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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

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wavelet

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
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
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
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
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

Applied Fields Using Wavelets
fourier transforms
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
  • 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
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

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
applications of wavelets in use1
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.

Reproducing the note requires reloading those coefficients into wavelet packet generator

Wavelet-packet-based music synthesizer

Applications of Wavelets In Use
future
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 thoroughlyworked out in applications such as practical data analysis where for example, discretely sampled time-series data might need to be analyzed.