Tasneem Shafiq

Denoising of Magnetic Resonance Images using Wavelets- a Comparative Study S Satheesh1Dr.KVSVR Prasad2 P.Vasuda3 1Asst.Prof., Dept. of ECE, G Narayanamma Institute of Technology and Science, Hyderabad, , India 2Prof. & Head Dept. of ECE, D.M.S.S.V.H. College of Engineering, Machilipatnam, India 3Asst.prof., Dept. of ECE, G Narayanamma Institute of Technology and Science, Hyderabad, India Abstract: Image denoising has become an essential exercise in medical imaging especially the Magnetic Resonance Imaging (MRI). As additive white Gaussian noise (AWGN) exhibits finest grain property of noise, multi resolution analysis using wavelet transform is gaining popularity. The aim of the work is to compare the effectiveness of three wavelet based denoising algorithms viz. Wiener filter, hard threshold and soft threshold using MRI images in the presence of AWGN. Wiener filter performs better visually and in terms of PSNR than the other thresholding techniques. Keywords: Denoising, Wavelet, MRI, Wiener filtering, Threshold 1. Introduction Image denoising is a procedure in digital image processing aiming at the removal of noise, which may corrupt an image during its acquisition or transmission, while retaining its quality. Medical images obtained from MRI are the most common tool for diagnosis in medical field. These images are often affected by random noise arising in the image acquisition process. The presence of noise not only produces undesirable visual quality but also lowers the visibility of low contrast objects. Noise removal is essential in medical imaging applications in order to enhance and recover anatomical details that may be hidden in the data. The wavelet transform has recently entered the arena of image denoising and it has firmly established its stand as a powerful denoising tool. There has been a fair amount of research on filtering and wavelet coefficients thresholding [6], because wavelets provide an appropriate basis for separating noisy image from the original image. These wavelet based methods mainly rely on thresholding the discrete wavelet transform (DWT) coefficients, which have been affected by AWGN. There has been much research by Donoho & Johnstone [3, 4, 5] on finding thresholds, however few are specifically designed for images. One of the most popular method consists of thresholding the wavelet coefficient (using the hard threshold or the soft threshold) as introduced by Donoho. Another denoising method in the wavelet domain consists of Wiener filtering the wavelet coefficients. In this paper, the performance of this method is done on a degraded image X such that X=S+N where S is the original image and N is an AWGN. The performance of three denoising techniques such as hard thresholding, soft thresholding and Wiener filter is compared both visually and in the PSNR sense. 2. Wavelet Based Image Denoising DWT has attracted more interest in image denoising [7]. The DWT can be interpreted as image decomposition in a set of independent, spatially oriented frequency channels. The image is passed through two complementary filters and emerges as two images, Approximation and Details. This is called Decomposition or Analysis. The components can be assembled back into the original image without loss of information. This process is called Reconstruction or Synthesis. The mathematical manipulation, which implies analysis and synthesis, is called DWT and inverse DWT. For a 2D image, an N level decomposition can be performed resulting in 3N+1 different frequency bands namely, LL, LH, HL, HH. Denoising algorithms that use the wavelet transform consist of three steps: 1. Calculate the wavelet transform of the noisy image 2. Modify the noisy wavelet coefficients according to some rule. 3. Compute the inverse transform using the modified coefficients. 2.1 Wiener Filter In signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s.Its purpose is to reduce the amount of noise present in a signal by comparison with an estimation of the desired noiseless signal. The discrete-time equivalent of Wiener's work was derived independently by Kolmogorov in 1941. Hence the theory is often called the Wiener-Kolmogorov filtering theory. The inverse filtering is a restoration technique for deconvolution, i.e., when the image is blurred by a known lowpass filter, it is possible to recover the image by inverse filtering or generalized inverse filtering. However, inverse filtering is very sensitive to additive noise. The approach of reducing one degradation at a time develops a restoration algorithm for each type of degradation and simply combines them. The Wiener filtering executes an optimal tradeoff between inverse filtering and noise smoothing. It removes the additive noise and inverts the blurring simultaneously. The Wiener filtering is optimal in terms of the mean square error. In other words, it minimizes the overall mean square error in the process of inverse filtering and noise smoothing. The Wiener filtering is a linear estimation of the original image. The approach is based on a stochastic framework [1, 8]. 2.1.1 Wiener Filter in the Wavelet Domain In the model we assume that the wavelet coefficients are conditionally independent Gaussian random variables. The noise is also modeled as stationary independent zero-mean Gaussian variable. Let us consider an image corrupted by a zero-mean Gaussian noise. The coefficients of the noisy image in the wavelet domain are given by [2]. (1) Where represent the coefficients of the noisy image in the wavelet domain, represent the coefficients of the undegraded image, represent the coefficients of the noise. Without loss of generality, we can assume that the’s can be determined by averaging the squared values of in a window centered at (i, j). This information can be expressed as (2) (3) (4) As a result, the coefficients of the Wiener filter can be expressed as (5) Restricting the values to only positive values, the numerator of the equation (4) takes the form and so (6) Where is the best linear estimate of the signal component The noise variance is estimated using the mean absolute deviation (MAD) method and is given by (7) (8) represents the wavelet coefficients. When using the Haar wavelet transform the steps for implementing denoising using the Wiener filter technique is as follows: i. Apply the Haar wavelet transform to the original image ii. is computed by convolving with a kernel of size 9. iii. The Wiener filter is then applied using the formula iv. Apply the inverse Haar wavelet transform. 2.2 Soft Thresholding In soft thresholding, the wavelet coefficients with magnitudes smaller than the threshold are set to zero, but the retained coefficients are also shrunk towards zero by the amount of the threshold value in order to decrease the effect of noise assumed to corrupt all the wavelet coefficients. Soft thresholding shrinks the coefficients above the threshold in absolute value. When using the Haar wavelet transform, the steps for implementing denoising using the soft thresholding technique is as follows: i. Apply the Haar wavelet transform to the original image ii. Apply the soft thresholding on the wavelet coefficients (9) (10) Where is the standard deviation of the noise, is the number of wavelet coefficients, are the de-noised wavelet coefficients, and is the universal threshold and the variance is estimated using MAD method. iii. Apply the inverse haar wavelet transform. 2.3 Hard Thresholding In hard thresholding, the wavelet coefficients with greater magnitudes than the threshold are retained unmodified as they are thought to comprise the informative part of data, while the rest of the coefficients are considered to represent noise and set to zero. However, it is reasonable to assume that coefficients are not purely either noise or informative but mixtures of those. The denoising method described in the previous subsection (soft thresolding) can be carried out using the hard threshold instead of the soft threshold on the wavelet coefficients in (ii). The hard thresholding formula is given as (11) 3. Results and Discussion In this section, simulation results are presented which is performed on the four MRI images i.e. Brain, Knee, Spine Abdomen. White Gaussian noise is added to the MRI images and denoised with the methods described previously. The performance of the three techniques is compared using, which is defined as (12) Where denotes the mean square error for two images & where one of the images is considered a noisy approximation of the other and is given as (13) From the simulation results it has been observed that the Wiener filter outperforms both thresholding methods visually and in terms of PSNR. More details were lost with the thresholding methods especially for the hard threshold wherein the background was not well denoised. If the Wiener filter could be thought as another thresholding function, it will perform better as its shape is smoother than the hard and soft thresholds. This can be clearly seen from Figure1 and Figure2 that the background of the denoised images with Wiener filter appears smoother. The Wiener filter removes the noise pretty well in the smooth regions but performs poorly along the edges. The comparison of PSNR of the three wavelet filters for different MRI images are tabulated in Table1 and is observed that the Wiener filter gives better values compared to soft and hard thresholding for different noise variances () such as 15, 20, 25, 30. Table 1: Comparison of PSNR of different wavelet filters for different MRI images corrupted by AWGN Image Noise () Peak Signal to Noise Ratio in dB(PSNR) Hard Thresholding Soft Thresholding Wiener Filter Brain- - - - Knee- - - - Spine- - - - Abdomen- - - - (a) (b) (c) (d) (e) Figure1. Denoising of Brain MRI image for variance=20 (a) Original image (b)Noisy image (c)Denoised image with hard threshold (d) Denoised image with soft threshold (e) Denoised image with Wiener filter (a) (b) (c) (d) (e) Figure2. Denoising of Spine MRI image for variance=30 (a) Original image (b)Noisy image (c)Denoised image with hard threshold (d) Denoised image with soft threshold (e) Denoised image with Wiener filter 4. Conclusions The paper presents a comparative analysis of three image denoising techniques using wavelet transforms. The analysis of all the experimental results demonstrates that Wiener filter surpasses other methods that have been discussed. There are a couple of areas which would like to be improved on. One area is in improving the denoising along the edges as the Wiener method did not perform so well along the edges. Another area of improvement would be to develop a better optimality criterion as the MSE is not always the best optimality criterion. Acknowledgements We wish to express our sincere thanks to Dr. K. Jitender Reddy, Consultant Radiologist, Dept. of Radiology & Imaging Sciences, Apollo Health City, Hyderabad for providing us with different MRI image datasets. References [1] Image Denoising using Wavelet Thresholding by Lakhwinder Kaur , Savita Gupta , R.C. Chauhan. 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[8] Image Denoising In The Wavelet Domain Using Wiener Filtering, Nevine Jacob and Aline Martin,December 17,2004.