Wavelets: a preview Acknowledgements: Material compiled from the MATLAB Wavelet Toolbox User Guide and Chapter 7, DIP 3e. READING: Chapter 7: 7.1.1.ECE 178: a wavelet tour 2 MotivationECE 178: a wavelet tour 3 Problem with Fourier… Fourier analysis -- breaks down a signal into constituent sinusoids of different frequencies. a serious drawback In transforming to the frequency domain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place.ECE 178: a wavelet tour 4 Gabor’s proposalECE 178: a wavelet tour 5 Fourier – Gabor – Wavelet Scale-space decompositionECE 178: a wavelet tour 6 Localization (or the lack of it)ECE 178: a wavelet tour 7 Fourier decomposition = + +ECE 178: a wavelet tour 8 and the Wavelet decomposition Fourier transform: Similarly, 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:"The result of the CWT are many wavelet coefficients C, which are a function of scale and position.ECE 178: a wavelet tour 9 Wavelet decomposition –contd. Multiplying each coefficient by the appropriately scaled and shifted wavelet yields the constituent wavelets of the original signal:ECE 178: a wavelet tour 10 What do we mean by scale?ECE 178: a wavelet tour 11 The scale factorECE 178: a wavelet tour 12 Shifting Wavelet function Shifted Wavelet functionECE 178: a wavelet tour 13 Computing a wavelet transformECE 178: a wavelet tour 14 Computing the WT (2)ECE 178: a wavelet tour 15 The discrete wavelet transform Calculating 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).ECE 178: a wavelet tour 16 Approximations and Details The 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.ECE 178: a wavelet tour 17 Downsampling Unfortunately, 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 of downsampling. This simply means throwing away every second data point. While doing this introduces aliasing in the signal components, it turns out we can account for this later on in the process.ECE 178: a wavelet tour 18 Downsampling (2) The process on the right, which includes downsampling, produces DWT coefficients.ECE 178: a wavelet tour 19 An exampleECE 178: a wavelet tour 20 Wavelet Decomposition Multiple-Level Decomposition The 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.ECE 178: a wavelet tour 21 Wavelet decomposition…ECE 178: a wavelet tour 22 Image Pyramids..ECE 178: a wavelet tour 23 Laplacian PyramidECE 178: a wavelet tour 24 IDWT: reconstructionECE 178: a wavelet tour 25 Analysis vs Synthesis Where 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:ECE 178: a wavelet tour 26 Perfect reconstructionECE 178: a wavelet tour 27 Quadrature Mirror FiltersECE 178: a wavelet tour 28 Reconstructing Approximation & DetailsECE 178: a wavelet tour 29ECE 178: a wavelet tour 30 Reconstructing As and Ds..contd.. Note that the coefficient vectors cA1 and cD1 — because they were produced by downsampling, contain aliasing distortion, 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.ECE 178: a wavelet tour 31 Reconstructing the signalECE 178: a wavelet tour 32 Multiscale AnalysisECE 178: a wavelet tour 33 Haar Wavelet TransformECE 178: a wavelet tour 34 Haar-- scaling function (approximations)ECE 178: a wavelet tour 35 Haar -- wavelet functionsECE 178: a wavelet tour 36 Haar decompositionECE 178: a wavelet tour 37 Haar