Synchrosqueezed Curvelet Transforms for 2D mode decomposition

Transcription

1 Synchrosqueezed Curvelet Transforms for 2D mode decomposition Haizhao Yang Department of Mathematics, Stanford University Collaborators: Lexing Ying and Jianfeng Lu Department of Mathematics and ICME, Stanford University Department of Mathematics, Duke University SIAM Conference on Imaging Science, May 24

2 Geophysics A superposition of several wave-like components. Wave field separation and ground roll removal problems

3 Materials science Atomic crystal analysis Observation: an assemblage of wave-like components; Goal: Crystal segmentations, crystal rotations, crystal defects, crystal deformations. Figure : Left: An atomic crystal image. Middle: Estimated crystal rotation. Right: Identified crystal defects. Takes about s for a 24*24 image, while other methods using GPU computing take at least 4s.

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

5 D synchrosqueezed wavelet transform (SSWT) Continuous wavelet transform of a signal s(t): W s (a, b) = s(t), φ ab (t) = s(t)a /2 φ( t b ) dt. a EX: s(t) = A cos(ωt). W s (a, b) = ŝ(ξ)a /2 φ(aξ)e ibξ dξ = A 2π 4π a/2 φ(aω)e ibω. Synchrosqueezing for better readability Figure : Numerical examples by Daubechies et al, signal f (t) = sin(8t).

6 Definitions of SSWT EX: s(t) = A cos(ωt). W s (a, b) = A 4π a/2 φ(aω)e ibω bw s (a,b) iw s (a,b) = ω Definition: Instantaneous frequency estimate ω s (a, b) = bw s (a, b) iw s (a, b). Definition: Synchrosqueezed wavelet transform T s (ω, b) = W s (a, b)a 3/2 δ(ω s (a, b) ω) da. {a:w s (a,b) }

7 Theory of SSWT Theorem: (Daubechies, Lu, Wu ACHA) If K K f (t) = f k (t) = α k (t)e 2πiN k φ k (t) k= and f k (t) are well-separated, then k= T f (a, b) has well-separated supports Z k concentrating (N k φ k (b), b); 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.

8 2D mode decomposition Known: A superposition of wave-like components K K f (x) = f k (x) = α k (x)e 2πiN k φ k (x). k= Unknown: Number K, components f k (x), local amplitudes α k (x), local wave vectors N k φ k (x). Existing methods: k=. 2D EMD methods (Huang et al.); 2. 2D Empirical wavelet, curvelet transforms (Gilles, Tran, Osher 3); 3. 2D SS wave packet and curvelet transforms (Y. and Ying 3, 4).

9 2D wavelet transform or wave packet transform? Consider two plane waves e ip x and e iq x again. ξ ξ ξ ξ ξ ξ Left: Supports of continuous wavelets and plane waves with different wave numbers in the Fourier domain. Radial separation. Middle: Supports of wavelets and plane waves with the same wave number. No angular separation. Right: Supports of wave packets and plane waves with the same wave number. Angular separation.

10 Definition: Wave Packets Given the mother wave packet w(x) and s (/2, ), the family of wave packets {w pb (x), p, b R 2 } is defined as or equivalently in Fourier domain w pb (x) = p s w( p s (x b))e 2πi(x b) p, ŵ pb (ξ) = p s e 2πib ξ ŵ( p s (ξ p)). Definition: The 2D wave packet transform of a function f (x) is a function of p, b R 2 W f (p, b) = w pb (x)f (x) dx.

11 Definition: The local wavevector estimation of a function f (x) at (p, b) is v f (p, b) = bw f (p, b) 2πiW f (p, b) for p, b R 2 such that W f (p, b). Definition: Given f (x), for v, b R 2, W f (p, b), and v f (p, b), the synchrosqueezed energy distribution T f (v, b) is T f (v, b) = W f (p, b) 2 δ(v f (p, b) v) dp. {p:w f (p,b) } Remark: The support of T f (v, b) is concentrating around local wavevectors.

12 p 2 v 2 v 2 v 2 Sketch of 2D mode decomposition Two components: f (x) = α (x)e 2πiNφ(x) + α 2 (x)e 2πiNφ2(x) p.5 b v.5 b v.5 b v.5 b 2 Fast algorithms for forward and inverse transforms. O(L 2 log L) for a L L image.

13 Theory of 2D SS wave packet transform (SSWPT) Theorem [Y.,Ying 3, SIAM Imaging Science] For a well-separated superposition f (x) = K k= α k(x)e 2πiNφ k (x) and ɛ >, we define and R f,ɛ = {(p, b) : W f (p, b) p s ɛ} Z f,k = {(p, b) : p N φ k (b) p s } for k K. There exists a constant ɛ (K) > such that for any ɛ (, ɛ ) there exists a constant N (K, ɛ) > such that for any N > N (K, ɛ) the following statements hold. (i) {Z f,k : k K} are disjoint and R f,ɛ k K Z f,k; (ii) For any (p, b) R f,ɛ Z f,k, v f (p, b) N φ k (b) N φ k (b) ɛ.

14 Banded wave-like components x x x x x x x x.2. Figure : Top-left: A banded deformed plane wave. Top-right: Number of wave packet coefficients W f (a, b) > δ. Bottom-left: Relative error of local wave-vector estimates using SSWPT. Bottom-right: Relative error of local wave-vector estimates using SSCT.

15 General curvelet transform: Some notations. The scaling matrix A a = ( ) a t a s. 2. The rotation angle θ and rotation matrix ( cos θ sin θ R θ = sin θ cos θ ). 3. The unit vector e θ = (cos θ, sin θ) T of rotation angle θ. 4. θ α represents the argument of given vector α. Definition: 2D general curvelets For 2 < s < t <, define w aθb (x) = a t+s 2 e 2πia(x b) e θ w(a a R (x b)). θ A family of curvelets {w aθb (x), a [, ), θ [, 2π), b R 2 }.

16 Definition: 2D general curvelet transform W f (a, θ, b) = w aθb (x)f (x) dx R 2 for a [, ), θ [, 2π), b R 2. Definition: local wave vector estimation Local wave-vector estimation at (a, θ, b) is v f (a, θ, b) = bw f (a, θ, b) 2πiW f (a, θ, b) for a [, ), θ [, 2π), b R 2 such that W f (a, θ, b). Definition: synchrosqueezed energy distribution Synchrosqueezed energy distribution T f (v, b) is T f (v, b) = W f (a, θ, b) 2 δ(rv f (a, θ, b) v)a da dθ for v R 2, b R 2. {(a,θ):w f (a,θ,b) }

17 Theory of the SS curvelet transform (SSCT) Theorem [Y., Ying SIAM Mathematical Analysis 4] Suppose 2 < s < η < t are fixed, and a well-separated superposition K f (x) = e (φ k (x) c k ) 2 /σ 2 k αk (x)e 2πiNφ k (x), k= where σ k N η. For any ɛ >, define { R f,ɛ = (a, θ, b) : W f (a, θ, b) a } s+t 2 ɛ and Z f,k = { (a, θ, b) : A a R θ (a e θ N φ k (b)) } for k K. For any ɛ, there exists N (ɛ) > such that for any N > N (ɛ) the following statements hold. (i) {Z f,k : k K} are disjoint and R f,ɛ k K Z f,k; (ii) For any (a, θ, b) R f,ɛ Z f,k, v f (a, θ, b) N φ k (b) N φ k (b) ɛ.

18 Applications in Geophysics Noisy Example for SS curvelet transform Figure : Top: A noisy superposition of two components. Left: First recovered component. Right: Second recovered component.

19 Noisy Example 2 for SS curvelet transform Figure : Left: A noisy superposition of two components. Right: The recovered main component.

20 A real example comment Figure : A superposition of several components.

21 Results of the real example Figure : Four recovered components.

22 Toy example of an atomic crystal image x x Energy 3.5 x Frequency Figure : Left: Crystal image. Crystal rotations, 7.5 degrees on the left and 45 on the right. Middle: 2D Fourier power spectrum. Right: Radially average Fourier power spectrum.

23 Results of this toy example x x Figure : Left: Crystal image. Middle: The angle estimates of crystal rotations. Right: The boundary indicator function.

24 Phase field crystal image Figure : Left: A crystal image and its zoomed-in image. Middle: The crystal rotation and its zoomed-in result. Right: The crystal defects and its zoomed-in result. Takes about s, while other methods take at least 4s.

25 Conclusion Developed 2D synchrosqueezed transforms to analyze images. A method with rigorous mathematical analysis. A method with both radial and angular separations. The SS curvelet transform can adapt different data structure: s and t. Fast algorithms to compute the forward and inverse transforms. Succesful applications in geophysics and materials science.

26 Results of 2D EEMD x x x x x x x x x x x x x Figure : A superposition of two components with similar local wave number.

27 Results of 2D Empirical curvelet transforms 2D Fourier power spetrum of a superposition of two components and Fourier domain partitions using empirical curvelet transforms.