Hau-Tieng Wu

The Synchrosqueezing algorithm for time-varying spectral analysis

We analyze the stability properties of the Synchrosqueezing transform, a time-frequency signal analysis method that can identify and extract oscillatory components with time-varying frequency and amplitude. We show that Synchrosqueezing is robust to bounded perturbations of the signal and to Gaussian white noise. These results justify its applicability to noisy or nonuniformly sampled data that is ubiquitous in engineering and the natural sciences. We also describe a practical implementation of Synchrosqueezing and provide guidance on tuning its main parameters. As a case study in the geosciences, we examine characteristics of a key paleoclimate change in the last 2.5 million years, where Synchrosqueezing provides significantly improved insights. Highlights? We study the stability of the Synchrosqueezing transform for spectral analysis. ? Synchrosqueezing is shown to be robust to bounded errors and Gaussian white noise. ? We describe a numerical implementation and compare it with other techniques. ? We apply Synchrosqueezing to proxies describing the evolution of the global climate.

Time-Frequency Reassignment and Synchrosqueezing: An Overview

This article provides a general overview of time-frequency (T-F) reassignment and synchrosqueezing techniques applied to multicomponent signals, covering the theoretical background and applications. We explain how synchrosqueezing can be viewed as a special case of reassignment enabling mode reconstruction and place emphasis on the interest of using such T-F distributions throughout with illustrative examples.

Vector Diffusion Maps and the Connection Laplacian

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low-dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low-dimensional manifold ℳ d embedded in ℝ p , we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold.

Synchrosqueezing-Based Recovery of Instantaneous Frequency from Nonuniform Samples

We propose a new approach for studying the notion of the instantaneous frequency of a signal. We build on ideas from the Synchrosqueezing theory of Daubechies, Lu, and Wu [Appl. Comput. Harmonic Anal., 30 (2010), pp. 243–261] and consider a variant of Synchrosqueezing, based on the short-time Fourier transform, to precisely define the instantaneous frequencies of a multicomponent AM-FM signal. We describe an algorithm to recover these instantaneous frequencies from the uniform or nonuniform samples of the signal and show that our method is robust to noise. We also consider an alternative approach based on the conventional, Hilbert transform-based notion of instantaneous frequency to compare to our new method. We use these methods on several test cases and apply our results to a signal analysis problem in electrocardiography.

Assess Sleep Stage by Modern Signal Processing Techniques

In this paper, two modern adaptive signal processing techniques, empirical intrinsic geometry and synchrosqueezing transform, are applied to quantify different dynamical features of the respiratory and electroencephalographic signals. We show that the proposed features are theoretically rigorously supported, as well as capture the sleep information hidden inside the signals. The features are used as input to multiclass support vector machines with the radial basis function to automatically classify sleep stages. The effectiveness of the classification based on the proposed features is shown to be comparable to human expert classification-the proposed classification of awake, REM, N1, N2, and N3 sleeping stages based on the respiratory signal (resp. respiratory and EEG signals) has the overall accuracy 81.7% (resp. 89.3%) in the relatively normal subject group. In addition, by examining the combination of the respiratory signal with the electroencephalographic signal, we conclude that the respiratory signal consists of ample sleep information, which supplements to the information stored in the electroencephalographic signal.

Using synchrosqueezing transform to discover breathing dynamics from ECG signals

The acquisition of breathing dynamics without directly recording the respiratory signals is beneficial in many clinical settings. The electrocardiography (ECG)-derived respiration (EDR) algorithm enables data acquisition in this manner. However, the EDR algorithm fails in analyzing such data for patients with atrial fibrillation (AF) because of their highly irregular heart rates. To resolve these problems, we introduce a new algorithm, referred to as SSTEDR, to extract the breathing dynamics directly from the single lead ECG signal; it is based on the EDR algorithm and the time-frequency representation technique referred to as the synchrosqueezing transform. We report a preliminary result about the relationship between the anesthetic depth and breathing dynamics. To the best of our knowledge, this is the first algorithm allowing us to extract the breathing dynamics of patients with obvious AF from the single lead ECG signal.

Alternating Projection, Ptychographic Imaging and Phase Synchronization

Abstract We demonstrate necessary and sufficient conditions of the local convergence of the alternating projection algorithm to a unique solution up to a global phase factor. Additionally, for the ptychography imaging problem, we discuss phase synchronization and graph connection Laplacian, and show how to construct an accurate initial guess to accelerate convergence speed to handle the big imaging data in the coming new light source era.

ConceFT: concentration of frequency and time via a multitapered synchrosqueezed transform

A new method is proposed to determine the time–frequency content of time-dependent signals consisting of multiple oscillatory components, with time-varying amplitudes and instantaneous frequencies. Numerical experiments as well as a theoretical analysis are presented to assess its effectiveness.

Augmented projections for ptychographic imaging

Ptychography is a popular technique to achieve diffraction limited resolution images of a two- or three-dimensional sample using high frame rate detectors. We introduce a relaxation of common projection algorithms to account for instabilities given by intensity and background fluctuations, position errors, or poor calibration using multiplexing illumination. This relaxation introduces an additional phasing optimization at every step that enhances the convergence rate of common projection algorithms. Numerical tests exhibit the exact recovery of the object and the perturbations when there is high redundancy in the data.

Evaluating Physiological Dynamics via Synchrosqueezing: Prediction of Ventilator Weaning

Oscillatory phenomena abound in many types of signals. Identifying the individual oscillatory components that constitute an observed biological signal leads to profound understanding about the biological system. The instantaneous frequency (IF), the amplitude modulation (AM), and their temporal variability are widely used to describe these oscillatory phenomena. In addition, the shape of the oscillatory pattern, repeated in time for an oscillatory component, is also an important characteristic that can be parametrized appropriately. These parameters can be viewed as phenomenological surrogates for the hidden dynamics of the biological system. To estimate jointly the IF, AM, and shape, this paper applies a novel and robust time-frequency analysis tool, referred to as the synchrosqueezing transform (SST). The usefulness of the model and SST are shown directly in predicting the clinical outcome of ventilator weaning. Compared with traditional respiration parameters, the breath-to-breath variability has been reported to be a better predictor of the outcome of the weaning procedure. So far, however, all these indices normally require at least 20 min of data acquisition to ensure predictive power. Moreover, the robustness of these indices to the inevitable noise is rarely discussed. We find that based on the proposed model, SST and only 3 min of respiration data, the ROC area under curve of the prediction accuracy is 0.76. The high predictive power that is achieved in the weaning problem, despite a shorter evaluation period, and the stability to noise suggest that other similar kinds of signal may likewise benefit from the proposed model and SST.