Lecture 26 Wavelet Transform

• 與FFT的差別

• Fourier Transform:represents an array of pixel intensities in terms of pure frequency functions

• Wavelet transform: expresses an image array in terms of functions which are restricted both in terms of frequency and spatial extent

• 基本上Wavelet transform 的定義是inner product 的一個概念 且associated function space

Definition

• If f and g are two functions on the set of real number R, then their inner product is given by

• Function space L2(R ) is the collection of all funcitons f:R->R with the property

Haar wavelet

• Haar scaling function is the function

• if 0≤x≤1

• 
  

• otherwise

• We will use it to represent pixel intensities

Image compression

• The fact that many of the wavelet coefficients area close to zero makes the wavelet transform useful for image compression.

Multiresolution analysis

• Standard basic functions

• These are functions which are zero everywhere outside a closed, bounded interval

• Z is all integers

• Orthornormal basis for

• 
  

MRA

• The dilation equation and refinement coefficients

• The cascade algorithm: iterate the refinement coefficients for a scaling function, 可以近似到任何想要的精確度

• The mother wavelet：與 Haar mother 近似

• The Daubechies D4 scaling function比Haar更好用

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