Image reconstruction by nonconvex inexact half-quadratic optimization

We consider the problem of reconstructing an image from noisy projection data. The problem is formulated as the minimization of a general objective function that can be particularized to the standard objectives used in medical applications (the most common choice is the sum of a convex data-fidelity term and a gradient-based regularization term). Our main focus is on half-quadratic (HQ) optimization, which amounts to solving a sequence of positive definite systems, or inner systems, and has the advantages of simplicity and versatility. HQ algorithms have been meticulously studied, but two difficulties remain: first, the inner systems are solved approximately in practice, which may hamper convergence; second, pointwise convergence is not guaranteed when the set of stationary points of the objective contains a continuum. We present new results that do not suffer from these limitations and we propose an efficient implementation based on conjugate gradient. The resulting inexact HQ algorithm sets a threshold on the accuracy of the computed solutions to the inner systems; it converges globally to a stationary point of the objective under a minimal set of conditions ubiquitous in image reconstruction. We investigate the behavior of the inexact HQ algorithm in the context of limited-data tomography. Our experiments show that it performs well in various nonconvex scenarios and converges to solutions accurate to full machine precision.