Total variation regularization for large-scale X-ray tomography

Abstract

A new large-scale computational total variation regularization algorithm is introduced and tested with examples arising from X-ray tomography with sparsely sampled data. The total variation penalty term is discretized using a basis of discontinuous functions. The approach is motivated by discontinuous Galerkin methods and leads to an additional term of the jump part of total variation. The proposed algorithm combines the usage of the jump part with a subgradient descent scheme. A comparison is provided with the gradient-based projected Barzilai-Borwein method which uses a smoothly approximated total variation penalty. The above two methods are examples of total variation regularization algorithms that can be applied to large-scale tomographic problems in reasonable computation time. A comparison between the methods shows that they use roughly equal computational resources and that the new method produces somewhat blockier reconstructions. Although the test problems are two-dimensional, both methods can be applied to three-dimensional situations as well.