Duality between noise and spatial resolution in linear systems

It is shown that in a broad class of linear systems, including general linear shift-invariant systems, the spatial resolution and the noise satisfy a duality relationship, resembling the uncertainty principle in quantum mechanics. The product of the spatial resolution and the standard deviation of output noise in such systems represents a type of phase-space volume that is invariant with respect to linear scaling of the point-spread function, and it cannot be made smaller than a certain positive absolute lower limit. A corresponding intrinsic 'quality' characteristic is introduced and then evaluated for the cases of some popular imaging systems, including computed tomography, generic image convolution and phasecontrast imaging. It is shown that in the latter case the spatial resolution and the noise can sometimes be decoupled, potentially leading to a substantial increase in the imaging quality.