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