Transmission-less attenuation estimation from time-of-flight PET histo-images using consistency equations

In positron emission tomography (PET) imaging, attenuation correction with accurate attenuation estimation is crucial for quantitative patient studies. Recent research showed that the attenuation sinogram can be determined up to a scaling constant utilizing the time-of-flight information. The TOF-PET data can be naturally and efficiently stored in a histo-image without information loss, and the radioactive tracer distribution can be efficiently reconstructed using the DIRECT approaches. In this paper, we explore transmission-less attenuation estimation from TOF-PET histo-images. We first present the TOF-PET histo-image formation and the consistency equations in the histo-image parameterization, then we derive a least-squares solution for estimating the directional derivatives of the attenuation factors from the measured emission histo-images. Finally, we present a fast solver to estimate the attenuation factors from their directional derivatives using the discrete sine transform and fast Fourier transform while considering the boundary conditions. We find that the attenuation histo-images can be uniquely determined from the TOF-PET histo-images by considering boundary conditions. Since the estimate of the attenuation directional derivatives can be inaccurate for LORs tangent to the patient boundary, external sources, e.g. a ring or annulus source, might be needed to give an accurate estimate of the attenuation gradient for such LORs. The attenuation estimation from TOF-PET emission histo-images is demonstrated using simulated 2D TOF-PET data.