Exact and approximate Fourier rebinning of PET data from time-of-flight to non time-of-flight

Abstract

The image reconstruction problem for fully 3D TOF PET is challenging because of the large data sizes involved. One approach to this problem is to first rebin the data into one of the following lower dimensional formats: 2D TOF, 3D non TOF or 2D non TOF. Here we present a unified framework based on a generalized projection slice theorem for TOF data that can be used to compute each of these mappings. We use this framework to develop approaches for rebinning into non TOF formats without significant loss of information. We first derive the exact mappings and then describe approximations which address the missing data problem for oblique sinograms. We evaluate the performance of approximate rebinning using Monte Carlo simulations. Our results show that rebinning into non TOF sinograms retains significant SNR advantages over sinograms collected without TOF information.

Figure 1

Using the generalized projection slice theorem, the mappings shown above can be derived. See table 1 for details.

Figure 2

(a) Transverse and (b) 3D 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 3

Graphical interpretation of the exact and approximate inverse Fourier rebinning mappings from 2D non TOF data to 3D non TOF data. Inverse Fourier rebinning is equivalent to finding the coordinates ( ${\omega }_s^{\prime }$, ϕ′) of a point (for example, point B) on a trajectory (for example, the line joining the points A and B) mapped from the 3D sinogram. In the approximate inverse rebinning, |CB| and ∠ABC are approximated by |AB| and −δω_z/ω_s respectively.

Figure 4

Graphical interpretation of the exact and approximate inverse Fourier rebinning mappings from 3D non TOF data to 3D TOF data. The exact inverse Fourier rebinning is equivalent to finding the coordinate information of the line segment CB (|CB| and ∠ABC) for the line segment AB. By the approximation in (16), |CB| and ∠ABC are approximated by |DB| and ∠ABD, respectively.

Figure 5

Averaged mapping errors of approximate rebinnings for different ring differences: (a) Ring difference (RD) = 10 (oblique angle θ = 2.7°), (b) RD = 25 (θ = 6.8°), (c) RD = 35 (θ = 9.4°), (d) RD = 50 (θ = 13.4°). The error value is normalized by the image frequency sample interval (Δω_x) in the transverse plane where Δω_x = 1/2/256 mm⁻¹ in our simulation. Notice the error is very large when ω_s is small, so special consideration will be required in this region, similarly to the FORE implementation of (Defrise et al. 1997).

Figure 6

A transverse and a sagittal plane of the NCAT torso phantom (Segars 2001) used for simulation studies. The resolution and variance of reconstructions were studied at the four points denoted by A, B, C and D.

Figure 7

A comparison of the mean and variance of rebinned sinograms, obtained by FORET-3D, and non TOF sinograms for two ring differences: (a) Mean of rebinned sinograms for RD (ring difference) = 0, (b) Variance profiles of rebinned and non TOF sinograms (RD=0), (c) Mean profiles of rebinned and non TOF sinograms (RD=54), (d) Variance profiles of rebinned and non TOF sinograms (RD=54). The profiles were taken at the 160-th angle shown as a dashed line in (a).

Figure 8

A comparison of the mean and variance of rebinned direct 2D sinograms obtained by 1) FORET-3D+FORE, 2) FORET-2D and 3) FORE of 3D non TOF data acquired summing the 3D TOF data over the TOF bins: (a) Mean of rebinned sinograms at axial center, (b) Variance of rebinned sinograms at axial center. The profiles were taken at the 160-th angle shown as a dashed line in Figure 7(a).

Figure 9

MC simulation for 2D image reconstruction: Resolution (FWHM) versus pixel variance plot for four different locations. (a) Pixel location A, (b) Pixel location B, (c) Pixel location C, (d) Pixel location D (see Figure 6).

Figure 10

MC simulation for 3D image reconstruction: Resolution (FWHM) versus voxel variance plot for four different locations in a transverse plane at axial center as shown in Figure 6. (a) Voxel location A, (b) Voxel location B, (c) Voxel location C, (d) Voxel location D.

Figure 11

A comparison of 3D TOF data reconstruction by ‘FORET-3D+MAP’ (top row) and ‘non TOF+MAP’ (bottom row) in the transverse view (first column), coronal view (second column) and sagittal view (third column).