EE123 Digital Signal Processing

Transcription

1 EE123 Digital Signal Processing Lecture 12 Introduction to Wavelets

2 Last Time Started with STFT Heisenberg Boxes Continue and move to wavelets Ham exam -- see Piazza post Please register at

3 Discrete STFT X[r, k] = LX 1 m=0 optional x[rr + m]w[m]e j2 km/n!! = 2 L t = L one STFT coefficient t

4 Limitations of Discrete STFT Need overlapping Not orthogonal Computationally intensive O(MN log N) Same size Heisenberg boxes

5 From STFT to Wavelets Basic Idea: low-freq changes slowly - fast tracking unimportant Fast tracking of high-freq is important in many apps. Must adapt Heisenberg box to frequency Back to continuous time for a bit...

6 From STFT to Wavelets Continuous time t t!! Sf(u, ) = Z 1 1 f(t)w(t u)e j t dt u t! Wf(u, s) = Z 1 1 f(t) 1 p s ( t u s )dt u *Morlet - Grossmann

7 From STFT to Wavelets Wf(u, s) = Z 1 1 f(t) 1 p s ( t u s )dt The function Must satisfy: Z 1 is called a mother wavelet Z (t) 2 dt =1 (t)dt =0 unit norm Band-Pass

8 STFT and Wavelets Atoms STFT Atoms (with hamming window) Wavelet Atoms w(t u)e j t 1 ps ( t u ) s hi s =1 u u lo s =3 u u

9 Examples of Wavelets Mexican Hat (t) =(1 t 2 )e t2 /2 Haar (t) = 8 < : 1 0 apple t< apple t<1 0 otherwise

10 Example: Wavelet of Chirp s u u

11 Wavelets VS STFT s f t

12 Example 2: Bumpy Signal SombreroWavelet log(s) u

13 Wavelets Transform Can be written as linear filtering Wf(u, s) = 1 p s Z 1 1 f(t) ( t s u )dt = f(t) s(t) (u) s = 1 p s ( t s ) Wavelet coefficients are a result of bandpass filtering

14 Wavelet Transform Many different constructions for different signals Haar good for piece-wise constant signals Battle-Lemarie : Spline polynomials Can construct Orthogonal wavelets For example: dyadic Haar is orthonormal i,n(t) = 1 p 2 i ( t 2i n 2 i ) i =[1, 2, 3, ]

15 Orthonormal Haar Same scale non-overlapping Orthogonal between scales

16 Scaling function i,n(t) = 1 p 2 i ( t 2i n 2 i ) i=m+3 i=m+2 i=m+1 i=m Problem: Every stretch only covers half remaining bandwidth recall, for chirp: Need Infinite functions

17 Scaling function i,n(t) = 1 p 2 i ( t 2i n 2 i ) i=m+1 i=m Problem: Every stretch only covers half remaining bandwidth Need Infinite functions Solution: Plug low-pass spectrum with a scaling function

18 Haar Scaling function 8 < 1 0 apple t< (t) = 1 : 2 apple t<1 0 otherwise (t) = 1 0 apple t<1 0 otherwise

19 Back to Discrete Early 80 s, theoretical work by Morlett, Grossman and Meyer (math, geophysics) Late 80 s link to DSP by Daubechies and Mallat. From CWT to DWT not so trivial! Must take care to maintain properties

20 Discrete Wavelet Transform d s,u = a s,u = X N 1 n=0 N 1 X n=0 x[n] s,u[n] x[n] s,u[n]! d00 d01 d02 d03 d10 d11 finest scale d20 a20 n

21 Discrete Wavelet Transform d s,u = a s,u = X N 1 n=0 N 1 X n=0 x[n] s,u[n] x[n] s,u[n]! d00 d01 d02 d03 stop here: d10 a10 d11 a11 n

22 Example: Discrete Haar Wavelet Haar for n=2 1 p 2 1 p 2 0,0 0,0 scaling function mother wavelet approximation detail! d00 a00 Equivalent to DFT2! t

23 Discrete Orthogonal Haar Wavelet Haar for n=8 scaling Φ20 1 p 8 Ψ00 1 p 2 1 p 2 Ψ20 1 p 8 Ψ01 1 p 2 Ψ Ψ02 1 p 2 Ψ Ψ03! t

24 Fast DWT with Filter Banks (more Later!) h0[n] h1[n] x[n] h0[n] h1[n] a0n? d0n? not quite... too many coefficients

25 Fast DWT with Filter Banks h0[n] h1[n] x[n] h0[n] 2 a0n h1[n] 2 d0n

26 Fast DWT with Filter Banks h0[n] h1[n] complexity: N + N/2 + N/4 + N/ = 2N =O(N) x[n] h0[n] 2 h0[n] a0n 2 a1n h1[n] 2 d0n h1[n] 2 d1n

27 Decomposition h0[n] h1[n] x[n] h0[n] 2 h0[n] a0n 2 a1n h1[n] 2 d0n h1[n] 2 d1n

28 Reconstruction h0[n] h1[n] g0[n] g1[n] x[n] g0[n] 2 g0[n] a0n 2 a1n g1[n] 2 d0n Just flip arrows, replace h with g g1[n] 2 d1n

29 Haar DWT Example x[n] Haar a2n d2n d1n d0n

30 Approximation from 25/256 coefficients Haar DFT

31 Example: Denoising Noisy Signals Haar

32 Example: Denoising by Thresholding noisy denoised largest 25 coefficients

33 Compression - JPEG2000 vs JPEG Jpeg Wavelet Jpeg - 66 fold compression ratio

34 Compression - JPEG2000 vs JPEG Jpeg Wavelet Jpeg - 66 fold compression ratio

35

36 a2 d2x d1x d0x d2y d2xy d1y d1xy d0y d0xy

37 Noisy Wavelet Denoised

38 Approximation/Compression

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41 Example in Research Robust 4D Flow Denoising using Divergence-free Wavelet Transform Frank Ong1, Martin Uecker1, Umar Tariq2, Albert Hsiao2, Marcus T Alley2, Shreyas S Vasanawala2, Michael Lustig1 0 cm/s 100 cm/s Vector visualization Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California. 2 Department of Radiology, Stanford University, Stanford, California. Original 1 Running head: 4D Flow Denoising with Divergence-free Wavelet Transform Address correspondence to: Michael Lustig 506 Cory Hall University of California, courtesy, Frank Ong Berkeley and Marcus Alley Berkeley, CA 94720

42 0 cm/s 100 cm/s Original Noisy Flow Data Vector visualization

43 Divergence Free Wavelets a Linear spline Φ0 Quadratic spline Φ1 Linear spline ψ0 Quadratic spline ψ1 b c vx wcx vx

44 Divergence-Free Wavelet Denoising