Image reconstruction in emission tomography using canonical origin ensembles

Tomographic reconstruction based on origin ensembles (OE) is a new procedure for estimation of emission intensity distributions from projection measurements. The advantages of this method over standard iterative approaches are shorter computation time and easier implementation of complex system models in emission tomography (ET) inverse problems. The approach is not based on data likelihood, as are the standard iterative methods, but uses 3N-dimensional probability density functions (PDFs) of origin locations, where N is the number of detected events. In this work, the theory of OE was extended by introducing a parameter, β, which can be viewed as an image regularization parameter. With this extension, the system is called the canonical origin ensemble (COE), because of the resemblance to the canonical ensemble in statistical mechanics. The acquired dataset is considered a macroscopic state of the system, and locations of emissions (event origins) are viewed as unknown microscopic states described by the PDF. Monte Carlo simulations of the realistic geometry of 3D Positron Emission Tomography (PET) were generated using Simset. Photon attenuation was simulated, but randoms and Compton scatter were ignored. By measuring the image entropy, we found that the statistical system described by the COE undergoes a phase transition at β=1. Notably, the bias of the reconstructed values undergoes a similar transition at β=1. This indicates that, for realistic noise levels, COE reconstruction with β close to, but less than, 1 should be used for optimal image quality. A remarkable similarity between COE reconstructed image for β=1 and the image obtained using 10,000 iterations of list-mode 3D maximum likelihood-expectation maximization (ML-EM) was discovered. This work provides guidelines for improving the accuracy of the OE reconstruction method and demonstrates the use of attenuation correction for OE 3D list-mode PET.