Synchrosqueezed Transforms and Applications

Transcription

1 Synchrosqueezed Transforms and Applications Haizhao Yang Department of Mathematics, Stanford University Collaborators: Ingrid Daubechies,JianfengLu ] and Lexing Ying Department of Mathematics, Duke University ] Department of Mathematics and Chemistry and Physics, Duke University Department of Mathematics and ICME, Stanford University January 9, 25

2 Medical study (Y., ACHA, 4) I A superposition of two ECG signals. f (t) = (t)s (2 (t)) + 2 (t)s 2 (2 2 (t)). I Spike wave shape functions s (t) ands 2 (t) x x 3 Figure : Complicated wave shape functions Figure : Good decomposition.

3 Geophysics (Y. and Ying, SIIMS 3, SIMA 4) I A superposition of several wave fields. I Nonlinear components, bounded supports Figure : One target component with structure noise and Gaussian random noise. Courtesy of Fomel and Hu for providing data.

4 Materials science (Y., Lu and Ying, preprint) Atomic crystal analysis I Observation: an assemblage of wave-like components; I Goal: Crystal segmentation, crystal rotations, crystal defects, crystal deformations

5 Art forensics (Y., Lu, Brown, Daubechies, Ying, preprint) Painting canvas analysis I Observation: a superposition of wave-like components; I Goal: count threads and estimate texture deformation Figure : Top: a X-ray image of canvas. Left: horizontal thread count. Right: horizontal thread angle.

6 D mode decomposition Known: A superposition of wave-like components KX KX f (t) = f k (t) = k (t)e 2 in k k (t). k= k= Unknown: Number K, components f k (t), smooth instantaneous amplitudes k (t), smooth instantaneous frequencies N k k (t). Existing methods: I Empirical mode decomposition methods (Huang et al. 98, 9); I Synchrosqueezed wavelet transform (Daubechies et al. 9, ); Synchrosqueezed wave packet transform (Y. 4); I Data-driven time-frequency analysis (Hou et al., 2, 3); I Regularized nonstationary autoregression (Fomel 3);

7 D wave packets Given a mother wave packet w(t) andascalingparameters 2 (/2, ), the family of wave packets {w ab (t) :a, b 2 R} is defined as w ab (t) =a s/2 w(a s (t b))e 2 i(t b)a, or equivalently, in the Fourier domain as dw ab ( ) =a s/2 e 2 ib bw(a s ( a)). D wave packet transform The D wave packet transform of a function f (t) is a function Z W f (a, b) =hw ab, f i = w ab (t)f (t)dt for a, b 2 R.

8 Asimpleexample A plane wave with an instantaneous frequency N: f (t) =e 2 int. Its wave packet transform: Z W f (a, b) = e 2 int a s/2 w(a s (t b))e 2 i(t b)a dt R = a s/2 e 2 inb ŵ(a s (N a)). The oscillation of W f (a, b) inb reveals b W f (a, b) 2 iw f (a, b) = a b e 2 inbŵ(a s (N a)) 2 ia s/2 e 2 inb ŵ(a s (N a)) = N.

9 Definition: Instantaneous frequency estimate for W f (a, b) 6=.! f (a, b) bw f (a, b) 2 iw f (a, b) Definition: Synchrosqueezed wave packet transform (SSWPT) Z T f (!, b) = Comparison of supports Aplanewavef (t) =e 2 int,forafixedb, R W f (a, b) 2 (<! f (a, b)!) da. suppw f (a, b) (N N s, N + N s ); suppt f (!, b) concentrates at! = N.

10 SS for sharpened representation k k x x Figure : The supports of the D wave packet transform and D SSWPT of a synthetic benchmark signal.

11 Theory of D SSWPT Theorem: (Y. 4 ACHA) If KX KX f (t) = f k (t) = k (t)e 2 in k k= k= and f k (t) arewell-separated,then I T f (a, b) haswell-separatedsupportsz k concentrating (N k k (t) k (b), b); I f k (t) can be accurately recovered by applying an inverse transform on I Zk (a, b)t f (a, b). where I Zk (a, b) is an indication function.

12 Robustness properties of D SSWPT I Bounded perturbation; I Gaussian random noise (colored); I Possible compactly supported in space. Theorem: (Y. and Ying, 4, preprint) I A non-linear wave f (x) = (x)e 2 in (x), (x) =O(); A zero mean Gaussian random noise e with covariance q for some q > ; Awavepacketw ab (x) compactly supported in the Fourier domain; I Main results: if s(x) = f (x) + e, then after thresholding, with a probability at least e O(N2 3s q ) e O(N 2 s q ),! s (a, b) bw s (a, b) 2 iw s (a, b) N (b)

13 Properties of D SSWPT 2 I When s =, wave packets become wavelets; I When s = /2, wave packets become wave atoms; I Larger s, more accurate to estimate instantaneous frequencies; I Smaller s, more robust to estimate instananeous frequencies; I Smaller s, better resolution to distinguish wave-like components in the high frequency domain. I Smaller s, better for the general mode decomposition problem: KX KX f (t) = f k (t) = k (t)s k (2 N k k (t)) k= = k= KX k= k (t) X n bs k (n)e 2 in k n k (t) Y. ACHA, 4. 2 Y. and Ying, arxiv:4.5939, 4.

14 Di erence of wavelets and wave packets The size of the essential support of dw ab ( ) iso(a s ) Figure : In the frequency domain: s =,wavelettiling(blue);samplingbump functions (black); Fourier transforms of plane waves (red) Figure : s <, wave packets.

15 Di erence of SS wavelets and SS wave packets 5 t=/2, SS wave atom 5 t = /2+/8 5 t=, SS wavelet Frequency (Hz) 5 Frequency (Hz) 5 Frequency (Hz) t=/2, SS wave atom 5 t = /2+/8 5 t=, SS wavelet Frequency (Hz) Frequency (Hz) Frequency (Hz) Figure : Seismic trace benchmark signal: s =.5; s =.625; s =. Top: whole domain. Bottom: high frequency part.

16 k k k Robustness properties of D SSTs Smaller scaling parameter s in the SSWPT, better robustness x x x Figure : Noisy synthetic benchmark signal. From left to right: s =.625, s =.75, and s =.875.

17 k k k Robustness properties of D SSTs Higher redundancy in the time-frequency transform, better robustness x x x Figure : 6 times redundancy. From left to right: s =.625, s =.75, and s =.875.

18 Frequency (Hz) 3 Frequency (Hz) Frequency (Hz) Frequency (Hz) 3 Frequency (Hz) Frequency (Hz) Volcanic signal tremor Figure : From left to right: s = (SSWT); s =.75; s =.625. Top: Normal SST. Bottom: Enhance the energy in the high frequency part.

19 2D Synchrosqueezed (SS) transforms 2D wave packets +SS = 2D SS wave packet (SSWPT) 2D general curvelets 2D SS curvelet (SSCT) 2D wave packets 2D wave packets {w ab (x) :a, b 2 R 2, a or equivalently in Fourier domain } are defined as w ab (x) = a s w( a s (x b))e 2 i(x b) a, dw ab ( ) = a s e 2 ib ŵ( a s ( a)).

20 Notations:. The scaling matrix A a = a t a s. 2. The rotation angle and rotation matrix cos sin R = sin cos. 3. Aunitvectore = (cos, sin ) T with a rotation angle. 2D general curvelets 2D general curvelets {w a b (x), a 2 [, ), 2 [, 2 ), b 2 R 2 } are defined as w a b (x) =a t+s 2 e 2 ia(x b) e w(a a R (x b)), or equivalently in Fourier domain dw a b ( ) =bw(a a R ( a e ))e 2 ib a t+s 2.

21 2D wave packets and 2D general curvelets p p a a t a s p p Figure : Essential support of the Fourier transform of: continuous wave packets; continuous general curvelets; a discrete general curvelet with parameters (s, t), roughly of size a s a t.

22 Theory for 2D SS wave packet transforms Theroem : (Y. and Ying SIIMS 3) A non-linear wave f (x) = (x)e 2 i (x),awavepacketw ab (x), define a transform: Z W f (a, b) =hs(x), w ab (x)i = s(x)w ab (x) dx.! f (a, b) = r bw f (a, b) r (b) 2 iw f (a, b) Theorem 2: (Y. and Ying preprint 4) A zero mean Gaussian random noise e with covariance q for some q >. If s(x) = f (x) + e, then after thresholding, with a probability at least e O(N2 2s q ) e O(N 2s q ) e O(N 2 q ),! s (a, b) = r bw s (a, b) Nr (b) 2 iw s (a, b)

23 Theory for 2D SS curvelet transforms Theroem : (Y. and Ying SIMA 4) A non-linear wave f (x) = (x)e 2 i (x), a general curvelet w a b (x), define atransform: Z W f (a,,b) =hs(x), w a b (x)i = s(x)w a b (x) dx.! f (a,,b) = r bw f (a,,b) r (b) 2 iw f (a,,b) Theroem 2: (Y. and Ying preprint 4) A zero mean Gaussian random noise e with covariance q for some q >. If s(x) = f (x) + e, then after thresholding, with a probability at least e O(N2 2t q ) e O(N 2s q ) e O(N 2 q ),! f (a,,b) = r bw f (a,,b) r (b) 2 iw f (a,,b)

24 Synchrosqueezing for sharpened representation: Z T f (!, b) = W f (a, b) (! f (a, b) {a:w f (a,b)6=}!) da. 2 x p p.5 b v x v.5 b 2 Figure : An example of a superposition of two 2D waves using 2D SSWPT.

25 2 2 2 v 2 v 2 v v.5 b v.5 b v.5 b 2 Figure : T f (!, b) of the same example in the last figure. Left: noiseless. Middle: SNR= 3. Right: SNR= 3.

26 Di erence of 2D SSWPT and 2D SSCT Usually s = t is better than s < t, exceptforthebandedwave-like components. Figure : Left: A superposition of two banded waves; Middle: 2D SSWPT; Right: 2D SSCT.

27 SynLab: a MATLAB toolbox I Available at haizhao/codes.htm. I D SS Wave Packet Transform 3 I 2D SS Wave Packet/Curvelet Transform 45 Applications: I Geophysics: seismic wave field separation and ground-roll removal. I Atomic crystal image analysis. I Art forensic. 3 Synchrosqueezed Wave Packet Transforms and Di eomorphism Based Spectral Analysis for D General Mode Decompositions, Applied and Computational Harmonic Analysis, Synchrosqueezed Wave Packet Transform for 2D Mode Decomposition,SIAM Journal on Imaging Science, Synchrosqueezed Curvelet Transform for 2D Mode Decomposition, SIAM Journal on Mathematical Analysis, 24.