It is well known that while spectral methods yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious
oscillations developing near the discontinuities and a much reduced overall convergence rate.
This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks
in the application of spectral methods. Recently spectral reprojection reconstruction methods have been developed to reconstruct piecewise smooth images in their smooth subintervals
and restore the exponential convergence properties of spectral methods. Specifically, unlike
standard filtering, the convergence does not deteriorate as the discontinuities are approached.
The most familar and easily analyzed spectral reprojection method is the Gegenbauer reconstruction method. However it is apparent that the Gegenbauer reconstruction method
is not robust and in particular it suffers from round off error. Methods to alleviate these
difficulties have been recently developed.
All high order reconstruction methods require apriori knowledge of the jump discontinuity locations, since these edges determine the intervals of smoothness in which the spectral
reprojection reconstruction method can be applied. The local edge detection method has
been recently been developed to determine the location of edges on scattered grid point data.
It can also be applied to detect the discontinuities in the derivatives of functions.
In this talk I discuss recent advances in edge detection and high order reconstruction
methods. Examples include applications from medical imaging, where noise is an additional
impediment to the reconstruction. Other spectral reprojection methods are briefly discussed.