**5.6 Wavelet bases**

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

Using (4) and a standard Fourier transform routine, one can calculate the continuous

To ensure that the wavelet transforms at each scale *s* are directly comparable to each other and to the transforms of other time series, the wavelet function at each scale *s* is normalized

> <sup>2</sup> () () *k k s s s t*

0

(19)

(21)

*<sup>n</sup> n* .

(20)

 

1/2 ^ ^

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

( )

*k*

where *N* is the number of points. Thus, the wavelet transform is weighted only by the amplitude of the Fourier coefficients *x*\$*k* and not by the wavelet function. If one is using the

' ' 1/2

( ) *nnt n nt t ss s*

One criticism of wavelet analysis is the arbitrary choice of the wavelet function, ( )

(It should be noted that the same arbitrary choice is made in using one of the more traditional transforms such as the Fourier, Bessel, Legendre, etc.) In choosing the wavelet

1. *Orthogonal or nonorthogonal.* In orthogonal wavelet analysis, the number of convolutions at each scale is proportional to the width of the wavelet basis at that scale. This produces a wavelet spectrum that contains discrete "blocks" of wavelet power and is useful for signal processing as it gives the most compact representation of the signal. Unfortunately for time series analysis, an a periodic shift in the time series produces a different wavelet spectrum. Conversely, a nonorthogonal analysis (such as used in this study) is highly redundant at large scales, where the wavelet spectrum at adjacent times is highly correlated. The nonorthogonal transform is useful for time series analysis,

2. *Complex or real.* A complex wavelet function will return information about both amplitude and phase and is better adapted for capturing oscillatory behavior. A real wavelet function

returns only a single component and can be used to isolate peaks or discontinuities. 3. *Width.* For concreteness, the width of a wavelet function is defined here as the *e*-folding time of the wavelet amplitude. The resolution of a wavelet function is determined by

where smooth, continuous variations in wavelet amplitude are expected.

 

0

 

0

*N*

*k s N*

 

wavelet transform (for a given *s*) at all *n* simultaneously and efficiently.

 

Using these normalizations, at each scale *s* one has

convolution formula (2), the normalization is

*<sup>n</sup> n* is normalized to have unit energy.

function, there are several factors which should be considered

**5.4 Normalization** 

to have unit energy:

where ( ) 

**5.5 Wavelet functions** 

Many wavelet families with various characteristics are known. Some wavelet families are especially useful for specific application. Most wavelets are based on FIR filters although research work is also done for wavelets constructed by IIR filters. The orthogonal and biorthogonal wavelets are the two basic categories of wavelets.


Table 1. Main properties of wavelet families

Table 1 summarizes the main properties of most well-known wavelets. The primary consideration in the use of wavelets for surface profile analysis is the amplitude and phase transmission characteristics of the wavelet basis. A combination of good amplitude and linear phase transmission is always desired to achieve minimum distortion of surface features. Among the listed wavelets in Table 1, Haar wavelet is the oldest and simplest wavelet that is not continuous. The Symlet and Coiflet wavelet come from Daubechie wavelet, but are more symmetric (Daubechies,1989,1992). Both the scaling function and wavelet of Meyer are defined in frequency domain. Although the scaling function and wavelet of Meyer are symmetric, no fast algorithm is available for its wavelet transform.The Morlet and Mexican wavelets only have wavelet functions and the corresponding scaling functions don't exist Four wavelets in three categories are selected for study here. They are orthogonal Daubechies wavelets and

Image Denoising Based on Wavelet Analysis for Satellite Imagery 467

**Peak signal to Noise Ratio**

32 34 36 38 40

Image3 Image2 Image1

**Range of Values (db)**

Fig. 7. PSNR values for shrinkage method based on coiflet Wavelet family

**5.8.2 Objective evaluation** 

Ne

**Shrinkage Methods** 

ighshrink

Surelet

Fig. 5. Peak to Signal noise ratio for wavelet methods

Fig. 6. Mean square error rate for wavelet methods

Butterworth wavelets with nonlinear phase, orthogonal Coiflets wavelets with near linear phase, and biorthogonal Spline wavelets with linear phase (Daubechies, 1989, 1992). All of them are associated with FIR filters except Butterworth wavelets.
