**3.3 Wavelet denoising in images**

The fundamental objective in image enhancement is to improve or accentuate subsequent processing tasks such as detection or recognition (Chang,2000,2006). Classical image enhancement techniques consider the use of spatial-invariant operators either in the spatial or in the fourier domain. Examples of techniques in the spatial domain are related with the histogram modification by a predetermined transformation as in histogram equalization and stretching. These methods are global in the sense that the pixels are modified in the entire image. However, it is often necessary to perform the enhancement process over small patches of the image. Examples of such techniques include local histogram stretching (in overlapping or non-overlapping windows), smoothing and sharpening . In the fourier domain, most methods are based in supressing low spatial frequencies relative to high spatial frequencies as in homomorphic filtering. Local image enhancement can also be performed by means of a multiscale image representation.Fourier transform based spectral analysis is the dominant analytical tool for frequency domain analysis. However, fourier transform cannot provide information of the spectrum changes with respect to time. Fourier transform assumes the signal as stationary, but PD signal is always non-stationary. To overcome this deficiency, a modified method-short time fourier transform allows representing the signal in both time and frequency domain through time windowing functions (Akansu et al., 1992). The window length determines a constant time and frequency resolution. Thus, a shorter time windowing is used in order to capture the transient behavior of a signal by sacrificing the frequency resolution.

A continuous-time wavelet transform of f(t) is defined as:

$$\text{CVV}T\_v f(a,b) = \text{VV}\_f(b,a) = \left| a \right|^{-1/2} \int\_{-a}^{a} f(t) \nu^\* \left( \frac{t-b}{a} \right) dt \tag{1}$$

Image Denoising Based on Wavelet Analysis for Satellite Imagery 457

This diagram is to be understood as representing the following sequence of operations

2. Downsample the filter output by 2 to give output coefficients y0 (n). 3. Filter the input signal x(n) with the filter whose Z-transform in H1(z). 4. Downsample the filter output by 2 to give output co-efficients y1(n)

1. Filter an input signal (whose value at time n is x (n)) with the filter whose Z-transform

The two main confines in image accuracy are categorized as blur and noise. Blur is intrinsic to image acquisition systems, as digital images have a finite number of samples and must respect the sampling conditions. The second main image perturbation is noise. Image denoising is used to remove the additive noise while retaining as much as possible the important signal features. Currently a reasonable amount of research is done on wavelet thresholding and threshold selection for signal de-noising, because wavelet provides an appropriate basis for separating noisy signal from the image signal. Two shrinkage methods are used over here to calculate new pixel values in a local neighborhood. Shrinkage is a well known and appealing denoising technique. The use of shrinkage is known to be optimal for Gaussian white noise, provided that the sparsity on the signal's representation is enforced using a unitary transform. Here a new approach to image denoising, based on the imagedomain minimization of an estimate of the mean squared error-Stein's unbiased risk estimate (SURE) is proposed and equation (3.1) specifies the same. Surelet method directly parameterizes the denoising process as a sum of elementary nonlinear processes with unknown weights. Unlike most existing denoising algorithms, using the SURE makes it needless to hypothesize a statistical model for the noiseless image. A key of it is, although the (nonlinear) processing is performed in a transformed domain-typically, an undecimated discrete wavelet transform, but nonorthonormal transformsis also addressed and this

2

(2)

1

*d i i i*

( ; ) 2.#{ : } ( )

*sure t x d i x t x t*

where *d* is the number of elements in the noisy data vector and *xi* is the wavelet coefficients. This procedure is smoothness-adaptive, meaning that it is suitable for denoising a wide range of functions from those that have many jumps to those that are essentially smooth.

It have high characteristics as it out performs Neigh shrink method. Comparison is done over these two methods to prove the elevated need of Surelet shrinkage for the denoising the SAR images. The experimental results are projected in graph format which shows that the Surelet shrinkage minimizes the objective function the fastest, while being as cheap as

Fig. 1. Building block for wavelet transform

minimization is performed in the image domain.

is H0(z).

Here a,bεR , 0 *a* since they are dilating and translating coefficients respectively. The asterisk denotes a complex conjugate. This multiplication of 1/2 *a* is for energy normalization purposes so that the transforms signal will have the same energy at every scale. The analysis function ( ), *t* the so called mother wavelet, is scaled by a, so a wavelet analysis is often called a time scale analysis rather than a time frequency analysis. The wavelet transform decomposes the signal into different scales with different levels of resolution by dilating a single prototype function, the mother wavelet. Furthermore, a mother wavelet has to satisfy that it has a zero net area, which suggest that the transformation kernel of the wavelet transform is a compactly support function (localized in time), thereby offering the potential to capture the PD spikes which normally occur in a short period of time. The general wavelet denoising procedure is as follows:


### **4. Denoising using shrinkage methods**

Conservative methods based on wavelet transforms have been emerged for removing Gaussian random noise from images. This local preprocessing speckle reduction technique is necessary prior to the processing of SAR images. Wavelet Shrinkage or thresholding as denoising method is the best identified method here. It is well known that increasing the redundancy of wavelet transforms can significantly improve the denoising performances. Thus a thresholding process which passes the coarsest approximation sub-band and attenuates the rest of the sub-bands should decrease the amount of residual noise in the overall signal after the denoising process (Achim et al.,2003). One dimensional dyadic discrete time wavelet transform is a transform similar to the discrete Fourier transform in that the input is a signal containing *N* numbers, say, and the output is a series of *M* numbers that describe the time-frequency content of the signal. The Fourier transform uses each output number to describe the content of the signal at one particular frequency, averaged over all time. In contrast, the outputs of the wavelet transform are localised in both time and frequency. The wavelet transform is based upon the building block shown in figure 1. This block is crucial for both understanding and implementing the wavelet transform.

Fig. 1. Building block for wavelet transform

representing the signal in both time and frequency domain through time windowing functions (Akansu et al., 1992). The window length determines a constant time and frequency resolution. Thus, a shorter time windowing is used in order to capture the

> 1/2 \* ( , ) ( , ) () ( ) *v f t b CWT f a b W b a a f t dt*

Here a,bεR , 0 *a* since they are dilating and translating coefficients respectively. The asterisk denotes a complex conjugate. This multiplication of 1/2 *a* is for energy normalization purposes so that the transforms signal will have the same energy at every

analysis is often called a time scale analysis rather than a time frequency analysis. The wavelet transform decomposes the signal into different scales with different levels of resolution by dilating a single prototype function, the mother wavelet. Furthermore, a mother wavelet has to satisfy that it has a zero net area, which suggest that the transformation kernel of the wavelet transform is a compactly support function (localized in time), thereby offering the potential to capture the PD spikes which normally occur in a

Apply wavelet transform to the noisy signal to produce the noisy wavelet coefficients to

Select appropriate threshold limit at each level and threshold method (hard or soft

Inverse wavelet transform of the thresholded wavelet coefficients to obtain a denoised

Conservative methods based on wavelet transforms have been emerged for removing Gaussian random noise from images. This local preprocessing speckle reduction technique is necessary prior to the processing of SAR images. Wavelet Shrinkage or thresholding as denoising method is the best identified method here. It is well known that increasing the redundancy of wavelet transforms can significantly improve the denoising performances. Thus a thresholding process which passes the coarsest approximation sub-band and attenuates the rest of the sub-bands should decrease the amount of residual noise in the overall signal after the denoising process (Achim et al.,2003). One dimensional dyadic discrete time wavelet transform is a transform similar to the discrete Fourier transform in that the input is a signal containing *N* numbers, say, and the output is a series of *M* numbers that describe the time-frequency content of the signal. The Fourier transform uses each output number to describe the content of the signal at one particular frequency, averaged over all time. In contrast, the outputs of the wavelet transform are localised in both time and frequency. The wavelet transform is based upon the building block shown in figure 1. This

block is crucial for both understanding and implementing the wavelet transform.

short period of time. The general wavelet denoising procedure is as follows:

the level which it can properly distinguish the PD occurrence.

 

*a*

(1)

( ), *t* the so called mother wavelet, is scaled by a, so a wavelet

transient behavior of a signal by sacrificing the frequency resolution.

A continuous-time wavelet transform of f(t) is defined as:

thresholding) to best remove the noises.

**4. Denoising using shrinkage methods** 

scale. The analysis function

signal.

This diagram is to be understood as representing the following sequence of operations


The two main confines in image accuracy are categorized as blur and noise. Blur is intrinsic to image acquisition systems, as digital images have a finite number of samples and must respect the sampling conditions. The second main image perturbation is noise. Image denoising is used to remove the additive noise while retaining as much as possible the important signal features. Currently a reasonable amount of research is done on wavelet thresholding and threshold selection for signal de-noising, because wavelet provides an appropriate basis for separating noisy signal from the image signal. Two shrinkage methods are used over here to calculate new pixel values in a local neighborhood. Shrinkage is a well known and appealing denoising technique. The use of shrinkage is known to be optimal for Gaussian white noise, provided that the sparsity on the signal's representation is enforced using a unitary transform. Here a new approach to image denoising, based on the imagedomain minimization of an estimate of the mean squared error-Stein's unbiased risk estimate (SURE) is proposed and equation (3.1) specifies the same. Surelet method directly parameterizes the denoising process as a sum of elementary nonlinear processes with unknown weights. Unlike most existing denoising algorithms, using the SURE makes it needless to hypothesize a statistical model for the noiseless image. A key of it is, although the (nonlinear) processing is performed in a transformed domain-typically, an undecimated discrete wavelet transform, but nonorthonormal transformsis also addressed and this minimization is performed in the image domain.

$$\text{s.s.}\\\text{vec}(t; \mathbf{x}) = d - 2. \#\{i : |\mathbf{x}\_i| \le t\} + \sum\_{i=1}^{d} \left( |\mathbf{x}\_i| \ln t \right)^2 \tag{2}$$

where *d* is the number of elements in the noisy data vector and *xi* is the wavelet coefficients. This procedure is smoothness-adaptive, meaning that it is suitable for denoising a wide range of functions from those that have many jumps to those that are essentially smooth.

It have high characteristics as it out performs Neigh shrink method. Comparison is done over these two methods to prove the elevated need of Surelet shrinkage for the denoising the SAR images. The experimental results are projected in graph format which shows that the Surelet shrinkage minimizes the objective function the fastest, while being as cheap as neighshrink method. Measuring the amount of noise equation (3) is done by its standard deviation ,( ) *n ,* one can define the signal to noise ratio (SNR) as

$$\text{SNR} = \frac{\sigma(\mu)}{\sigma(n)},\tag{3}$$

Image Denoising Based on Wavelet Analysis for Satellite Imagery 459

( ) 0,

*Sx x*

( ( ( )) ( )) *sign S x sign x* 

functional.

by variational analysis.

**4.2 Applying shrinkage methods** 

been scaled to the interval [0, 1].

Fig. 2. The image eye.

,

(6)

*x x*

 

*x x*

to the coefficients. Beside these two possibilities there are many others (semi-soft shrinkage, firm shrinkage,….) and as long as the shrinkage function preserves the sign

and shrinks the magnitude ,one can expect a denoising effect.

mathematics in a natural way. Four places where shrinkage appears naturally are :

1. As the subgradient descent along the absolute value.

signal and the noise are distributed in a certain way.

The interesting thing about wavelet shrinkage is, that it appears in very different fields of

2. As the function which maps an initial value onto the minimizer of a variational

3. As the function "identity minus projection onto a convex set" which is also motivated

4. As the maximum a posteriori estimator for an additively disturbed signal, where the

The effect of wavelet shrinkage is illustrated in this section. An image of an eye is taken which is a closeup on a man's eye. It is a suitable image for illustrative purposes because it provides very different regions: small and sharp details like the eyelashes, texture-like parts of different contrast like the eyebrows or the skin below the eye, smooth parts like the eyeball or the skin above the eye and sharp edges like the edge of the lower lid. The image has a resolution of 256 times 256 pixels, 256 gray levels For calculations the gray levels have

,

Where ( ) in equation (3) denotes the empirical standard deviation of ( ), *i*

$$\sigma(\mu) = \left(\frac{1}{|I|}\sum\_{i} \left(\mu(i) - \overline{\mu)^2}\right)^{1/2}\right)^{1/2} \tag{4}$$

And <sup>1</sup> ( ) *i I <sup>i</sup> I* is the average grey level value. The standard deviation of the noise can

also be obtained as an empirical measurement or formally computed when the noise model and parameters are known. This parameter measures the degree of filtering applied to the image It also demonstrates the PSNR rises faster using the proposed method than the former. Hence the resulted denoised image is conceded to the next segment for the transformation to be applied and it is also proved to improve detection process.

#### **4.1 Wavelet shrinkage - Description and short history**

Wavelet shrinkage is a quite recent denoising method compared to classical methods like the Wiener filter or convolution filters and is applied very successfully to various denoising problems (Liu &Raja, 1996).. A very interesting thing about wavelet shrinkage is that it can be motivated from very different fields of mathematics, namely partial differential equations, the calculus of variations, harmonic analysis or statistics.

A heuristic way to wavelet shrinkage goes as follows. A signal *f* is considered which is distributed by additive white noise: *g f* .Since the discrete wavelet transform is linear and orthogonal, the wavelet transform of g has the form ( ) ( ) *g f* where the coefficients of the noise are given white noise. Usually the signal *f* results in a few number of large wavelet coefficients and most of the coefficients are zero or nearly zero (Chen &Bui 2003). The noise on the other hand leads to a large number of small coefficients on all scales. Thus, the small coefficients *g* mostly contain noise. Hence, it seems to be a good idea to set all the coefficients which are small to zero. But what shall happen to the large coefficients? The two most popular ones are hard and soft shrinkage. By application of hard shrinkage one leaves the large coefficients unchanged and sets the coefficients below a certain threshold to zero. Mathematically speaking one applies the function

$$S\_{\mathcal{A}}(\mathbf{x}) = \begin{cases} \mathbf{x}, |\mathbf{x}| > \mathcal{X} \\ \mathbf{0}, |\mathbf{x}| \le \mathcal{X} \end{cases} \tag{5}$$

to the wavelet coefficients. Another famous way is soft shrinkage where the magnitude of all coefficients is reduced by the threshold in which one applies the function

neighshrink method. Measuring the amount of noise equation (3) is done by its standard

( ) , ( )

1/2

(3)

(4)

.Since the discrete wavelet transform is linear

mostly contain noise. Hence, it seems to be a

 

(5)

  where the

 ( ), *i*

*n* 

 

is the average grey level value. The standard deviation of the noise can

<sup>1</sup> <sup>2</sup> ( ) ( () ) *i u i*

also be obtained as an empirical measurement or formally computed when the noise model and parameters are known. This parameter measures the degree of filtering applied to the image It also demonstrates the PSNR rises faster using the proposed method than the former. Hence the resulted denoised image is conceded to the next segment for the

Wavelet shrinkage is a quite recent denoising method compared to classical methods like the Wiener filter or convolution filters and is applied very successfully to various denoising problems (Liu &Raja, 1996).. A very interesting thing about wavelet shrinkage is that it can be motivated from very different fields of mathematics, namely partial differential

A heuristic way to wavelet shrinkage goes as follows. A signal *f* is considered which is

number of large wavelet coefficients and most of the coefficients are zero or nearly zero (Chen &Bui 2003). The noise on the other hand leads to a large number of small coefficients

good idea to set all the coefficients which are small to zero. But what shall happen to the large coefficients? The two most popular ones are hard and soft shrinkage. By application of hard shrinkage one leaves the large coefficients unchanged and sets the coefficients below a

, ( ) 0, *x x*

*x*

 

to the wavelet coefficients. Another famous way is soft shrinkage where the magnitude of

of the noise are given white noise. Usually the signal *f* results in a few

and orthogonal, the wavelet transform of g has the form ( ) ( ) *g f*

certain threshold to zero. Mathematically speaking one applies the function

*S x*

all coefficients is reduced by the threshold in which one applies the function

( ) *n ,* one can define the signal to noise ratio (SNR) as

*SNR*

in equation (3) denotes the empirical standard deviation of

*I*

transformation to be applied and it is also proved to improve detection process.

 

**4.1 Wavelet shrinkage - Description and short history** 

distributed by additive white noise: *g f*

on all scales. Thus, the small coefficients *g*

equations, the calculus of variations, harmonic analysis or statistics.

deviation ,

Where

( ) 

And <sup>1</sup> ( ) *i I <sup>i</sup> I*

coefficients

 

 
$$S\_{\mathcal{A}}(\mathbf{x}) = \begin{cases} \mathbf{x} - \mathcal{X}, \mathbf{x} \ge \mathcal{X} \\ \mathbf{0}, \|\mathbf{x}\| \le \mathcal{X} \\ \mathbf{x} + \mathcal{X}, \mathbf{x} \le -\mathcal{X} \end{cases} \tag{6}$$

to the coefficients. Beside these two possibilities there are many others (semi-soft shrinkage, firm shrinkage,….) and as long as the shrinkage function preserves the sign ( ( ( )) ( )) *sign S x sign x* and shrinks the magnitude ,one can expect a denoising effect.

The interesting thing about wavelet shrinkage is, that it appears in very different fields of mathematics in a natural way. Four places where shrinkage appears naturally are :


#### **4.2 Applying shrinkage methods**

The effect of wavelet shrinkage is illustrated in this section. An image of an eye is taken which is a closeup on a man's eye. It is a suitable image for illustrative purposes because it provides very different regions: small and sharp details like the eyelashes, texture-like parts of different contrast like the eyebrows or the skin below the eye, smooth parts like the eyeball or the skin above the eye and sharp edges like the edge of the lower lid. The image has a resolution of 256 times 256 pixels, 256 gray levels For calculations the gray levels have been scaled to the interval [0, 1].

Fig. 2. The image eye.

Image Denoising Based on Wavelet Analysis for Satellite Imagery 461

( ) ( |1 ) ( ) *t t t S f Sf*

( ) 0, (0) *<sup>t</sup>*

 

2 2 <sup>2</sup> <sup>2</sup> *L R rangeL L R dadb a* ( ) ( , /) 

But the range of the wavelet transform is a proper subspace of the space 22 2 *L R dadb a* ( , /) . In

Unfortunately, a subgradient descent in such a subspace is in general not a subgradient

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

which is a diffusivity. It is decreasing and according to the above proposition it hols g(0)=1

 

1 2 ( ) <sup>0</sup> *for x g x else*

This is a "piecewise linear diffusion" where diffusion is forbidden if the derivative has

justifications and arguments in its favor remain highly compelling. The procedure does not require any assumptions about the nature of the signal, permits discontinuities and spatial variation in the signal, and exploits the spatially adaptive multiresolution features essential to the wavelet transform. Furthermore, the procedure exploits the fact that the wavelet transform maps white noise in the signal domain to white noise in the transform domain. Thus, while signal energy becomes more concentrated into fewer coeffcients in the transform domain, noise energy does not. It is this important principle that enables the separation of signal from noise. Wavelet shrinkage denoising has been theoretically proven to be nearly optimal from the following perspectives: spatial adaptation, estimation when local smoothness is unknown, and estimation when global smoothness is unknown. This section presents an extensive study for wavelet denoising methods and shows that many of them are leading to the idea of

[ ] *Sx x x* ( ) (1 ( ))

 

*x* 

is not invariant under shrinkage.

**Soft Shrinkage :** The well known soft shrinkage function *S x x si* () ( ) *gnx*

*g x*

**Hard Shrinkage:** The Hard shrinkage function (or hard thresholding)

is a solution of the descent equation

**4.4 Continuous wavelet shrinkage** 

particular range *L*

and g(x) 0 for *x*

absolute value larger that 2

shrinkage in a general sense.

The wavelet transform is an isometry and explained as

descent in the original Hilbert space and vice versa.

 

 

leads to

.With regard to wavelet shrinkage denoising, the theoretical

 

(12)

(13)

(9)

*f* (10)

(11)

gives

#### **4.3 Discrete wavelet shrinkage**

This is the place where shrinkage methods have their origin and where they are used the most.

Fig. 3. Illustration of discrete wavelet shrinkage. The wavelet used here is the coiflet. Top row, from left right: the noisy image, wavelet shrinkage for different values of λ. The second row shows the discrete wavelet transform of the upper row.

Consider an orthogonal periodic wavelet base { ) of <sup>2</sup>*L I*( )

Define the orthogonal mapping as

$$\mathcal{W}: \mathcal{L}^2(I) \to \mathcal{l}^2(\mathbb{1} \cup \Gamma) \text{ via } f \to (\{f \mid 1\}\_{\prime}(f\_{\gamma})\_{\text{pf}}) \tag{7}$$

The mapping W is invertible and isometrical.

Define <sup>2</sup> *onl* (1 ) by 1 (1 ) ( ) *<sup>l</sup> a a* and the functional need is <sup>2</sup> : () *LI R* defined by

$$\Psi(f) = \Phi(\mathsf{V}\mathsf{V}f) = \begin{cases} \left\| \left< f \mid \mathbf{1} \right> + \left\| \left( f\_{\mathcal{I}} \right)\_{\mathsf{y}\mathsf{d}\mathsf{I}} \right\|\_{\mathsf{I}^{1}(\mathsf{I}^{\mathsf{T}})} \\ \infty \end{cases} \tag{8}$$

The result obtained that the discrete wavelet shrinkage

$$
\mu(t) = S\_t(\{f \mid 1\}) + \sum\_{\gamma \in \Gamma} S\_t(f\_\gamma) \mu\_\gamma \tag{9}
$$

is a solution of the descent equation

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

This is the place where shrinkage methods have their origin and where they are used the

Fig. 3. Illustration of discrete wavelet shrinkage. The wavelet used here is the coiflet. Top row, from left right: the noisy image, wavelet shrinkage for different values of λ. The second

> 

2 2 *WLI l* : ( ) (1 ) via *fff* ( |1 ,( ) )

<sup>1</sup> ( ) |1 ( ) () ( ) *<sup>l</sup> f f f Wf*

Define <sup>2</sup> *onl* (1 ) by 1 (1 ) ( ) *<sup>l</sup> a a* and the functional need is <sup>2</sup> : () *LI R*

of <sup>2</sup>*L I*( )

 

 

(7)

defined by

(8)

row shows the discrete wavelet transform of the upper row.

Consider an orthogonal periodic wavelet base { )

The mapping W is invertible and isometrical.

The result obtained that the discrete wavelet shrinkage

Define the orthogonal mapping as

**4.3 Discrete wavelet shrinkage** 

most.

$$
\partial\_t \mu + \partial \Psi(\mu) \ni 0, \mu(0) = f \tag{10}
$$

#### **4.4 Continuous wavelet shrinkage**

The wavelet transform is an isometry and explained as

$$\text{L}^2(\mathbb{R}) \to \text{rangeL}\_{\mathbb{M}} \subset \text{L}^2(\mathbb{R}^2, \text{dadb } / \text{a}^2) \tag{11}$$

But the range of the wavelet transform is a proper subspace of the space 22 2 *L R dadb a* ( , /) . In particular range *L*is not invariant under shrinkage.

Unfortunately, a subgradient descent in such a subspace is in general not a subgradient descent in the original Hilbert space and vice versa.

**Soft Shrinkage :** The well known soft shrinkage function *S x x si* () ( ) *gnx* gives

$$\log(|\mathbf{x}| = (1 - \frac{(|\mathbf{x}| - \sqrt{2\lambda})\_+}{|\mathbf{x}|}) \tag{12}$$

which is a diffusivity. It is decreasing and according to the above proposition it hols g(0)=1 and g(x) 0 for *x*

**Hard Shrinkage:** The Hard shrinkage function (or hard thresholding)

$$S\_{\lambda}(\mathbf{x}) = \mathbf{x}(1 - \mathbb{1}\_{[-\lambda \lambda]}(\mathbf{x})) \qquad \text{leads to}$$

$$g(\|\mathbf{x}\|) = \begin{cases} 1 \, for \, |\mathbf{x}| \le \sqrt{2\lambda} \\ 0 \, else \end{cases} \tag{13}$$

This is a "piecewise linear diffusion" where diffusion is forbidden if the derivative has absolute value larger that 2 .With regard to wavelet shrinkage denoising, the theoretical justifications and arguments in its favor remain highly compelling. The procedure does not require any assumptions about the nature of the signal, permits discontinuities and spatial variation in the signal, and exploits the spatially adaptive multiresolution features essential to the wavelet transform. Furthermore, the procedure exploits the fact that the wavelet transform maps white noise in the signal domain to white noise in the transform domain. Thus, while signal energy becomes more concentrated into fewer coeffcients in the transform domain, noise energy does not. It is this important principle that enables the separation of signal from noise. Wavelet shrinkage denoising has been theoretically proven to be nearly optimal from the following perspectives: spatial adaptation, estimation when local smoothness is unknown, and estimation when global smoothness is unknown. This section presents an extensive study for wavelet denoising methods and shows that many of them are leading to the idea of shrinkage in a general sense.

Image Denoising Based on Wavelet Analysis for Satellite Imagery 463

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

2/2 1/4 <sup>0</sup> ( ) *i n <sup>n</sup>*

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

'

*n nt Ws x*

( ) () \*

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

1 '

*s* 

12 /

*kn N*

*i t*

 

2

*k N <sup>k</sup>*

*k N <sup>k</sup>*

2

'

^ 1

1 *<sup>N</sup>*

*k n n x xe N*

0

1 ^ ^

 

0 ( ) \*( ) *<sup>k</sup>*

> 2 :

*k*

*k*

2 :

*N t*

*N t*

*n kk k Ws x s e*

*N*

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

*N n n n*

0

  (14)

*<sup>n</sup> n* :

(15)

(16)

(17)

(18)

Morlet wavelet, consisting of a plane wave modulated by a Gaussian:

*o*

defined as the convolution of *xn* with a scaled and translated version of ( )

space using a discrete Fourier transform (DFT). The DFT of *xn* is

the inverse Fourier transform of the product:

where the angular frequency is defined as
