A noise property analysis of single-photon emission computed tomography data

We give formulae describing the simplest statistical properties of single-photon emission computed tomography (SPECT) data modelled as the attenuated ray transform with Poisson noise. To obtain some of these formulae we obtain and use an inequality relating the even and odd parts of the attenuated ray transform without noise. Precise equations relating the even and odd parts of this transform are also discussed. Using these results we propose new possibilities for improving the stability of SPECT imaging based on the explicit inversion formula for the attenuated ray transformation with respect to the Poisson noise in the emission data. Numerical examples illustrating some of the theoretical conclusions of the present work are given.