Computational and Applied Math Proseminar

Monday, March 26, 2:00 p.m. GWC 487

Russ Park

Dept Math & Stats

Optimal Compression and Numerical Stability of
Gegenbauer Reconstructions

Abstract
Image reconstruction methods are characterized by the source data space on which they operate, the range space and the degree to which objectives such as artifact suppression, compression and numerical stability are optimized. The Gegenbauer reconstruction method operates on Fourier source data, projecting it onto a finite set of Gegenbauer polynomials.

Gegenbauer data is then expanded on sub-domains of physical space segmented by presumed jump discontinuities in the source data. The absence of jump discontinuities within each sub-domain assures spectral convergence as long as reconstruction parameters lambda and m linearly track the resolution N as it approaches infinity. The implicit benefit of Gegenbauer reconstruction is source data compression, unfortunately the process is also limited by numerical instability as either lambda or m, or both, increase.

Early studies on this issue assumed lambda and m to be linearly tied to N and then characterized the bounds of instability as well as recommended safe reconstruction parameter combinations. Subsequent work demonstrated how to automatically predict the source data smoothness parameters, of which apriori knowledge is required for accurate reconstruction. This study performs asymptotic analyses on the predicted error bounds as N goes to infinity while fixing either m or lambda, leading to the discovery of reconstruction parameters optimized for an objective of either compression or numerical stability. Finally, the effectiveness of this new approach is illustrated by extensive numerical experiments.

For further information please contact: mittelmann@asu.edu