Space-Frequency Atoms
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1 Space-Frequency Atoms FREQUENCY FREQUENCY SPACE SPACE FREQUENCY FREQUENCY SPACE SPACE Figure 1: Space-frequency atoms.
2 Windowed Fourier Transform 1 line Figure 2: A Gabor function. 1 line Figure 3: A second Gabor function.
3 Windowed Fourier Transform (contd.) Analysis F(u,b)= w(x b)e j2πux, f = f(x)w(x b)e j2πux dx Synthesis f(x) = F(u,b)w(x b)e j2πux du db
4 What is a Wavelet? All basis functions (daughter wavelets) are generated by translation and dilation of a mother wavelet: Ψ a,b (x) = 1 ( ) x b Ψ a a when a<1 it shrinks the wavelet. The a factor keeps the norm constant: ( ) x b ( ) f = x b 2 a f dx a = a f(x).
5 What is a Wavelet? (contd.) The mother wavelet, Ψ, must satisfy the admissibility criterion: C Ψ = Ψ(s) 2 ds< s where Ψ is the Fourier transform of Ψ. This means that: Ψ(s) 2 decays faster than 1/ s Ψ(0)=0.
6 What is a Wavelet? (contd.) 1 line Figure 4: A Morlet wavelet. 0.8 line Figure 5: A second Morlet wavelet.
7 Vanishing Moments The n-th moment of Ψ is defined to be: M n {Ψ}= t n Ψ(t)dt. If M 0 {Ψ}=0 then Ψ has one vanishing moment. Because M 0 {Ψ}= Ψ(x)dx= Ψ(0)=0 all wavelets have at least one vanishing moment. If M 0 {Ψ} = M 1 {Ψ} = 0, then Ψ has two vanishing moments, etc.
8 Vanishing Moments (contd.) If Ψ has one vanishing moment, then Ψ a,b, a 0 =0. If Ψ has two vanishing moments, then Ψ a,b, a 1 x+a 0 =0. If Ψ has n vanishing moments, then Ψ a,b, a n 1 x n 1 + +a 1 x+a 0 =0, i.e., the daughter wavelets are orthogonal to any polynomial of degree less than n. Vanishing moments are the reason why smooth signals have sparse representations in wavelet bases.
9 Continuous Wavelet Transform Analysis Synthesis f(x) = 1 C Ψ where F(a,b) = Ψ a,b, f 0 = 1 a 2 C Ψ = f(x)ψ a,b (x) dx F(a,b)Ψ a,b (x) db da Ψ(s) 2 ds s
10 Continuous Wavelet Transform (Example) Figure 6: Continuous wavelet transform of time-series using derivative of Gaussian wavelet (from Vialar, T., Complex and Chaotic Nonlinear Dynamics, Springer, 2009).
11 Two Dimensional Continuous Wavelet Transform Analysis F(a,b x,b y ) = Ψ a,bx,b y, f Synthesis = f(x,y)ψ a,bx,b y (x,y) dx dy f(x,y)= 1 1 a 3 C Ψ where and 0 Ψ a,bx,b y (x,y) = 1 a Ψ C Ψ = F(a,b x,b y )Ψ a,bx,b y (x,y) db x db y da ( x bx a, y b ) y a Ψ(u,v) 2 u 2 + v 2dudv
12 Wavelet Transform as Convolution Recall that the relationship between daughter wavelet Ψ a,b and mother wavelet Ψ involves both translation and dilation: Ψ a,b (x)= 1 ( ) x b Ψ. a a Let s define a function Ψ a to represent a daughter which is dilated by a factor a but is not translated: Ψ a (x b)=ψ a,b (x)= 1 ( ) x b Ψ a a and a function Ψ a (x) to represent a reflected and conjugated instance of Ψ a : Ψ a (x)=ψ a ( x).
13 Wavelet Transform as Convolution (contd.) Using Ψ a and Ψ a the forward and inverse continuous wavelet transforms can be expressed as follows: Analysis Synthesis F(a,b) = Ψ a,b, f f(x) = 1 C Ψ = 1 C Ψ = 1 C Ψ = = = { f Ψ a }(b) 1 0 a a 2 0 f(x)ψ a (x b) dx f(x)ψ a (b x) dx Ψ a,b, f Ψ a,b (x) db da { f Ψ a }(b)ψ a (x b) db da 1 a 2{ f Ψ a Ψ a }(x) da
14 Wavelet Series Transform Is it possible to replace the integrals over a and b in the synthesis formula with sums? Can we represent any f in a Hilbert space, H, using a discrete set, S, of wavelet coefficients? If for all f H there exist A>0 and B< such that A f 2 Ψ a,b, f 2 B f 2 (a,b) S then Ψ a,b for (a,b) S form a frame for H. Furthermore, there exists a set of functions Ψ a,b for(a,b) S which form a dual frame for H : 1 B f 2 Ψ a,b, f 2 1 A f 2. (a,b) S
15 Wavelet Series Transform (contd.) The wavelets, Ψ a,b, are used for analysis: Ψ a,b, f = f(x)ψ a,b (x) dx and the wavelets, Ψ a,b, are used for synthesis: f(x)= Ψ a,b, f Ψ a,b (x). (a,b) S
16 Self-inverting Wavelet Series If A=B, then Ψ a,b, f 2 = A f 2 (a,b) S and the Ψ a,b for (a,b) S form a tight-frame for H, in which case f(x)= 1 A Ψ a,b, f Ψ a,b (x). (a,b) S Such frames are said to be self-inverting because Ψ a,b (x)= 1 A Ψ a,b (x).
17 Redundancy Recall that for a tight-frame A= (a,b) S Ψ a,b, f 2. f 2 Assuming that Ψ =1, then A provides a measure of the redundancy of the expansion, i.e., the degree of overcompleteness. If A = 1 there is no redundancy, and the expansion is orthonormal. How can one find wavelet series transforms with no redundancy?
18 Dyadic Sampling A sampling pattern is dyadic if the daughter wavelets are generated by dilating the mother wavelet by 2 j and translating it by k2 j : Ψ j,k (x) = 1 ( ) x k2 j Ψ 2 j Dyadic sampling is optimal because the space variable is sampled at the Nyquist rate for any given frequency. 2 j
19 Dyadic Sampling (contd.) FREQUENCY SPACE Figure 7: Dyadic sampling pattern.
Space-Frequency Atoms
Space-Frequency Atoms FREQUENCY FREQUENCY SPACE SPACE FREQUENCY FREQUENCY SPACE SPACE Figure 1: Space-frequency atoms. Windowed Fourier Transform 1 line 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 0 100 200
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