Linear response theory for detectors consisting of discrete arrays

The optical transfer function (OTF) and the noise power or Wiener spectrum are defined for detectors consisting of a lattice of discrete elements with the assumptions of linear response, Gaussian statistics, and stationarity under the discrete group of translations which leave the lattice fixed. For the idealized classification task of determining the presence or absence of a signal under signal known exactly/background known exactly (SKE/BKE) conditions, the Wiener spectrum, the OTF, along with an analog of the gray-scale transfer characteristic, determine the signal-to-noise ratio (SNR), which quantifies the ability of an ideal observer to perform this task. While this result is similar to the established result for continuous detectors, such as screen-film systems, the theory of discrete lattices of detectors must take into account the fact that the lattice only supports a bounded but (in the limit of a detector of arbitrarily great extent) continuous range of frequencies. Incident signals with higher spatial frequencies appear in the data at lower aliased frequencies, and there are pairs of signals which are not distinguishable by the detector (the SNR vanishes for the task of distinguishing such signals). Further, the SNR will in general change if the signal is spatially displaced by a fraction of the lattice spacing, although this change will be small for objects larger than a single pixel. Some of the trade-offs involved in detectors of this sort, particularly in dealing with signal frequencies above those supported by the lattice, are studied in a simple model.