**7. References**

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

method can also distinct between homogeneous and heterogeneous areas as well as separate

a) b)

c) d)

This book chapter has presented the proposed methods for despeckling a synthetic and real SAR images using second-generation wavelets. The Bayesian approach in incorporated into second generation wavelets using the wavelet domain. The prior and likelihood pdfs are modeled using GGMRF and Gaussian distribution. The second order Bayesian inference is used to better estimate model parameters and to find the best values possible. The evidence

Fig. 19. Classification of texture parameter *θ* using the *K*-means algorithm and the a)

original, b) bandelet, c) contourlet, and d) MBD-based algorithm

**6. Conclusion** 

different textures in the scene, as shown in Fig. 19 d).


**17** 

*1Algeria 2UK* 

**The Wavelet Transform** 

*University of the West of England* 

Bouden Toufik1 and Nibouche Mokhtar2

**for Image Processing Applications** 

*1Automatic Department, Laboratory of Non Destructive Testing, Jijel University 2Bristol Robotic Laboratory, Department of Electrical and Computer Engineering,* 

In recent years, the wavelet transform emerged in the field of image/signal processing as an alternative to the well-known Fourier Transform (FT) and its related transforms, namely, the Discrete Cosine Transform (DCT) and the Discrete Sine Transform (DST). In the Fourier theory, a signal (an image is considered as a finite 2-D signal) is expressed as a sum, theoretically infinite, of sines and cosines, making the FT suitable for infinite and periodic signal analysis. For several years, the FT dominated the field of signal processing, however, if it succeeded well in providing the frequency information contained in the analysed signal; it failed to give any information about the occurrence time. This shortcoming, but not the only one, motivated the scientists to scrutinise the transform horizon for a "messiah" transform. The first step in this long research journey was to cut the signal of interest in several parts and then to analyse each part separately. The idea at a first glance seemed to be very promising since it allowed the extraction of time information and the localisation of different frequency components. This approach is known as the Short-Time Fourier Transform (STFT). The fundamental question, which arises here, is how to cut the signal? The best solution to this dilemma was of course to find a fully scalable modulated window in which no signal cutting is needed anymore. This goal was achieved successfully by the

Formally, the wavelet transform is defined by many authors as a mathematical technique in which a particular signal is analysed (or synthesised) in the time domain by using different versions of a dilated (or contracted) and translated (or shifted) basis function called the wavelet prototype or the mother wavelet. However, in reality, the wavelet transform found its essence and emerged from different disciplines and was not, as stated by Mallat, totally new to mathematicians working in harmonic analysis, or to computer vision researchers

At the beginning of the 20th century, Haar, a German mathematician introduced the first wavelet transform named after him (almost a century after the introduction of the FT, by the French J. Fourier). The Haar wavelet basis function has compact support and integer coefficients. Later, the Haar basis was used in physics to study Brownian motion (Graps,

**1. Introduction** 

use of the wavelet transform.

studying multiscale image processing (Mallat, 1989).

