WAVELET(Article Presentation)

by : Tilottama Goswami

Sources:

www.amara.com/IEEEwave/IEEEwavelet.htm

www.mat.sbg.ac.at/~uhl/wav.html

www.mathsoft.com/wavelets.html

Rivier College, CS699 Professional Seminar

OVERVIEW

• What is wavelet?

– Wavelets are mathematical functions

• What does it do?

– Cut up data into different frequency components , and then study each component with a resolution matched to its scale

• Why it is needed?

– Analyzing discontinuities and sharp spikes of the signal

– Applications as image compression, human vision, radar, and earthquake prediction

What existed before this technique?

• Approximation using superposition of functions has existed since the early 1800's

• Joseph Fourier discovered that he could superpose sines and cosines to represent other functions , to approximate choppy signals

• These functions are non-local (and stretch out to infinity)

• Do a very poor job in approximating sharp spikes

Terms and Definitions

• Mother Wavelet : Analyzing wavelet , wavelet prototype function

• Temporal analysis : Performed with a contracted, high-frequency version of the prototype wavelet

• Frequency analysis : Performed with a dilated, low-frequency version of the same wavelet

• Basis Functions : Basis vectors which are perpendicular, or orthogonal to each other The sines and cosines are the basis functions , and the elements of Fourier synthesis

Terms and Definitions (Continued)

• Scale-Varying Basis Functions : A basis function varies in scale by chopping up the same function or data space using different scale sizes. – Consider a signal over the domain from 0 to 1 – Divide the signal with two step functions that range

from 0 to 1/2 and 1/2 to 1 – Use four step functions from 0 to 1/4, 1/4 to 1/2, 1/2 to

3/4, and 3/4 to 1.– Each set of representations code the original signal with

a particular resolution or scale. • Fourier Transforms: Translating a function in the time

domain into a function in the frequency domain

Applied Fields Using Wavelets

• Astronomy

• Acoustics

• Nuclear engineering

• Sub-band coding

• Signal and Image processing

• Neurophysiology

• Music

• Magnetic resonance imaging

• Speech discrimination,

• Optics

• Fractals,

• Turbulence

• Earthquake-prediction

• Radar

• Human vision

• Pure mathematics applications such as solving partial differential equations

Fourier Transforms

• Fourier transform have single set of basis functions

– Sines

– Cosines

• Time-frequency tiles

• Coverage of the time-frequency plane

Wavelet Transforms

• Wavelet transforms have a infinite set of basis functions

• Daubechies wavelet basis functions

• Time-frequency tiles

• Coverage of the time-frequency plane

How do wavelets look like?

• Trade-off between how compactly the basis functions are localized in space and how smooth they are.

• Classified by number of vanishing moments

• Filter or Coefficients

– smoothing filter (like a moving average)

– data's detail information

Applications of Wavelets In Use

Computer and Human

Vision

AIM: Artificial vision for robots

• Marr Wavelet:intensity changes at different scales in an image

• Image processing in the human has hierarchical structure of layers of processing

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Compression

AIM:Compression of 6MB for pair of hands

• Choose the best wavelets

• Truncate coefficients below a threshold

• Sparse coding makes wavelets valuable tool in data compression.

Applications of Wavelets In Use

Denoising Noisy Data

AIM:Recovering a true signal from noisy data

• Wavelet shrinkage and Thresholding methods

• Signal is transformed using Coiflets , thresholded and inverse-transformed

• No smoothing of sharp structures required, one step forward

Musical Tones

AIM: Sound synthesis

• Notes from instrument decomposed into wavelet packet coefficients.

• Reproducing the note requires reloading those coefficients into wavelet packet generator

• Wavelet-packet-based music synthesizer

FUTURE

• Basic wavelet theory is now in the refinement stage

• The refinement stage involves generalizations and extensions of wavelets, such as extending wavelet packet techniques

• Wavelet techniques have not been thoroughly worked out in applications such as practical data analysis where for example, discretely sampled time-series data might need to be analyzed.