Wavelet Transform

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Introducing Wavelet Transform

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Presented by, Dhanraj.A.Ghanti M.Tech(DCE) A Seminar on

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CONTENTS INTRODUCTION WHY WAVELET TRANSFORM WAVELETS CONTINUOUS WAVELET TRANSFORM DISCRETE WAVELET TRANSFORM APPLICATIONS CONCLUSION REFERENCES

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INTRODUCTION WHAT IS TRANSFORM? - Transform of a signal is just another form of representing the signal. It does not change the information content present. WHY TRANSFORM? - Mathematical transform are applied to signal to obtain further information which is not present in raw signal

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WHY WAVELET TRANSFORM FOURIER TRANSFORM SHORT TIME FOURIER TRANSFORM WAVELET TRANSFORM

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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.

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ULTIMATE SOLUTION: WAVELET TRANSFORM 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”.

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SHORT TIME FOURIER TRANSFORM Short time fourier transform provides time frequency representation of a signal. UNCERTAINITY PRINCIPLE “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

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WAVE Demonstration of wave A wave is an oscillating function of time or space and is periodic.

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Wavelets are localized waves they have finite energy. They are suited for analysis of transient signal WAVELETS

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PROPERTIES OF WAVELETS: Consider a real or complex value continuous time function (t) with the following properties ---- (1) 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.

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Shifting operation gives time represntation of the spectral component. Scaling operation gives frequency.

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WAVELET FAMILIES (a) Haar Wavelet (b) Daubechies4 Wavelet (c)Coiflet1 Wavelet

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(d) Symlet2 Wavelet (e) MexicanHat Wavelet (f) Meyer Wavelet (g) Morlet Wavelet

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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

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DISCRETE WAVELET TRANSFORM SUB BAND CODING

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MULTIRESOLUTION ANALYSIS USING FILTER BANK Three-level wavelet decomposition tree Three-level wavelet reconstruction tree.

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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

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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

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Denoising "Before" and "after" illustrations of a nuclear magnetic resonance signal.

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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.

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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.

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(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

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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.amara.com/ieeewave/iw_ref.html#ten. http://www.thewavelet tutorial by ROBI Polikar.htm

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