Wavelet Transform Fourier Transform Wavelet Transform Gives only frequency component No time information We don’t know where the frequency change during the time Good for stationary signal (no sudden change in frequency) Wavelet Transform Gives both frequency component + time We know where the frequency change in time

Fourier Transform Weakness

Introduction to Wavelet Transform (1) Objectives Gives information about what frequencies contain in the signal (frequency component) Where the frequency change in time (time information) Allow us to detect features that might go underdetected at one resolution but may be easy to spot at another Basis function Short-period wave (wavelet) Mother wavelet Scale and Translation wavelet Example of Wavelet Application Music example Extract the frequency of notes and when to play that note

Time-Frequency Division

Introduction to Wavelet Transform (2) is similar to Fourier Transform Fourier Transform

Introduction to Wavelet Transform (3) Example of Wavelet Basis Function Mother wavelet Scale & Translated wavelet

Introduction to Wavelet Transform (4) Ex. Haar Wavelet Mother wavelet Scaled and Translated Wavelet

Forward Wavelet Transform Scaling function (Approximation function) Wavelet function (Detail function)

DWT Example f (x) = [1 2 3 4] j0 = 0 Scaling function

DWT Example j = j0 = 0; k = 0 Wavelet function

DWT Example j = 1; k = 0 j = 1; k = 1 Wavelet function

Wavelet Coefficients Scaling function Wavelet function

Inverse DWT Scaling function

Inverse DWT Wavelet function

Generation of New Mother wavelet from simple rectangular pulse Using two-scale relation Mother wavelet New Mother Wavelet

Wavelet (Proof for fast wavelet) Mother wavelet Wavelet

Wavelet Coefficients n=2k, k>=0

Wavelet Blockdiagram 2

Approximation and Detail Wavelet 2

Multi-resolution Wavelet Implementation

2D Wavelet (Separable) Row Wavelet Column Wavelet (LL) (LH) (HL) (HH)

Example of Wavelet Coefficients LL LH HL HH

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