Conditioning of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples.

Abstract

The problem of reconstructing an image from irregular samples of its 2-D DTFT arises in synthetic aperture radar (SAR), magnetic resonance imaging (MRI), computed tomography (CT), limited angle tomography, and 2-D filter design. The problem of determining a configuration of a limited number of 2-D DTFT samples also arises in magnetic resonance spectroscopic imaging (MRSI) and 3-D MRI. This work first focuses on the selection of the measurement data. Since there is no 2-D Lagrange interpolation formula, sufficient conditions for the uniqueness and conditioning of the reconstruction problem are both not apparent. Kronecker substitutions, such as the Good-Thomas FFT, the helical scan FFT, and the 45-degree rotated support, unwrap the 2-D problem into a 1-D problem, resulting in uniqueness and insights into the problem conditioning. The variance of distances between the adjacent unwrapped 1-D DTFT samples was developed as a sensitivity measure to quickly and accurately estimate of the condition number of the system matrix. A well-conditioned configuration of DTFT samples, restricted to radial lines in CT or spirals in MRI, is found by simulated annealing with the variance sensitivity measure as the objective function. The preconditioned conjugate gradient method reconstructs the 1-D solution that is then rewrapped to a 2-D image. In unrestricted cases, 2-D DTFT configurations like a regular hexagonal pattern can be unwrapped to uniformly-spaced and perfectly conditioned 1-D configurations and quickly solved using an inverse 1-D DFT. The next focus is on developing fast reconstruction algorithms. A non-iterative DFT-based method of reconstructing an image is presented, by first masking the 2-D DTFT samples with the frequency response of a filter that is zeroed at the unknown 2-D DFT locations, and then quickly deconvolving the filtered image using three 2-D DFTs. The masking filter needs to be precomputed only once per DTFT configuration. A divide-and-conquer image reconstruction method is also presented using subband decomposition and Gabor filters to solve smaller subband problems, leading to a quick unaliased low-resolution image or later to be recombined into the full solution. All methods are applied to actual CT data resulting in faster reconstructions than POCS and FBP with equivalent errors.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77912/1/leebc_1.pd