Donghwan Kim

Statistical X-ray computed tomography (CT) reconstruction can improve image quality from reduced dose scans, but requires very long computation time. Ordered subsets (OS) methods have been widely used for research in X-ray CT statistical image reconstruction (and are used in clinical PET and SPECT reconstruction). In particular, OS methods based on separable quadratic surrogates (OS-SQS) are massively parallelizable and are well suited to modern computing architectures, but the number of iterati

This paper provides a new way of developing the "Fast Iterative Shrinkage/Thresholding Algorithm (FISTA)" [3] that is widely used for minimizing composite convex functions with a nonsmooth term such as the ℓ₁ regularizer. In particular, this paper shows that FISTA corresponds to an optimized approach to accelerating the proximal gradient method with respect to a worst-case bound of the cost function. This paper then proposes a new algorithm that is derived by instead optimizing the st

Statistical image reconstruction algorithms in X-ray computed tomography (CT) provide improved image quality for reduced dose levels but require substantial computation time. Iterative algorithms that converge in few iterations and that are amenable to massive parallelization are favorable in multiprocessor implementations. The separable quadratic surrogate (SQS) algorithm is desirable as it is simple and updates all voxels simultaneously. However, the standard SQS algorithm requires many iterat

This paper generalizes the optimized gradient method (OGM) that achieves the optimal worst-case cost function bound of first-order methods for smooth convex minimization. Specifically, this paper studies a generalized formulation of OGM and analyzes its worst-case rates in terms of both the function value and the norm of the function gradient. This paper also develops a new algorithm called OGM-OG that is in the generalized family of OGM and that has the best known analytical worst-case bound wi

This paper optimizes the step coefficients of first-order methods for smooth convex minimization in terms of the worst-case convergence bound (i.e., efficiency) of the decrease in the gradient norm. This work is based on the performance estimation problem approach. The worst-case gradient bound of the resulting method is optimal up to a constant for large-dimensional smooth convex minimization problems, under the initial bounded condition on the cost function value. This paper then illustrates t

First-order methods are used widely for large scale optimization problems in signal/image processing and machine learning, because their computation depends mildly on the problem dimension. Nesterov's fast gradient method (FGM) has the optimal convergence rate among first-order methods for smooth convex minimization; its extension to non-smooth case, the fast iterative shrinkage-thresholding algorithm (FISTA), also satisfies the optimal rate; thus both algorithms have gained great interest. We r

This paper proposes an accelerated proximal point method for maximally monotone operators. The proof is computer-assisted via the performance estimation problem approach. The proximal point method includes various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers, and thus the proposed acceleration has wide applications. Numerical experiments are presented to demonstrate the accelerating behaviors.

Statistical image reconstruction methods provide improved image quality in low-dose X-ray CT. However, the long computation time of iterative algorithms limits their clinical use. Ordered subsets algorithms based on separable quadratic surrogates (OS-SQS) are attractive as they are simple and amenable for massive parallelization in modern computing architecture, but require many iterations to converge. Here, we further accelerate OS algorithms by using momentum techniques. We use real patient CT

Statistical image reconstruction in X-ray CT can provide decent images even with low dose, but requires substantial computation time. Recently, we have proposed combining ordered subsets (OS) methods and Nesterov's momentum technique for accelerated X-ray CT image reconstruction. We have observed rapid convergence speed of the proposed algorithms in our experiments, but sometimes encountered unstable behavior. Therefore, we introduce a diminishing step size rule, called a relaxed momentum approa

Iterative algorithms for X-ray CT image reconstruction offer the possibility of reduced dose and/or improved image quality, but require substantial compute time. Reducing the time will likely require algorithms that can be massively parallelized. Ordered subsets (OS) algorithms update all voxels simultaneously and thus are amenable to such parallelization. We present an new monotonic algorithm for regularized image reconstruction that is derived using optimization transfer with separable quadrat

We consider minimizing the composite function that consists of a strongly convex function and a convex function. The fast dual proximal gradient (FDPG) method decreases the dual function with a rate $O(1/k^2)$, leading to a rate $O(1/k)$ for decreasing the primal function. We propose a generalized FDPG method that guarantees an $O(1/k^{1.5})$ rate for the dual proximal gradient norm decrease. By relating this to the primal function decrease, the proposed approach decreases the primal function wi