Analytical properties of time-of-flight PET data

Abstract

We investigate the analytical properties of time-of-flight (TOF) positron emission tomography (PET) sinograms, where the data are modeled as line integrals weighted by a spatially invariant TOF kernel. First, we investigate the Fourier transform properties of 2D TOF data and extend the 'bow-tie' property of the 2D Radon transform to the time-of-flight case. Second, we describe a new exact Fourier rebinning method, TOF-FOREX, based on the Fourier transform in the time-of-flight variable. We then combine TOF-FOREX rebinning with a direct extension of the projection slice theorem to TOF data, to perform fast 3D TOF PET image reconstruction. Finally, we illustrate these properties using simulated data.

Figure 1

(a) Top and (b) side view of a cylindrical 3D PET scanner. For each line of response (LOR), the object is multiplied by the TOF kernel h and integrated along the line to form the TOF data.

Figure 2

2D TOF sinograms and their magnitudes after taking 1D Fourier transform in t; The horizontal axis is the sinogram angle ϕ ∈ [0, 2π] and the vertical axis is the radial variable s.

Figure 3

2D Fourier transform ℘_2D(ω_s, ω_ϕ; ω_t) of ℘_2D(s, ϕ; ω_t) in (12) for four different ω_t frequency indices. The horizontal axis represents ω_ϕ and the vertical axis ω_s. Also shown in (d) is the hyperbolic curve bounding the effective support of the bow-tie for TOF data in the 5th frequency bin.

Figure 4

Comparison of rebinned TOF sinograms with corresponding directly computed TOF sinograms (TOF timing resolution 500 ps); (a) Columns show TOF bin indices from 4 to 7; rows show the calculated direct sinogram (Reference), the rebinned sinogram (TOF-FOREX), and their difference (Difference); (b) radial profile through the sinograms for 4th TOF bin at 40-th angle; (c) radial profile through the sinograms for 7th TOF bin at 40-th angle

Figure 5

Comparison of (top) noisy 2D direct sinogram with (bottom) noise reduction resulting from rebinning of oblique data into direct planes for (a) 500 ps, and (b) 250 ps resolution.

Figure 6

3D reconstructions from 200 million counts for the central axial slice: (a) true image; (b) 3D non-TOF reconstruction: FOREX followed by direct 2D Fourier reconstruction; (c) 3D TOF reconstruction using TOF-FOREX followed by 2D TOF direct Fourier reconstruction with TOF timing resolutions of 500 ps: (d) as for (c) but with 250 ps resolution. The 3D Hoffman digital brain phantom was scaled to the size of 512mm×512mm×164mm to better demonstrate the effects of TOF information.