Non Parametric Regression of Biological Signals using Wavelets

Non Parametric Regression of Biological Signals using Wavelets

Dr. V. V. K. D. V. Prasad

Professor of ECE,

Gudlavalleru Engineering college, Krishna dt, Andhra Pradesh, INDIA

A. Ravi Naga Teja

M.Tech Student

Gudlavalleru Engineering College, Krishna dt, Andhra Pradesh, INDIA

Abstract – To estimate the biological signals from noisy environment wavelet shrinkage method is popularly used. In this method hard and garrote filters are commonly used. A hybrid thresholding filter is proposed in this paper to estimate the biological signals. This hybrid filter is applied to noisy EEG signal contaminated with additive white Gaussian noise using FDR method and Hypothesis method. These results are compared with existing hard and garrote filters and also compared with Block James stein method using Mean square error (MSE) and Signal to noise ratio (SNR) parameters. From the results, the hybrid thresholding filter performs better than both hard and garrote filters with FDR method and Hypothesis method and also performs better than existing Block James stein method.

Keywords: Wavelet Transform, EEG signal, Wavelet shrinkage method, Denoising, hybrid thresholding filter

1. INTRODUCTION

   and Block James stein method are proposed and coiflet wavelet is used in wavelet transform and inverse wavelet transform.

   1. FDR method

      The minFDR (minimizing false discovery rate) method was introduced by B. Vidakovic for one-dimensional data. It determines the same global threshold for all shrinkage functions by keeping the expected value of the fraction of coefficients erroneously included in the reconstruction below a given fraction p. Given the M wavelet coefficients (n, n = 1, 2… M), first it computes x-values.

      xn=2[1- (|wn|/)]

      Where (.) is the cumulative distribution function of the standard normal distribution and is an estimation of the noise-standard deviation. Then xn values are ordered as x(1)x(2)x(3).x(M). Starting with n = 1, let q be the largest index such that

      In present days there has been phenominal growth in collection of signals or data. During the transmission, it is

      x (q)

      p

      contaminated with noise. Before processing the signal at the destination, denoising is required. This is applicable to biological signal also. The random noises uncorrelated with biological signals can be approximated by additive gaussian

      The threshold value is obtained as

      = -1(1- (x

   2. Hypothesis method

      (q)

      /2))

      noise There are so many methods for denoising the signal, Wavelet transform is very popular method because of its good localization properties of time and frequency domain.

      Wavelet shrinkage denoising is widely used technique for denoising of biological signal and a hybrid thresholding filter is proposed in this paper. This filter is evaluated by using FDR method and Hypothesis method to denoising the EEG signal. These results are compared with existing filters

      The threshold estimation in this method is independent of thresholding filter used. It calculates level dependant thresholds after performing wavelet transformation on the signal (Ogden, 1997).

      Let the wavelet coefficients are M in number at a particular level and assume that they are normally distributed.

      and Block James stein method using MSE and SNR parameters.

      V M= {

      -1[((1-)

      1/M+1)/2]} 2

2. WAVELET SHRINKAGE DENOISING

   ( ) is cumulative distribution function of standard normal density. Where is error probability parameter. Then find the largest of the squared wavelet coefficients at that level,

   In this method, first the noisy EEG signal is applied to Wavelet transform to get wavelet coefficients. After fixing the threshold value using thresholding method, the wavelet

   denoted by 2

   (M-1) and compare it to the above value V M. If

   M M

   2 /2> V

   coefficients are modified subjecting to threshold value using a thresholding filter. The inverse wavelet transform is applied on modified wavelet coefficients to get the denoised EEG signal. In this paper, FDR method, Hypothesis method

   Where, is an estimate of the standard deviation of noise, 2M is retained as signal. Next repeat the process with the square of second largest (in absolute value) wavelet

   coefficient 2 (M-1). If

   2 (M-1)

   /2> V

   Garrote filter is defined as

   , = 2 for all ||>

   M-1

   The procedure continues until at some point the bth largest (in absolute value) coefficient satisfies

   = 0 otherwise

   2 (b)

   /2> V

   ,

   b

   The threshold at that level is then set as = 2 (b).

   The recommended value for is 0.05.

   1. Block James stein method

      Block James stein method was proposed by CAI (1999). It is also a thresholding method containing blocks but in this thesholding filter is not used. Fix a block size L and a threshold level and divide the empirical wavelet coefficients j,k at any given resolution level j into nonoverlapping blocks of size L. In each case, the first few empirical wavelet coefficients might be re-used to fill the last block which is called the Augmented case or the last few remaining empirical wavelet coefficients might not be used in the inference which is called the Truncated case, should L not divide 2^j exactly.

      j,k

      S

      Denote (jb) the indices of the coefficients in the b-th block at level j, i.e. (jb) = {(j, k): (b 1) L + 1 k bL}.Let S2jb= k(jb)2 denote the sum of squared empirical wavelet coefficients in the block. A block (jb) is deemed important if

      jb

      2 is larger than the threshold and then all the coefficients in the block are retained otherwise the block is considered negligible and all the coefficients in the block are estimated by zero.

   2. Thresholding Filters

   Thresholding filters are used to apply the threshold value. In this paper Hard and Garrote filters are consider.

   Hard filter is proposed by Donoho and Johnstone and it is defined as

   , = for || >

   = 0 otherwise

   ,

   – 0

   – 0

   Fig. 2 Garrote Thresholding Filter

   represents wavelet coefficients and represents threshold value

3. HYBRID THRESHOLDING FILTER

   For filtering the noisy wavelet coefficients, the hybrid thresholding filter is proposed in this paper. This filter is designed by taking the average of outputs of Garrote and Hard thresholding filters for above threshold value. For below threshold value, 20 % of coefficient value is considered. It is represented as

   2

   +

   , = for all || >

   2

   = 0.2* otherwise

   represents wavelet coefficients and represents threshold value.

4. SIMULATION RESULTS AND DISCUSSION

In this section the results obtained using hard, garrote and hybrid thresholding Filter on noisy EEG signal are presented. The EEG signal of sample size is 2048 and it is contaminated with white Gaussian noise of different standard deviation values are simulated. Wavelet transform with coiflet wavelet is used to decompose the EEG signal. FDR method and Hypothesis method are used to fix the threshold value. Wavelet coefficients are modified using a thresholding filter. The inverse wavelet transform is applied on modified wavelet coefficients. The results are compared using MSE and SNR parameters.

Fig 1 Hard Thresholding Filter

MSE = 1

=1

2

SNR = 10 log10

=1

2

2

=1

n represents no.of samples, is original signal and is denoised signal. The simulation experiment is repeated 100 times and average values of MSE and SNR values are taken. This process is conducted on different EEG signals and the results are same. This simulation is implemented in MATLAB environment. The results of EEG signal for =10, 20 and 30 using hard, garrote and hybrid thresholding filter with FDR method and Hypothesis method are shown in Table 1 and Table 2. Table 3 indicates the denoising results of EEG signal using Block James stein method. The original and denoised EEG signals using hybrid thresholding filter with FDR method and Hypothesis method are shown in Figs 3-6. Fig 7 shows denoised signal of EEG signal using Block James stein method. Graph 1-6 shows the comparision of the results in FDR, Hypothesis and Block James stein methods.

For =10, MSE is 60.3710 and SNR is 17.8923 are obtained on denoising of noisy EEG signal with hard thresholding filter and MSE is 70.9529 and SNR is 17.1982 are obtained with garrote thresholding filter using FDR method. For hybrid thresholding filter, MSE is 48.2747 and

SNR is 18.8645 are obtained (Table 1). This indicates that the hybrid thresholding filter performs better than both hard and garrote thresholding filters. The same behavior is found for =20 and 30 (Table 1).

Using Hypothesis method, for =10, MSE is 69.1445 and SNR is 17.3054 are obtained on denoising of noisy EEG signal with hard thresholding filter and MSE is 86.1091 and SNR is 16.3554 are obtained with garrote thresholding filter. For hybrid thresholding filter, MSE is 55.2715 and SNR is 18.2779 are obtained (Table 2). This indicates that the hybrid thresholding filter performs better than both hard and garrote thresholding filters. The same behavior is found for =20 and 30 (Table 2) .

Using Block James stein method, for =10, MSE is 60.1850 and SNR is 17.9113 are obtained on denoising of noisy EEG signal (Table 3). It is clear that the MSE and SNR values of hybrid thresholding filter using FDR method and Hypothesis method are better than Block James stein method. For =20 and 30 also the same result obtained.

Table 1: Denoising Results of EEG signal using Hard, garrote and hybrid Thresholding Filters: FDR method

	=10		=20		=30
	MSE	SNR	MSE	SNR	MSE	SNR
Noisy signal	100.4997	15.6767	397.6322	9.7044	899.4194	6.1594
Hard filter	60.3710	17.8923	164.2737	13.5472	291.6928	11.0574
Garrote filter	70.9529	17.1982	191.9717	12.8735	355.9487	10.1958
Hybrid filter	48.2747	18.8645	135.8736	14.3714	249.6676	11.7317

Table 2: Denoising Results of EEG signal using Hard, garrote and hybrid Thresholding Filters: Hypothesis method

	=10		=20		=30
	MSE	SNR	MSE	SNR	MSE	SNR
Noisy signal	100.4997	15.6767	397.6322	9.7044	899.4194	6.1594
Hard filter	69.1445	17.3054	166.2564	13.4932	278.9246	11.2512
Garrote filter	86.1091	16.3554	199.8411	12.6974	334.6151	10.4638
Hybrid filter	55.2715	18.2779	138.8248	14.2779	242.9398	11.8502

Table 3: Denoising Results of EEG signal: James stein method, length: 2048

	=10		=20		=30
	MSE	SNR	MSE	SNR	MSE	SNR
Noisy signal	100.4997	15.6767	397.6322	9.7044	899.4194	6.1594
Denoised signal	60.1850	17.9113	157.7887	13.7248	273.7316	11.3370

Fig 3: Original EEG signal

20

18

16

SNR

14

12

10

8

6

10 20 30

satandard deviation

noisy signal hard filter

garrote filter hybrid filter

Graph 2: SNR values of different filters in FDR method.

Fig 4: Noisy EEG signal

Fig 5: Denoised EEG signal using hybrid thresholding filter with FDR method

800

MSE

600

400

200

0

10 20 30

standard deviation

noisy signal hard filter

garrote filter hybrid filter

Graph 3: MSE values of different filters in Hypothesis method

Fig 6: Denoised EEG signal using hybrid thresholding filter with Hypothesis method

Fig 7: Denoised EEG signal with James Stein method

20

18

16

SNR

14

12

10

8

6

10 20 30

standard deviation

noisy signal hard filter

garrote filter hybrid

filter

800

MSE

600

400

200

0

10 20 30

standard deviation

noisy signal hard filter

garrote filter hybrid

filter

Graph 4: SNR values of different filters in Hypothesis method

Graph 1: MSE values of different filters in FDR method.

800

MSE

600

400

200

0

10 20 30

standard deviation

noisy signal

denoised signal

REFERENCES

[1]. Mallat, S. G., (1989), A theory for multiresolution signal decomposition: The Wavelet representation, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 11, pp 674-693.

[2]. Dr. V.V.K.D.V.Prasad (2013) Denoising of Biological signals using wavelets, International Journal of Current Engineering and Technology, Vol.3, No.3, pp.863-866

[3]. Donoho, D.L., and Johnstone, I.M., (1995), Adapting to unknown smoothness via Wavelet Shrinkage, Journal of the American Statistical Association, Vol. 90, No. 432, pp 1200-1224.

[4]. Donoho , D. L., and Johnstone, I. M., (1994), Ideal spatial adaptation via Wavelet Shrinkage, Biometrika, Vol. 81, pp 425- 455.

[5]. Donoho, D. L., (1995) ,Denoising by Soft Thresholding, IEEE Trans. Information Theory, Vol. 41, No. 3, pp 613-627

Graph 5: MSE values in Block James stein method

20

18

[6]. Fodor, I. K., and Kamath, C., (2003) ,Denoising through wavelet shrinkage: An empirical study, Journal of Electronic Imaging, vol. 12, pp. 151-160.

[7]. Carl Taswell, (2000) The what, how and why of wavelet shrinkage denoising, Computing in Science and Engineering, pp 12-

16

SNR

14

12

10

8

6

10 20 30

standard deviation

noisy signal

denoised signal

19

[8]. Dr. V.V.K.D.V.Prasad (2008) Denosing of Biological Signals Using Different Wavelet Based Methods and Their Comparison, Asian Journal of Information Technology, Vol.7, No.4, pp.146-149

[9]. Dr. V.V.K.D.V.Prasad (2013) A New Wavelet Packet Based Method for Denoising of Biological Signals, International Journal of Research in Computer and Communication Technology, Vol.2, Issue.10, pp.1056-1062.

[10]. Akhilesh Bijalwan, Aditya Goyal, Nidhi Sethi (2012),Wavelet Transform Based Image Denoise Using Threshold Approaches, International Journal of Engineering and Advanced Technology (IJEAT), Vol.1, Issue.5,pp.218-221.

Graph 6: SNR values in Block James stein method.

5. CONCLUSION

In this paper, a hybrid thresholding filter for wavelet shrinkage denoising of Biological signals is proposed. The performance of this filter is evaluated by using EEG signals. The results are compared with existing hard and garrote filters and Block James stein method. It is found that the hybrid thresholding filter performs better than both hard and garrote filters with FDR method and Hypothesis method and also performs better than existing Block James stein method.

ACKNOWLEDGEMENTS

The authors place on record their thanks to the authorities of Gudlavalleru Engineering College, A.P for the facilities they provided.