Efficient edge-preserving regularization for limited-angle tomography

We demonstrate that the constraint of piecewise smoothness, applied through the use of edge-preserving regularization, can provide excellent tomographic reconstructions from limited-angle data. The tomography problem is formulated as a regularized least-squares optimization problem, and is then solved using a generalization of a recently proposed deterministic relaxation algorithm. This algorithm has been shown to converge under certain conditions when the original cost functional being minimized is convex. We have proven that our more general algorithm is globally convergent under less restrictive conditions, even when the original cost functional is nonconvex. Simulation results demonstrate the effectiveness of the algorithm, and show that for moderate to high photon counts, spectrally weighted error norms perform as well as, or better than a standard error norm that is commonly used for Poisson-distributed data. This suggests that a recently proposed fast Fourier algorithm, which is restricted to using a spectrally weighted error norm, can be used in many practical limited-angle problems to perform the minimization needed by the deterministic relaxation algorithm.