Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors

Abstract

Discrete models of imagers with orthogonal detectors are developed and aspects of their singular-value decompositions (SVDs) specified. The models have been developed as a step in attempts to specify the SVDs of multiple-pinhole transaxial imagers in terms of the SVDs of single-pinhole transaxial imagers. The models presented here do not fully represent orthogonal-pinhole imagers; however, the models do have key symmetries of such systems and the models are sufficiently tractable as to allow for the analytic determination of their SVDs. Key properties of the SVDs of the models correlate well with those reported elsewhere for related multiple-pinhole imagers. Each model is a 2n/spl times/n/sup 2/ matrix H mapping an n/spl times/n-pixel object space to two n-pixel orthogonal detectors. The models assume parallel projection onto weighted detectors. For each model, HH/sup T/=[nW/sup 2//ww/sup T/ ww/sup T//nW/sup 2/], where w=(w/sub 1/,w/sub 2/,...,w/sub n/) is a vector of detector weights and W is a diagonal matrix with diagonal w; For the case of equal weights (w/sub 1/=w/sub 2/=...=w/sub n/), the singular values of the orthogonal-detector system have been specified and related to the singular values of the related single-detector imager. For the case of symmetric weights, for n even (w/sub 1/=w/sub n/,...,w/sub n/2/=w/sub (n/2)+1/, expressions for n of the 2n-1 nonzero singular values have been determined and for n odd (w/sub 1/=w/sub n/,...,w/sub (n-1)/2/=w/sub (n+3)/2/), w/sub (n+1)/2/), expressions for n-1 of the 2n nonzero singular values have been determined. Additionally, for an example of the case of symmetric weights, the singular vectors of the orthogonal-detector system have been specified in terms of the related single-detector imager.