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Axioms
Special Issues
Wavelets and Applications
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Wavelets and Applications

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (15 April 2013) | Viewed by 65038

Special Issue Editor

Prof. Dr. Palle E.T. Jorgensen

E-Mail Website
Guest Editor
Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA 52242-1419, USA
Interests: mathematical physics; Euclidean field theory; reflection positivity; representation theory; operators in Hilbert space; harmonic analysis; fractals; wavelets; stochastic processes; financial mathematics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Recently, there has been an enormous amount of interest in the theory and applications of wavelets. The applications include signal processing, data compression, turning fingerprints into digital data files, subdivision algorithms for graphics, and the JPEG 2000 encoding of images. As a mathematical subject, wavelet theory involves tools from a host of neighboring fields, functional and harmonic analysis, numerical analysis, mathematics of computation, and operator theory.

Wavelets now serve as an alternative to classical Fourier methods, Fourier series and integrals. The reasons for this is that they are better localized, they are better adapted to discontinuities; they have a certain
form of self-similarity, which makes the theory suited for the analysis of fractals and non-linear dynamical systems. The self-similarity properties of the scaling functions connect them to fractals and non-linear dynamical systems.

The multiresolutions offer fast algorithms. The feature of localization for wavelets is shared by related recursive basis constructions from multi-resolutions in Hilbert spaces, for example for fractals and iterated function systems in dynamics. The multiresolutions and locality yield much better pointwise approximations than is possible for traditional Fourier bases.

In signal or image-processing one is interested in subdividing analogue-signals into frequency bands. This idea goes back to Norbert Wiener, but it is of relevance in modern-day wireless signal and image processing. This suggests a representation theoretic framework. The idea is thus to build a representation theory which creates Hilbert spaces H and specific families of closed subspaces in H in such a way that "non-overlapping frequency bands" in our model correspond to orthogonal subspaces in H; or equivalently to systems of orthogonal projections. Since the different frequency bands must exhaust the signals for the entire system, one looks for orthogonal projections which add to the identity operator in H. Since time/frequency analysis is non-commutative, one is further faced with a selection of special families of commuting orthogonal projections. When an iteration scheme is applied to the initial generators, one generates new bases and frames by repeated subdivision sequences; wavelet families as recursive scheme.

Professor Palle Jorgensen
Guest Editor

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Keywords

  • wavelets
  • computations
  • applied harmonic analysis
  • signal and image processing
  • fourier methods
  • multi-resolutions
  • operators in Hilbert space
  • approximation
  • wavelet filters
  • filter bank

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Published Papers (12 papers)

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952 KiB  
Article
Nonnegative Scaling Vectors on the Interval
by David K. Ruch and Patrick J. Van Fleet
Axioms 2013, 2(3), 371-389; https://doi.org/10.3390/axioms2030371 - 9 Jul 2013
Viewed by 4706
Abstract
In this paper, we outline a method for constructing nonnegative scaling vectors on the interval. Scaling vectors for the interval have been constructed in [1–3]. The approach here is different in that the we start with an existing scaling vector ϕ that generates [...] Read more.
In this paper, we outline a method for constructing nonnegative scaling vectors on the interval. Scaling vectors for the interval have been constructed in [1–3]. The approach here is different in that the we start with an existing scaling vector ϕ that generates a multi-resolution analysis for L2(R) to create a scaling vector for the interval. If desired, the scaling vector can be constructed so that its components are nonnegative. Our construction uses ideas from [4,5] and we give results for scaling vectors satisfying certain support and continuity properties. These results also show that less edge functions are required to build multi-resolution analyses for L2 ([a; b]) than the methods described in [5,6]. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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1348 KiB  
Article
Wavelet-Based Monitoring for Biosurveillance
by Galit Shmueli
Axioms 2013, 2(3), 345-370; https://doi.org/10.3390/axioms2030345 - 9 Jul 2013
Cited by 5 | Viewed by 5362
Abstract
Biosurveillance, focused on the early detection of disease outbreaks, relies on classical statistical control charts for detecting disease outbreaks. However, such methods are not always suitable in this context. Assumptions of normality, independence and stationarity are typically violated in syndromic data. Furthermore, outbreak [...] Read more.
Biosurveillance, focused on the early detection of disease outbreaks, relies on classical statistical control charts for detecting disease outbreaks. However, such methods are not always suitable in this context. Assumptions of normality, independence and stationarity are typically violated in syndromic data. Furthermore, outbreak signatures are typically of unknown patterns and, therefore, call for general detectors. We propose wavelet-based methods, which make less assumptions and are suitable for detecting abnormalities of unknown form. Wavelets have been widely used for data denoising and compression, but little work has been published on using them for monitoring. We discuss monitoring-based issues and illustrate them using data on military clinic visits in the USA. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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563 KiB  
Article
Complexity L0-Penalized M-Estimation: Consistency in More Dimensions
by Laurent Demaret, Felix Friedrich, Volkmar Liebscher and Gerhard Winkler
Axioms 2013, 2(3), 311-344; https://doi.org/10.3390/axioms2030311 - 9 Jul 2013
Cited by 1 | Viewed by 5799
Abstract
We study the asymptotics in L2 for complexity penalized least squares regression for the discrete approximation of finite-dimensional signals on continuous domains—e.g., images—by piecewise smooth functions. We introduce a fairly general setting, which comprises most of the presently popular partitions of signal or [...] Read more.
We study the asymptotics in L2 for complexity penalized least squares regression for the discrete approximation of finite-dimensional signals on continuous domains—e.g., images—by piecewise smooth functions. We introduce a fairly general setting, which comprises most of the presently popular partitions of signal or image domains, like interval, wedgelet or related partitions, as well as Delaunay triangulations. Then, we prove consistency and derive convergence rates. Finally, we illustrate by way of relevant examples that the abstract results are useful for many applications. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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882 KiB  
Article
Some Notes on the Use of the Windowed Fourier Transform for Spectral Analysis of Discretely Sampled Data
by Robert W. Johnson
Axioms 2013, 2(3), 286-310; https://doi.org/10.3390/axioms2030286 - 24 Jun 2013
Cited by 2 | Viewed by 6012
Abstract
The properties of the Gabor and Morlet transforms are examined with respect to the Fourier analysis of discretely sampled data. Forward and inverse transform pairs based on a fixed window with uniform sampling of the frequency axis can satisfy numerically the energy and [...] Read more.
The properties of the Gabor and Morlet transforms are examined with respect to the Fourier analysis of discretely sampled data. Forward and inverse transform pairs based on a fixed window with uniform sampling of the frequency axis can satisfy numerically the energy and reconstruction theorems; however, transform pairs based on a variable window or nonuniform frequency sampling in general do not. Instead of selecting the shape of the window as some function of the central frequency, we propose constructing a single window with unit energy from an arbitrary set of windows that is applied over the entire frequency axis. By virtue of using a fixed window with uniform frequency sampling, such a transform satisfies the energy and reconstruction theorems. The shape of the window can be tailored to meet the requirements of the investigator in terms of time/frequency resolution. The algorithm extends naturally to the case of nonuniform signal sampling without modification beyond identification of the Nyquist interval. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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607 KiB  
Article
Change Detection Using Wavelets in Solution Monitoring Data for Nuclear Safeguards
by Claire Longo, Tom Burr and Kary Myers
Axioms 2013, 2(2), 271-285; https://doi.org/10.3390/axioms2020271 - 18 Jun 2013
Cited by 5 | Viewed by 6421
Abstract
Wavelet analysis is known to be a good option for change detection in many contexts. Detecting changes in solution volumes that are measured with both additive and relative error is an important aspect of safeguards for facilities that process special nuclear material. This [...] Read more.
Wavelet analysis is known to be a good option for change detection in many contexts. Detecting changes in solution volumes that are measured with both additive and relative error is an important aspect of safeguards for facilities that process special nuclear material. This paper qualitatively compares wavelet-based change detection to a lag-one differencing option using realistic simulated solution volume data for which the true change points are known. We then show quantitatively that Haar wavelet-based change detection is effective for finding the approximate location of each change point, and that a simple piecewise linear optimization step is effective to refine the initial wavelet-based change point estimate. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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431 KiB  
Article
Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces
by Sergey Ajiev
Axioms 2013, 2(2), 224-270; https://doi.org/10.3390/axioms2020224 - 23 May 2013
Cited by 5 | Viewed by 4975
Abstract
We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions [...] Read more.
We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hölder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration. Full article
(This article belongs to the Special Issue Wavelets and Applications)
1336 KiB  
Article
Time Scale Analysis of Interest Rate Spreads and Output Using Wavelets
by Marco Gallegati, James B. Ramsey and Willi Semmler
Axioms 2013, 2(2), 182-207; https://doi.org/10.3390/axioms2020182 - 23 Apr 2013
Cited by 8 | Viewed by 5255
Abstract
This paper adds to the literature on the information content of different spreads for real activity by explicitly taking into account the time scale relationship between a variety of monetary and financial indicators (real interest rate, term and credit spreads) and output growth. [...] Read more.
This paper adds to the literature on the information content of different spreads for real activity by explicitly taking into account the time scale relationship between a variety of monetary and financial indicators (real interest rate, term and credit spreads) and output growth. By means of wavelet-based exploratory data analysis we obtain richer results relative to the aggregate analysis by identifying the dominant scales of variation in the data and the scales and location at which structural breaks have occurred. Moreover, using the “double residuals” regression analysis on a scale-by-scale basis, we find that changes in the spread in several markets have different information content for output at different time frames. This is consistent with the idea that allowing for different time scales of variation in the data can provide a fruitful understanding of the complex dynamics of economic relationships between variables with non-stationary or transient components, certainly richer than those obtained using standard time domain methods. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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907 KiB  
Article
A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations
by Donald A. McLaren, Lucy J. Campbell and Rémi Vaillancourt
Axioms 2013, 2(2), 142-181; https://doi.org/10.3390/axioms2020142 - 23 Apr 2013
Cited by 1 | Viewed by 5377
Abstract
This paper describes and tests a wavelet-based implicit numerical method for solving partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time-discretization into multiple smaller [...] Read more.
This paper describes and tests a wavelet-based implicit numerical method for solving partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time-discretization into multiple smaller systems that can be solved sequentially. Included is a test on a basic non-linear problem, with both the results of the test, and the time required to calculate them, compared with control results based on a single system with fine resolution. The method is then tested on a non-trivial problem, its computational time and accuracy checked against control results. In both tests, it was found that the method requires less computational expense than the control. Furthermore, the method showed convergence towards the fine resolution control results. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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260 KiB  
Article
Construction of Multiwavelets on an Interval
by Ahmet Altürk and Fritz Keinert
Axioms 2013, 2(2), 122-141; https://doi.org/10.3390/axioms2020122 - 17 Apr 2013
Cited by 3 | Viewed by 5140
Abstract
Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article [...] Read more.
Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article we show how an algorithm of Madych for scalar wavelets can be generalized to multiwavelets, given an extra assumption. We then develop a new algorithm that does not require this additional condition. Finally, we apply results from a previous paper to resolve the non-uniqueness of the algorithm by imposing regularity conditions (including approximation orders) on the boundary functions. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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528 KiB  
Article
Divergence-Free Multiwavelets on the Half Plane
by Joseph Lakey and Phan Nguyen
Axioms 2013, 2(2), 100-121; https://doi.org/10.3390/axioms2020100 - 11 Apr 2013
Cited by 2 | Viewed by 4623
Abstract
We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal [...] Read more.
We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R2 +. Such wavelets are suitable to study the Navier–Stokes equations on a half plane when imposing a Navier boundary condition. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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626 KiB  
Article
Mollification Based on Wavelets
by Tohru Morita and Ken-ichi Sato
Axioms 2013, 2(2), 67-84; https://doi.org/10.3390/axioms2020067 - 25 Mar 2013
Cited by 1 | Viewed by 5306
Abstract
The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case [...] Read more.
The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case when the fractional derivative of a function is calculated. Noting that the estimations are applicable even when the orthogonality of the wavelets is not satisfied, we study mollifications using unorthogonalized wavelets, as well as those using orthogonal wavelets. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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499 KiB  
Article
Signal Estimation Using Wavelet Analysis of Solution Monitoring Data for Nuclear Safeguards
by Tom Burr and Claire Longo
Axioms 2013, 2(1), 44-57; https://doi.org/10.3390/axioms2010044 - 20 Mar 2013
Cited by 1 | Viewed by 4734
Abstract
Wavelets are explored as a data smoothing (or de-noising) option for solution monitoring data in nuclear safeguards. In wavelet-smoothed data, the Gibbs phenomenon can obscure important data features that may be of interest. This paper compares wavelet smoothing to piecewise linear smoothing and [...] Read more.
Wavelets are explored as a data smoothing (or de-noising) option for solution monitoring data in nuclear safeguards. In wavelet-smoothed data, the Gibbs phenomenon can obscure important data features that may be of interest. This paper compares wavelet smoothing to piecewise linear smoothing and local kernel smoothing, and illustrates that the Haar wavelet basis is effective for reducing the Gibbs phenomenon. Full article
(This article belongs to the Special Issue Wavelets and Applications)
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