1 A New, Fast, Relaxation-Free, Convergent Ordered-Subset Algorithm for Emission Tomography Ing-Tsung Hsiao 1 , Anand Rangarajan 3 , Parmeshwar Khurd 2 , Gene Gindi 2 1 School of Medical Technology, Chang Gung University, Tao-Yuan 333, Taiwan 2 Departments of Electrical & Computer Engineering and Radiology, SUNY Stony Brook, Stony Brook, NY 11784, USA 3 Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611, USA Abstract — In this paper, we propose a fast, convergent OS-type (ordered-subset) reconstruction algorithm for emission tomography (ET) by taking into account the Hessian information in the ML Poisson objective. Most importantly, our proposed algorithm does not have a relaxation parameter and it is fundamentally not based on EM-ML algorithm in 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 likelihood. After using a separable surrogate 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 likelihood. Preliminary results show that this new algorithm is faster than our previous COSEM algorithm after some initial iterations and compatible with RAMLA. However, in contrast to RAMLA, and similar to COSEM, the new algorithm does not require any user-specified relaxation parameters. I. Introduction Emission tomography in nuclear medicine, including PET and SPECT, is a useful diagnostic tool for investigating functions in an organ of interest in- vivo. To obtain a tomographic image of the injected radiotracer distribution in the body, one needs to apply a recontruction algorithm to collected data. Statistical reconstructin methods are capable of modelling photon noise and system phsyics more accurately than the traditional FBP method, and thus have drawn much attention in recent years. However, statistical reconstruction has the drawback of being slow when used for clinical studies as comparied to the FBP. To improve the speed of the statistical reconstruction, many fast reconstruction algorithms have been proposed in the past few years. Among them, the OS-type (ordered subsets) algorithms are the most popular approaches since the introduction of the OSEM algorithm in 1994[?]. We can rougly classify the OS-type algorithms for maximum likelihood-based (ML) reconstruction into three broad categories; i) heuristic OS-EM type algorithms which are not provably convergent, ii) convergent ordered subsets algorithms requiring a user-specified relaxation schedule and iii) convergent OS incremental EM type algorithms (COSEM). Our previous work on fast, provably convergent ML-EM algorithms is based on the third (COSEM) category above. Since EM algorithms (OS, incremental or otherwise) typically do not take into account the Hessian information in the ML objective, we now embark upon a new, fourth category of provably convergent, fast ordered subsets algorithms; iv) fast, convergent OS Hessian-based algorithms which do not have a relaxation parameter and which are fundamentally not based on EM. To derive our new method, we expand 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 likelihood. After using a separable surrogate 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 likelihood. This paper is to introduce and access the performance of the proposed algorithm. Section II we start on introducing the negative Poisson likelihood function, and then by the change of variables under the use of second order Taylor approximation, the new algorithm is derived step- by-step. In Section III, we run some empirical simulations to illustrate the speed of the proposed algorithm in comparison to ML-EM, COSEM and RAMLA. Section IV contains the discussion and the conclusion.