**5.2 Using Wavelet for SAR speckle denoising**


#### **5.3 Wavelet transform**

The *wavelet transform* can be used to analyze time series that contain nonstationary power at many different frequencies. Assume that one has a time series, *xn*, with equal time spacing d*t* and *n* = 0 … *N* - 1. Also assume that one has a *wavelet function,* ( ) *<sup>n</sup> n* , that depends on

The wavelet transform is a mathematical tool widely used in image processing. Some applications of the transform to remote sensing images have been investigated in the literature. It was found useful for texture analysis , image compression and noise reduction . The transform allows representation of a signal onto an orthonormal basis. Each term of the basis represents the signal at a given scale. In order to decompose the signal onto the basis, the algorithm developed is applied to the signal. It consists of iterations of one-dimensional highpass and low-pass filtering steps. The algorithm creates a pyramid of low-resolution approximations as well as a wavelet pyramid in which the details a stored as wavelet coefficients. This representation is called wavelet representation(Jiang et al., 2000).. One way of image analysis, is to choose the wavelet for speckle in SAR images which is always problematic(Ali,2007). Often, it is impenitent to reduce noise-before trying to extract scene features. Many filters have been developed to improve image quality by conserving the intrinsic scene features and textures. Interpretation of SAR images by human is possible in the presence of speckle. The wavelet transform, as the mammal visual system, provides and allows for a multiscale analysis of images. This section presents how the wavelet transform can be used for extraction of linear features such as edges and thin stripes. It will also show how speckle can be relaxed by taking into account the speckle contribution to wavelet coefficients.

A proportionality relation exists between Speckle noise and the wavelet coefficients . Since Speckle is approached as a multiplicative noise, this contribution will be larger for higher reflectivity regions. It is also known that speckle in SAR images is spatially correlated. That is to say the noise is colored. Therefore its behavior in the Fourier domain is such that there is a peak of a given width around the zero frequency. This is readily observed as a wavelet coefficient variance plateau when decomposing a correlated noise model on a wavelet basis. As decomposition gets to scales larger than the correlation length, the contribution from

 The main advantage of wavelet analysis is that it allows the use of long time intervals where more precise low frequency information is wanted, and shorter intervals where

 Wavelet analysis is therefore capable of revealing aspects of data that other image analysis techniques miss, such as trends, breakdown points, and discontinuities in

Wavelets are also capable of compressing or de-noising a image without appreciable

The *wavelet transform* can be used to analyze time series that contain nonstationary power at many different frequencies. Assume that one has a time series, *xn*, with equal time spacing

*<sup>n</sup> n* , that depends on

d*t* and *n* = 0 … *N* - 1. Also assume that one has a *wavelet function,* ( )

**5. Finding a solution for denoising in sar imagery using wavelet** 

**5.1 Noise in SAR imagery** 

speckle decreases linearly.

**5.3 Wavelet transform** 

**5.2 Using Wavelet for SAR speckle denoising** 

high frequency information is sought.

higher derivatives and self-similarity.

degradation of the original image.

a nondimensional "time" parameter h. To be "admissible" as a wavelet, this function must have zero mean and be localized in both time and frequency space. An example is the Morlet wavelet, consisting of a plane wave modulated by a Gaussian:

$$
\Psi\_o(\eta) = \pi^{-1/4} \ell^{i\alpha\_0 n} \ell^{-n^{2/2}} \tag{14}
$$

where w0 is the nondimensional frequency, here taken to be 6 to satisfy the admissibility condition . The term "wavelet function" is used generically to refer to either orthogonal or nonorthogonal wavelets. The term "wavelet basis" refers only to an orthogonal set of functions. The use of an orthogonal basis implies the use of the *discrete wavelet transform,* while a nonorthogonal wavelet function can be used with either the discrete or the continuous wavelet transform (Grossman et al.,1989). The continuous wavelet transform of a discrete sequence *xn* is defined as the convolution of *xn* with a scaled and translated version of ( ) *<sup>n</sup> n* :

$$\mathcal{W}\_n(s) = \sum\_{n'=0}^{N-1} \mathbf{x}\_{n'} \boldsymbol{\nu}^\* \left[ \frac{(n'-n)\delta t}{s} \right] \tag{15}$$

where the (\*) indicates the complex conjugate. By varying the *wavelet scale s* and translating along the *localized time index n*, one can construct a picture showing both the amplitude of any features versus the scale and how this amplitude varies with time. The subscript 0 on y has been dropped to indicate that this y has also been normalized (see next section). Although it is possible to calculate the wavelet transform using (2), it is considerably faster to do the calculations in Fourier space. To approximate the continuous wavelet transform, the convolution (2) should be done *N* times for each scale, where *N* is the number of points in the time series (Kaiser 1994). (The choice of doing all *N* convolutions is arbitrary, and one could choose a smaller number, say by skipping every other point in *n.*) By choosing *N*  points, the convolution theorem allows us do all *N* convolutions simultaneously in Fourier space using a discrete Fourier transform (DFT). The DFT of *xn* is

$$\stackrel{\frown}{\mathbf{x}\_k} = \frac{1}{N} \sum\_{n=0}^{N-1} \mathbf{x}\_n e^{12\pi kn/N} \tag{16}$$

where *k* = 0 … *N* - 1 is the frequency index. In the continuous limit, the Fourier transform of a function y(*t*/*s*) is given by y\$ (*s*w). By the convolution theorem, the wavelet transform is the inverse Fourier transform of the product:

$$\mathcal{W}\_n(\mathbf{s}) = \sum\_{k=0}^{N=1} \mathbf{x}\_k \overset{\wedge}{\boldsymbol{\nu}} \boldsymbol{\nu}^\*(\mathbf{s}o\_k) e^{i o\_k \eta \delta t} \tag{17}$$

where the angular frequency is defined as

$$\begin{aligned} o\_k &= \left\{ \frac{2\pi k}{N\delta t} : k \le \frac{N}{2} \right\} \\ o\_k &= \left\{ -\frac{2\pi k}{N\delta t} : k \le \frac{N}{2} \right\} \end{aligned} \tag{18}$$

Using (4) and a standard Fourier transform routine, one can calculate the continuous wavelet transform (for a given *s*) at all *n* simultaneously and efficiently.

#### **5.4 Normalization**

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 to have unit energy:

$$
\stackrel{\circ}{\psi}(s o\_k) = \left(\frac{2\pi s}{\delta t}\right)^{1/2} \stackrel{\circ}{\psi}\_0(s o\_k) \tag{19}
$$

Image Denoising Based on Wavelet Analysis for Satellite Imagery 465

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

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

broad function will have poor time resolution, yet good frequency resolution. 4. *Shape.* The wavelet function should reflect the type of features present in the time series. For time series with sharp jumps or steps, one would choose a boxcar-like function such as the Harr, while for smoothly varying time series one would choose a smooth function such as a damped cosine. If one is primarily interested in wavelet power spectra,then the choice of wavelet function is not critical, and one function will give the

same *qualitative* results as another.

Table 1. Main properties of wavelet families

biorthogonal wavelets are the two basic categories of wavelets.

**5.6 Wavelet bases** 

the balance between the width in real space and the width in Fourier space. A narrow (in time) function will have good time resolution but poor frequency resolution, while a

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

$$\sum\_{k=0}^{N-1} \left| \nu(s o\_k) \right|^2 = N \tag{20}$$

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 convolution formula (2), the normalization is

$$
\nu \left[ \frac{(n^{\dot{\cdot}} - n) \delta t}{s} \right] = \left( \frac{\delta t}{s} \right)^{1/2} \nu\_0 \left| \frac{\left( n^{\dot{\cdot}} - n \right) \delta t}{s} \right| \tag{21}
$$

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

#### **5.5 Wavelet functions**

One criticism of wavelet analysis is the arbitrary choice of the wavelet function, ( ) *<sup>n</sup> n* . (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 function, there are several factors which should be considered


the balance between the width in real space and the width in Fourier space. A narrow (in time) function will have good time resolution but poor frequency resolution, while a broad function will have poor time resolution, yet good frequency resolution.

4. *Shape.* The wavelet function should reflect the type of features present in the time series. For time series with sharp jumps or steps, one would choose a boxcar-like function such as the Harr, while for smoothly varying time series one would choose a smooth function such as a damped cosine. If one is primarily interested in wavelet power spectra,then the choice of wavelet function is not critical, and one function will give the same *qualitative* results as another.
