WAVELET (Article Presentation) by : Tilottama Goswami Sources: 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 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 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.