Inverse optical imaging viewed as a backward channel communication problem

TL;DR

This paper models inverse optical imaging as a communication channel problem, evaluating the maximum distinguishable messages based on noise bounds, and links classical information theory with topological concepts to understand imaging limits.

Contribution

It introduces a novel perspective by applying Kolmogorov's $oldsymbol{oldsymbol{ extit{ ext{ε}}}}$-capacity to inverse optical imaging, connecting information theory with optical resolution limits.

Findings

Maximum number of distinguishable messages grows exponentially as noise decreases.

$oldsymbol{ extit{ ext{ε}}}$-capacity approximates the information content in the image.

Classical and topological information theories are effectively compared in this context.

Abstract

The inverse problem in optics, which is closely related to the classical question of the resolving power, is reconsidered as a communication channel problem. The main result is the evaluation of the maximum number $M_\epsilon$ of $\epsilon$-distinguishable messages ($\epsilon$ being a bound on the noise of the image) which can be conveyed back from the image to reconstruct the object. We study the case of coherent illumination. By using the concept of Kolmogorov's $\epsilon$-capacity, we obtain: $M_\epsilon ~ 2^{S \log(1/\epsilon)} \to \infty$ as $\epsilon \to 0$, where S is the Shannon number. Moreover, we show that the $\epsilon$-capacity in inverse optical imaging is nearly equal to the amount of information on the object which is contained in the image. We thus compare the results obtained through the classical information theory, which is based on the probability theory, with those derived from a form of topological information theory, based on Kolmogorov's $\epsilon$-entropy and $\epsilon$-capacity, which are concepts related to the evaluation of the massiveness of compact sets.