Statistics – Computation
Scientific paper
Jan 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995phdt........18r&link_type=abstract
Thesis (PH.D.)--UNIVERSITY OF CENTRAL FLORIDA, 1995.Source: Dissertation Abstracts International, Volume: 56-12, Section: B, pag
Statistics
Computation
Signal Recovery, Wavelets
Scientific paper
Signal recovery and image reconstruction have been a topic of interest for the last decade. The problem of reconstructing images from the magnitude of its Fourier transform arises in different applications. Currently, the most efficient and practical algorithms employed to the image reconstruction problem are the transform iterative algorithms that seek numerical solutions that minimize the distance between the measurement and the estimate rather than finding an exact solution. These algorithms suffer from several drawbacks that set a limit to the size and complexity of images to be reconstructed. The major disadvantage of the current iterative algorithm is the stagnation problem where an algorithm becomes trapped in local minima. The second problem is the slow convergence of the algorithms and the third is their high computational costs. In this dissertation, two new multiresolution adaptations of the iterative algorithm are introduced that enable the algorithm to avoid stagnation, improve global convergence and dramatically reduce the computational complexity. The first approach is the pyramid decomposition approach and the second is the Multiresolution Error-Reduction (MRER) approach that is based on a wavelet decomposition of the problem. These two methods can provide a rough and quick solution at a low resolution that may be refined as the algorithm progress. This coarse-to-find strategy enables the algorithm to avoid stagnation by providing a better initial guess and give the algorithm a higher likelihood of arriving at a global minimum. Second, they can improve the convergence rate by decomposing the search space into orthogonal subspaces that can reduce the low frequency component of the error responsible for the slow convergence of the algorithm. Finally, since the number of independent variables to be processed at each coarser level is less than that at the full resolution grid, this will dramatically reduce the computational cost of the algorithm and ensure faster convergence rate. Our computer simulation results demonstrate a significant advantage of the multiresolution approaches, in terms of convergence rate, computational complexity and robustness in the presence of noise, over its single-grid counterparts.
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