. Filtering performance of various filters operated on Test_pattern image corrupted with Gaussian noise of variance =0.5 Figure 3.40: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 93

35 Lena Image Results I. Filtering performance of various filters operated on Lena image corrupted with Gaussian noise of variance =0.05 Figure 3.41: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) II. Filtering performance of various filters operated on Lena image corrupted with Gaussian noise of variance =0.1 Figure 3.42: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 94

36 III. Filtering performance of various filters operated on Lena image corrupted with Gaussian noise of variance =0.25 Figure 3.43: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) IV. Filtering performance of various filters operated on Lena image corrupted with Gaussian noise of variance =0.5 Figure 3.44: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 95

37 Cameraman Image Results I. Filtering performance of various filters operated on Cameraman Image corrupted with Gaussian noise of variance=0.05 Figure 3.45: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) II. Filtering performance of various filters operated on Cameraman image corrupted with Gaussian noise of variance=0.1 Figure 3.46: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 96

38 III. Filtering performance of various filters operated on Cameraman image corrupted with Gaussian noise of variance=0.25 Figure 3.47: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) IV. Filtering performance of various filters operated on Cameraman image corrupted with Gaussian noise of variance=0.5 Figure 3.48: (a) Original Image (b) Noisy image (c) Mean (d) Disk (e) Gaussian (f) 97

39 3.4.3 Quantitative Results of Different Filters on Images Corrupted With Gaussian Noise Cameraman Image Results Table 3.1: Results of different quantitative parameters for Cameraman image with Gaussian noise of variance=0.05 Average Disk Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost Table 3.2: Results of different quantitative parameters for Cameraman image with Gaussian noise of variance=0.1 Average Disk e Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost Table 3.3: Results of different quantitative parameters for Cameraman image with Gaussian noise of variance=0.25 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e

40 Table 3.4: Results of different quantitative parameters for Cameraman image with Gaussian noise of variance=0.5 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e Lena Image Results Table 3.5: Results of different quantitative parameters for Lena with Gaussian noise of variance=0.05 Average Disk Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost Table 3.6: Results of different quantitative parameters for Lena with Gaussian noise of variance=0.1 Average Disk Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost

41 Table 3.7: Results of different quantitative parameters for Lena with Gaussian noise of variance=0.25 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e Table 3.8: Results of different quantitative parameters for Lena with Gaussian noise of variance=0.5 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e Test_pattern Image Results Table 3.9: Results of different quantitative parameters for Test_pattern image with Gaussian noise of variance=0.05 Average e Disk e Gaussian Laplacian e LoG e Unsharp e Median e Wiener Lee e Frost e

42 Table 3.10: Results of different quantitative parameters for Test_pattern image with Gaussian noise of variance=0.1 Average e Disk e Gaussian Laplacian e LoG e Unsharp e Median e Wiener Lee e Frost e Table 3.11: Results of different quantitative parameters for Test_pattern image with Gaussian noise of variance=0.25 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e Table 3.12: Results of different quantitative parameters for Test_pattern image with Gaussian noise of variance=0.5 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e

43 Test_corners Image Results Table 3.13: Results of different quantitative parameters for Test_corners Image with Gaussian noise of variance=0.05 Average Disk e Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost Table 3.14: Results of different quantitative parameters for Test_corners Image with Gaussian noise of variance=0.1 Average Disk e Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee e Frost Table 3.15: Results of different quantitative parameters for Test_corners Image with Gaussian noise of variance=0.25 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e

44 Table 3.16: Results of different quantitative parameters for Test_corners Image with Gaussian noise of variance=0.5 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e

45 3.4.4 Quantitative Results of Different Filters on Images Corrupted with Salt and Pepper Noise Cameraman Image Results Table 3.17: Results of different quantitative parameters for Cameraman Image with salt and pepper noise density=0.05 Average Disk Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost Table 3.18: Results of different quantitative parameters for Cameraman Image with salt and pepper noise density=0.1 Average Disk Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost Table 3.19: Results of different quantitative parameters for Cameraman Image with salt and pepper noise density=0.25 Average e Disk Gaussian e Laplacian e LoG e Unsharp e Median Wiener e Lee e Frost e

46 Table 3.20: Results of different quantitative parameters for Cameraman Image with salt and pepper noise density=0.5 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e Lena Image Results Table 3.21: Results of different quantitative parameters for Lena image with salt and pepper noise density=0.05 Average Disk Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost Table 3.22: Results of different quantitative parameters for Lena image with salt and pepper noise density=0.1 Average Disk Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee Frost

47 Table 3.23: Results of different quantitative parameters for Lena image with salt and pepper noise density=0.25 Average Disk Gaussian e Laplacian e LoG e Unsharp e Median Wiener e Lee Frost e Table 3.24: Results of different quantitative parameters for Lena image with salt and pepper noise density=0.5 Average e Disk Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e Test_pattern Image Results Table 3.25: Results of different quantitative parameters for Test_pattern with salt and pepper noise density=0.05 Average e Disk e Gaussian Laplacian e LoG e Unsharp e Median Wiener Lee e Frost e

48 Table 3.26: Results of different quantitative parameters for Test_pattern with salt and pepper noise density=0.1 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median Wiener e Lee e Frost e Table 3.27: Results of different quantitative parameters for Test_pattern with salt and pepper noise density=0.25 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e Table 3.28: Results of different quantitative parameters for Test_pattern with salt and pepper noise density=0.5 Average e Disk e Gaussian e Laplacian e LoG e Unsharp e Median e Wiener e Lee e Frost e