Wavelets: a previewFebruary 6, 2003Acknowledgements: Material compiled from the MATLAB Wavelet Toolbox UG.02/6/03ECE 178: a wavelet tour 2Problem with Fourier…Fourier analysis -- breaks down a signal into constituent sinusoids of different frequencies.a serious drawback In transforming to the frequencydomain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place.02/6/03ECE 178: a wavelet tour 3Gabor’s proposal02/6/03ECE 178: a wavelet tour 4Fourier – Gabor – Wavelet 02/6/03ECE 178: a wavelet tour 5Sinusoid with a small discontinuity02/6/03ECE 178: a wavelet tour 6in the transform domain02/6/03ECE 178: a wavelet tour 7Localization (or the lack of it)02/6/03ECE 178: a wavelet tour 8Fourier decomposition=++02/6/03ECE 178: a wavelet tour 9and the Wavelet decompositionFourier transform:() ()jtFfte dtww•--•=ÚSimilarly, the continuous wavelet transform (CWT) is defined as the sum over all time of the signal multiplied by scaled, shifted versions of the wavelet function ψ:-(scale, position) = f(t) (scale, position, )ctdty••ÚThe result of the CWT are many wavelet coefficients C, which are a function of scale and position.02/6/03ECE 178: a wavelet tour 10Wavelet decomposition –contd.Multiplying each coefficient by the appropriately scaled and shifted wavelet yields the constituent wavelets of the original signal:02/6/03ECE 178: a wavelet tour 11What do we mean by scale?02/6/03ECE 178: a wavelet tour 12The scale factor02/6/03ECE 178: a wavelet tour 13Shifting02/6/03ECE 178: a wavelet tour 14Computing a wavelet transform02/6/03ECE 178: a wavelet tour 15Computing the WT (2)02/6/03ECE 178: a wavelet tour 16Visualizing the WT coefficients02/6/03ECE 178: a wavelet tour 17..in 3-D02/6/03ECE 178: a wavelet tour 18The discrete wavelet transformCalculating wavelet coefficients at every possible scale is a fair amount of work, and it generates an awful lot of data. What if we choose only a subset of scales and positions at which to make our calculations?It turns out, rather remarkably, that if we choose scales and positions based on powers of two — so-called dyadic scales and positions — then our analysis will be much more efficient and just as accurate. We obtain just such an analysis from the discrete wavelet transform (DWT).02/6/03ECE 178: a wavelet tour 19Approximations and DetailsThe approximations are the high-scale, low-frequency components of the signal. The details are the low-scale, high-frequency components. The filtering process, at its most basic level, looks like this: The original signal, S, passes through two complementary filters and emerges as two signals.02/6/03ECE 178: a wavelet tour 20DownsamplingUnfortunately, if we actually perform this operation on a real digital signal, we wind up with twice as much data as we started with. Suppose, for instance, that the original signal S consists of 1000 samples of data. Then the approximation and the detail will each have 1000 samples, for a total of 2000.To correct this problem, we introduce the notion ofdownsampling. This simply means throwing away every second data point. While doing this introduces aliasingin the signal components, it turns out we can account for this later on in the process.02/6/03ECE 178: a wavelet tour 21Downsampling (2)The process on the right, which includes downsampling, produces DWT coefficients.02/6/03ECE 178: a wavelet tour 22An example02/6/03ECE 178: a wavelet tour 23Wavelet DecompositionMultiple-Level DecompositionThe decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken down into many lower-resolution components. This is called the wavelet decomposition tree.02/6/03ECE 178: a wavelet tour 24Wavelet decomposition…02/6/03ECE 178: a wavelet tour 25IDWT: reconstruction02/6/03ECE 178: a wavelet tour 26Analysis vs SynthesisWhere wavelet analysis involves filtering and downsampling, the wavelet reconstruction process consists of upsampling and filtering. Upsampling is the process of lengthening a signal component by inserting zeros between samples:02/6/03ECE 178: a wavelet tour 27Perfect reconstruction02/6/03ECE 178: a wavelet tour 28Quadrature Mirror Filters02/6/03ECE 178: a wavelet tour 29Reconstructing Approximation & Details02/6/03ECE 178: a wavelet tour 3002/6/03ECE 178: a wavelet tour 31Reconstructing As and Ds..contd..Note that the coefficient vectors cA1 and cD1 — because they were produced by downsampling, contain aliasingdistortion, and are only half the length of the original signal — cannot directly be combined to reproduce the signal. It is necessary to reconstruct the approximations and details before combining them.02/6/03ECE 178: a wavelet tour 32Reconstructing the signal02/6/03ECE 178: a wavelet tour 33Multiscale Analysis02/6/03ECE 178: a wavelet tour 34Wavelet Toolbox■ See the wavelet demo■ Wavemenu – compression