Performance Analysis of Spatial and Transform Filters for Efficient Image Noise Reduction
Total Page：16
File Type：pdf, Size：1020Kb
Performance Analysis of Spatial and Transform Filters for Efficient Image Noise Reduction Santosh Paudel Ajay Kumar Shrestha Pradip Singh Maharjan Rameshwar Rijal Computer and Electronics Computer and Electronics Computer and Electronics Computer and Electronics Engineering Engineering Engineering Engineering KEC, TU KEC, TU KEC, TU KEC, TU Lalitpur, Nepal Lalitpur, Nepal Lalitpur, Nepal Lalitpur, Nepal [email protected] [email protected] [email protected] [email protected] Abstract—During the acquisition of an image from its systems such as laser, acoustics and SAR (Synthetic source, noise always becomes integral part of it. Various Aperture Radar) images [3]. algorithms have been used in past to denoise the images. Image denoising still has scope for improvement. Visual This paper introduces different types of noise to be information transmitted in the form of digital images has considered in an image and analyzed for various spatial become a considerable method of communication in the and transforms domain filters by considering the image modern age, but the image obtained after transmission is metrics such as mean square error (MSE), root mean often corrupted due to noise. In this paper, we review the squared error (RMSE), Peak Signal to Noise Ratio existing denoising algorithms such as filtering approach (PSNR) and universal quality index (UQI). and wavelets based approach, and then perform their comparative study with bilateral filters. We use different II. BACKGROUND noise models to describe additive and multiplicative noise The Wavelet Transform (WT) is a powerful tool of in an image. Based on the samples of degraded pixel signal and image processing, which has been neighborhoods as inputs, the output of an efficient filtering successfully used in many scientific fields such as signal approach has shown a better image denoising performance. processing, image compression, computer graphics, and This yields promising qualitative and quantitative results pattern recognition. WT represents image energy in of the degraded noisy images in terms of Peak Signal to compact form and representation helps in determining Noise Ratio, Mean Square Error and Universal Quality threshold between noisy features and important image Identifier. feature [4]. The Continuous WT (CWT) technique Keywords— Image noise; Image filtering; Wavelet expands the signal on basis functions created by transform; PSNR; MSE; MRBF expanding, shrinking and shifting a single prototype function, which is named as mother wavelet, specially selected for the signal under considerations. This I. INTRODUCTION transformation decomposes the signal into different Noise in an image is a very common problem. An scales with different levels of resolution. Since a scale image gets corrupted with noise during acquisition, parameter shrinks or expands the mother wavelet in transmission, storage and retrieval processes. Noise may CWT, the result of the transform appears as time-scale be classified as substitutive or impulsive noise (e.g., salt representation. The scale parameter is indirectly related and pepper noise, random-valued impulse noise), to frequency when the center frequency of mother additive noise (e.g., additive white Gaussian noise) and wavelet is considered. A mother wavelet has a zero multiplicative noise (e.g. speckle noise) [1]. The simple mean value, which requires the transformation kernel of median filter works efficiently to suppress impulse noise the wavelet transform to compactly support localization of low density. However, many denoising schemes have in time, thereby offering the potential to capture the been proposed recently which are efficient in spikes occurring instantly in a short period of time suppressing impulse noise of moderate and high noise [2],[5]. A wavelet expansion is a representation of a densities. In many occasions, noise in digital images is signal in terms of an orthogonal collection of real-values found to be additive in nature with uniform power in the generated by applying suitable transformation to the whole bandwidth along with Gaussian probability original selected wavelets. The main difference between distribution and is termed as Additive White Gaussian mother wavelet functions such as Haar, Daubechies, Noise (AWGN). It is difficult to suppress AWGN since Symlets, Coiflets and Bi-orthogonal lies on how their it corrupts almost all pixels in an image. The arithmetic scaling signals and the wavelets are defined. mean filter, commonly known as Mean filter, can be employed to suppress AWGN but it introduces a D. L. Donoho has done a lot of work on filtering of blurring effect [2]. Multiplicative or speckle noise is an additive Gaussian noise using wavelet soft thresholding inherent property of medical ultrasound imaging. [6]. Wavelets play a major role in image compression Speckle noise occurs in almost all coherent imaging and image denoising [7]. These Wavelet coefficients calculated by a wavelet transform represent change in the time series at a particular resolution. It is always A. Visu Shrink possible to filter out the noise by considering the time series at various resolutions. The small coefficients are Visu Shrink is also called as the universal threshold dominated by noise after applying wavelet transform. method. The method was introduced by D. L. Donoho et However, coefficients with a larger absolute value carry al. [8],[9]. It uses a threshold value t that is proportional more signal information than noise. Replacing the to the standard deviation of the noise. It follows the hard smallest, noisy coefficients by zero and a backward thresholding rule. It is defined by Eq. (1). wavelet transform on the result may lead to tn 2log .......................................................................(1) reconstruction with the essential signal characteristics and reduced noise. For thresholding, there are three Here, σ2 is the noise variance and n is the number of observations and assumptions which are given as: samples. 1. The decorrelating property of a wavelet transform An estimate of the noise level σ̃ was defined based creates a sparse signal in which most of the on the median absolute deviation given by Eq. (2). untouched coefficients are zero or close to zero. median g: k 0,1,2.........2j 1 2. Noise is spread out equally over all the jk1, coefficients. ..................(2) 0.625 3. The noise level is not too high, therefore we can recognize the signal and the signal wavelet B. Sure Shrink coefficients. Sure Shrink is based on Stein’s Unbiased Risk Thus, the choice of threshold level is an important Estimator (SURE) and was proposed by Donoho and task. The coefficients having magnitude greater than Johnston [6],[8]. It is a combination of the universal threshold are considered as signal of interest and the threshold and the SURE threshold. This method same or modified coefficients are kept according to the specifies a threshold value for each resolution level j in selected threshold, whereas other coefficients become the wavelet transform which is referred as level zero [8]. Then the image is reconstructed from the dependent thresholding [10]. The objective of this modified coefficients. Usually the selection of threshold method is to minimize the mean square error, defined by and the preservation of the edges of the denoised images Eq. (3). are important points of interest. n 1 2 MSE ( Z ( xi , y i ) S ( x i , y i )) ..................................(3) n xy,1 III. OVERVIEW This paper basically focuses on the wavelet Here, Z(xi,yj) is the estimate of the signal, S(xi,yj) is transform filtering method. All wavelet transform the original signal without noise and n is the size of the denoising algorithms involve the following three steps in signal. Sure Shrink suppresses noise by thresholding the general as shown in Figure 1. empirical wavelet coefficients. The Sure Shrink threshold t* is defined by Eq. (4). Forward Wavelet Transform: Wavelet coefficients are obtained by applying the wavelet transform t* min( t , 2log n ).........................................................(4) Estimation. Here, t denotes the value that minimizes SURE, σ2 is Clean coefficients are estimated from the noisy ones. the noise variance and n is the number of samples. Inverse Wavelet Transform: A clean image is obtained by applying the inverse wavelet transform. C. Bayes Shrink There are various methods for wavelet thresholding, Bayes Shrink was proposed by Chang, Yu and Vetter which rely on the choice of a threshold value. The [11]. The purpose of this method is to minimize the typically used threshold methods for denoising an image Bayesian risk. It uses soft thresholding and is sub band- are Visu Shrink, Sure Shrink, Bayes Shrink, Neigh dependent where the thresholding is done at each band Shrink, Oracle Shrink, Smooth Shrink and Fuzzy based of resolution in the wavelet decomposition. It is also a Shrink. smooth adaptive method as similar as the Sure Shrink procedure. The Bayes threshold is defined by Eq. (5). Compression-based ε 22 Denoising tb n/ s ..................................................................(5) f g Wavelet Y Quantize X̂ Inverse f̂ 2 2 Transform (T.∆.m) Transform Here, σn is noise variance and σs is signal variance without noise. The definition of additive noise gives the following Eq. (6) and Eq. (7). Estimate Parameter w( x , y ) s ( x , y ) n ( x , y ).................................................(6) s Here, w(x,y) is noisy image, s(x,y) is original image Fig. 1. Denoising using Wavelet Transform. and n(x,y) is added noise. 2 2 2 2 σw = σs + σn ; where σw is total variance. F. Multi Resolution Bilateral Filter (MRBF) 2 σn are computed as Multi resolution analysis has been proven to be an important tool for eliminating noise in signals. It is nn 211 2 2 possible to distinguish between noise and image wnw( x , y ) , n ( x , y ).............................(7) nnx, y 1 x , y 1 information better at one resolution level than another. Therefore, the bilateral filter is used for noise reduction 2 The variance (σs ) of the signal is computed as in a multi-resolution framework [15]. An image is shown in Eq.