A backprojection kernel (KRNL3D) for very-wide-aperture 3D tomography applied to PET with Multigrid for precise use of time-of-flight data

Abstract

Abstract In ‘KRNL3D’ we derive a kernel function K(y 1, y 2, φ) whose backprojections from all directions (θ, φ) in the spherical band ∣ φ ∣ < φ ¯ max on the celestial sphere, when integrated with respect to solid angle, yield ρ, the 3D Gaussian point response function (PRF) of radius 1. This K, when convolved against line integral data from an unknown density function f, yields an integral formula for the ‘mollification’ ff = ρ ∗ f, which is a slightly blurred version of f, and which stabilizes the mild ill-posedness. Applied to positron emission tomography that backprojection reconstruction occurs stochastically and one emission event at a time, after needed data corrections. We describe Octave (≈Matlab) codes to tabulate K and to test its use with a large aperture φ ¯ max = π / 3 or π/6. ‘KRNL3D- TOF’ truncates backprojection to a cylindrical patch about the TOF approximate location of each event. These ‘backplacements’ decrease the computational cost and limit noise and streaking in one region from contaminating the reconstruction in more distant regions. They also retain the ability to count emission events in an isolated blob despite very low event counts, a valuable feature for dynamic studies of metabolic processes. ‘Multigrid’ allows further reduction in the radius and lengths of the cylinders, thereby enabling even more precise use of the TOF information. This precision should be especially important as researchers decrease the TOF uncertainty in newer generation scanners. Finally, we discuss ‘further work’ that needs to be done. Our codes are being made freely available at https://github.com/keithmillerberkeley/PET-codes.