A unified Fourier theory for time-of-flight PET data

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 for cylindrical PET scanners and make photon-limited TOF-PET imaging more robust than non-TOF PET imaging. 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 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 the 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 measurement. 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 equations (FORCEs) 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 data. The new FORCEs can be used as new Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. Further, we give a geometric interpretation of the general solutions—the two families of characteristic curves can be obtained by respectively changing the azimuthal and co-polar angles of the biorthogonal coordinates in Fourier space. We conclude the unified Fourier theory by showing that the Fourier consistency equations are necessary and sufficient for 3D X-ray transform with TOF measurement. Finally, we give numerical examples of inverse rebinning for a 3D TOF PET and Fourier-based rebinning for a 2D TOF PET using the FORCEs to show the efficacy of the unified Fourier solutions.

Fourier rebinning and consistency equations for time-of-flight PET planograms

“…We derived two independent consistency equations (23) and (24) to characterize the two degrees of redundancy. We also derived two Fourier consistency equations (FCEs) and the Fourier-John equation (FJE) for 3D TOF PET based on the generalized Fourier-slice theorem, which are the duals of the spatial-domain consistency equations and John's equation, respectively [18, 39]. We then solved the three equations using the method of characteristics and obtained the Fourier rebinning and consistency equations (FORCEs).…”

“…Equation (4) is the generalized Fourier-slice theorem for 3D TOF data. For 3D TOF sinogram parametrization, the corresponding Fourier-slice theorem is given by (2) in Cho et al [9] and (7) in Li et al [39]. For histo-image parametrization [46], the transfer matrix is an identity matrix, i.e., A ( n̂ ) = I , an equivalent form is given by (A.4) in Li et al [37].…”

“…Figure 4 shows the basis conversion scheme among different types of PET planograms. A similar mapping scheme was obtained for cylindrical scanners [39], and a simple mapping scheme was given in [9]. A combination of these conversions can also be applied.…”

“…So the two consistency equations are necessary and sufficient for 3D planogram transform with TOF measurement. The sufficiency of the two consistency equations in the sinogram format was proven for 3D X-ray transform with TOF measurement in Li et al [39]. For the non-TOF case, the sufficiency was proven by [29].…”

Attenuation correction in emission tomography using the emission data—A review

“…101) or TOF data 97,98 from 3-D to 2-D, restoration of missing sinogram data, 102,103 rebinning of TOF data to non-TOF data, 99 and attenuation correction. 60 In fact, consistent TOF PET data satisfy the following two independent consistency equations: 96,99 ∂p ∂φ +t cosθ ∂p ∂s…”