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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. What is wavelet? Wavelets are mathematical functions

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WAVELET

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. Fourier Transforms • Fourier transform have single set of basis functions • Sines • Cosines • Time-frequency tiles • Coverage of the time-frequency plane

  8. Wavelet Transforms • Wavelet transforms have a infinite set of basis functions • Daubechies wavelet basis functions • Time-frequency tiles • Coverage of the time-frequency plane

  9. 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

  10. 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

  11. 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

  12. 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.

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