Synchrosqueezed Curvelet Transform for Two-Dimensional Mode Decomposition

Haizhao Yang, Lexing Ying
2014 SIAM Journal on Mathematical Analysis  
This paper introduces the synchrosqueezed curvelet transform as an optimal tool for two-dimensional mode decomposition of wavefronts or banded wave-like components. The synchrosqueezed curvelet transform consists of a generalized curvelet transform with application dependent geometric scaling parameters, and a synchrosqueezing technique for a sharpened phase space representation. In the case of a superposition of banded wave-like components with well-separated wave-vectors, it is proved that
more » ... t is proved that the synchrosqueezed curvelet transform is capable of recognizing each component and precisely estimating local wave-vectors. A discrete analogue of the continuous transform and several clustering models for decomposition are proposed in detail. Some numerical examples with synthetic and real data are provided to demonstrate the above properties of the proposed transform. SYNCHROSQUEEZED CURVELET TRANSFORM two components sharing the same wave-number but having different wave-vectors, because of the isotropic character of the high dimensional wavelet transform. In fact, this is a common phenomenon in many applications of high frequency wave propagation. To specify this problem, let us consider a simple superposition of two plane waves e 2πip·x and e 2πiq·x with the same wave-number (|p| = |q|) but different wavevectors (p = q). In the Fourier domain, the gray region in Figure 1 (left) shows the support of one continuous wavelet. The wavelet cannot distinguish these two plane waves in the sense that the gray region has to cover two dots p and q simultaneously, or has to exclude them simultaneously. To overcome this inherent limitation of the synchrosqueezed wavelet transform in high dimensional space, the synchrosqueezed wave packet transform (SSWPT) was developed in [33] , inspired by the localized support of wave packets in the Fourier domain. The finer supports result in the better resolution for wave-number separation and, more importantly, the anisotropic supports contribute to the angular separation of wave-vectors. As shown in Figure 1 (middle), in the Fourier domain, the supports of e 2πip·x and e 2πiq·x are in the supports of two different wave packets, as long as p and q are well-separated. Yang and Ying [33] proved that SSWPT could identify different nonlinear and nonstationary high frequency wave-like components with different wavevectors in high dimensional space in a general case, even with severe noise. In some applications such as wave field separation problems [28, 30] and ground roll removal problems [2, 12, 34] in geophysics, it is required to separate overlapping wavefronts or banded wave-like components. In this case, the boundary of these components gives rise to many nonzero coefficients of wave packet transform, which results in unexpected interferential synchrosqueezed energy distribution (see Figure 2 top-right). This would dramatically reduce the accuracy of local wave-vector estimation, because the locations of nonzero energy provide estimation of local wave-vectors. As shown in Figure 2 (top-right), there exists misleading local wave-vector estimates at the location where the signal is negligible. Even if at the location where the signal is relevant, the relative error is still unacceptable. To solve this problem, an empirical idea is that good basis elements in the synchrosqueezed transform should look like the components, i.e., they should appear in a needle-like shape. An optimal solution is curvelets. The curvelet transform is anisotropic (as shown in Figure 1 right) and is designed for optimally representing curved edges [4, 29] and banded wavefronts [3] . This motivates the design of the synchrosqueezed curvelet transform (SSCT) as an optimal tool to estimate local wave-vectors of wavefronts or banded wave-like components in Downloaded 08/18/14 to Redistribution subject to SIAM license or copyright; see
doi:10.1137/130939912 fatcat:2m4cuw7nxrhwjb6cq4vemoqxki