Efficient reconstruction of band-limited sequences from nonuniformly decimated versions by use of polyphase filter banks

An efficient polyphase structure for the reconstruction of a band-limited sequence from a nonuniformly decimated version is developed. Theoretically, the reconstruction involves the implementation of a bank of multilevel filters, and it is shown that how all these reconstruction filters can be obtained at the cost of one Mth band low-pass filter and a constant matrix multiplier. The resulting structure is therefore more general than previous schemes. In addition, the method offers a direct means of controlling the overall reconstruction distortion T(z) by appropriate design of a low-pass prototype filter P(z). Extension of these results to multiband band-limited signals and to the case of nonconsecutive nonuniform subsampling are also summarized, along with generalizations to the multidimensional case. Design examples are included to demonstrate the theory, and the complexity of the new method is seen to be much lower than earlier ones

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

EMD-based filtering (EMDF) of low-frequency noise for speech enhancement

An Empirical Mode Decomposition based filtering (EMDF) approach is presented as a post-processing stage for speech enhancement. This method is particularly effective in low frequency noise environments. Unlike previous EMD based denoising methods, this approach does not make the assumption that the contaminating noise signal is fractional Gaussian Noise. An adaptive method is developed to select the IMF index for separating the noise components from the speech based on the second-order IMF statistics. The low frequency noise components are then separated by a partial reconstruction from the IMFs. It is shown that the proposed EMDF technique is able to suppress residual noise from speech signals that were enhanced by the conventional optimallymodified log-spectral amplitude approach which uses a minimum statistics based noise estimate. A comparative performance study is included that demonstrates the effectiveness of the EMDF system in various noise environments, such as car interior noise, military vehicle noise and babble noise. In particular, improvements up to 10 dB are obtained in car noise environments. Listening tests were performed that confirm the results

Stochastic dynamics and wavelets techniques for system response analysis and diagnostics: Diverse applications in structural and biomedical engineering

In the first part of the dissertation, a novel stochastic averaging technique based on a Hilbert transform definition of the oscillator response displacement amplitude is developed. In comparison to standard stochastic averaging, the requirement of “a priori” determination of an equivalent natural frequency is bypassed, yielding flexibility in the ensuing analysis and potentially higher accuracy. Further, the herein proposed Hilbert transform based stochastic averaging is adapted for determining the time-dependent survival probability and first-passage time probability density function of stochastically excited nonlinear oscillators, even endowed with fractional derivative terms. To this aim, a Galerkin scheme is utilized to solve approximately the backward Kolmogorov partial differential equation governing the survival probability of the oscillator response. Next, the potential of the stochastic averaging technique to be used in conjunction with performance-based engineering design applications is demonstrated by proposing a stochastic version of the widely used incremental dynamic analysis (IDA). Specifically, modeling the excitation as a non-stationary stochastic process possessing an evolutionary power spectrum (EPS), an approximate closed-form expression is derived for the parameterized oscillator response amplitude probability density function (PDF). In this regard, IDA surfaces are determined providing the conditional PDF of the engineering demand parameter (EDP) for a given intensity measure (IM) value. In contrast to the computationally expensive Monte Carlo simulation, the methodology developed herein determines the IDA surfaces at minimal computational cost. In the second part of the dissertation, a novel multiple-input/single-output (MISO) system identification technique is developed for parameter identification of nonlinear and time-variant oscillators with fractional derivative terms subject to incomplete non-stationary data. The technique utilizes a representation of the nonlinear restoring forces as a set of parallel linear sub-systems. Next, a recently developed L1-norm minimization procedure based on compressive sensing theory is applied for determining the wavelet coefficients of the available incomplete non-stationary input-output (excitation-response) data. Several numerical examples are considered for assessing the reliability of the technique, even in the presence of incomplete and corrupted data. These include a 2-DOF time-variant Duffing oscillator endowed with fractional derivative terms, as well as a 2-DOF system subject to flow-induced forces where the non-stationary sea state possesses a recently proposed evolutionary version of the JONSWAP spectrum. In the third part of this dissertation, a joint time-frequency analysis technique based on generalized harmonic wavelets (GHWs) is developed for dynamic cerebral autoregulation (DCA) performance quantification. DCA is the continuous counter-regulation of the cerebral blood flow by the active response of cerebral blood vessels to the spontaneous or induced blood pressure fluctuations. Specifically, various metrics of the phase shift and magnitude of appropriately defined GHW-based transfer functions are determined based on data points over the joint time-frequency domain. The potential of these metrics to be used as a diagnostics tool for indicating healthy versus impaired DCA function is assessed by considering both healthy individuals and patients with unilateral carotid artery stenosis. Next, another application in biomedical engineering is pursued related to the Pulse Wave Imaging (PWI) technique. This relies on ultrasonic signals for capturing the propagation of pressure pulses along the carotid artery, and eventually for prognosis of focal vascular diseases (e.g., atherosclerosis and abdominal aortic aneurysm). However, to obtain a high spatio-temporal resolution the data are acquired at a high rate, in the order of kilohertz, yielding large datasets. To address this challenge, an efficient data compression technique is developed based on the multiresolution wavelet decomposition scheme, which exploits the high correlation of adjacent RF-frames generated by the PWI technique. Further, a sparse matrix decomposition is proposed as an efficient way to identify the boundaries of the arterial wall in the PWI technique

Data-driven time-frequency analysis of multivariate data

Empirical Mode Decomposition (EMD) is a data-driven method for the decomposition and time-frequency analysis of real world nonstationary signals. Its main advantages over other time-frequency methods are its locality, data-driven nature, multiresolution-based decomposition, higher time-frequency resolution and its ability to capture oscillation of any type (nonharmonic signals). These properties have made EMD a viable tool for real world nonstationary data analysis. Recent advances in sensor and data acquisition technologies have brought to light new classes of signals containing typically several data channels. Currently, such signals are almost invariably processed channel-wise, which is suboptimal. It is, therefore, imperative to design multivariate extensions of the existing nonlinear and nonstationary analysis algorithms as they are expected to give more insight into the dynamics and the interdependence between multiple channels of such signals. To this end, this thesis presents multivariate extensions of the empirical mode de- composition algorithm and illustrates their advantages with regards to multivariate non- stationary data analysis. Some important properties of such extensions are also explored, including their ability to exhibit wavelet-like dyadic filter bank structures for white Gaussian noise (WGN), and their capacity to align similar oscillatory modes from multiple data channels. Owing to the generality of the proposed methods, an improved multi- variate EMD-based algorithm is introduced which solves some inherent problems in the original EMD algorithm. Finally, to demonstrate the potential of the proposed methods, simulations on the fusion of multiple real world signals (wind, images and inertial body motion data) support the analysis

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Learning Theory and Approximation

Learning theory studies data structures from samples and aims at understanding unknown function relations behind them. This leads to interesting theoretical problems which can be often attacked with methods from Approximation Theory. This workshop - the second one of this type at the MFO - has concentrated on the following recent topics: Learning of manifolds and the geometry of data; sparsity and dimension reduction; error analysis and algorithmic aspects, including kernel based methods for regression and classification; application of multiscale aspects and of refinement algorithms to learning