In reconstructing an object function F(r) from finitely many noisy linear-functional values ∫F(r)Gn(r)dr we face the problem that finite data, noisy or not, are insufficient to specify F(r) uniquely. Estimates based on the finite data may succeed in recovering broad features of F(r), but may fail to resolve important detail. Linear and nonlinear, model-based data extrapolation procedures can be used to improve resolution, but at the cost of sensitivity to noise. To estimate linear-functional values of F(r) that have not been measured from those that have been, we need to employ prior information about the object F(r), such as support information or, more generally, estimates of the overall profile of F(r). One way to do this is through minimum-weighted-norm (MWN) estimation, with the prior information used to determine the weights. The MWN approach extends the Gerchberg-Papoulis band-limited extrapolation method and is closely related to matched-filter linear detection, the approximation of the Wiener filter, and to iterative Shannon-entropy-maximization algorithms. Nonlinear versions of the MWN method extend the noniterative, Burg, maximum-entropy spectral-estimation procedure.
© 2006 Optical Society of AmericaPDF Article