Near optimal energy selective x‐ray imaging system performance with simple detectors

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This article describes a method to achieve near optimal performance with low energy resolution detectors. Tapiovaara and Wagner [Phys. Med. Biol. 30, 519-529 (1985)] showed that an energy selective x-ray system using a broad spectrum source can produce images with a larger signal to noise ratio (SNR) than conventional systems using energy integrating or photon counting detectors. They showed that there is an upper limit to the SNR and that it can be achieved by measuring full spectrum information and then using an optimal energy dependent weighting. A performance measure is derived by applying statistical detection theory to an abstract vector space of the line integrals of the basis set coefficients of the two function approximation to the x-ray attenuation coefficient. The approach produces optimal results that utilize all the available energy dependent data. The method can be used with any energy selective detector and is applied not only to detectors using pulse height analysis (PHA) but also to a detector that simultaneously measures the total photon number and integrated energy, as discussed by Roessl et al. [Med. Phys. 34, 959-966 (2007)]. A generalization of this detector that improves the performance is introduced. A method is described to compute images with the optimal SNR using projections in a "whitened" vector space transformed so the noise is uncorrelated and has unit variance in both coordinates. Material canceled images with optimal SNR can also be computed by projections in this space. The performance measure is validated by showing that it provides the Tapiovaara-Wagner optimal results for a detector with full energy information and also a conventional detector. The performance with different types of detectors is compared to the ideal SNR as a function of x-ray tube voltage and subject thickness. A detector that combines two bin PHA with a simultaneous measurement of integrated photon energy provides near ideal performance across a wide range of operating conditions. Low energy resolution detectors can be used in energy selective x-ray imaging systems to produce images with near optimal performance.