Rapid calculation of detectability in Bayesian single photon emission computed tomography

We consider the calculation of lesion detectability using a mathematical model observer, the channelized Hotelling observer (CHO), in a signal-known-exactly/background-known-exactly detection task for single photon emission computed tomography (SPECT). We focus on SPECT images reconstructed with Bayesian maximum a posteriori methods. While model observers are designed to replace time-consuming studies using human observers, the calculation of CHO detectability is usually accomplished using a large number of sample images, which is still time consuming. We develop theoretical expressions for a measure of detectability, the signal-to-noise-ratio (SNR) of a CHO observer, that can be very rapidly evaluated. Key to our expressions are approximations to the reconstructed image covariance. In these approximations, we use methods developed in the PET literature, but modify them to reflect the different nature of attenuation and distance-dependent blur in SPECT. We validate our expressions with Monte Carlo methods. We show that reasonably accurate estimates of the SNR can be obtained at a computational expense equivalent to approximately two projection operations, and that evaluating SNR for subsequent lesion locations requires negligible additional computation.