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

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.