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45
The Chirplet Transform: Physical Considerations
, 1995
"... We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call qchirps for short), giving rise to a parameter space that includes both the timef ..."
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Cited by 29 (3 self)
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We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call qchirps for short), giving rise to a parameter space that includes both the timefrequency plane and the timescale plane as twodimensional subspaces. The parameter space contains a "timefrequencyscale volume ", and thus encompasses both the shorttime Fourier transform (as a slice along the time and frequency axes), and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shearintime (obtained through convolution with a qchirp) and shearin frequency (obtained through multiplication by a qchirp). Signals in this multidimensional space can be obtained by a new transform which we call the "qchirplet transform", or simply the "chiplet transform". ...
Wave propagation using bases for bandlimited functions
 Wave Motion
, 2005
"... We develop a twodimensional solver for the acoustic wave equation with spatially varying coefficients. In what is a new approach, we use a basis of approximate prolate spheroidal wavefunctions and construct derivative operators that incorporate boundary and interface conditions. Writing the wave eq ..."
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Cited by 13 (1 self)
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We develop a twodimensional solver for the acoustic wave equation with spatially varying coefficients. In what is a new approach, we use a basis of approximate prolate spheroidal wavefunctions and construct derivative operators that incorporate boundary and interface conditions. Writing the wave equation as a firstorder system, we evolve the equation in time using the matrix exponential. Computation of the matrix exponential requires efficient representation of operators in two dimensions and for this purpose we use short sums of onedimensional operators. We also use a partitioned lowrank representation in one dimension to further speed up the algorithm. We demonstrate that the method significantly reduces numerical dispersion and computational time when compared with a fourthorder finite difference scheme in space and an explicit fourthorder Runge–Kutta solver in time.
Uncertainty Principles in Hilbert Spaces
, 2002
"... In this paper we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities a ..."
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Cited by 8 (5 self)
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In this paper we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities are sharp and apply our results to concrete examples of importance in the literature.
NonBoolean Descriptions for MindMatter Problems
"... A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmat ..."
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Cited by 6 (0 self)
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A framework for the mindmatter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as elements for a more comprehensive nonBoolean description of a world without an apriorigiven mindmatter distinction. Such a description in terms of a locally Boolean but globally nonBoolean structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept the insight that there are no ultimate building blocks, the existence of holistic correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely nonBoolean description is to find an appropriate partition of the universe of discourse. If we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain. In the sense of a regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged position, the Now. Since this partition refers to two complementary descriptions which are not given apriori,wehavetoexpectcorrelations between these two domains. In physics it corresponds to Newton’s separation of universal laws of nature and contingent initial conditions. Both descriptions have a nonBoolean structure and can be encompassed into a single nonBoolean description. Tensed and tenseless time can be synchronized by holistic correlations. 1.
Sampling of functions and sections for compact groups
 in Modern Signal Processing
, 1999
"... Abstract. In this paper we investigate quadrature rules for functions on compact Lie groups and sections of homogeneous vector bundles associated with these groups. First a general notion of bandlimitedness is introduced which generalizes the usual notion on the torus or translation groups. We deve ..."
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Cited by 5 (1 self)
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Abstract. In this paper we investigate quadrature rules for functions on compact Lie groups and sections of homogeneous vector bundles associated with these groups. First a general notion of bandlimitedness is introduced which generalizes the usual notion on the torus or translation groups. We develop a sampling theorem that allows exact computation of the Fourier expansion of a bandlimited function or section from sample values and quantifies the error in the expansion when the function or section is not bandlimited. We then construct specific finitely supported distributions on the classical groups which have nice error properties and can also be used to develop efficient algorithms for the computation of Fourier transforms
A new friendly method of computing prolate spheroidal wave functions and wavelets
, 2005
"... Prolate spheroidal wave functions, because of their many remarkable properties leading to new applications, have recently experienced an upsurge of interest. They may be defined as eigenfunctions of either a differential operator or an integral operator (as observed by Slepian in the 1960’s). There ..."
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Cited by 5 (0 self)
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Prolate spheroidal wave functions, because of their many remarkable properties leading to new applications, have recently experienced an upsurge of interest. They may be defined as eigenfunctions of either a differential operator or an integral operator (as observed by Slepian in the 1960’s). There are various ways of calculating their values based on both approaches. The standard one uses an approximation based on Legendre polynomials, which, however, is valid only on a finite interval. An alternative, valid in a neighborhood of infinity, uses a Bessel function approximation. In this paper we present a new method based on an eigenvalue problem for a matrix operator equivalent to that of the integral operator. Its solution gives the values of these functions on the entire real line and is computationally more efficient. Key words: sampling theory, prolate spheroidal wave functions. 1
MULTISTATIC IMAGING OF EXTENDED TARGETS ∗
"... Abstract. In this paper we develop iterative approaches for imaging extended inclusions from multistatic response measurements at single or multiple frequencies. Assuming measurement noise, we perform a detailed stability and resolution analysis of the proposed algorithms in two different asymptotic ..."
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Cited by 5 (1 self)
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Abstract. In this paper we develop iterative approaches for imaging extended inclusions from multistatic response measurements at single or multiple frequencies. Assuming measurement noise, we perform a detailed stability and resolution analysis of the proposed algorithms in two different asymptotic regimes. We consider both the Born approximation in the nonmagnetic case and a highfrequency regime in the general case. Based on a highfrequency asymptotic analysis of the measurements, an algorithm for finding a good initial guess for the illuminated part of the inclusion is provided and its optimality is shown. We illustrate our main findings with a variety of numerical examples.
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz ..."
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Cited by 4 (0 self)
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We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
Estimating the Energy Source and Reflectivity By Seismic Inversion
 Inverse Problems, 11:383395
, 1995
"... Data produced by a reproducible source contains redundant information which allows seismic inversion to simultaneously determine the highfrequency fluctuation in the Pwave velocity (or reflectivity) as well as the input energy source. The seismogram model is the planewave convolutional model deri ..."
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Cited by 3 (2 self)
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Data produced by a reproducible source contains redundant information which allows seismic inversion to simultaneously determine the highfrequency fluctuation in the Pwave velocity (or reflectivity) as well as the input energy source. The seismogram model is the planewave convolutional model derived from the constant density, variable sound velocity acoustic wave equation. The first step is to analyze this linearized model when the background velocity is constant. Then perturbations in the seismic data stably determine corresponding perturbations in the source and reflectivity. The stability of this determination improves as the slowness aperture over which the data is defined increases. Further, the normal operator for the convolutional seismogram model is continuous with respect to velocity. Thus the stability result for constant background velocities may be extended to more realistic background velocity models which vary slowly and smoothly with depth. The theory above is illustr...
Tirao, An invitation to matrixvalued spherical functions: Linearization of products in the case of complex projective space
 P2(C), Modern Signal Processing
, 2004
"... Abstract. The classical (scalarvalued) theory of spherical functions, put forward by Cartan and others, unifies under one roof a number of examples that were very wellknown before the theory was formulated. These examples include special functions such as like Jacobi polynomials, Bessel functions, ..."
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Cited by 3 (2 self)
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Abstract. The classical (scalarvalued) theory of spherical functions, put forward by Cartan and others, unifies under one roof a number of examples that were very wellknown before the theory was formulated. These examples include special functions such as like Jacobi polynomials, Bessel functions, Laguerre polynomials, Hermite polynomials, Legendre functions, which had been workhorses in many areas of mathematical physics before the appearance of a unifying theory. These and other functions have found interesting applications in signal processing, including specific areas such as medical imaging. The theory of matrixvalued spherical functions is a natural extension of the wellknown scalarvalued theory. Its historical development, however, is different: in this case the theory has gone ahead of the examples. The purpose of this article is to point to some examples and to interest readers in this new aspect in the world of special functions. We close with a remark connecting the functions described here with the theory of matrixvalued orthogonal polynomials. 1.