Browsing by Author "Luke, D. Russell"
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Item Dynamics of generalizations of the AGM continued fraction of Ramanujan. Part I: divergence.(Department of Mathematical Sciences, 2004-11-19) Borwein, J. M.; Luke, D. RussellWe study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters [4, 3, 2]. A study of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in [2], we determine the convergence properties of deterministic difference equations and so divergence of their corresponding continued fractions.Item Dynamics of Random Continued Fractions(Department of Mathematical Sciences, 2004-11-16) Borwein, J. M.; Luke, D. RussellWe study a generalization of a continued fraction of Ramanujan with random coefficients. A study of the continued fraction is equivalent to an analysis of the convergence of certain stochastic difference equations and the stability of random dynamical systems. We determine the convergence properties of stochastic difference equations and so divergence of their corresponding continued fractions.Item Generalized Filtered Backprojection in Nonlinear Acoustics(Department of Mathematical Sciences, 2004) Luke, D. RussellThe filtered backprojection algorithm (FBP) is a standard image reconstruction algorithm used in many imaging modalities. The FBP algorithm is suited for high-frequency imaging applications, that is, applications where the specimen is much larger than the wavelengths of the incident fields. However, when data is scarce due to limited view or dosage considerations, or when super-resolution is required, different techniques are needed to adequately resolve the specimen. We present a generalization of the filtered backprojection algorithm that extends the capabilities of acoustic imaging systems to limited aperture and wavelength resolution. Our principal interest is applications to acoustic (scalar) scattering, though the methodology we develop can be extended to general electromagnetic (vector) settings.Item A New Generation of Iterative Transform Algorithms for Phase Contrast Tomography(Department of Mathematical Sciences, 2004) Bauschke, H. H.; Combettes, P. L.; Luke, D. RussellIn recent years, improvements in electromagnetic sources, detectors, optical components, and computational imaging have made it possible to achieve three-dimensional atomicscale resolution using tomographic phase-contrast imaging techniques. These greater capabilities have placed a premium on improving the efficiency and stability of phase retrieval algorithms for recovering the missing phase information in diffraction observations. In some cases, . so called direct methods suffice, but for large macromolecules and nonperiodic structures one must rely on numerical techniques for reconstructing the missing phase. This is the principal motivation of our work. We report on recent progress in algorithms for iterative phase retrieval. The theory of convex optimisation is used to develop and to gain insight into counterparts for the .nonconvex problem of phase retrieval. We propose a relaxation of averaged alternating reflectors and determine the fundamental mathematical properties of the related operator in the convex case. Numerical studies support our theoretical observations and demonstrate the effectiveness of the newer generation of algorithms compared to the current state of the art.