This paper introduces the synchrosqueezed curvelet transform as an optimal tool for 2D 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 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.
There is another interesting line of work for mode decomposition, which is the empirical mode decomposition (EMD) initiated and refined by in @cite_21 @cite_15 . Starting from the most oscillatory mode, the EMD method decomposes a signal into a collection of intrinsic mode functions (IMFs) and estimates instantaneous frequencies via the Hilbert transform. However, the dependence on local extrema limits its applications in noisy cases. To address the robustness problem, some variants were proposed in @cite_6 @cite_27 . Following the idea of EMD, there are two existing methods for high dimensional mode decomposition. The first one is based on high dimensional interpolation @cite_26 @cite_8 @cite_19 @cite_7 and the second one applies a 1D decomposition to each dimension and then combines the results with a proper combination strategy @cite_4 @cite_10 @cite_20 . In spite of their considerable success, these existing methods in this research line are not suitable to separate two modes with similar wave-numbers but different wave-vectors due to the lack of anisotropic angular separation as discussed in @cite_13 .
@cite_13 @cite_26 @cite_4 @cite_7 @cite_8 @cite_21 @cite_6 @cite_19 @cite_27 @cite_15 @cite_10 @cite_20
2019900743 1976836781 2110983594 2113193347
Some recent methods, like the empirical mode decomposition (EMD), propose to decompose a signal accordingly to its contained information. Even though its adaptability seems useful for many applications, the main issue with this approach is its lack of theory. This paper presents a new approach to build adaptive wavelets. The main idea is to extract the different modes of a signal by designing an appropriate wavelet filter bank. This construction leads us to a new wavelet transform, called the empirical wavelet transform. Many experiments are presented showing the usefulness of this method compared to the classic EMD. Recent developments in analysis methods on the non-linear and non-stationary data have received large attention by the image analysts. In 1998, Huang introduced the empirical mode decomposition (EMD) in signal processing. The EMD approach, fully unsupervised, proved reliable monodimensional (seismic and biomedical) signals. The main contribution of our approach is to apply the EMD to texture extraction and image filtering, which are widely recognized as a difficult and challenging computer vision problem. We developed an algorithm based on bidimensional empirical mode decomposition (BEMD) to extract features at multiple scales or spatial frequencies. These features, called intrinsic mode functions, are extracted by a sifting process. The bidimensional sifting process is realized using morphological operators to detect regional maxima and thanks to radial basis function for surface interpolation. The performance of the texture extraction algorithms, using BEMD method, is demonstrated in the experiment with both synthetic and natural images. We introduce a new adaptive method for analyzing nonlinear and nonstationary data. This method is inspired by the empirical mode decomposition (EMD) method and the recently developed compressed sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form a(t )c os(θ(t)) ,w herea ≥ 0i s assumed to be smoother than cos(θ(t)) and θ is a piecewise smooth increasing function. We formulate this problem as a nonlinear L 1 optimization problem. Further, we propose an iterative algorithm to solve this nonlinear optimization problem recursively. We also introduce an adaptive filter method to decompose data with noise. Numerical examples are given to demonstrate the robustness of our method and comparison is made with the EMD method. One advantage of performing such a decomposition is to preserve some intrinsic physical property of the signal, such as trend and instantaneous frequency. Our method shares many important properties of the original EMD method. Because our method is based on a solid mathematical formulation, its performance does not depend on numerical parameters such as the number of shifting or stop criterion, which seem to have a major effect on the original EMD method. Our method is also less sensitive to noise perturbation and the end effect compared with the original EMD method. Previous work on empirical mode decomposition in two dimensions typically generates a residue with many extrema points. In this paper we propose an improved method to decompose an image into a number of intrinsic mode functions and a residue image with a minimum number of extrema points. We further propose a method for the variable sampling of the two-dimensional empirical mode decomposition. Since traditional frequency concept is not applicable in this work, we introduce the concept of empiquency, shortform for empirical mode frequency, to describe the signal oscillations. The very special properties of the intrinsic mode functions are used for variable sampling in order to reduce the number of parameters to represent the image. This is done blockwise using the occurrence of extrema points of the intrinsic mode function to steer the sampling rate of the block. A method of using overlapping 7 × 7 blocks is introduced to overcome blocking artifacts and to further reduce the number of parameters required to represent the image. The results presented here shows that an image can be successfully decomposed into a number of intrinsic mode functions and a residue image with a minimum number of extrema points. The results also show that subsampling offers a way to keep the total number of samples generated by empirical mode decomposition approximately equal to the number of pixels of the original image.