Slide 1: Presented by,
M.Tech(DCE) A Seminar on Slide 2: CONTENTS INTRODUCTION
WHY WAVELET TRANSFORM
CONTINUOUS WAVELET TRANSFORM
DISCRETE WAVELET TRANSFORM
REFERENCES Slide 3: INTRODUCTION WHAT IS TRANSFORM?
- Transform of a signal is just another form of representing the signal. It does not change the information content present.
- Mathematical transform are applied to signal to obtain further information which is not present in raw signal Slide 4: WHY WAVELET TRANSFORM FOURIER TRANSFORM
SHORT TIME FOURIER TRANSFORM
WAVELET TRANSFORM Slide 5: FOURIER TRANSFORM:
Fourier Transform of a time domain signal gives frequency domain representation.
LIMITATION OF FOURIER TRANSFORM:
When we are in time domain fourier transform will not give information regarding frequency and when we are in frequency domain it will not provide information regarding time. Slide 6: ULTIMATE SOLUTION:
Wavelet transform provides time frequency representation simultaneosly.
It provides variable resolution as follows:
“At high frequency wavelet transform gives good time resolution and poor frequency resolution”
“At low frequency wavelet transform gives good frequency resolution and poor time resolution”. Slide 7: SHORT TIME FOURIER TRANSFORM
Short time fourier transform provides time frequency representation of a signal.
“Which states that we cannot exactly know what frequency exist at what time instance but we can know only what frequency band exists at what time. DECOMPOSITION OF SIGNAL 0-500 Hz 500-1000Hz 250-500 Hz 0-250 Hz 0-125 Hz 125-250 Hz Slide 8: WAVE Demonstration of wave
A wave is an oscillating function of time or space and is periodic. Slide 9: Wavelets are localized waves they have finite energy. They are suited for analysis of transient signal WAVELETS Slide 10: PROPERTIES OF WAVELETS:
Consider a real or complex value continuous time function (t) with the following properties
In equation (1) ( ) stands for Fourier transform of (t) .
The admissibility condition implies that the Fourier transform of (t) vanishes at the zero frequency i.e A zero at the zero frequency also means that the average value of the wavelet in the time domain must be zero (t) must be oscillatory. In other words (t) must be a wave. Slide 11: Shifting operation gives time represntation of the spectral component.
Scaling operation gives frequency. Slide 12: WAVELET FAMILIES (a) Haar Wavelet (b) Daubechies4 Wavelet (c)Coiflet1 Wavelet Slide 13: (d) Symlet2 Wavelet (e) MexicanHat Wavelet (f) Meyer Wavelet (g) Morlet Wavelet Slide 14: THE CONTINUOUS WAVELET TRANSFORM where * denotes complex conjugation of
f(t) is the signal to be analyzed S is the scaling factor is the translation factor Inverse wavelet transform is given by Slide 15: DISCRETE WAVELET TRANSFORM SUB BAND CODING Slide 16: MULTIRESOLUTION ANALYSIS USING FILTER BANK Three-level wavelet decomposition tree Three-level wavelet reconstruction tree. Slide 17: CONDITION FOR PERFECT RECONSTRUCTION
To achieve perfect reconstruction analysis and synthesis filter have to satisfy following conditions: G0 (-z) G1 (z) + H0 (-z). H1 (z) = 0 -------- (1)
G0 (z) G1 (z) + H0 (z). H1 (z) = 2z-d ------- (2) Where G0(z) be the low pass analysis filter,
G1(z) be the low pass synthesis filter,
H0(z) be the high pass analysis filter,
H1(z) be the high pass synthesis filter.
First condition implies that reconstruction is aliasing free
Second condition implies that amplitude distortion has amplitude of unity Slide 18: APPLICATIONS In the field of digital image processing
In FBI Finger Print Compression An FBI-digitized left thumb fingerprint The image on the left is the original; the one on the right is reconstructed from a compression Slide 19: Denoising "Before" and "after" illustrations of a nuclear magnetic resonance signal. Slide 20: Denoising an image The top left image is the original. At top right is a close-up image of her left eye. At bottom left is a close-up image with noise added. At bottom right is a close-up image, denoised. Slide 21: CONCLUSION Fourier transform provided information regarding frequency.
Short time fourier transform gives only constant resolution.
So, wavelet transform is preferred over fourier transform and short time fourier transform since it provided multiresolution. Slide 22: (a) Original Image256x256Pixels, 24-BitRGB (b) JPEG (DCT) Compressed with compression ratio 43:1(c) JPEG2000 (DWT) Compressed with compression ratio 43:1 a b c Slide 23: REFERENCES Wavelet Transform, Introduction to Theory and Applications, By Raghaveer M.Rao Ajit.S., Bopardikar
Digital Image Processing, 2nd edition, Rafael.C.Gonzalez , Richard E.Woods.
http://www.thewavelet tutorial by ROBI Polikar.htm