Theory and application of Fourier crosstalk: An evaluator for digital-system design

Persistent Link:
http://hdl.handle.net/10150/282554
Title:
Theory and application of Fourier crosstalk: An evaluator for digital-system design
Author:
Gifford, Howard Carl, 1961-
Issue Date:
1997
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
A large class of digital imaging systems can be modeled mathematically by linear, continuous-to-discrete operators. In this dissertation, we present the theory of Fourier crosstalk as a means of analyzing such systems and demonstrate its use through several imaging applications. Crosstalk employs a Fourier-series representation for compactly supported objects and defines the aliasing associated with a sampling geometry in terms of the action of the system operator on the Fourier basis. This information is quantified within the infinite-dimensional Fourier crosstalk matrix, elements of which record how pairs of basis functions alias in data space. Several parallels exist between crosstalk and other system evaluation tools: the analysis of a system in terms of a specially defined basis is similar to Singular-value Decomposition; for linear operators with shift-invariant kernels, any finite-dimensional crosstalk submatrix becomes diagonal in the limit of infinite sampling, and the diagonal elements are proportional to the squared moduli of the Fourier coefficients of the modulation transfer function; for the task of estimating Fourier coefficients from noisy data, the crosstalk matrix is proportional to the Fisher information matrix for certain Gaussian and Poisson noise models. This last relation is an example of how crosstalk applies to objective task-based assessment of image quality. In an investigation of 1D sampling, we examine the aliasing characteristics of homogeneous and stochastic sampling. The merits for these sampling schemes depend on the task required of the data. Estimation problems benefit from the relatively low-magnitude aliasing created by homogeneous sampling. A detection problem involving a low-pass signal in high-frequency background and noise suggests that stochastic sampling can sometimes perform better. This result emphasizes the shortcomings of image quality measures that are not task-related. In an application of crosstalk to the x-ray projective transform, we demonstrate that there is a consistency between efficient sampling geometries as defined by crosstalk theory and a recognized Nyquist sampling definition. Applied to the cone-beam transform, crosstalk indicates that symmetries in the placement of orbit points are detrimental since they preclude adequate sampling of all elements in a bandlimited set of Fourier basis functions. An extension of the sampling problem that considers features of a pinhole aperture cone-beam system includes studies of the effects of pinhole size and detector spacing. These show that pinhole radius has a greater impact on resolution of the Fourier basis than does detector spacing. In an accompanying evaluation of numerical methods for calculating crosstalk, Monte Carlo techniques are shown to be an essential tool for developing other, more efficient, methods.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.; Engineering, Biomedical.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Applied Mathematics
Degree Grantor:
University of Arizona
Advisor:
Barrett, Harrison H.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleTheory and application of Fourier crosstalk: An evaluator for digital-system designen_US
dc.creatorGifford, Howard Carl, 1961-en_US
dc.contributor.authorGifford, Howard Carl, 1961-en_US
dc.date.issued1997en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractA large class of digital imaging systems can be modeled mathematically by linear, continuous-to-discrete operators. In this dissertation, we present the theory of Fourier crosstalk as a means of analyzing such systems and demonstrate its use through several imaging applications. Crosstalk employs a Fourier-series representation for compactly supported objects and defines the aliasing associated with a sampling geometry in terms of the action of the system operator on the Fourier basis. This information is quantified within the infinite-dimensional Fourier crosstalk matrix, elements of which record how pairs of basis functions alias in data space. Several parallels exist between crosstalk and other system evaluation tools: the analysis of a system in terms of a specially defined basis is similar to Singular-value Decomposition; for linear operators with shift-invariant kernels, any finite-dimensional crosstalk submatrix becomes diagonal in the limit of infinite sampling, and the diagonal elements are proportional to the squared moduli of the Fourier coefficients of the modulation transfer function; for the task of estimating Fourier coefficients from noisy data, the crosstalk matrix is proportional to the Fisher information matrix for certain Gaussian and Poisson noise models. This last relation is an example of how crosstalk applies to objective task-based assessment of image quality. In an investigation of 1D sampling, we examine the aliasing characteristics of homogeneous and stochastic sampling. The merits for these sampling schemes depend on the task required of the data. Estimation problems benefit from the relatively low-magnitude aliasing created by homogeneous sampling. A detection problem involving a low-pass signal in high-frequency background and noise suggests that stochastic sampling can sometimes perform better. This result emphasizes the shortcomings of image quality measures that are not task-related. In an application of crosstalk to the x-ray projective transform, we demonstrate that there is a consistency between efficient sampling geometries as defined by crosstalk theory and a recognized Nyquist sampling definition. Applied to the cone-beam transform, crosstalk indicates that symmetries in the placement of orbit points are detrimental since they preclude adequate sampling of all elements in a bandlimited set of Fourier basis functions. An extension of the sampling problem that considers features of a pinhole aperture cone-beam system includes studies of the effects of pinhole size and detector spacing. These show that pinhole radius has a greater impact on resolution of the Fourier basis than does detector spacing. In an accompanying evaluation of numerical methods for calculating crosstalk, Monte Carlo techniques are shown to be an essential tool for developing other, more efficient, methods.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
dc.subjectEngineering, Biomedical.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorBarrett, Harrison H.en_US
dc.identifier.proquest9814458en_US
dc.identifier.bibrecord.b37745050en_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.