2,859 research outputs found

    2-D Prony-Huang Transform: A New Tool for 2-D Spectral Analysis

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    This work proposes an extension of the 1-D Hilbert Huang transform for the analysis of images. The proposed method consists in (i) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure, and (ii) providing a local spectral analysis of the obtained IMFs in order to get the local amplitudes, frequencies, and orientations. For the decomposition step, we propose two robust 2-D mode decompositions based on non-smooth convex optimization: a "Genuine 2-D" approach, that constrains the local extrema of the IMFs, and a "Pseudo 2-D" approach, which constrains separately the extrema of lines, columns, and diagonals. The spectral analysis step is based on Prony annihilation property that is applied on small square patches of the IMFs. The resulting 2-D Prony-Huang transform is validated on simulated and real data.Comment: 24 pages, 7 figure

    Non-linear system identification in structural dynamics: advances in characterisation of non-linearities and non-linear modal analysis

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    Many new methods for theoretical modelling, numerical analysis and experimental testing have been developed in non-linear dynamics in recent years. Although the computational power has greatly improved our ability to predict non-linear behaviour, non-linear system identification, a central topic of this thesis, still plays a key role in obtaining and quantifying structural models from experimental data. The first part of the thesis is motivated by the industrial needs for fast and reliable detection and characterisation of structural non-linearities. For this purpose a method based on the Hilbert transform in the frequency domain is proposed. The method detects and characterises structural non-linearities from a single frequency response function and does not require a priori knowledge of the system. The second part of the thesis is driven by current research trends and advances in non-linear modal analysis and adaptive time series processing using the Hilbert-Huang transform. Firstly, the alternatives of the Hilbert transform, which is commonly used in structural dynamics for the estimation of the instantaneous frequency and amplitude despite suffering from a number of numerical issues, are compared to assess their potential for non-linear system identification. Then, a possible relation between the Hilbert-Huang transform and complex non-linear modes of mechanical systems is investigated. Based on this relation, an approach to experimental non-linear modal analysis is proposed. Since this approach integrates the Hilbert-Huang transform and non-linear modes, it allows not only to detect and characterise structural non-linearities in a non-parametric manner, but also to quantify the parameters of a selected model using extracted non-linear modes. Lastly, a new method for the identification of systems with asymmetric non-linear restoring forces is proposed. The application of all proposed methods is demonstrated on simulated and experimental data.Open Acces

    Towards Real-Time Non-Stationary Sinusoidal Modelling of Kick and Bass Sounds for Audio Analysis and Modification

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    Sinusoidal Modelling is a powerful and flexible parametric method for analysing and processing audio signals. These signals have an underlying structure that modern spectral models aim to exploit by separating the signal into sinusoidal, transient, and noise components. Each of these can then be modelled in a manner most appropriate to that component's inherent structure. The accuracy of the estimated parameters is directly related to the quality of the model's representation of the signal, and the assumptions made about its underlying structure. For sinusoidal models, these assumptions generally affect the non-stationary estimates related to amplitude and frequency modulations, and the type of amplitude change curve. This is especially true when using a single analysis frame in a non-overlapping framework, where biased estimates can result in discontinuities at frame boundaries. It is therefore desirable for such a model to distinguish between the shape of different amplitude changes and adapt the estimation of this accordingly. Intra-frame amplitude change can be interpreted as a change in the windowing function applied to a stationary sinusoid, which can be estimated from the derivative of the phase with respect to frequency at magnitude peaks in the DFT spectrum. A method for measuring monotonic linear amplitude change from single-frame estimates using the first-order derivative of the phase with respect to frequency (approximated by the first-order difference) is presented, along with a method of distinguishing between linear and exponential amplitude change. An adaption of the popular matching pursuit algorithm for refining model parameters in a segmented framework has been investigated using a dictionary comprised of sinusoids with parameters varying slightly from model estimates, based on Modelled Pursuit (MoP). Modelling of the residual signal using a segmented undecimated Wavelet Transform (segUWT) is presented. A generalisation for both the forward and inverse transforms, for delay compensations and overlap extensions for different lengths of Wavelets and the number of decomposition levels in an Overlap Save (OLS) implementation for dealing with convolution block-based artefacts is presented. This shift invariant implementation of the DWT is a popular tool for de-noising and shows promising results for the separation of transients from noise

    Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient

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    Backtracking linesearch is the de facto approach for minimizing continuously differentiable functions with locally Lipschitz gradient. In recent years, it has been shown that in the convex setting it is possible to avoid linesearch altogether, and to allow the stepsize to adapt based on a local smoothness estimate without any backtracks or evaluations of the function value. In this work we propose an adaptive proximal gradient method, adaPG, that uses novel estimates of the local smoothness modulus which leads to less conservative stepsize updates and that can additionally cope with nonsmooth terms. This idea is extended to the primal-dual setting where an adaptive three term primal-dual algorithm, adaPD, is proposed which can be viewed as an extension of the PDHG method. Moreover, in this setting the ``essentially'' fully adaptive variant adaPD+^+ is proposed that avoids evaluating the linear operator norm by invoking a backtracking procedure, that, remarkably, does not require extra gradient evaluations. Numerical simulations demonstrate the effectiveness of the proposed algorithms compared to the state of the art

    Wavelet analysis of non-stationary signals with applications

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    The empirical mode decomposition (EMD) algorithm, introduced by N.E. Huang et al in 1998, is arguably the most popular mathematical scheme for non-stationary signal decomposition and analysis. The objective of EMD is to separate a given signal into a number of components, called intrinsic mode functions (IMF\u27s), after which the instantaneous frequency (IF) and amplitude of each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet transform (SST), introduced by I. Daubechies and S. Maes in 1996 and further developed by I. Daubechies, J. Lu and H.-T. Wu in 2011, is first applied to estimate the IF\u27s of all signal components of the given signal, based on one single frequency reassignment rule, under the assumption that the signal components satisfy certain strict properties of the so-called adaptive harmonic model, before the signal components of the model are recovered, based on the estimated IF\u27s. The objective of this dissertation is to develop a hybrid EMD-SST computational scheme by applying a modified SST to each IMF produced by a modified EMD, as an alternative approach to the original EMD-HSA method. While our modified SST assures non-negative instantaneous frequencies of the IMF\u27s, application of the EMD scheme eliminates the dependence on a single frequency reassignment rule as well as the guessing work of the number of signal components in the original SST approach. Our modification of the SST consists of applying analytic vanishing moment wavelets (introduced in a recent paper by C.K. Chui, Y.-T. Lin and H.-T. Wu) with stacked knots to process signals on bounded or half-infinite time intervals, and spline curve fitting with optimal smoothing parameter selection through generalized cross-validation. In addition, we modify EMD by formulating a local spline interpolation scheme for bounded intervals, for real-time realization of the EMD sifting process. This scheme improves over the standard global cubic spline interpolation, both in quality and computational cost, particularly when applied to bounded and half-infinite time intervals

    B-splines in EMD and Graph Theory in Pattern Recognition

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    With the development of science and technology, a large amount of data is waiting for further scientific exploration. We can always build up some good mathematical models based on the given data to analyze and solve the real life problems. In this work, we propose three types of mathematical models for different applications.;In chapter 1, we use Bspline based EMD to analysis nonlinear and no-stationary signal data. A new idea about the boundary extension is introduced and applied to the Empirical Mode Decomposition(EMD) algorithm. Instead of the traditional mirror extension on the boundary, we propose a ratio extension on the boundary.;In chapter 2 we propose a weighted directed multigraph for text pattern recognition. We set up a weighted directed multigraph model using the distances between the keywords as the weights of arcs. We then developed a keyword-frequency-distance-based algorithm which not only utilizes the frequency information of keywords but also their ordering information.;In chapter 3, we propose a centrality guided clustering method. Different from traditional methods which choose a center of a cluster randomly, we start clustering from a LEADER - a vertex with highest centrality score, and a new member is added into an existing community if the new vertex meet some criteria and the new community with the new vertex maintain a certain density.;In chapter 4, we define a new graph optimization problem which is called postman tour with minimum route-pair cost. And we model the DNA sequence assembly problem as the postman tour with minimum route-pair cost problem

    Empirical Bayesian Smoothing Splines for Signals with Correlated Errors: Methods and Applications

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    Smoothing splines is a well stablished method in non-parametric statistics,  although the selection of the smoothness degree of the regression function is rarely addressed and, instead, a two times differentiable function, i.e.  cubic smoothing spline, is assumed. For a general regression function there is no known method that can identify the smoothness degree under the presence of correlated errors. This apparent disregard in the literature can be justified because the condition number of the solution increases with the smoothness degree of the function, turning the estimation unstable. In this thesis we introduce  an exact expression for the Demmler-Reinsch basis constructed as the solution of an ordinary differential equation, so that the estimation can be carried out for an arbitrary smoothness degree, and under the presence of correlated errors, without affecting the condition number of the solution. We provide asymptotic properties of the proposed estimators and conduct simulation experiments to study their finite sample properties. We expect this new approach to have a direct impact on related methods that use smoothing splines as a building block. In this direction, we present extensions of the method to signal extraction and functional principal component analysis. The empirical relevance to our findings in these areas of statistics is shown in applications for agricultural economics and biophysics. R packages of the implementation of the developed methods are also provided.
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