Reconstruction of attenuation map using discrete consistency conditions

Methods of quantitative emission computed tomography require compensation f or linear photon attenuation. A current trend in single-photon emission com puted tomography (SPECT) and positron emission tomography (PET) is to emplo y transmission scanning to reconstruct the attenuation map. Such an approac h, however, considerably complicates both the scanner design and the data a cquisition protocol. A dramatic simplification could be made if the attenua tion map could be obtained directly from the emission projections, without the use of a transmission scan. This can be done by applying the consistenc y conditions that enable us to identify the operator of the problem and, th us, to reconstruct the attenuation map. In this paper, we propose a new app roach based on the discrete consistency conditions. One of the main advanta ges of the suggested method over previously used continuous conditions is t hat it can easily be applied in various scanning configurations, including fully three-dimensional (3-D) data acquisition protocols. Also, it provides a stable numerical implementation, allowing us to avoid the crosstalk betw een the attenuation map and the source function. A computationally efficien t algorithm is implemented by using the QR and Cholesky decompositions. App lication of the algorithm to computer-generated and experimentally measured SPECT data is considered.