An information theory of image gathering

Abstract

Shannon's mathematical theory of communication is extended to image gathering. The performance of image gathering, as well as of natural vision, is critically constrained by the realizability of the spatial-frequency response of optical apertures, the sampling passband of the photon-detection mechanism, and the signal-to-noise ratio. Hence, whereas Shannon could assume sufficient sampling, we have to deal with insufficient sampling. This extension requires a careful examination of the basic principles of information theory to obtain an unambiguous quantitative description of information for undersampled systems. In this paper we obtain expressions for the total information that is received with a single image-gathering channel and with parallel channels. Some of these expressions challenge intuition, but, when properly interpreted, have clear and precise meaning. The thrust of this meaning leads inexorably to the conclusion that the aliased signal components carry information even though these components interfere with the within-passband signal components in conventional image gathering and restoration, thereby degrading the fidelity and visual quality of the restored image. But a close examination of the expression for minimum mean-square-error, or Wiener-matrix, restoration from parallel image-gathering channels also reveals a method for unscrambling the within-passband and aliased signal components to restore spatial frequencies beyond the sampling passband out to the spatial-frequency response cutoff of the optical aperture. This method requires us to gather K discrete images, each with a different image-gathering response, i.e., channel as obtained, for example, by changing the objective lens aperture, to restore images with a resolution that is √ K times finer than the sampling interval.