An investigation on the property and fast implementation of a ray-driven method for inversion of the attenuated Radon transform with variable focusing fan-beam collimators

Single photon emission computed tomography (SPECT) is based on the measurement of radiation emitted by a radiotracer injected into the patient. Because of photoelectric absorption and Compton scatter, the gamma photons are attenuated inside the body before arriving at the detector. A quantitative reconstruction must consider the attenuation, which is usually non-uniform. Novikov had derived an explicit inversion formula for the non-uniformly attenuated Radon transform for SPECT reconstruction of parallel-beam collimated projections. In our previous work, we extended his work to variable focusing fan-beam (VFF) collimators. A ray-driven analytical inversion formula for VFF reconstruction with non-uniform attenuation was derived. The drawback of ray-driven based reconstruction is time consuming. In this work, we proposed a fast implementation method. Based on the property of ray-driven strategy, we calculated the reconstruction contribution of every ray for every reconstruction point in advance and saved them into database. When performing the reconstruction, we loaded the saved database, multiplied it by the corresponding rays and added the contributions of all rays together. The final image is then obtained efficiently by computing a divergence equation from the contributions. In addition to the software optimization, we utilized the texture mapping architecture of PC graphics/video card to accelerate the speed of the image reconstruction. We further investigated the property of the associated artifacts in the analytical inversion of the attenuated Radon transform. The artifacts were remarkably reduced when more projections were sampled to mitigate the problem of wide bandwidth of discrete Hilbert transform.