Space-Frequency Atoms

Transcription

1 Space-Frequency Atoms

2 FREQUENCY FREQUENCY SPACE SPACE FREQUENCY FREQUENCY SPACE SPACE Figure 1: Space-frequency atoms.

3 Windowed Fourier Transform 1 line Figure 2: A Gabor function. 1 line Figure 3: A second Gabor function.

4 Windowed Fourier Transform (contd.) Analysis F(u,b) = f,w(x b)e j2πux = f(x)w(x b)e j2πux dx Synthesis f(x) = F(u,b)w(x b)e j2πux du db

5 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).

6 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.

7 What is a Wavelet? (contd.) 1 line Figure 4: A Morlet wavelet. 0.8 line Figure 5: A second Morlet wavelet.

8 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.

9 Vanishing Moments (contd.) If Ψ has one vanishing moment, then a 0,Ψ a,b = 0. If Ψ has two vanishing moments, then a 1 x+a 0,Ψ a,b = 0. If Ψ has n vanishing moments, then a n 1 x n 1 + +a 1 x+a 0,Ψ a,b = 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.

10 Three Kinds of Wavelet Transform: Continuous wavelet transform analysis synthesis input output discrete continuous Wavelet series transform analysis synthesis input output discrete continuous Discrete wavelet transform analysis synthesis input output discrete continuous

11 Continuous Wavelet Transform Analysis Synthesis F(a,b) = f,ψ a,b f(x) = 1 C Ψ where = C Ψ = f(x)ψ a,b (x) dx F(a,b)Ψ a,b (x) db da a 2 Ψ(s) 2 ds s

12 Two Dimensional Continuous Wavelet Transform Analysis F(a,b x,b y ) = f,ψ a,bx,b y Synthesis = f(x,y)ψ a,bx,b y (x,y) dx dy f(x,y) = Z 1 C Ψ where and 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 a 3 ( x bx a, y b ) y a Ψ(u,v) 2 u 2 + v 2dudv

13 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).

14 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) = f,ψ a,b f(x) = 1 C Ψ = 1 C Ψ = 1 C Ψ = = Z = { f Ψ a }(b) Z Z f(x)ψ a (x b) dx f(x)ψ a (b x) dx f,ψ a,b Ψ a,b (x) db da a 2 { f Ψ a }(b)ψ a (x b) db da { f Ψ a Ψ a }(x) da a 2 a 2

15 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 f,ψ a,b 2 B f 2 (a,b) S then Ψ a,b for (a,b) S form a frame forh. Furthermore, there exists a set of functions Ψ a,b for (a,b) S which form a dual frame forh : 1 B f 2 f, Ψ a,b 2 1 A f 2. (a,b) S

16 Wavelet Series Transform (contd.) The wavelets, Ψ a,b, are used for analysis: f,ψ a,b = f(x)ψ a,b (x) dx and the wavelets, Ψ a,b, are used for synthesis: f(x) = f,ψ a,b Ψ a,b (x). (a,b) S

17 Self-inverting Wavelet Series If A = B, then f,ψ a,b 2 = A f 2 (a,b) S and the Ψ a,b for (a,b) S form a tight-frame forh, in which case f(x) = 1 A f,ψ a,b 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).

18 Redundancy Recall that for a tight-frame A = (a,b) S f,ψ a,b 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?

19 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

20 Dyadic Sampling (contd.) FREQUENCY SPACE Figure 6: Dyadic sampling pattern.