A new, fast, relaxation-free, convergent, Hessian-based, ordered-subsets algorithm for emission tomography

We propose a fast, convergent, positivity preserving, OS-type (ordered-subsets) maximum likelihood (ML) reconstruction algorithm for emission tomography (ET) which takes into account the Hessian information in the ML Poisson objective. In contrast to recent approaches, our proposed algorithm is fundamentally not based on the well known EM-ML algorithm for ET. Our new algorithm is based on an expansion of the ML objective using a second order Taylor series approximation w.r.t. the projection of the source distribution. Defining the projection of the source as an independent variable, we construct a new objective function in terms of the source distribution and the projection. This new objective function contains the Hessian information of the original Poisson negative log-likelihood. After using a separable surrogates transformation of the new Hessian-based objective, we derive an ordered subsets, positivity preserving algorithm which is guaranteed to asymptotically reach the maximum of the original ET log-likelihood. Preliminary results show that this new algorithm is about as fast as RAMLA after a few initial iterations. However, in contrast to RAMLA, the new algorithm does not require any user-specified, relaxation parameters.