Resolution, linear filtering, and positivity

Abstract

Ultimately, the band-limited nature of imaging systems restricts image quality in measured data. However, prior knowledge can be employed to improve image quality beyond that available from measured data. Kinds of prior knowledge include knowledge of the support of the object and knowledge that the object has only positive intensities. In previous work it has been shown that prior knowledge increases image quality by two means: superresolution and improvements in the signal-to-noise ratio in the Fourier domain. However, after prior knowledge is enforced, the resulting filter that multiplies the Fourier data may unduly limit resolution in the constrained image. Here maximum achievable resolutions are derived for one- and two-dimensional filters. In addition, it is shown that requiring a signal to be positive results in lowering its maximum achievable resolution by as much as a factor of 2. As a result, algorithms that use positivity to improve the quality of Fourier-domain data may benefit from a final postprocessing step to increase resolution.