CHAPTER 4 WAVELET TRANSFORM-GENETIC ALGORITHM DENOISING TECHNIQUE

Transcription

1 102 CHAPTER 4 WAVELET TRANSFORM-GENETIC ALGORITHM DENOISING TECHNIQUE 4.1 INTRODUCTION This chapter introduces an effective combination of genetic algorithm and wavelet transform scheme for the denoising of electrocardiogram (ECG) signals, corrupted by non-stationary noises, using genetic algorithm (GA) and wavelet transform (WT). The wavelet theory denoising has been widely exploited in the noisy ECG filtering. Several wavelet denoising ECG signal algorithms were developed, each exploring a particular parameter; the wavelet function, threshold calculus and level decomposition. Xiao-Ping Zhang et al (1998) proposed a new adaptive denoising method based on stein s unbiased risk estimate (SURE) and on a new class of thresholding functions. Unlike the standard soft-thresholding function, these functions have continuous derivatives. The new thresholding functions do the similar manipulations as the standard soft thresholding function and they make it possible to search for optimal thresholds using gradient based adaptive algorithms. This method is very effective in adaptively finding the optimal solution in mean square error (MSE) than that of conventional wavelet shrinkage methods. An effective technique for the denoising of electrocardiogram signals corrupted by non stationary noises (Ercelebi 2004) is based on a

2 103 second generation wavelet transform and level-dependent threshold estimator. Here, the wavelet coefficients of ECG signals were obtained with liftingbased wavelet filters. A lifting scheme is used to construct second-generation wavelets and is an alternative and faster algorithm for a classical wavelet transform. Numerical results comparing the performance of this method with that of the nonlinear filtering techniques (median filter) demonstrate consistently superior denoising performance of this method over median filtering. Kania et al (2007) were investigated the application of wavelet denoising in noise reduction of multichannel high resolution ECG signals. In particular, the influences of the selection of wavelet function and the choice of decomposition level on efficiency of denoising process were considered and whole procedures of noise reduction were implemented in the Matlab environment. The Fast Wavelet Transform was used. The advantage of used denoising method is that the noise level decreasing in ECG signals, in which noise reduction occurs by averaging and has limited application. Manikandan and Dandapat (2007) proposed a novel Wavelet Energy based diagnostic distortion (WEDD) measure to assess the reconstructed signal quality for ECG compression algorithms. WEDD is evaluated from the Wavelet coefficients of the original and the reconstructed ECG signals. For each ECG segment, a Wavelet energy weight vector is computed via five-level biorthogonal discrete wavelet transform (DWT). WEDD provides a better prediction accuracy and exhibits a statistically better monotonic relationship with the Mean Opinion Score(MOS) ratings than wavelet based weighted percentage root mean square difference measure (WWPRD), PRD and other objective measures. Prasad et al (2008) proposed a shrinkage method based on a New Thresholding filter for denoising of biological signals The efficacy of this filter is evaluated by applying this filter for denoising of ECG signals

3 104 contaminated with additive Gaussian noise. The performance of this filter is compared with that of hard and soft thresholding filters using Mean Square Error and Signal to Noise ratio (SNR). The New Thresholding filter is significantly more efficient than Hard and Soft filters in denoising the signals. It embodies the features of both Hard and Soft filters. Alfaouri and Daqrouq (2008) proposed a new approach based on the threshold value of ECG signal determination using Wavelet Transform coefficients. Electrocardiography has had a profound influence on the practice of medicine. The ECG signal allows for the analysis of anatomic and physiologic aspects of the whole cardiac muscle. This method is compared with Donoho's method for signal denoising where in better results are obtained for ECG signals by this algorithm. Sumithra and Thanuskodi (2009) proposed a new thresholding algorithm called trimmed thresholding algorithm. However, the soft thresholding is best in reducing noise but worst in preserving edges and hard thresholding is best in preserving edges but worst in de-noising. Motivated by finding a more general case that incorporates the soft and hard thresholding to achieve a compromise between the two methods, the trimmed thresholding method is proposed to enhance the speech from background noise. Umamaheswara Reddy et al (2009) proposed a new thresholding technique for denoising of ECG signal. This new de-noising method called as improved thresholding de-noising method could be regarded as a compromising between hard- and soft-thresholding de-noising methods. The advantage of the improved thresholding de-noising method is that it retains both the geometrical characteristics of the original ECG signal and variations in the amplitudes of various ECG waveforms effectively. Mahesh et al (2010) proposed a wavelet denoising algorithm. This method implemented Haar and Daubechies wavelets are on speech signals

4 105 and performance is evaluated. Haar wavelet is not suitable for speech denoising application. As Haar is not smooth when compared to other wavelets, it has limitations when applied to non stationary signal such as speech. Higher order Daubechies can be used and are found to be suitable for the work done. Also soft thresholding is better than hard thresholding. Sayed and Ei-Dahshan (2010) proposed an effective hybrid scheme for the denoising of electrocardiogram signals corrupted by non-stationary noises using genetic algorithm (GA) and wavelet transform. Selection of a suitable wavelet denoising parameter is critical for the success of ECG signal filtration in wavelet domain. Therefore, in this noise elimination method, the genetic algorithm has been used to select the optimal wavelet denoising parameters which lead to maximize the filtration performance. Efficient selection of wavelet denoising parameters, such as wavelet function, threshold function (method), and threshold selection rules are critical to the success of signal denoising. Usually, these parameters are selected empirically; which leads to low noise elimination performance. So the contribution of this work is to introduce an evolutionary optimization method based on the Genetic Algorithm to search the wavelet denoising parameters in order to obtain the optimal ECG signal filtration efficiency. The efficiency performance of our scheme is evaluated using percentage root mean square difference and signal to noise ratio. 4.2 APPLICATIONS OF WAVELET TRANSFORM IN ECG SIGNAL The wavelet transform is a powerful and promising method for time and frequency signal analysis. A signal is decomposed into building blocks that are well represented in time and frequency. In the search for significant features of the ECG signal, it is filtered using wavelet filtering based on the wavelet transform.

5 106 While the set of decomposition functions of the Fourier transform are the functions of sin(k 0 t) and cos(k 0 t) only, the set of decomposition functions of the wavelet transform are wider and different sets of decomposition functions are used. Virtually all wavelet systems have these very general characteristics. Where the Fourier transform maps a onedimensional signal to a one-dimensional sequence of coefficients, the wavelet transform maps it into two dimensional arrays of coefficients. This allows localizing the signal in both time and frequency. The concept of the wavelet transform is usually introduced by the resolution concept to define the effect of changing scale. The application of wavelet noise suppression requires the selection of different parameters. The wavelet noise reduction performance of the ECG signal is conditioned by three processing parameters named wavelet denoising parameters, namely Type of wavelet basis function, Thresholding function, Threshold selection rules, 4.3 SELECTION OF THE WAVELET This is the most interesting question for most of the users. The wavelet has one or two parameters. Because wavelets have so many constraints, that are not associated with the signal, but more with math and calculation limitations, it is virtually impossible to blindly select a wavelet. The most general-purpose usable wavelet is Daubechies. The Haar wavelet is actually a differential operator. The Daubechies1 equals Haar.

6 107 As mentioned, the wavelets have one primary parameter. This parameter defines two things: region of support and the number of vanishing moments. The region of support means, how long the wavelet is. This will affect the localization capabilities. The longer the wavelet, the larger the part of the time series that will be taken into account for calculating the amplitude at any time position. And more averaging will occur, similar to that in DFT. The number of vanishing moments is always the same as the region of support level. The number of vanishing moments defines the order of the polynomial that will be ignored if present in the time series. The attention of researchers has gradually turned from frequencybased analysis using Fourier transforms to scale-based analysis using wavelet transforms when it started to become clear that an approach measuring average fluctuations at different scales might prove less sensitive to noise. Based on experimental results, any one kind of wavelet has to be chosen for usage. The mother wavelet DB1 (Daubechies One) may be used because its detail coefficients indicate sharp changes in a signal indicating transition state (acceleration or deceleration) and implement it. To segment a signal automatically using wavelets, an algorithm may be developed and implemented. The different wavelet families make different trade-offs between how compactly the basis functions are localized in space and how smooth they are. Within each family of wavelets are wavelet subclasses distinguished by the number of coefficients and by the level of iteration. Wavelets are classified within a family most often by the number of vanishing moments. 4.4 DIFFERENT FAMI LIES OF WAVELETS FUNCTION wavelet Several families of wavelets have proven to be useful. Some families are Meye (meyr), Mexican hat (mexh), Morlet (morl),

7 108 Gaussian (gaus1-gaus8), Symlet (sym1-sym45), Coiflet (coif1-coif5), Daubechies (db1-db45), and Biorthogonal (bior1.1-bior1.5 and bior2.2- bior2.8 and bior3.1-bior3.9). In this proposed method the following four wavelet transforms namely Harr, Daubechies (db1-db45), Symlet (sym1- sym45) and Biorthogonal (bior1.1-bior1.5 and bior2.2-bior2.8 and bior3.1- bior3.9) are chosen. 4.5 THRESHOLD SELECTION RULES The choice of the thresholding functions and threshold values plays an important role in the global performance of a wavelet processor for noise reduction. Threshold selection Rules are based on the underlying model. There are mainly four threshold selection rules. 1. Rigrsure Threshold is selected using the principle of Stein s Unbiased Risk Estimate (quadrature loss function). One gets an estimate of the risk for a particular threshold value t. Minimizing the risks in t gives a selection of the threshold value. 2. Sqtwolog Fixed form threshold yielding minimax performance multiplied by a small factor proportional to log (length(s)). It is usually equal to sqrt (2* log (length (s))) 3. Heursure Threshold is selected using a mixture of first two methods.. As a result, if the signal-to-noise ratio is very small, the SURE estimate is very noisy. Hence, if such a situation is detected, the fixed form threshold is used.

8 Minimaxi This method uses a fixed threshold, chosen to yield minimax performance for mean square error against an ideal procedure. The minimax principle is used in statistics in order to design estimators. Since the de-noised signal can be assimilated to the estimator of the unknown regression function, the minimax estimator is the one that realizes the minimum of the maximum mean square error obtained for the worst function in a given set. 4.6 GENETIC ALGORITHM APPROACH IN FITNESS FUNCTION GA works with a set of candidate solutions called a population. Based on the principle of survival of the fittest, the GA obtains the optimal solution after a series of iterative computations on its operators: the reproduction, the crossover, and the mutation. The size of the population and the probability rates for crossover and mutation are called the control parameters of the genetic algorithm. GA generates successive populations of alternate solutions that are represented by a chromosome, i.e. a solution to the problem, until acceptable results are obtained based on the fitness function. The fitness function has to provide some measures of the GA s performance in a particular environment and assess the quality of a solution in the evaluation step. The objectives of denoising are to suppress effectively the noise and restore the original ECG signal. A common goal of optimization in ECG noise suppression is to minimize the mean square error between the original ECG signal and the denoisy version of this ECG signal, and so the MSE has been chosen as the fitness function. Given an original signal x (n), consisting of N samples, and a reconstructed approximation to this signal x ˆ( n), the MSE is given by

9 110 N 1 MSE ( ) ( ) (4.1) N n 1 2 x n xˆ n 4.7 ECG GA-WAVELET BASED DENOISING Consider an ECG signal corrupted by standard white Gaussian noise. The GA was used to search for the optimum wavelet denoising parameters for ECG signal noise elimination problems. The proposed GAwavelet based denoising is shown in Figure 4.1 and can be explained in the following steps. Step 1. The inputs for the proposed technique are noisy ECG signal and wavelet denoising parameters Step 2. Set the proper wavelet thresholding denoising parameter ranges for ECG signal and construct the objective functions, including the mean square error. Step 3. Optimize the wavelet denoising parameters using GA, by means of selection, crossover and mutation a satisfied termination criteria is reached (according to the noise suppression performance) and select the optimal denoising parameters. Step 4. Perform a 1-D discrete wavelet transform for the noisy ECG signal to get all the wavelet coefficients. Step 5. Step 6. Threshold the noisy coefficients in ECG signal with the optimal thresholds and get the modified new ECG components. Reconstruct the denoising ECG signal.

10 111 Figure 4.1 The GA-wavelet denoising technique Table 4.1 shows the denoising results of ECG signal obtained using the GA-wavelet denoising technique for Input SNR: 0 45 db. Here the decomposition level is chosen as 3 for all the methods. Among the chosen wavelet functions (Harr, Daubechies (db1-db45), Symlet (sym1-sym45) and Biorthogonal (bior1.1-bior1.5 and bior2.2-bior2.8 and bior3.1-bior3.9)) and the selection rules (Rigrsure, Sqtwolog, Heursure and Minimaxi), one which provides the best performance in terms of SNR and PRD for the soft threshold method and proposed method are tabulated. For instance, as seen the Table 4.1, for input SNR of 10 db, GA selects the wavelet function Daubechies6 and Rigrsure for soft threshold method that gives the output SNR of db and Biorthogonal 3.9 and Rigrsure for proposed method to give output SNR of db.

11 Table 4.1 The performance of denoising the ECG signals in terms of SNR and PRD Input SNR (db) Output SNR (db) Improvement SNR (db) PRD % Wavelet Function Soft Threshold Proposed Soft Threshold Proposed Soft Threshold Proposed Soft Threshold Proposed Threshold Selection Rule Soft Threshold Proposed Harr Biorthogonal 3.9 Sqtwolog Minimaxi Biorthogonal 3.9 Biorthogonal 3.9 Sqtwolog Rigrsure Daubechies6 Biorthogonal 3.9 Rigrsure Rigrsure Biorthogonal 3.9 Biorthogonal 3.9 Minimaxi Minimaxi Biorthogonal 3.9 Biorthogonal 3.9 Rigrsure Heursure Biorthogonal 3.9 Biorthogonal 3.9 Heursure Rigrsure Biorthogonal 3.9 Biorthogonal 3.9 Rigrsure Rigrsure Biorthogonal 3.9 Biorthogonal 3.9 Rigrsure Sqtwolog Biorthogonal 3.9 Biorthogonal 3.9 Rigrsure Sqtwolog Biorthogonal 3.9 Biorthogonal 3.9 Heursure Sqtwolog 112

12 113 It is observed from the Table 4.1, improvement in SNR obtained increases for the proposed method than that of soft threshold method as the input SNR increases that is from the input SNR value of 20 db in the Table RESULTS AND DISCUSSION This section presents the simulation performed to verify the effectiveness of the proposed method. The performance of the proposed modified method on the basis of two performance measures; 1) Percent root mean square difference and 2) Signal to noise ratio. Case 1 A denoising technique for ECG signals is proposed based on genetic algorithm and wavelet transform. The noise reduction of a signal depends on the optimum value of the level of decomposition, the suitable forms of wavelet family and the thresholding techniques. The original ECG signal and the corrupted ECG with noise is shown in Figure 4.2. The noisy signal is decomposed using DWT into wavelet coefficients. Thresholding technique is applied and reconstructed using IDWT to obtain denoised signal. Figure 4.3 and Figure 4.4 show the signal obtained using soft threshold and proposed method (modified soft threshold method) respectively. The percent root mean-square difference and the signal-to-noise ratio SNR are used as measures of noise reduction performance. The PRD and SNR (in db) are calculated as follows by using the equations (2.49) and (2.50) respectively, PRD N N 2 2 x( n) xˆ( n) / x( n) 100 (2.49) n 1 n 1

13 114 SNR 10 (2.50) n 1 n 1 N N 2 2 log x n x n x n 10 ( ) / ( ) ˆ( ) Case 2 This section represents the simulation of various parameters performed to verify the effectiveness of the proposed method (modified soft threshold) when compared with the soft threshold method. The improved simulated result in signal to noise ratio (SNR) for proposed method is represented in Figure 4.5. The output signal to noise ratio for proposed method is initially low when compared with soft threshold method. When the input SNR is above 20 db, signal to noise ratio in the proposed method increased when compared with the soft threshold method. Similarly the PRD of proposed method initially increases and then decreases as input SNR increases when compared with the soft threshold method as shown in Figure 4.6. Finally the bar chart is plotted for signal to noise ratio for different types of wavelet functions and threshold selection rules are plotted in Figures 4.7 and 4.8. Figure 4.2 (a) Original signal (b) The corrupted ECG with noise at input SNR 20 db

14 115 Figure 4.3 The denoised ECG signal resulting from the soft threshold technique ( = soft, = Rigrsure, = Biorthogonal 3.9) Figure 4.4 The denoised ECG signal resulting from the modified soft threshold (proposed method) technique ( = soft, = Heursure, = Biorthogonal 3.9) Figure 4.5 Comparision of signal to noise ratio for soft threshold and proposed method (modified soft threshold method)

15 116 Figure 4.6 Percent root mean square difference (PRD) for various methods Figure 4.7 Signal to noise ratio for different types of wavelet function Figure 4.8 Signal to noise ratio for different types of threshold selection rule

16 CONCLUSION A denoising technique for ECG signals is proposed based on genetic algorithm and wavelet transform. Selection of wavelet denoising parameters is critical to the success of noise elimination process for the ECG signal. For efficient selection of wavelet denoising parameters, besides experience, GA is proposed to optimize the entire range set of wavelet denoising parameters leading to an efficient ECG signal filtration. The noise reduction of a signal depends on suitable forms of wavelet family and the thresholding techniques. This varies for different kinds of input signals. In spite of hard thresholding being the simplest method, soft thresholding can produce better results than hard thresholding. This is because hard thresholding may cause discontinuities in the signals. In this work, soft thresholding is compared with a new thresholding algorithm called modified soft thresholding in terms of SNR and PRD. Modified soft thresholding gives better results than the soft thresholding. Taken into consideration that GA is a powerful tool for parameters selection and optimization, therefore the combination between the GA and wavelet transform makes this denoising technique more powerful than the available systems.