Advances in Adaptive Data Analysis

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 most robust and accurate although limited to the mean frequency over a quarter wavelength of temporal resolution. With these results, we believe most of the problems associated with the IF determination are resolved, and a true time–frequency analysis is thus taking another step toward maturity.

Empirical mode decomposition (EMD) is an adaptive, data-driven algorithm that decomposes any time series into its intrinsic modes of oscillation, which can then be used in the calculation of the instantaneous phase and frequency. Ensemble EMD (EEMD), where the final EMD is estimated by averaging numerous EMD runs with the addition of noise, was an advancement introduced by Wu and Huang (2008) to try increasing the robustness of EMD and alleviate some of the common problems of EMD such as mode mixing. In this work, we test the performance of EEMD as opposed to normal EMD, with emphasis on the effect of selecting different stopping criteria and noise levels. Our results indicate that EEMD, in addition to slightly increasing the accuracy of the EMD output, substantially increases the robustness of the results and the confidence in the decomposition.

In this study, we develop empiric and dynamic methods to detect the underlying sea bottom topography from synthetic aperture radar (SAR) images. The ocean bottom features can be seen from SAR images due to the modulation of the surface waves by the nonuniform currents, which, by motion equations, should be a function of depth. We first derive the SAR image equation at a given time from Valenzuela's formula and the expression of micro-scale wave spectrum. By integrating the SAR image equation and the governing equations for ocean currents, we establish the "forward and inverse problems" for dynamic detection of topography. To utilize the SAR images, we separate the current modulation scale from the surface waves by a two-dimensional empirical mode decomposition method based on the Delaunay triangulation with the most protruding principle and the Berstein–Bezier fitting and interpolation with the most optimum principle. Examples of bottom topography detection from SAR images are presented.

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.

To investigate whether the first intrinsic mode function, obtained from Hilbert–Huang transform (HHT), of heart rate variability is respiratory related. Electrocardiogram and chest circumference signals were recorded from 10 healthy subjects at supine rest. The HHT was applied to both R-R interval and chest circumference signals to figure out their first intrinsic mode functions (C1_RR and C1_RESP, respectively) from which the instantaneous amplitude, phase, and frequency were calculated. Although the instantaneous amplitudes and frequencies of C1_RR and C1_RESP were variable, linear regression analysis indicated a phase lock between C1_RR and C1_RESP. Intake of 500 ml water significantly elevated the amplitude ratio of C1_RR to C1_RESP; however, the phase difference of C1_RR to C1_RESP was still unchanged. The data indicate that the first intrinsic mode function of heart rate variability is respiratory related and may be equivalent to respiratory sinus arrhythmia. As compared to fast Fourier transform, HHT of respiratory sinus arrhythmia provides a comparative spatial measurement with a much higher temporal resolution.

Scattered data interpolation is an essential part of bidimensional empirical mode decomposition (BEMD) of an image. In the decomposition process, local maxima and minima of the image are extracted at each iteration and then interpolated to form the upper and the lower envelopes, respectively. The number of two-dimensional intrinsic mode functions resulting from the decomposition and their properties are highly dependent on the method of interpolation. Though a few methods of interpolation have been tested and/or applied to the BEMD process, many others remain to be tested. This paper evaluates the performance of some of the widely used surface interpolation techniques to identify one or more good choices of such methods for envelope estimation in BEMD. The interpolation techniques studied in this paper include various radial basis function interpolators and Delaunay triangulation based interpolators. The analysis is done first using a synthetic texture image and then using two different real texture images. Simulations are made to focus mainly on the effect of interpolation methods by providing less or negligible control on the other parameters or factors of the BEMD process.