Wen-Liang Hwang

Singularity detection and processing with wavelets

The mathematical characterization of singularities with Lipschitz exponents is reviewed. Theorems that estimate local Lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed. It is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately. The local frequency of such oscillations is measured from the wavelet transform modulus maxima. It has been shown numerically that one- and two-dimensional signals can be reconstructed, with a good approximation, from the local maxima of their wavelet transform modulus. As an application, an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales. In two dimensions, the wavelet transform maxima indicate the location of edges in images. >

Characterization of signals by the ridges of their wavelet transforms

The characterization and the separation of amplitude and frequency modulated signals is a classical problem of signal analysis and signal processing. We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of one-dimensional (1-D) signals. These detection procedures are shown to be robust to additive white noise. We also derive and test a new reconstruction procedure. The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides a very efficient way to code the information contained in the signal.

Practical Time-Frequency Analysis

The purpose of the book is to give a self-contained presentation of the techniques of time-frequency/time-scale analysis of 1-D signals and provide a set of useful tools to perform the analyses. Such a package should be especially attractive to the part of the scientific community interested in mathematical and practical issues, especially if they involve random or noisy signals with possibly nonstationary features. The use of the S language is a reflection of the intent to reach the statistical community that, despite the traditional interest in the spectral analysis of time series and the pervasive use of orthonormal wavelet bases, has seen very few attempts to understand the benefits of the continuous transforms. The first part of the book is intended to be a hands-on crash course on some of the major components of the time-frequency analysis of signals. A special emphasis is put on the analyses of noisy signals, and great care is taken to address the stationarity issue and describe the statistical significance of the spectral analyses and the denoizing procedures. The second part of the book is a reference manual for the library of S functions, which is written to perform all the computations relative to the examples described in the first part of the monograph.

Multiridge detection and time-frequency reconstruction

The ridges of the wavelet transform, the Gabor transform, or any time-frequency representation of a signal contain crucial information on the characteristics of the signal. Indeed, they mark the regions of the time-frequency plane where the signal concentrates most of its energy. We introduce a new algorithm to detect and identify these ridges. The procedure is based on an original form of Markov chain Monte Carlo algorithm especially adapted to the present situation. We show that this detection algorithm is especially useful for noisy signals with multiridge transforms. It is a common practice among practitioners to reconstruct a signal from the skeleton of a transform of the signal (i.e., the restriction of the transform to the ridges). After reviewing several known procedures, we introduce a new reconstruction algorithm, and we illustrate its efficiency on speech signals and its robustness and stability on chirps perturbed by synthetic noise at different SNRs.

EMD Revisited: A New Understanding of the Envelope and Resolving the Mode-Mixing Problem in AM-FM Signals

Empirical mode decomposition (EMD) is an adaptive and data-driven approach for analyzing multicomponent nonlinear and nonstationary signals. The stop criterion, envelope technique, and mode-mixing problem are the most important topics that need to be addressed in order to improve the EMD algorithm. In this paper, we study the envelope technique and the mode-mixing problem caused by separating multicomponent AM-FM signals with the EMD algorithm. We present a new necessary condition on the envelope that questions the current assumption that the envelope passes through the extreme points of an intrinsic mode function (IMF). Then, we present a solution to the mode-mixing problem that occurs when multicomponent AM-FM signals are separated. We experiment on several signals, including simulated signals and real-life signals, to demonstrate the efficacy of the proposed method in resolving the mode-mixing problem.

Adaptive Signal Decomposition Based on Local Narrow Band Signals

We propose an operator-based method of adaptive signal decomposition, whereby a local narrow band signal is defined in the null space of a singular local linear operator. Based on the definition and the algorithm, we propose two types of local narrow band signals and two singular .operator estimation methods for adaptive signal decomposition. We show that our approach can solve a special case of Huang s empirical-mode decomposition algorithm. For signals that cannot be resolved by our method or the empirical-mode decomposition algorithm, we propose a hybrid approach. Conceptually, the approach applies the empirical-mode decomposition algorithm, followed by our algorithms. Our experiments show that the proposed hybrid approach can solve a wide range of complex signals effectively.

Analysis on multiresolution mosaic images

Image mosaicing is the act of combining two or more images and is used in many applications in computer vision, image processing, and computer graphics. It aims to combine images such that no obstructive boundaries exist around overlapped regions and to create a mosaic image that exhibits as little distortion as possible from the original images. In the proposed technique, the to-be-combined images are first projected into wavelet subspaces. The images projected into the same wavelet space are then blended. Our blending function is derived from an energy minimization model which balances the smoothness around the overlapped region and the fidelity of the blended image to the original images. Experiment results and subjective comparison with other methods are given.

Null Space Pursuit: An Operator-based Approach to Adaptive Signal Separation

The operator-based signal separation approach uses an adaptive operator to separate a signal into additive subcomponents. The approach can be formulated as an optimization problem whose optimal solution can be derived analytically. However, the following issues must still be resolved: estimating the robustness of the operators parameters and the Lagrangian multipliers, and determining how much of the information in the null space of the operator should be retained in the residual signal. To address these problems, we propose a novel optimization formula for operator-based signal separation and show that the parameters of the problem can be estimated adaptively. We demonstrate the effectiveness of the proposed method by processing several signals, including real-life signals.

Shape from texture: estimation of planar surface orientation through the ridge surfaces of continuous wavelet transform

In this correspondence, a method is proposed for estimating the surface orientation of a planar texture under perspective projection based on the ridge of a two-dimensional (2-D) continuous wavelet transform (CWT). We show that an analytical solution of the surface orientation can be derived from the scales of the ridge surface. A comparative study with an existing method is given.

Spatially-varying out-of-focus image deblurring with L1-2 optimization and a guided blur map

In this paper, we propose a spatially-varying deblurring method to remove the out-of-focus blur. Our proposed method mainly contains three parts: blur map generation, image deblurring, and scale selection. First, we derive a blur map using local contrast prior and the guided filter. Second, we propose our image deblurring method with L1-2 optimization to obtain a better image quality. Finally, we adopt the scale selection to ensure our output free from ringing artifacts. The experimental results demonstrate our proposed method is promising.