Efficient edge-preserving and multiresolution algorithms for limited-data tomography
Delaney, Alexander Hector
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Permalink
https://hdl.handle.net/2142/19455
Description
Title
Efficient edge-preserving and multiresolution algorithms for limited-data tomography
Author(s)
Delaney, Alexander Hector
Issue Date
1995
Doctoral Committee Chair(s)
Bresler, Yoram
Department of Study
Electrical and Computer Engineering
Discipline
Electrical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Electronics and Electrical
Language
eng
Abstract
This thesis presents and analyzes several novel algorithms and techniques that efficiently produce high-quality reconstructions for certain classes of limited-data tomographic reconstruction problems.
First, a new multiresolution reconstruction algorithm based on the two-dimensional wavelet transform is presented. This algorithm is similar to the conventional filtered backprojection algorithm, except that the filters are now angle dependent, and the backprojection produces wavelet coefficients of the reconstruction, instead of samples of the reconstruction. By collecting a special set of truncated projections, this algorithm can be used to reduce radiation exposure and computation when full resolution is required only over small localized regions of the reconstruction.
Next, an operator framework is used to develop an efficient and accurate Fourier-based iterative reconstruction algorithm for sampled parallel-beam projections. The tomography problem is formulated as a regularized least-squares optimization problem and is solved using the conjugate gradient (CG) algorithm. It is proven that if the radial sampling period of the projections is sufficiently small so that no aliasing occurs, the main step required at each iteration of the CG algorithm can be computed exactly by a discrete 2-D convolution, which can be cheaply implemented via the FFT.
The constraint of piecewise smoothness is then applied through the use of edge-preserving regularization, and is shown to produce excellent tomographic reconstructions from limited-angle data. The tomography problem is again formulated as a least-squares optimization problem, which is solved with a new deterministic relation algorithm. The deterministic relaxation algorithm converts the (possibly) nonconvex optimization problem into a series of convex optimization problems, and is proven to converge to a stationary point of the original cost function, even when the cost function is nonconvex. Numerous simulation results are presented to analyze algorithm performance.
Finally, the least-squares formulation is extended to the three-dimensional case, and the iterative algorithms are extended to handle the the special case of electron microscopy tomography. Numerical examples with real EM data are presented to demonstrate algorithm performance.
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