**3. Discrete Wavelet Transform (DWT) and Wavelet Packet Decomposition**

The wavelet transform has become an essential tool for many applications. However, the wavelet transform has been presented a method representing a time-frequency method, continuous wavelets transform (CWT), and the wavelet transform generally has used for the decomposition of the signal into high and low frequency components. The wavelet coefficient represents a measure of similarity in the frequency content between a signal and a chosen wavelet function. These coefficients are computed as a convolution of the signal

Signal and Image Denoising Using Wavelet Transform 499

(a) (b)

In this figure, ↓2 and ↑2 refers to down sampling and up sampling, respectively. This decomposition sometimes called as sub-band coding. The low pass filter produces the approximation of the signal, and the high pass filers represent the details or its high frequency components. The decomposition successively can be applied on the low

Whereas the successive decomposition is applied on the approximation coefficients only as in the DWT, the decomposition may be applied on both sub part of the signal, approximation coefficients and detail coefficients. If the decomposition is applied on the both sides, approximation and details, this kind of decomposition called as wavelet packet transform or wavelet packet tree decomposition. Fig. 2 represents wavelet packet

(a) (b) Fig. 2. The wavelet packet decomposition and reconstruction steps of a 1D signal for level of 2;

In 2D case, the image signal is considered as rows and columns as if they are one dimentional signals. In DWT, firstly the each rows of the image is filtered, then the each columns are filtered as in 1D case. Figure 3 demonstare the decompositon of an image for one level. As in signal decomposition, after each filtering, the subsampling is realized. The result of this process gives four images; approximation, horizantal details, vertical details and diagonal details. Because of subsampling after each filtering, the result subimages of the

Fig. 1. The DWT decomposition and reconstruction steps of a 1D signal for level of 2;

a. Decomposition, b. Reconstruction

decomposition and reconstruction.

a. Decomposition, b. Reconstruction

original image has the quarter size of the original image.

frequency components, approximation coefficients, in DWT.

and the scaled wavelet function, which can be interpreted as a dilated band-pass filter because of its band-pass like spectrum (Valens ; Rioul and Vetterli 1991) .

In practice, the wavelet transform is implemented with a perfect reconstruction filter bank using orthogonal wavelet family. The idea is to decompose the signal into sub-signals corresponding to different frequency contents. In the decomposition step, a signal is decomposed on to a set of orthonormal wavelet function that constitutes a wavelet basis (Misiti, Misiti et al.). The most common wavelets providing the ortogonality properties are daubechies, symlets, coiflets and discrete meyer in order to provide reconstruction using the fast algorithms (Beylkin, Coifman et al. 1991; Cohen, Daubechies et al. 1993).

The use of wavelet transform as filter bank called as DWT (Discrete Wavelet Transform). The DWT of a signal produces a non-redundant restoration, which provides better spatial and spectral localization of signal formation, compared with other multi-scale representation such as Gaussian and Laplacian pyramid. The result of the DWT is a multilevel decomposition, in which the signal is decomposed in *'approximation'* and *'detail'* coefficients at each level (Mallat 1989). This is made through a process that is equivalent to low-pass and high passes filtering, respectively.

As stated previous section, the wavelet transform is firstly introduced for the timefrequency analysis of transient continuous signals, and then extended to the theory of multi-resolution wavelet transform using FIR filter approximation. This managed using the dyadic form of CWT. In dyadic form, the scaling function is chosen as power of two. And then, the discrete wavelets /2 , ( ) 2 (2 ) *m m m n t t n* used in multi-resolution analysis constituting an orthonormal basis for <sup>2</sup>*L* ( ) (Vetterli and Herley 1992; Donoho and Johnstone 1994).

If a signal, *x t*( ) , decomposed into low and high frequency components, that they are respectively named as approximation coefficients and detail coefficients, *x t*( ) reconstructed as;

$$\mathbf{x}(t) = \sum\_{m=1}^{L} \left[ \sum\_{k=-\infty}^{\infty} D\_{m}(k)\boldsymbol{\nu}\_{m,k}(t) + \sum\_{k=-\infty}^{\infty} A\_{l}(k)\boldsymbol{\phi}\_{l,k}(t) \right] \tag{2}$$

Where , *m k* ( )*t* is discrete analysis wavelet, and , ( ) *l k t* is discrete scaling, *D k <sup>m</sup>*( ) is the detailed signal at scale 2*<sup>m</sup>* , and ( ) *A k <sup>l</sup>* is the approximated signal at scale 2*<sup>l</sup>* . *D k <sup>m</sup>*( ) and ( ) *A k <sup>l</sup>* is obtained using the scaling and wavelet filters (Mallat 1999).

$$\begin{aligned} h(n) &= 2^{-1/2} \left\langle \phi(t), \phi(2t - n) \right\rangle \\ g(n) &= 2^{-1/2} \left\langle \psi(t), \phi(2t - n) \right\rangle \\ &= (-1)^n h(1 - n) \end{aligned} \tag{3}$$

The wavelet coefficient can be computed by means of a pyramid transfer algorithm. The algorithms refer to a FIR filter bank with low-pass filter **h**, high-pass filter **g**, and down sampling by a factor 2 at each stage of the filter bank. Fig. 1 shows the tree structure of DWT decomposition for three levels. DWT decomposition leads to a tree structure as shown in Fig. 1, where approximation and detail coefficients are presented.

and the scaled wavelet function, which can be interpreted as a dilated band-pass filter

In practice, the wavelet transform is implemented with a perfect reconstruction filter bank using orthogonal wavelet family. The idea is to decompose the signal into sub-signals corresponding to different frequency contents. In the decomposition step, a signal is decomposed on to a set of orthonormal wavelet function that constitutes a wavelet basis (Misiti, Misiti et al.). The most common wavelets providing the ortogonality properties are daubechies, symlets, coiflets and discrete meyer in order to provide reconstruction using the

The use of wavelet transform as filter bank called as DWT (Discrete Wavelet Transform). The DWT of a signal produces a non-redundant restoration, which provides better spatial and spectral localization of signal formation, compared with other multi-scale representation such as Gaussian and Laplacian pyramid. The result of the DWT is a multilevel decomposition, in which the signal is decomposed in *'approximation'* and *'detail'* coefficients at each level (Mallat 1989). This is made through a process that is equivalent to

As stated previous section, the wavelet transform is firstly introduced for the timefrequency analysis of transient continuous signals, and then extended to the theory of multi-resolution wavelet transform using FIR filter approximation. This managed using the dyadic form of CWT. In dyadic form, the scaling function is chosen as power of two.

analysis constituting an orthonormal basis for <sup>2</sup>*L* ( ) (Vetterli and Herley 1992; Donoho

If a signal, *x t*( ) , decomposed into low and high frequency components, that they are respectively named as approximation coefficients and detail coefficients, *x t*( ) reconstructed as;

( ) ( ) () ( ) ()

detailed signal at scale 2*<sup>m</sup>* , and ( ) *A k <sup>l</sup>* is the approximated signal at scale 2*<sup>l</sup>* . *D k <sup>m</sup>*( ) and

( ) 2 ( ), (2 ) ( ) 2 ( ), (2 )

 

 

 

The wavelet coefficient can be computed by means of a pyramid transfer algorithm. The algorithms refer to a FIR filter bank with low-pass filter **h**, high-pass filter **g**, and down sampling by a factor 2 at each stage of the filter bank. Fig. 1 shows the tree structure of DWT decomposition for three levels. DWT decomposition leads to a tree structure as

*hn t t n g n t tn h n*

*m k k xt D k t A k t* 

> 1/2 1/2 n

 

(-1) (1 )

shown in Fig. 1, where approximation and detail coefficients are presented.

*m n* 

, ( ) 2 (2 ) *m m*

*t t n* used in multi-resolution

*t* is discrete scaling, *D k <sup>m</sup>*( ) is the

(3)

 

, ,

 

(2)

*m mk l lk*

because of its band-pass like spectrum (Valens ; Rioul and Vetterli 1991) .

fast algorithms (Beylkin, Coifman et al. 1991; Cohen, Daubechies et al. 1993).

low-pass and high passes filtering, respectively.

And then, the discrete wavelets /2

1

*m k* ( )*t* is discrete analysis wavelet, and , ( ) *l k*

( ) *A k <sup>l</sup>* is obtained using the scaling and wavelet filters (Mallat 1999).

*L*

and Johnstone 1994).

Where , 

Fig. 1. The DWT decomposition and reconstruction steps of a 1D signal for level of 2; a. Decomposition, b. Reconstruction

In this figure, ↓2 and ↑2 refers to down sampling and up sampling, respectively. This decomposition sometimes called as sub-band coding. The low pass filter produces the approximation of the signal, and the high pass filers represent the details or its high frequency components. The decomposition successively can be applied on the low frequency components, approximation coefficients, in DWT.

Whereas the successive decomposition is applied on the approximation coefficients only as in the DWT, the decomposition may be applied on both sub part of the signal, approximation coefficients and detail coefficients. If the decomposition is applied on the both sides, approximation and details, this kind of decomposition called as wavelet packet transform or wavelet packet tree decomposition. Fig. 2 represents wavelet packet decomposition and reconstruction.

Fig. 2. The wavelet packet decomposition and reconstruction steps of a 1D signal for level of 2; a. Decomposition, b. Reconstruction

In 2D case, the image signal is considered as rows and columns as if they are one dimentional signals. In DWT, firstly the each rows of the image is filtered, then the each columns are filtered as in 1D case. Figure 3 demonstare the decompositon of an image for one level. As in signal decomposition, after each filtering, the subsampling is realized. The result of this process gives four images; approximation, horizantal details, vertical details and diagonal details. Because of subsampling after each filtering, the result subimages of the original image has the quarter size of the original image.

Signal and Image Denoising Using Wavelet Transform 501

*y x if x Hard threshold :* 

*Soft threshold : y sign x x*

which is critical as the estimator leading to destruction, reduction, or increase in the value of

(a) (b)

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

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

> 1, <sup>1</sup> ( ) , 0,1,...,2 1 0.6745 *median dL k <sup>L</sup>*

where L denotes the number of decomposition levels. As mentioned above, this median

*k* (6)

Input Signal

Where *x* is the input signal, y is the signal after threshold and

Thresholded Signal

a wavelet coefficient.

0

 0.5

1


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

soft thresholding techniques (Donoho 1995).

selection made on the detail coefficient of the analyzed signal.

for the wavelet denoising given as;


0

*y if x*

  (4)

(5)

Thresholded Signal

is the threshold value,

Input Signal

0

 0.5

1



Fig. 3. The DWT decomposition and reconstruction steps of a 2D image signal for level of 2; a.Decomposition, b. Reconstruction
