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.
This paper introduces the synchrosqueezed wave packet transform as a method for analyzing two-dimensional images. This transform is a combination of wave packet transforms of a certain geometric scaling, a reallocation technique for sharpening phase space representations, and clustering algorithms for modal decomposition. For a function that is a superposition of several wave-like components with a highly oscillatory pattern satisfying certain separation conditions, we prove that the synchrosqueezed wave packet transform identifies these components and estimates their local wavevectors. A discrete version of this transform is discussed in detail, and numerical results are given to demonstrate the properties of the proposed transform. 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.  Image empirical mode decomposition (IEMD) is an empirical mode decomposition concept used in Hilbert–Huang transform (HHT) expanded into two dimensions for the use on images. IEMD provides a tool for image processing by its special ability to locally separate superposed spatial frequencies. The tendency is that the intrinsic mode functions (IMFs) other than the first are low-frequency images. In this study we give an overview of the state-of-the-art methods to decompose an image into a number of IMFs and a residue image with a minimum number of extrema points, together with the use of the method. Ideas and open problems are presented. This study introduces a new approach based on Bidimensional Empirical Mode Decomposition (BEMD) to extract texture features at multiple scales or spatial frequencies. Moreover, it can resolve the intrawave frequency modulation provided the frequency modulation. This decomposition, obtained by the bidimensional sifting process, plays an important role in the characterization of regions in textured images. The sifting process is realized using morphological operators to analyze the spatial frequencies and thanks to radial basis functions (RBF) for surface interpolation. We modified the original sifting algorithm to permit a pseudo bandpass decomposition of images by inserting scale criterion. Its effectiveness is demonstrated on synthetic and natural textures. In particular, we show that many different elements in textures can be extracted through the bidimensional empirical mode decomposition, which is fully unsupervised. A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the ‘empirical mode decomposition’ method with which any complicated data set can be decomposed into a finite and often small number of ‘intrinsic mode functions’ that admit well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes. With the Hilbert transform, the ‘instrinic mode functions’ yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert spectrum. In this method, the main conceptual innovations are the introduction of ‘intrinsic mode functions’ based on local properties of the signal, which make the instantaneous frequency meaningful; and the introduction of the instantaneous frequencies for complicated data sets, which eliminate the need for spurious harmonics to represent nonlinear and non-stationary signals. Examples from the numerical results of the classical nonlinear equation systems and data representing natural phenomena are given to demonstrate the power of this new method. Classical nonlinear system data are especially interesting, for they serve to illustrate the roles played by the nonlinear and non-stationary effects in the energy-frequency-time distribution. In this paper, we propose a variant of the Empirical Mode Decomposition method to decompose multiscale data into their intrinsic mode functions. Under the assumption that the multiscale data satisfy certain scale separation property, we show that the proposed method can extract the intrinsic mode functions accurately and uniquely. 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. A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall... Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the m...  A multi-dimensional ensemble empirical mode decomposition (MEEMD) for multi-dimensional data (such as images or solid with variable density) is proposed here. The decomposition is based on the applications of ensemble empirical mode decomposition (EEMD) to slices of data in each and every dimension involved. The final reconstruction of the corresponding intrinsic mode function (IMF) is based on a comparable minimal scale combination principle. For two-dimensional spatial data or images, f(x,y), we consider the data (or image) as a collection of one-dimensional series in both x-direction and y-direction. Each of the one-dimensional slices is decomposed through EEMD with the slice of the similar scale reconstructed in resulting two-dimensional pseudo-IMF-like components. This new two-dimensional data is further decomposed, but the data is considered as a collection of one-dimensional series in y-direction along locations in x-direction. In this way, we obtain a collection of two-dimensional components. Thes...