**1. Introduction**

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The wavelet transform (WT) a powerful tool of signal and image processing that have been successfully used in many scientific fields such as signal processing, image compression, computer graphics, and pattern recognition (Daubechies 1990; Lewis and Knowles 1992; Do and Vetterli 2002; Meyer, Averbuch et al. 2002; Heric and Zazula 2007). On contrary the traditional Fourier Transform, the WT is particularly suitable for the applications of nonstationary signals which may instantaneous vary in time (Daubechies 1990; Mallat and Zhang 1993; Akay and Mello 1998). It is crucial to analyze the time-frequency characteristics of the signals which classified as non-stationary or transient signals in order to understand the exact features of such signals (Rioul and Vetterli 1991; Ergen, Tatar et al. 2010). For this reason, firstly, researchers has concentrated on continuous wavelet transform (CWT) that gives more reliable and detailed time-scale representation rather than the classical short time Fourier transform (STFT) giving a time-frequency representation (Jiang 1998; Qian and Chen 1999).

The CWT technique expands the signal onto basis functions created by expanding, shrinking and shifting a single prototype function, which named as mother wavelet, specially selected for the signal under considerations. This transformation decomposes the signal into different scales with different levels of resolution. Since a scale parameter shrinking or expanding the mother wavelet in CWT, the result of the transform is time-scale representation. The scale parameter is indirectly related to frequency, when considered the center frequency of mother wavelet.

A mother wavelet has satisfy that it has a zero mean value, which require that the transformation kernel of the wavelet transform compactly supports localization in time, thereby offering the potential to capture the spikes occurring instantly in a short period of time (Mallat 1989; Rioul and Vetterli 1991; Akay and Mello 1998).

A wavelet expansion is representation of a signal in terms of an orthogonal collection of real-valued generated by applying suitable transformation to the original selected wavelet. The properties and advantages of a family of wavelets depend upon the mother wavelet features. The expansion is formed by two dimensional expansion of a signal and thus provides a time-frequency localization of the input signal. This implies that most of the energy of the signal will be captured a few coefficient. The basis functions in a wavelet

Signal and Image Denoising Using Wavelet Transform 497

the processing. The denoising process can be described as to remove the noise while retaining and not distorting the quality of processed signal or image (Chen and Bui 2003; Portilla, Strela et al. 2003; Buades, Coll et al. 2006). The traditional way of denoising to remove the noise from a signal or an image is to use a low or band pass filter with cut off frequencies. However the traditional 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. Therefore,

The algorithms and processing techniques used for signals can be also used for images because an image can be considered as a two dimensional signal. Therefore, the digital signal processing techniques for a one dimensional signal can be adapted to process two

Because the origin and nonstationarity of the noise infecting in the signal, it is difficult to model it. Nevertheless, if the noise assumed as stationary, an empirically recorded signal

modeled as stationary independent zero-mean white Gaussian variables (Moulin and Liu

When this model is used, the objective of noise removal is to reconstruct the original signal *x i*( ) from a finite set of *y*( )*i* values without assuming a particular structure for the signal. The usual approach to noise removal models noise as high frequency signal added to an original signal. These high frequencies can be bringing out using traditional Fourier transform, ultimately removing them by adequate filtering. This noise removal technique conceptually clear and efficient since depends only calculating DFT (Discrete Fourier

However, there is some issue that must be under consideration. The most prominent having same frequency as the noise has important information in the original signal. Filtering out these frequency components will cause noticeable loss of information of the desired signal when considered the frequency representation of the original signal. It is clear that a method is required in order to conserve the prominent part of the signal having relatively high frequencies as the noise has. The wavelet based noise removal techniques have provided

**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

, *i n* 0,1,..., 1 (1)

( )*i* are independently normal

represents the intensity of the noise in *y*( )*i* . The noise is usually

many denoising techniques are proposed to overcome this problem.

*y*() () () *i xi i* 

that is corrupted by additive noise can be represented as;

Where *y i*( ) noisy signal, *x i*( )is noise free actual signal and

dimensional signals or images.

random variables and

1999; Alfaouri and Daqrouq 2008).

Transform)(Wachowiak, Rash et al. 2000).

this conservation of the prominent part.

transform are produced from the mother wavelet by scaling and translation operations. When the scaling is chosen as power of two, this kind of wavelet transform is called dyadicorthonormal wavelet transform, which makes a way for discrete wavelet transform (Zou and Tewfik 1993; Blu 1998) . If the chosen mother wavelet has orthonormal properties, there is no redundancy in the discrete wavelet transforms. In addition, this provides the multiresolution algorithm decomposing a signal into scales with different time and frequency resolution (Mallat 1989; Daubechies 1990).

The fundamental concept involved in mutiresolution is to find average features and details of the signal via scalar products with scaling signals and wavelets. The spikes in signal are typically of high frequency and it is possible discriminate the spikes with other noises through the decomposition of multiresolution into different levels. The differences between mother wavelet functions (e.g. Haar, Daubechies, Symlets, Coiflets, Biorthogonal and etc.) consist in how these scaling signals and the wavelets are defined (Zou and Tewfik 1993; Blu 1998; Ergen, Tatar et al. 2010).

The continuous wavelet transform is computed by changing the scale of the mother wavelet, shifting the scaled wavelet in time, multiplying by the signal, and integrating over all times. When the signal to be analyzed and wavelet function are discredited, the CWT can be realized on computer and the computation time can be significantly reduced if the redundant samples removed respect to sampling theorem. This is not a true discrete wavelet transform. The fundamentals of discrete wavelet transform goes back to sub-band coding theorem (Fischer 1992; Vetterli and Kova evi 1995; Vetterli and Kovacevic 1995). The subband coding encodes each part of the signal after separating into different bands of frequencies. Some studies have made use of wavelet transform as a filter bank in order to separate the signal.

After discovering the signal decomposition of a signal into frequency bands using discrete wavelet transform, the DWT has found many application area, from signal analysis to signal compression (Chang and Kuo 1993; Qu, Adam et al. 2003; He and Scordilis 2008).

The one of the first application of the DWT is the denoising process, which aims to remove the small part of the signal assumed as noise (Lang, Guo et al. 1996; Simoncelli and Adelson 1996; Jansen 2001). All kind of the signal obtained from the physical environment has contains more or less disturbing noise. Therefore, wavelet denoising procedure has applied many one or two dimensional signal after particularly soft or hard thresholding methods had proposed (Donoho and Johnstone 1994; Donoho 1995). Such signals some time are onedimensional simple power or control signals (Sen, Zhengxiang et al. 2002; Giaouris, Finch et al. 2008) as well as more complex medical images (Wink and Roerdink 2004; Pizurica, Wink et al. 2006). Especially, wavelet denoising has found an application field about image processing recently (Nasri and Nezamabadi-pour 2009; Chen, Bui et al. 2010; Jovanov, Pi urica et al. 2010).
