Abstract

In reconstructing an object function $F\left(r\right)$ from finitely many noisy linear-functional values $\int F\left(r\right)G_n\left(r\right)\mathrm{d}r$ we face the problem that finite data, noisy or not, are insufficient to specify $F\left(r\right)$ uniquely. Estimates based on the finite data may succeed in recovering broad features of $F\left(r\right)$, 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 $F\left(r\right)$ that have not been measured from those that have been, we need to employ prior information about the object $F\left(r\right)$, such as support information or, more generally, estimates of the overall profile of $F\left(r\right)$. 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 America

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