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Wavelet Transformations (Chapter 7)

Wavelet Transformations (Chapter 7). CSC 446 Lecturer: Nada ALZaben. Outline: . Differences between Wavelet and Fourier Transformations. What is Wavelet function ? Wavelet transformation types. Differences between Wavelet and Fourier Transformations. .

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Wavelet Transformations (Chapter 7)

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  1. Wavelet Transformations(Chapter 7) CSC 446 Lecturer: Nada ALZaben

  2. Outline: Differences between Wavelet and Fourier Transformations. What is Wavelet function? Wavelet transformation types

  3. Differences between Wavelet and Fourier Transformations. Wavelet transformation is easier than Fourier Transformation in Compressing ,Transmitting and Analyze many images. Unlike Fourier Transformation whose basis functions are sinusoids, Wavelet transformations are based on small waves called “wavelets” of varying frequency and limited duration.

  4. What is Wavelet function? Wavelet function is defined by The Wavelet function together with its integer translations and binary scaling's, spans the difference between any two adjacent scaling sub-spaces and

  5. Wavelet function.. for all this spans the spaces in the previous figure we write: and note that if then : The scaling and wavelet function sub-spaces are related by: The orthogonal complement of in is and all members of are orthogonal to the members of .

  6. Wavelet function.. We can express the spaces as: Then wavelet function can be expressed as : Wavelet function coefficient Scaling function

  7. Wavelet Transform types • Generalized wavelet series expansion • Discrete wavelet transform • Continuous wavelet transform

  8. Discrete Wavelet Transform (DWT) • DWTproperties: • The image after DWT have same number of pixels as the original image. • Its local statistics are relatively constant and easy modeled. • Many of its values are close to zero (good for image compression) • Resolution approximation of the original Image can be extracted from it.

  9. Discrete Wavelet Transform (DWT) • the DWT function is combined by two pairs: the are functions of the discrete variable , we let M = , and

  10. Discrete Wavelet Transform (DWT) • HAAR transformation matrix is used as M X M • Example: consider the discrete function of four points: f(0)=1, f(1)=4, f(2)=-3, f(3)=0. show the DWT? (hint use H4 matrix) Answer: Since M=4  J=2, by defualt, j=0,1 then k=0 if j=0 and k=0,1 for j=1 …

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