Simultaneous activity and attenuation emission tomography as a nonlinear ill-posed problem

In Single Photon Emission Computed Tomography (SPECT) one is interested in reconstructing the activity distribution f of some radiopharmaceutical. Alas the data gathered suffer from attenuation described by the tissue density μ. We only have noisy sample values of the Attenuated‐Radon‐Transform \documentclass{article}\pagestyle{empty}\begin{document}$$ A(f,\mu)(\omega,s) = \mathop \smallint \nolimits_{ - \infty }^\infty f(s\omega ^ \bot + t\omega)\exp (- \mathop \smallint \nolimits_t^\infty \mu (s\omega ^ \bot + \tau \omega)d\tau)dt $$\end{document} (which is nonlinear in μ) per imaged slice at hand. Traditional theory for SPECT reconstruction treats μ as a known parameter. In practical applications however μ is not known, but crudely estimated or neglected. We try to develop an algorithm that approximates both f and μ from SPECT data alone, in order to obtain quantitatively accurate SPECT images. This is done using Tikhonov regularization techniques developed for non‐linear parameter estimation problems in differential equations and an adapted Gauß‐Newton‐CG minimization algorithm.