FORCE: Fourier rebinning and consistency equations for time-of-flight PET

Abstract

Fully 3D time-of-flight (TOF) PET scanners offer the potential of previously unachievable image quality in clinical PET imaging. TOF measurements add another degree of redundancy and make count-limited TOF-PET imaging more robust. The data space for 3D TOF-PET data is five dimensional with two degrees of redundancy. Previously, consistency equations were used to characterize the redundancy of TOF-PET data. In this paper, we first derive two Fourier consistency equations and the Fourier-John equation for 3D TOF PET based on the generalized projection-slice theorem; the three partial differential equations (PDEs) are the dual of the sinogram consistency equations and John's equation. We then solve the three PDEs using the method of characteristics. The two degrees of entangled redundancy of TOF-PET data can be explicitly elicited and exploited by the solutions of the PDEs along the characteristic curves, which gives a complete understanding of the rich structure of the 3D X-ray transform with TOF measurements. Fourier rebinning equations and other mapping equations among different types of PET data are special cases of the general solutions. We also obtain new Fourier rebinning and consistency (FORCE) equations from other special cases of the general solutions, and thus we obtain a complete scheme to convert among different types of PET data: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF PET data. The new FORCE equations can be used as new Fourier rebinning algorithms for TOF-PET data reduction, or inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. Finally, we give a numerical example for a 2D TOF PET to show the efficacy of the unified Fourier solutions.