An analytical inversion of the nonuniformly attenuated Radon transform with variable focal-length 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 nonuniform. Novikov has derived an explicit inversion formula for the nonuniformly attenuated Radon transform for SPECT reconstruction of parallel-beam collimated projections. In this paper, we extend his research to variable focal-length fan-beam VFF collimator geometry. A ray-driven analytical formula for VFF reconstruction with nonuniform attenuation was derived. As a unified framework, this formula can be used for parallel-beam, fan-beam, and VFF collimators. Its accuracy is demonstrated by computer simulation experiments.