Applied Mathematics Commons^™

Hybrid Power Spectral And Wavelet Image Roughness Analysis, Basel White

Electronic Theses and Dissertations

The Two-Dimensional Wavelet Transform Modulus Maxima (2D WTMM) sliding window methodology has proven to be a robust approach, in particular for the extraction of the Hurst (H) roughness exponent from grayscale mammograms. The power spectrum is a computational analysis based on the Fourier transform that can be used to estimate the roughness of a scale-invariant image or region via the calculation of H. We aim to examine how the calculation of H in fractional Brownian motion (fBm) images and mammograms can be improved. fBm images are generated for H ∈ [0.00,1.00] for testing through the previous 2D …

Wavelet Anova Bisection Method For Identifying Simulation Model Bias, Andrew D. Atkinson, Raymond R. Hill, Joseph J. Pignatiello Jr., G. Geoffrey Vining, Edward D. White, Eric Chicken

Faculty Publications

High-resolution computer models can simulate complex systems and processes in order to evaluate a solution quickly and inexpensively. Many simulation models produce dynamic functional output, such as a set of time-series data generated during a process. These computer models require verification and validation (V&V) to assess the correctness of these simulations. In particular, the model validation effort evaluates if the model is an appropriate representation of the real-world system that it is meant to simulate. However, when assessing a model capable of generating functional output, it is useful to learn more than simply whether the model is valid or invalid. …

Construction Of Energy Preserving Qmf, Jian-Ao Lian, Yonghui Wang

Applications and Applied Mathematics: An International Journal (AAM)

Recently, a family of perfect reconstruction (PR) quadrature mirror filterbanks (QMF) with finite impulse response filters (FIR) from systems of biorthogonal refinable functions and wavelets were introduced and also applied to image processing. However, a detailed procedure was absent. The main objective of this paper is to present extensive examples that will provide a thorough process of construction of the new family of PR QMF with FIR filterbanks. These new filters are linearphase due to the symmetry property of their corresponding biorthogonal refinable functions and wavelets. In addition, these filters have odd lengths so that the symmetric extension can be …

Reduced-Order Modeling Using Orthogonal And Bi-Orthogonal Wavelet Transforms, Miguel Hernandez Iv

Open Access Theses & Dissertations

It is well known that model reduction methods borrow techniques typically found in data compression, and current state-of-the-art techniques for data compression are based on the wavelet transform. Given these facts, it is surprising that model reduction using wavelets has not received much attention and has not been adequately addressed in the literature. This research seeks to determine if wavelets can be used for model reduction and if wavelet model reduction is a viable alternative to existing model reduction methods.

In this work we propose a novel method for model reduction using wavelets. Specifically, we introduce techniques for deriving wavelet …

Wavelet-Based Analysis Of Neutron-Induced Photon Spectral Data, Bruce Kessler, Alexander Barzilov, Phillip Womble

Mathematics Faculty Publications

Neutron-based methods of non-destructive inter- rogation of objects for the purpose of their characterization are well-established techniques, employed in the field of bulk material analysis, contraband detection, unexploded ordnance, etc. The characteristic gamma rays produced in nuclear reactions initiated by neutrons in the volume of the irradiated object (inelastic neutron scattering, thermal neutron capture, and activation) are used for the elemental identification. In many real-world applications, an automated spectral analysis is needed, and many algorithms are used for that purpose. The Applied Physics Institute at Western Kentucky University has recently started to employ a mathematical spectrum analysis technique based on …

Sparse Representation For Detection Of Transients Using A Multi-Resolution Representation Of The Auto-Correlation Of Wavelets, Caroline Sieger

All Theses

This thesis seeks to detect damped sinusoidal transients, specifically capacitor switching transients, buried in noise and to answer the following questions: 1.) Can the transient s(t;q) be sparsely represented from s&delta(t) = s(t;q) + &epsilon(t) using sparsity methods, where &epsilon(t) is white Gaussian noise? 2.) Does computing the local auto-correlation of the signal around the transient improve detection? 3.) How does the auto-correlation shell representation compare to the wavelet representation? 4.) Which basis is ''best''? 5.) Which method and representation is best? This thesis explores detection schemes based on classical methods and newer sparsity methods. Classical methods considered include reconstruction …

Sparse Representations In Power Systems Signals, Jack Cooper

All Theses

This thesis seeks to detect transient disturbances in power system signals in a sparse framework. To this end, an overcomplete wavelet packet dictionary and damped sinusoid dictionary are considered, and for each dictionary Matching Pursuit is compared with Basis Pursuit. Previous work in developing waveform dictionary theory and sparse representation is reviewed, and simulations are run on a test signal in both noisy and noiseless environments. The solutions are viewed as time-frequency plane tilings to compare the accuracy and sparsity of these algorithms in properly resolving optimal representations of the disturbances. The advantages and disadvantages of each combination of dictionary …

Wavelet Decompositions For Quantitative Pattern Matching, Bruce Kessler

Mathematics Faculty Publications

The purpose of this paper is to provide an introduction to the concepts of wavelets and multiwavelets, and explain how these tools can be used by the analyst community to find patterns in quantitative data. Three multiwavelet bases are introduced, the GHM basis from \cite{GHM}, a piecewise polynomial basis with approximation order 4 from \cite{DGH}, and a smoother approximation-order-4 basis developed by the author in previous work \cite{K}. The technique of using multiwavelets to find patterns is illustrated in a traffic-analysis example. Acknowledgements: This work supported in part by the NACMAST consortium under contract EWAGSI-07-SC-0003.

Multiwavelets For Quantitative Pattern Matching, Bruce Kessler

Bruce Kessler

The purpose of this paper is to provide an introduction to the concepts of wavelets and multiwavelets, and explain how these tools can be used by the analyst community to find patterns in quantitative data. Three multiwavelet bases are introduced, the GHM basis from \cite{GHM}, a piecewise polynomial basis with approximation order 4 from \cite{DGH}, and a smoother approximation-order-4 basis developed by the author in previous work \cite{K}. The technique of using multiwavelets to find patterns is illustrated in a traffic-analysis example. Acknowledgements: This work supported in part by the NACMAST consortium under contract EWAGSI-07-SC-0003.

Local Behavior Of Distributions And Applications, Jasson Vindas

LSU Doctoral Dissertations

This dissertation studies local and asymptotic properties of distributions (generalized functions) in connection to several problems in harmonic analysis, approximation theory, classical real and complex function theory, tauberian theory, summability of divergent series and integrals, and number theory. In Chapter 2 we give two new proofs of the Prime Number Theory based on ideas from asymptotic analysis on spaces of distributions. Several inverse problems in Fourier analysis and summability theory are studied in detail. Chapter 3 provides a complete characterization of point values of tempered distributions and functions in terms of a generalized pointwise Fourier inversion formula. The relation of …

Function Spaces, Wavelets And Representation Theory, Jens Gerlach Christensen

LSU Doctoral Dissertations

This dissertation is concerned with the interplay between the theory of Banach spaces and representations of groups. The wavelet transform has proven to be a useful tool in characterizing and constructing Banach spaces, and we investigate a generalization of an already known technique due to H.G. Feichtinger and K. Gröchenig. This generalization is presented in Chapter 3, and in Chapters 4 and 5 we present examples of spaces which can be described using the theory. The first example clears up a question regarding a wavelet characterization of Bergman spaces related to a non-integrable representation. The second example is a wavelet …

A Wavelet-Based Method For Overcoming The Gibbs Phenomenon, Nataniel Greene

Publications and Research

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities. Here we describe a numerical procedure for overcoming the Gibbs phenomenon called the inverse wavelet reconstruction method. The method takes the Fourier coefficients of an oscillatory partial sum and uses them to construct the wavelet coefficients of a non-oscillatory wavelet series.

Statistical Issues In Proteomic Research, Jeffrey S. Morris

Jeffrey S. Morris

No abstract provided.

A New Family Of Pr Two Channel Filter Banks, Jian-Ao Lian

Applications and Applied Mathematics: An International Journal (AAM)

A new family of multidimensional dimensional (MD) perfect reconstruction (PR) two channel filter banks with finite impulse response (FIR) filters induced from systems of biorthogonal MD scaling functions and wavelets are introduced. One of the advantages of this construction is that the biorthogonal scaling functions and wavelets are easy to establish due to the interpolatory property of the scaling functions to start with. The other advantage is that all filters can be centrosymmetric or bi-linear phase. Examples of two dimensional (2D) bi-linear phase PR twochannel FIR filter banks will be demonstrated.

Wavelet-Based Functional Mixed Model Analysis: Computational Considerations, Richard C. Herrick, Jeffrey S. Morris

Jeffrey S. Morris

Wavelet-based Functional Mixed Models is a new Bayesian method extending mixed models to irregular functional data (Morris and Carroll, JRSS-B, 2006). These data sets are typically very large and can quickly run into memory and time constraints unless these issues are carefully dealt with in the software. We reduce runtime by 1.) identifying and optimizing hotspots, 2.) using wavelet compression to do less computation with minimal impact on results, and 3.) dividing the code into multiple executables to be run in parallel using a grid computing resource. We discuss rules of thumb for estimating memory requirements and computation times in …

Nonlinear Equations And Wavelets, Andrei Ludu

Andrei Ludu

No abstract provided.

The Rational Resolution Analysis: A Generalization Of Multi-Resolution Analyses With Application To To The Specific Emitter Identification Problem, Bruce P. Anderson

Theses and Dissertations

The rational resolution analysis RRA is introduced and developed as a generalization of the integer, dilation multiresolution analyses MRA developed by Mallat and Meyer. Rational dilation factors are achieved by relaxing the condition on MRAs that successive approximation spaces be embedded. Conditions for perfect reconstruction are discussed and it is shown that perfect reconstruction is possible with specific constraints on the scaling function the scaling filter must have its roots on the unit circle. Furthermore, the required arrangement of the roots indicate the scaling function must be derived from a B-spline of some degree. It is proven the only compactly …