Trapezoidal back projection for positron emission tomography reconstruction.

Background: In the back projection step of the 3D PET reconstruction, all Lines of Responses (LORs) that go through a given voxel need to be identified and included in an integral. The standard Monte Carlo solution to this task samples stochastically the surfaces of the detector crystals and the volume of the voxel to search for valid LORs. To get a low noise Monte Carlo estimate, the number of samples needs to be very high, making the computational cost of the projection significant. In this paper, a novel deterministic projection algorithm called trapezoidal back projection (TBP) is proposed that replaces the extensive Monte Carlo sampling. Its goal is to determine all LORs that contribute to a given voxel together with their exact contribution weights. This is achieved by trapezoidal rasterization and a pre-computed look-up table.

Results: The precision and speed of the proposed TBP algorithm were compared to that of the Monte Carlo back projection of 1000, 10,000 and 100,000 samples. Measurements were run on a National Electrical Manufacturers Association (NEMA) NU 4-2008 image quality phantom as well as on a mouse acquisition. Results show that the TBP algorithm achieves the same low noise level (2.5 Uniformity %STD) as the Monte Carlo method with the highest sample number, but 13 times faster-the highest-precision Monte Carlo back projection takes 31.3 s, while TBP takes only 2.3 s on the NEMA NU 4-2008 image quality phantom of $200\times 200\times 333$ voxels.

Conclusion: The proposed deterministic TBP algorithm achieves a low noise level in a short runtime, thus it can be a promising solution for the back projection of the 3D PET reconstruction. Its performance advantage could be used to reduce either the reconstruction time, the data acquisition time, or the noise level of the image.