Restoring with Maximum Likelihood and Maximum Entropy*

TLDR

The derivation adapts Jaynes’ unbiased estimation of positive probability functions to optics. The study seeks the most likely object given M sampled image values of an incoherent object. The authors test a positive, non‑band‑limited restoring formula—derived from a communication‑theory image‑formation model—via computer simulation on noisy random‑impulse images and apply it to experimental line‑spectra data. The method yields maximum‑entropy restorations that are positive, not band‑limited, and achieve resolution beyond the Rayleigh limit without spurious detail, provided input noise stays below ~40 % of the signal, and the experimental results confirm the simulations.

Abstract

Given M sampled image values of an incoherent object, what can be deduced as the most likely object? Using a communication-theory model for the process of image formation, we find that the most likely object has a maximum entropy and is represented by a restoring formula that is positive and not band limited. The derivation is an adaptation to optics of a formulation by Jaynes for unbiased estimates of positive probability functions. The restoring formula is tested, via computer simulation, upon noisy images of objects consisting of random impulses. These are found to be well restored, with resolution often exceeding the Rayleigh limit and with a complete absence of spurious detail. The proviso is that the noise in each image input must not exceed about 40% of the signal image. The restoring method is applied to experimental data consisting of line spectra. Results are consistent with those of the computer simulations.