Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography

We introduce a generalization of a recently proposed deterministic relaxation algorithm for edge-preserving regularization in linear inverse problems. This recently proposed algorithm transforms the original (possibly nonconvex) optimization problem into a sequence of quadratic optimization problems, and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex. We prove that our more general algorithm is globally convergent (i.e., converges to a local minimum from any initialization) under less restrictive conditions, even when the original cost functional is nonconvex. We apply this algorithm to tomographic reconstruction from limited-angle data by formulating the problem as one of regularized least-squares optimization. The results demonstrate that the constraint of piecewise smoothness, applied through the use of edge-preserving regularization, can provide excellent limited-angle tomographic reconstructions. Two edgepreserving regularizers—one convex, the other nonconvex—are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited-angle scenarios, and to explore how factors, such as choice of error norm, angular sampling rate and amount of noise, affect reconstruction quality and algorithm performance. These simulation results show that for this application, the nonconvex regularizer produces consistently superior results.