Accelerated Optimization Algorithms for Statistical 3d X-ray Computed Tomography Image Reconstruction

ACCELERATED OPTIMIZATION ALGORITHMS FOR STATISTICAL 3D X-RAY COMPUTED TOMOGRAPHY IMAGE RECONSTRUCTION by Donghwan Kim Chair: Jeffrey A. Fessler X-ray computed tomography (CT) has been widely celebrated for its ability to visualize the anatomical information of patients, but has been criticized for high radiation exposure. Statistical image reconstruction algorithms in X-ray CT can provide improved image quality for reduced dose levels in contrast to the conventional reconstruction methods like filtered back-projection (FBP). However, the statistical approach requires substantial computation time, more than half an hour for commercial 3D X-ray CT products. Therefore, this dissertation focuses on developing iterative algorithms for statistical reconstruction that converge within fewer iterations and that are amenable to massive parallelization in modern multiprocessor implementations. Ordered subsets (OS) methods have been used widely in tomography problems, because they reduce the computational cost by using only a subset of the measurement data per iteration. This dissertation first improves OS methods so that they better handle 3D helical cone-beam CT geometries. OS methods have been used in commercial positron emission tomography (PET) and single-photon emission CT (SPECT) since 1997. However, they require too long a reconstruction time in X-ray CT to be used routinely for every clinical CT scan. In this dissertation, two main approaches are proposed for accelerating OS algorithms, one that uses new optimization transfer approaches and one that combines OS with momentum algorithms. First, the separable quadratic surrogates (SQS) methods, a widely used optimization transfer method with OS methods yielding simple, efficient and massively parallelizable OSSQS methods, have been accelerated in three different ways; a nonuniform SQS (NU-SQS), xiv a SQS with bounded interval (SQS-BI), and a quasi-separable quadratic surrogates (QSQS) method. Among them, a new NU-SQS method that encourages larger step sizes for the voxels that are expected to change more between the current and the final image has highly accelerated the convergence, while the derivation guarantees monotonic descent. Next, we combined OS methods with momentum approaches that cleverly reuse previous updates with almost negligible increased computation. Using momentum approaches such as well-known Nesterov’s methods were found to be not fast enough for 3D X-ray CT reconstruction, but the proposed combination of OS methods and momentum approaches (OSmomentum) resulted in very fast convergence rates. OS-momentum algorithms sometimes suffered from instability, we adapted relaxed momentum schemes. This refinement improves stability without losing the fast rate of OS-momentum. To further accelerate OS-momentum algorithms, this dissertation proposes new momentum methods, called optimized gradient methods, which are twice as fast yet have remarkably simple implementations comparable to Nesterov’s methods. Finally, in addition to OS-type algorithms, one variant of the block coordinate descent (BCD) algorithm, called axial BCD (ABCD), is specifically designed for 3D cone-beam CT geometry. The chosen axial block of voxels in 3D CT geometry for the BCD algorithm enables both accelerated convergence and efficient computation, particularly when designed hand-in-hand with separable footprint (SF) projector. Overall, this dissertation proposes several promising accelerated iterative algorithms for 3D X-ray CT image reconstruction. Their performance are investigated on simulated and real patient 3D CT scans.