Fourier rebinning of time-of-flight PET data

Abstract

This paper explores fast reconstruction strategies for 3D time-of-flight (TOF) positron emission tomography (PET), based on 2D data rebinning. Starting from pre-corrected 3D TOF data, a rebinning algorithm estimates for each transaxial slice the 2D TOF sinogram that would have been acquired by a single-ring scanner. The rebinned sinograms can then be reconstructed using any algorithm for 2D TOF reconstruction. We introduce TOF-FORE, an approximate rebinning algorithm obtained by extending the Fourier rebinning method for non-TOF data. In addition, we identify two partial differential equations that must be satisfied by consistent 3D TOF data, and use them to derive exact rebinning algorithms and to characterize the degree of the approximation in TOF-FORE. Numerical simulations demonstrate that TOF-FORE is more accurate than two different TOF extensions of the single-slice rebinning method, and suggest that TOF-FORE will be a valuable tool for practical TOF PET in the range of axial apertures and time resolutions typical of current scanners.

Analytical properties of time-of-flight PET data

“…This result differs from the recent TOF rebinning methods in (Defrise et al 2005) and (Vandenberghe et al 2006). The latter is an approximate method that operates in the native coordinates of the TOF sinograms while that in (Defrise et al 2005) uses Fourier transform relationships and the stationary phase approximation. Defrise et al (2005) also describe two exact rebinning methods, both based on consistency conditions and requiring the calculation of partial derivatives.…”

“…Although TOF-FOREX is exact, it requires more computation compared to the approximated TOF-FORE in (Defrise et al 2005) due to the inverse rebinning step required to estimate missing planes. Consequently it is not clear that the exact rebinning operator will be of direct practical value, although further investigation in this direction is planned.…”

“…Following Defrise et al (2005), we represent the 3D TOF PET data p 3D for a cylindrical scanner geometry, illustrated in figure 1, as a line integral weighted by the TOF kernel h: (1) where f represents the 3D object, s and ϕ are the radial and angular coordinates, respectively, z is the axial midpoint of each LOR, δ is the tangent of the oblique angle θ, and t is the TOF variable. The TOF kernel h, often modeled as a Gaussian function , is assumed to be shift invariant, so that the integral in (1) is in the form of a convolution.…”

“…The latter is an approximate method that operates in the native coordinates of the TOF sinograms while that in (Defrise et al 2005) uses Fourier transform relationships and the stationary phase approximation. Defrise et al (2005) also describe two exact rebinning methods, both based on consistency conditions and requiring the calculation of partial derivatives. The result described here uses a Fourier transform in the TOF variable to derive an exact relationship.…”

Comparison of list-mode and DIRECT approaches for time-of-flight PET reconstruction

“…This approach has been extended to 3D data [7]. Axial single-slice and Fourier rebinning approaches followed by 2D reconstruction have been described [8]- [10]; these techniques include rebinning of the TOF data into nonTOF arrays [11]. Iterative reconstruction algorithms such as ordered subsets expectation maximization (OSEM) [12] and list-mode OSEM [13] have been modified to include TOF information: fully 3D implementations of the TOF-OSEM algorithm from list-mode data have been described by Popescu, et al [14] and Groiselle and Glick [15].…”