Optimal compression and numerical stability for Gegenbauer reconstructions with applications
Reconstruction methods are characterized by their respective domain and range spaces as well as the degree to which objectives such as artifact suppression, source data compression and numerical stability are optimized. The Gegenbauer reconstruction method operates on a variety of source data spaces, mapping the domain onto a finite set of Gegenbauer polynomial basis functions. The method then expands the Gegenbauer coefficients 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 are judiciously chosen and linearly track the resolution N as it grows without bound.
The explicit benefit of Gegenbauer reconstruction to eliminate Gibbs artifacts has been understood for nearly two decades. But an accompanying implicit benefit is the ability to significantly compress source data prior to reconstruction. Unfortunately, the choice of Gegenbauer reconstruction parameters is limited by regions of numerical instability as either parameter, lambda or m, increases.
Prior studies assumed lambda and m to be linearly tied to N then characterized the bounds of instability as well as recommended safe reconstruction parameter combinations. Subsequent work demonstrated how to predict source data analyticity, of which a priori knowledge is required to minimize reconstruction error. This thesis complements such previous studies and recommends new Gegenbauer reconstruction parameter guidelines based on a suite of parameter optimizations spanning seven unique objectives. The first three of these objectives are achieved using asymptotic analysis while the remaining four are met using traditional numerical objective minimization techniques.