Fast X-Ray CT Image Reconstruction Using a Linearized Augmented Lagrangian Method With Ordered Subsets

Abstract

Augmented Lagrangian (AL) methods for solving convex optimization problems with linear constraints are attractive for imaging applications with composite cost functions due to the empirical fast convergence rate under weak conditions. However, for problems such as X-ray computed tomography (CT) image reconstruction, where the inner least-squares problem is challenging and requires iterations, AL methods can be slow. This paper focuses on solving regularized (weighted) least-squares problems using a linearized variant of AL methods that replaces the quadratic AL penalty term in the scaled augmented Lagrangian with its separable quadratic surrogate (SQS) function, leading to a simpler ordered-subsets (OS) accelerable splitting-based algorithm, OS-LALM. To further accelerate the proposed algorithm, we use a second-order recursive system analysis to design a deterministic downward continuation approach that avoids tedious parameter tuning and provides fast convergence. Experimental results show that the proposed algorithm significantly accelerates the convergence of X-ray CT image reconstruction with negligible overhead and can reduce OS artifacts when using many subsets.

Fast 3D reconstruction method for differential phase contrast X-ray CT

“…(2) The fast method computes the kernel, r, via a discrete convolution, Eq. (27), and interpolation, then uses it to compute H T Hc via Eq. (26).…”

“…Again, we note for reference that computing the kernel, r in Eq. (27) . Accuracy (left) and computation time (right) as functions of input/output size for our fast algorithm for H T Hc with upsamping rates of 1, 2, and 4.…”

“…The challenge of fully 3D reconstruction is that it requires the entire reconstructed volume to fit in memory and can be computationally slow. Finally, we note that there is ongoing work on creating fast methods for X-ray CT reconstruction across a variety of scan geometries and problem formulations, including, e.g., [26], [27], and our own work, [28]. Broadly, these either propose faster optimization methods or accelerate the computations necessary for each iteration, including forward projection (H), back projections (H T ), and the normal operator (H T H).…”

Alternating Dual Updates Algorithm for X-ray CT Reconstruction on the GPU

“…When used with the ordered subsets (OS) approximation [10], [12], these algorithms can converge very rapidly. Unfortunately, without relaxation, OS-based algorithms have uncertain convergence properties.…”

“…Unfortunately, the circulant approximation is less useful in 3D CT. We partially overcame these difficulties in [19] by using a duality-based approach to solving problems involving the CT system matrix, but the resulting algorithm still used ADMM, which has difficult-to-tune penalty parameters and relatively high memory use. Gradient-based algorithms like OS [12] with acceleration [13] and the linearized augmented Lagrangian method with ordered subsets [10] (OS-LALM), can produce rapid convergence rates but use an approximation to the gradient of the data-fit term and have uncertain convergence properties. Some of these algorithms require generalizations to handle non-smooth regularizers.…”