Image reconstruction with a primal-dual algorithm

Objectives A major problem with Bayesian reconstruction problems is to determine weighting parameters of regularizing functions. This work implements a primal-dual (PD) optimization algorithm for SPECT applications that simultaneously reconstructs and optimizes weighting factors.

Methods The theory of duality provides the mathematical formalism for the development of PD algorithms for Bayesian reconstruction problems. The Bayesian formulation is treated as a Lagrangian function with the weighing parameters corresponding to Lagrange multipliers. Optimizing the Lagrangian function by minimization in terms of intensity values provides the reconstruction (primal problem) and maximization in terms of the Lagrange multipliers (dual problem) provides optimum weighting parameters. In this work, the PD algorithm developed by Chambolle and Pock is applied to convex Bayesian optimization problems in the reconstruction of cardiac SPECT images from simulated, canine, and pinhole rat data. Simulated data consisted of 72 projections of the NCAT phantom over 360°. Canine data (72 projections over 360°) were acquired after the injection of 15 mCi of ^99mTc-sestamibi using the GE Millennium VG3 Hawkeye SPECT/CT camera with low-energy high resolution parallel-hole collimators. Rat data (90 projections over 360°) were obtained after the injection of 5 mCi of ¹²³I-MIBG using the same camera with pinhole collimators. Fifteen iterations were performed and the results were compared with ML-EM and conjugate gradient (CG) algorithms.

Results The PD algorithm had comparable run-time per iteration as ML-EM and CG algorithms. The PD algorithm provided the reconstruction and the optimum weighting parameters without extensive experimentation.

Conclusions The PD algorithm simultaneously solves a primal problem with its dual problem in which the duality gap provides a robust, non-heuristic convergence check. This has potential for application to Bayesian reconstruction problems that include other external constraints.

Research Support This study was supported by National Institutes of Health Grants R01 HL050663 “Dynamic Cardiac SPECT Imaging” and R01 EB07219 "Molecular Imaging of Cardiac Hypertrophy Using microPET and Pinhole SPECT."