DIGITAL SIGNAL PROCESSING TECHNIQUES APPLIED TO OPTICAL PROPAGATION PROBLEMS
Abstract
In digital image processing, two-dimensional signals frequently are not sampled optimally. Hexagonal sampling offers substantial computational efficiency in processing circularly symmetric and/or circularly band-limited functions and also offers a significant reduction of required sampling points compared with rectangular sampling. The reduced space-bandwidth product required for error-free reconstruction makes hexagonal sampling an attractive sampling scheme for use in Fourier optics. Both Fraunhofer and Fresnel computer-generated holograms using hexagonal sampling are capable of exploiting this property. Hexagonal fast Fourier transform (HFFT) and inverse fast Fourier transform (IHFFT) algorithms have been developed along with efficient algorithms for generating the lens transmittance and free-space-transfer matrices. They comprise software packages designed to accept hexagonally sampled input arrays and generate hexagonally sampled output arrays suitable for direct encoding into digital holograms. The prevalence of circularly symmetric functions in Fourier optics led to the investigation of polar sampling and the corresponding Hankel transform defined as the two-dimensional Fourier transform of a circularly symmetric function. A new Hankel transform algorithm based on this definition is described. The new algorithm efficiently generates a rectangulary sampled two-dimensional output array by using the circular symmetry properties of the input array and two-dimensional vector radix fast Fourier transform techniques. The algorithm partitions the input matrix into smaller and smaller processing blocks while removing redundant blocks from data manipulations. For applications that require the output data to be sampled on a two-dimensional rectangular raster such as in computer-generated holography, the convenience and computational speed of the resulting algorithm offer advantages over the one-dimensional Hankel transform algorithms currently available. The aliasing errors prevalent in the computer-generated holograms are visually apparent as random speckle noise. Optical aberrations in an imaging system are generally assumed unimportant when deriving speckle output statistics. We show that the output speckle statistics are affected by deterministic aberrations present in an imaging system when the speckle is only partially developed. The speckle power spectral density, autocorrelation, and intensity contrast depend on these aberrations. The rough target surface is modeled as a perfectly transmitting (reflecting) surface with uniform phase distribution over (-(alpha),(alpha)) where (alpha) (LESSTHEQ) (pi). The effects of the type and severity of the primary aberrations on the normalized power spectral density and intensity contrast are examined.
Degree
Ph.D.
Subject Area
Electrical engineering
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