**4. Thresholding and threshold estimation techniques**

The simpler way to remove noise or to reconstruct the original signal from a contaminated signal, in case of 1D or 2D, using the wavelet coefficients which are the result of decomposition in wavelet transform, is to eliminate the small coefficient associated to the noise. After updating the coefficients by removing the small coefficients assuming as noise, the original signal can be obtained by the reconstruction algorithm using the noise free coefficients. Because it is usually considered that the noise has high frequency coefficients, the elimination of the small coefficient generally applied on the detail coefficients after the decomposition. Indeed, the main idea of the wavelet denoising to obtain the ideal components of the signal from the noisy signal requires the estimation of the noise level. The estimated noise level is used in order to threshold the small coefficient assumed as noise.

The procedure of the signal denoising based on DWT is consist of three steps; decomposition of the signal, thresholding and reconstruction of the signal. Several methods use this idea proposed and implements it in different ways. When attempting to decrease the influence of noise wavelets coefficient, it is possible to do this in particular ways, also the need of information of the underlying signal leads to different statistical treatments of the available information.

In the linear penalization method every wavelet coefficient is affected by a linear shrinkage particular associated to the resolution level of the coefficient. It can be said that linear thresholding is appropriate only for homogeny signals with important levels of regularity. The wavelet thresholding or shrinkage methods are usually more suitable. Since the work of Donoho and Johnstone (Donoho and Johnstone 1994), there has been a lot of research on the way of defining the threshold levels and their type. Donoho and Johnstone proposed a nonlinear strategy for thresholding. In their approaches, the thresholding can be applied by implementing either hard or soft thresholding method, which also called as shrinkage.

In the hard thresholding, the wavelet coefficient below a give value are stetted to zero, while in soft thresholding the wavelet coefficient are reduced be a quantity to the thresh value. The threshold value is the estimation of the noise level, which is generally calculated from the standard deviation of the detail coefficient (Donoho 1995). Fig. 4 indicates the two types of thresholding, which can be expressed analytically as;

Approximation

Horizantal Details

Vertical Details

Diagonal Details

(a) (b) Fig. 3. The DWT decomposition and reconstruction steps of a 2D image signal for level of 2;

The simpler way to remove noise or to reconstruct the original signal from a contaminated signal, in case of 1D or 2D, using the wavelet coefficients which are the result of decomposition in wavelet transform, is to eliminate the small coefficient associated to the noise. After updating the coefficients by removing the small coefficients assuming as noise, the original signal can be obtained by the reconstruction algorithm using the noise free coefficients. Because it is usually considered that the noise has high frequency coefficients, the elimination of the small coefficient generally applied on the detail coefficients after the decomposition. Indeed, the main idea of the wavelet denoising to obtain the ideal components of the signal from the noisy signal requires the estimation of the noise level. The estimated noise level is used in order to threshold the small coefficient assumed as noise.

The procedure of the signal denoising based on DWT is consist of three steps; decomposition of the signal, thresholding and reconstruction of the signal. Several methods use this idea proposed and implements it in different ways. When attempting to decrease the influence of noise wavelets coefficient, it is possible to do this in particular ways, also the need of information of the underlying signal leads to different statistical treatments of the

In the linear penalization method every wavelet coefficient is affected by a linear shrinkage particular associated to the resolution level of the coefficient. It can be said that linear thresholding is appropriate only for homogeny signals with important levels of regularity. The wavelet thresholding or shrinkage methods are usually more suitable. Since the work of Donoho and Johnstone (Donoho and Johnstone 1994), there has been a lot of research on the way of defining the threshold levels and their type. Donoho and Johnstone proposed a nonlinear strategy for thresholding. In their approaches, the thresholding can be applied by implementing either hard or soft thresholding method, which also called as shrinkage.

In the hard thresholding, the wavelet coefficient below a give value are stetted to zero, while in soft thresholding the wavelet coefficient are reduced be a quantity to the thresh value. The threshold value is the estimation of the noise level, which is generally calculated from the standard deviation of the detail coefficient (Donoho 1995). Fig. 4 indicates the two types

of thresholding, which can be expressed analytically as;

<sup>2</sup> h

<sup>2</sup> g

Column Filtering Row Filtering

g

g

<sup>2</sup> h

<sup>2</sup> h

2

2

X(m,n)

Reconstructed Image Signal

Approximation

Horizantal Details

Vertical Details

Diagonal Details

h <sup>2</sup>

g <sup>2</sup>

a.Decomposition, b. Reconstruction

available information.

Row Filtering Column Filtering

g

g

h <sup>2</sup>

h <sup>2</sup>

2

2

**4. Thresholding and threshold estimation techniques** 

X(m,n)

Original Image Signal

$$\begin{array}{c} \text{Hard threshold}: \begin{cases} y = \mathbf{x} & \text{if } \begin{|\mathbf{x}| > \mathcal{X}} \\ y = \mathbf{0} & \text{if } \begin{|\mathbf{x}| < \mathcal{X}} \end{|} \end{cases} \end{array} \tag{4}$$

$$\text{Softth threshold}: \begin{cases} y = \text{sign}(\mathbf{x})(|\mathbf{x}| - \mathcal{X}) \\ \end{cases} \tag{5}$$

Where *x* is the input signal, y is the signal after threshold and is the threshold value, which is critical as the estimator leading to destruction, reduction, or increase in the value of a wavelet coefficient.

Fig. 4. Threshold types; a. Hard, b. Soft.

Hard thresholding method does not affect on the detail coefficients that grater the threshold level, whereas the soft thresholding method to these coefficients. There are several considerations about the properties and limitation of these two strategies. However the hard thresholding may be unstable and sensitive even to small changes in the signal, the soft thresholding can create unnecessary bias when the true coefficients are large. Although more sophisticated methods has been proposed to overcome the drawbacks of the described nonlinear methods, it is still the most efficient and reliable methods are still the hard and soft thresholding techniques (Donoho 1995).

One important point in thresholding methods is to find the appropriate value for the threshold. Actually, many approaches have been proposed for calculating the threshold value. But, all the approaches require the estimation of noise level. However the standard deviation of the data values may be use as an estimator, Donoho proposed a good estimator for the wavelet denoising given as;

$$
\sigma = \frac{median(d\_{L-1,k})}{0.6745}, \quad k = 0, 1, \dots, 2^{L-1} - 1 \tag{6}
$$

where L denotes the number of decomposition levels. As mentioned above, this median selection made on the detail coefficient of the analyzed signal.

The most known threshold selection algorithms are minimax, universal and rigorous sure threshold estimation techniques (Donoho and Johnstone 1994; Donoho and Johnstone 1998).

The *minimax* threshold value *<sup>M</sup>* proposed by Donoho consists an optimal threshold that derived from minimizing the constant term in an upper bound of the risk involved in the estimation. The proposed threshold depends of the available data and also takes into account the noise level contaminating the signal. The optimal threshold is defined as;

$$
\mathcal{X}\_{\mathcal{M}} = \sigma \mathcal{X}\_n^\* \tag{7}
$$

Signal and Image Denoising Using Wavelet Transform 503

<sup>2</sup> ( , ) 2 : min( , *Sure X n i X*

*i i*

The best way to test the effect of noise on a signal is to add a Gaussian white noise, in which case its values independently and identically distributed (i.i.d) Gaussian real values. After the denoising process, the performance can be measure by comparing the denoised signal and the original signal. However, many methods have been proposed to measure the performance of denoising algorithms, the signal to noise ratio (SNR) and peak signal to noise ratio (PSNR) has generally accepted to measure the quality of signal and images, respectively. For one dimensional signal, measuring the performance of the denoising

> 0 0 10log ( ) ( ) ( ]) *<sup>r</sup> N N*

 

where *x n*( ) is the original signal, ( ) *<sup>r</sup> x n* is the denoised signal and *x n*( ) refers to the mean

0 0 10log (, ) (, ) *<sup>r</sup> N M*

where *L* denotes the quantized gray level of the images, *x n*( ) is original images, *xnm* (, )is the mean value of *x n*( ) , and (, ) *<sup>r</sup> x nm* refers to reconstructed image. In order to get visible alteration on signal, the power of noise should be chosen adequately. Indeed, SNR is usually the most important measure rather than the power of noise, when taking into consideration that the power of the signal to denoise can be varied. When the SNR is chosen above 3dB, it

The records of the acoustical vibrations produced by heart, acquired through microphones from human chest, called phonocardiogram (PCG), consist of the heart sounds and the murmurs. This records of acoustic signals are unfortunately disturbed by various factors which effecting as noise. These effects decrease the performance of visual and computerized

*n m PSNR L x n m x n m* 

*n n SNR x n xn x n* 

In order to measure the quality of image, it is generally used PSNR, which given as;

1 1 <sup>2</sup> <sup>2</sup>

 

(14)

(15)

*X* (13)

, it is found that Sure is

Where the operator ( ) returns the cardinality of the set *i X*: *<sup>i</sup>*

*l risk* .

Where Sure is defined as

an un biased estimate of the <sup>2</sup>

**5.1 Comparison assessments** 

value of *x n*( ) .

**5. Denoising application examples** 

method by calculation of the residual SNR given as;

is generally enough to get the visible corruption.

analysis (Akay, Semmlow et al. 1990; Ergen, Tatar et al. 2010).

**5.2 Phonocardiogram denoising** 

10

 1 1 <sup>2</sup> 10

where \* *n* is defined as the value of and satisfying as;

$$\mathcal{A}\_n^\* = \inf\_{\lambda} \sup\_{\lambda} \mathcal{A} \left\{ \frac{R\_{\lambda}(d)}{n^{-1} + R\_{\text{oracle}}(d)} \right\} \tag{8}$$

where <sup>2</sup> *Rd E d d* () ( () ) and ( ) *R d oracle* named as oracle used to account for the risk associated to the modification of the value of a given wavelet coefficient. Two oracles are considered, the diagonal liner projection (DLP) and the diagonal linear shrinker (DLS)(Donoho and Johnstone 1994). The ideal risks for these oracles are given by

$$R\_{oracle}^{DLP}(d) = \min(d^2, 1) \tag{9}$$

$$R\_{\text{oracle}}^{DLS}(d) = \frac{d^2}{d^2 + 1} \tag{10}$$

The minimax method is used in statistics to design estimator. The minimax estimator is realizes the minimum of the maximum mean square error, over a given set of functions. Another proposed threshold estimator by Donoho is the *universal* threshold, or global threshold, as an alternative to the minimax threshold, however it uses a fixed threshold form given as;

$$
\lambda\_{\rm til} = \sigma \sqrt{2 \log(n)} \tag{11}
$$

Where *n* denotes the length of the analyzed signal and is given by Eq. (6). The advantage of this thresholding appears in software implementation due to easy to remember and coding. Additionally, this threshold estimator ensures that every sample in the wavelet transform in which the underlying function is exactly zero will be estimated as zero.

Again another common estimator is Rigorous Sure (rigresure) threshold proposed by Donoho. This threshold describes a scheme which uses a threshold at each resolution level *l* of the wavelet coeffient. The Rigorous Sure, also known as *SureShrink*, uses the Stein's Unbiased Risk Estimate criterion to obtain unbiased estimate. The threshold is given as follows;

$$\mathcal{A}\_{\rm S} = \arg\min\_{0 < \mathcal{A} < \mathcal{A}\_{\rm U}} \operatorname{Sure} \left( \mathcal{A}\_{\prime} \frac{S(a, b)}{\sigma} \right) \tag{12}$$

Where Sure is defined as

502 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

The most known threshold selection algorithms are minimax, universal and rigorous sure threshold estimation techniques (Donoho and Johnstone 1994; Donoho and Johnstone 1998).

derived from minimizing the constant term in an upper bound of the risk involved in the estimation. The proposed threshold depends of the available data and also takes into

and satisfying as;

1 ( ) inf sup ( ) *n d*

associated to the modification of the value of a given wavelet coefficient. Two oracles are considered, the diagonal liner projection (DLP) and the diagonal linear shrinker

*oracle*

*R d nR d* 

2 <sup>2</sup> ( ) <sup>1</sup>

*<sup>d</sup>*

The minimax method is used in statistics to design estimator. The minimax estimator is realizes the minimum of the maximum mean square error, over a given set of functions. Another proposed threshold estimator by Donoho is the *universal* threshold, or global threshold, as an alternative to the minimax threshold, however it uses a fixed threshold

of this thresholding appears in software implementation due to easy to remember and coding. Additionally, this threshold estimator ensures that every sample in the wavelet

Again another common estimator is Rigorous Sure (rigresure) threshold proposed by

level *l* of the wavelet coeffient. The Rigorous Sure, also known as *SureShrink*, uses the Stein's Unbiased Risk Estimate criterion to obtain unbiased estimate. The threshold is given as

transform in which the underlying function is exactly zero will be estimated as zero.

0 (,) arg min , *<sup>S</sup> <sup>U</sup>*

  \*

account the noise level contaminating the signal. The optimal threshold is defined as;

*<sup>M</sup>* 

(DLS)(Donoho and Johnstone 1994). The ideal risks for these oracles are given by

*DLS oracle <sup>d</sup> R d*

 

Donoho. This threshold describes a scheme which uses a threshold

Where *n* denotes the length of the analyzed signal and

\*

*<sup>M</sup>* proposed by Donoho consists an optimal threshold that

(8)

<sup>2</sup> ( ) min( ,1) *DLP Rd d oracle* (9)

*<sup>U</sup>* 2log( ) *n* (11)

*Sab Sure*

 

 

(10)

is given by Eq. (6). The advantage

(12)

at each resolution

and ( ) *R d oracle* named as oracle used to account for the risk

*<sup>n</sup>* (7)

The *minimax* threshold value

where <sup>2</sup> *Rd E d d* () ( () ) 

 

*n* is defined as the value of

where \* 

form given as;

follows;

$$\text{Sure}(\mathcal{X}, \mathbf{X}) = n - 2 \cdot \Theta\left\{ \mathbf{i} : \left| \mathbf{X}\_i \le \mathcal{X} \right| \right\} + \left[ \min\{ \left| \mathbf{X}\_i \right|, \mathcal{X} \right]^2 \right\} \tag{13}$$

Where the operator ( ) returns the cardinality of the set *i X*: *<sup>i</sup>* , it is found that Sure is an un biased estimate of the <sup>2</sup> *l risk* .

## **5. Denoising application examples**

#### **5.1 Comparison assessments**

The best way to test the effect of noise on a signal is to add a Gaussian white noise, in which case its values independently and identically distributed (i.i.d) Gaussian real values. After the denoising process, the performance can be measure by comparing the denoised signal and the original signal. However, many methods have been proposed to measure the performance of denoising algorithms, the signal to noise ratio (SNR) and peak signal to noise ratio (PSNR) has generally accepted to measure the quality of signal and images, respectively. For one dimensional signal, measuring the performance of the denoising method by calculation of the residual SNR given as;

$$\text{SNR} = 10 \log\_{10} \left( \sum\_{n=0}^{N-1} \mathbf{x}^2(n) \bigg/ \sum\_{n=0}^{N-1} \left( \overline{\mathbf{x}}(n) - \mathbf{x}'(n] \right)^2 \right) \tag{14}$$

where *x n*( ) is the original signal, ( ) *<sup>r</sup> x n* is the denoised signal and *x n*( ) refers to the mean value of *x n*( ) .

In order to measure the quality of image, it is generally used PSNR, which given as;

$$PSNR = 10\log\_{10}\left(L\left\langle \sum\_{n=0}^{N-1} \sum\_{m=0}^{M-1} \left(\overline{\mathbf{x}}(n,m) - \mathbf{x}^\prime(n,m)\right)^2\right\rangle\right) \tag{15}$$

where *L* denotes the quantized gray level of the images, *x n*( ) is original images, *xnm* (, )is the mean value of *x n*( ) , and (, ) *<sup>r</sup> x nm* refers to reconstructed image. In order to get visible alteration on signal, the power of noise should be chosen adequately. Indeed, SNR is usually the most important measure rather than the power of noise, when taking into consideration that the power of the signal to denoise can be varied. When the SNR is chosen above 3dB, it is generally enough to get the visible corruption.

#### **5.2 Phonocardiogram denoising**

The records of the acoustical vibrations produced by heart, acquired through microphones from human chest, called phonocardiogram (PCG), consist of the heart sounds and the murmurs. This records of acoustic signals are unfortunately disturbed by various factors which effecting as noise. These effects decrease the performance of visual and computerized analysis (Akay, Semmlow et al. 1990; Ergen, Tatar et al. 2010).

Signal and Image Denoising Using Wavelet Transform 505

As an example of denoising process for PCG signal, the denoised signal and the difference between the original signal and the denoised signal are given in Fig. 7, respectively. '*symlet8'*, '*rigresure'* and *'soft thresholding'* parameters are used in the denoising process. When we compared even the original the original signal and the denoised signal visually,

*x*

*N/2*

*N*

the wavelet denoising process has a quite success.

*Ax*

*N/4*

*DAx*

Fig. 6. The approximation and the detailed coefficients in the tree structure of the DWT.

(c) (d)

a) Approximation coefficients at level one, b) Detail coefficients at level one, c) Approximation coefficients at level two, b) Detail coefficients at level two.

(a) (b)

*L1 Dx*

*AAx*

*L2*

*L3*

Fig. 7. Decomposition of the noisy signal,

*AAAx DAAx*

*N/8*

The respiration sounds by lung mechanical actions, patient movement, and improper contacts of microphone to the skin, and external noises from the environments are added as noise signal into PCG records. The traditional method to remove the noise from a PCG signal is to use a low or band pass filter with cut off frequencies. However the filtering techniques are able to remove a relevant of the noise, they are incapable if the noise in the band of the signal to be analyzed.

The frequency components of a normal PCG signals can be rise up 200Hz, and the energy of the most significant components concentrates around the frequency band 100-150Hz (Ergen, Tatar et al. 2010). The frequency bands of the signal are very important when we use the denoising technique using DWT approaches. Because the DWT approaches decomposes the signal into frequency bands to eliminate the small detail components assumed as noise, the decomposition level reflects directly on the frequency components that cause the smoothed version of the signal.

As stated previous section, the most reasonable way to determine the effectiveness of denoising method is to compare an original signal and the denoised signal obtained from its noise added form. Therefore, here, we will use the noise added signal to examine the effectiveness of wavelet denoising method through the comparison between the original signal and the denoised signal (or reconstructed) signal. Figure 5.a shows a PCG during cardiac period and its noise added form.

Fig. 5. Wavelet denoising of a PCG signal, a) Original signal, b) Noisy signal

The result of the DWT is a multilevel decomposition, in which the signal is decomposed in 'approximation' and 'detail' coefficients at each level. This is made through a process that is equivalent to low-pass and high passes filtering, respectively. DWT decomposition leads to a tree structure as shown in Fig. 6, where approximation and detail coefficients are presented.

The approximation coefficients and the detail coefficients of the noisy signal for the decomposition level of one and two are given Figure 7. In Fig. 7c and Fig. 7d are the results of the decomposition of the approximation coefficient at level one, which represented in Fig. 8a.

The respiration sounds by lung mechanical actions, patient movement, and improper contacts of microphone to the skin, and external noises from the environments are added as noise signal into PCG records. The traditional method to remove the noise from a PCG signal is to use a low or band pass filter with cut off frequencies. However the filtering techniques are able to remove a relevant of the noise, they are incapable if the noise in the

The frequency components of a normal PCG signals can be rise up 200Hz, and the energy of the most significant components concentrates around the frequency band 100-150Hz (Ergen, Tatar et al. 2010). The frequency bands of the signal are very important when we use the denoising technique using DWT approaches. Because the DWT approaches decomposes the signal into frequency bands to eliminate the small detail components assumed as noise, the decomposition level reflects directly on the frequency components that cause the smoothed

As stated previous section, the most reasonable way to determine the effectiveness of denoising method is to compare an original signal and the denoised signal obtained from its noise added form. Therefore, here, we will use the noise added signal to examine the effectiveness of wavelet denoising method through the comparison between the original signal and the denoised signal (or reconstructed) signal. Figure 5.a shows a PCG during

(a) (b)

The result of the DWT is a multilevel decomposition, in which the signal is decomposed in 'approximation' and 'detail' coefficients at each level. This is made through a process that is equivalent to low-pass and high passes filtering, respectively. DWT decomposition leads to a tree structure as shown in Fig. 6, where approximation and detail coefficients are

The approximation coefficients and the detail coefficients of the noisy signal for the decomposition level of one and two are given Figure 7. In Fig. 7c and Fig. 7d are the results of the decomposition of the approximation coefficient at level one, which represented in Fig. 8a.

Fig. 5. Wavelet denoising of a PCG signal, a) Original signal, b) Noisy signal

band of the signal to be analyzed.

cardiac period and its noise added form.

version of the signal.

presented.

As an example of denoising process for PCG signal, the denoised signal and the difference between the original signal and the denoised signal are given in Fig. 7, respectively. '*symlet8'*, '*rigresure'* and *'soft thresholding'* parameters are used in the denoising process. When we compared even the original the original signal and the denoised signal visually, the wavelet denoising process has a quite success.

Fig. 6. The approximation and the detailed coefficients in the tree structure of the DWT.

Fig. 7. Decomposition of the noisy signal, a) Approximation coefficients at level one, b) Detail coefficients at level one, c) Approximation coefficients at level two, b) Detail coefficients at level two.

Signal and Image Denoising Using Wavelet Transform 507

The other parameters to obtain best SNR level are the kind of the wavelet and the thresholding rule. Table 2 presents the SNR levels using different wavelet when the decomposition level is

Nevertheless, it is attracting that the mother wavelets having high oscillation number produces better SNR results. For instance, the symlet wavelet having eight oscillations in its mother wavelet produces better SNR level than the lower ones. In this case, it can be say that the choice of the very lower oscillation frequency to avoid the computational

When the performance of the noise estimation techniques is considered in the respect of the decomposition level and the initial SNR level, the estimation techniques show the same performance for the level five respects to the initial SNR level. For the comparison, the initial SNR level before denoising is increased from 1dB to 30dB, and the result SNR level after denoising is calculated. Fig. 9 presents a comparison of the four noise estimation methods

(a) (b)

Fig. 9. The SNR values after denoising before denoising for level 5 and 8.

five. In table 2, there is no significant difference in SNR in terms of wavelet types.

**Wavelet Type Hard Soft**  Daubechies2 16.5378 16.5057 Daubechies3 18.9391 18.8353 Daubechies4 19.8138 19.8002 Daubechies5 19.8747 19.7425 Symlet2 16.3487 16.4181 Symlet3 18.5401 18.7874 Symlet4 19.5732 19.8002 Symlet5 19.4795 19.5458 Coiflet1 16.7746 16.7658 Coiflet2 19.4866 19.4501 Coiflet 3 19.7812 19.6252 Discrete Meyer 19.9018 19.7154

Table 2. SNR values respect to wavelet types (Rigrsure, level=5)

complexity of the wavelet causes the lower SNR results.

for level five and level six when *'symlet8'* used.

Fig. 8. Denoised signal, a) Denoised signal, d) Difference between the original and the denoised signal.

Also, the effected components in DWT decomposition are related to not only decomposition level but also sampling frequency. The decomposition level influences the frequency bands by dividing the sampling frequency respect the power of two. When we choose the decomposition level is as five, the interested frequencies are about 300Hz while the sampling frequency is 11.5KHz.

Therefore, the most important factor determining the SNR level is the level of the decomposition. Table 1 presents the SNR results respect to the decomposition level by using *symlet8* and *rigresure* estimation for hard and soft thresholding as denoising parameter*.* For the both tresholding techniques, it is seen that the highest SNR value obtained when the decomposition level is five. If the decomposition level is chosen too high, the thresholding will effect on the main frequencies of the original signal. Thus, the SNR has lower values for the level higher than five.


Table 1. SNR level respect to the depth of decomposition.

(a) (b)

Also, the effected components in DWT decomposition are related to not only decomposition level but also sampling frequency. The decomposition level influences the frequency bands by dividing the sampling frequency respect the power of two. When we choose the decomposition level is as five, the interested frequencies are about 300Hz while the

Therefore, the most important factor determining the SNR level is the level of the decomposition. Table 1 presents the SNR results respect to the decomposition level by using *symlet8* and *rigresure* estimation for hard and soft thresholding as denoising parameter*.* For the both tresholding techniques, it is seen that the highest SNR value obtained when the decomposition level is five. If the decomposition level is chosen too high, the thresholding will effect on the main frequencies of the original signal. Thus, the SNR has lower values for

> **Level Hard Soft**  1 8.1209 7.8843 2 11.1471 10.9218 3 14.3251 14.0031 4 17.2973 16.9275 5 20.1305 19.4396 6 13.2248 13.2472 7 12.1531 9.8726 8 10.8010 8.3255 9 10.4986 8.1632 10 10.4912 8.1593

Table 1. SNR level respect to the depth of decomposition.

Fig. 8. Denoised signal, a) Denoised signal, d) Difference between the original and the

denoised signal.

sampling frequency is 11.5KHz.

the level higher than five.


The other parameters to obtain best SNR level are the kind of the wavelet and the thresholding rule. Table 2 presents the SNR levels using different wavelet when the decomposition level is five. In table 2, there is no significant difference in SNR in terms of wavelet types.

Table 2. SNR values respect to wavelet types (Rigrsure, level=5)

Nevertheless, it is attracting that the mother wavelets having high oscillation number produces better SNR results. For instance, the symlet wavelet having eight oscillations in its mother wavelet produces better SNR level than the lower ones. In this case, it can be say that the choice of the very lower oscillation frequency to avoid the computational complexity of the wavelet causes the lower SNR results.

When the performance of the noise estimation techniques is considered in the respect of the decomposition level and the initial SNR level, the estimation techniques show the same performance for the level five respects to the initial SNR level. For the comparison, the initial SNR level before denoising is increased from 1dB to 30dB, and the result SNR level after denoising is calculated. Fig. 9 presents a comparison of the four noise estimation methods for level five and level six when *'symlet8'* used.

Fig. 9. The SNR values after denoising before denoising for level 5 and 8.

Signal and Image Denoising Using Wavelet Transform 509

important if the oscillation number is not very low, the decomposition level is absolutely depends on the frequency band of the PCG signal and its sampling frequency, and rigresure

All digital images contain some degree of noise due to the corruption in its acquisition and transmission by various effects. Particularly, medical image are likely disturbed by a complex type of addition noise depending on the devices which are used to capture or store them. No medical imaging devices are noise free. The most commonly used medical images are received from MRI (Magnetic Resonance Imaging) and CT (Computed Tomography) equipments. Usually, the addition noise into medical image reduces the visual quality that

Because the wavelet transform has an ability to capture the energy of a signal in few energy transform values, the wavelet denoising technique is very effective as stated previous parts. As stated previous sections, when an image is decomposed using wavelet transform, the four subimages are produced, approximation, horizontal details, vertical details and diagonal details. Fig. 11 represents a sample medical image which belongs to a patient having cranial trauma and its four subimages when decomposed for one level using DWT. This image has acquired from a BT device. A noise added MRI image and its denoised form using wavelet denoising procedure is given Fig. 12. The added noise has Gaussian distribution, and symlet6, decomposition level of two, hard thresholding are chosen as wavelet denoising parameters.

Fig. 11. Decomposition of a sample medical image; original, approximation, horizontal

details, vertical details, and diagonal details in left to right.

Fig. 12. A noisy image having PSNR 62dB and its denoised version.

method is best of the noise estimation techniques.

complicates diagnosis and treatment.

**5.3 Image denoising** 

(c)

Fig. 10. The denoised signal using three different threshold rules at level eight.

We have observed no distinguishing evidence among the noise level estimation methods until level six. After this level, rigresure method has produced better SNR values. And it is observed that rigresure preserve the second heart sound in PCG signals while the other methods destroying. This situation is clearly seen in Fig. 10. The signal part related to second heart sound taking place at around 0.7s in Fig.10a is not able to seen in Fig. 10b and Fig. 10c. This shows that the rigresure preserve the main characteristic of the signal. Therefore, we can conclude that the rigresure is the better noise estimation method.

A level-dependent scaling of the thresholds was used to remove Gaussian white noise from the signal. Although it could not found evidence that a single wavelet was the best suited for denoising PCG signal, some wavelets used in this study were slightly better than the others. We conclude that reasonable decomposition level is absolutely depending on the sampling frequency and the frequency band of the signal. Just in this study, the decomposition level of 5 produced reasonable results because the frequency band of a normal PCG signal is around 150-200Hz and the sampling frequency is 11.5KHz. Since the noise level method is one of the important parameter in wavelet denoising, it is examined for different levels. We have not seen any noteworthy differences in the methods from level 1 to level 6. After this level, rigresure method has showed superiority to the other methods in terms of SNR level. Consequently, it is determined that the wavelet type is not very important if the oscillation number is not very low, the decomposition level is absolutely depends on the frequency band of the PCG signal and its sampling frequency, and rigresure method is best of the noise estimation techniques.
