WAVELET

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# WAVELET - PowerPoint PPT Presentation

Rivier College, CS699 Professional Seminar. 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. OVERVIEW. What is wavelet? Wavelets are mathematical functions

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## WAVELET

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Rivier College, CS699 Professional Seminar

### 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

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
Astronomy

Acoustics

Nuclear engineering

Sub-band coding

Signal and Image processing

Neurophysiology

Music

Magnetic resonance imaging

Speech discrimination,

Optics

Fractals,

Turbulence

Earthquake-prediction

Human vision

Pure mathematics applications such as solving partial differential equations

Applied Fields Using Wavelets
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
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

FBI Fingerprint

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 structuresrequired, one step forward

Musical Tones

AIM: Sound synthesis

Notes from instrument decomposed into wavelet packet coefficients.