The approximate inverse and conjugate gradient: non-symmetrical algorithms for fast attenuation correction in SPECT

Hybrid methods have been known for a long time as very efficient algorithms for attenuation correction in single-photon emission computed tomography, but only recently have efforts been made to formulate them with more rigorous mathematics. This has allowed us to explain their efficiency in terms of approximate inversion, and to establish a convergence condition. The present study focuses on the convergence problem and emphasizes the question of symmetry. Hybrid method operators are not symmetrical; therefore the convergence condition is not easily verified. New schemes based on a modified conjugate gradient method are presented. Convergence is proved and performances are shown to be at least as good as the standard hybrid schemes on perfect and noisy simulated data.