Recovering fine details from under-resolved electron tomography data using higher order total variation ℓ₁ regularization

Over the last decade or so, reconstruction methods using ℓ₁ regularization, often categorized as compressed sensing (CS) algorithms, have significantly improved the capabilities of high fidelity imaging in electron tomography. The most popular ℓ₁ regularization approach within electron tomography has been total variation (TV) regularization. In addition to reducing unwanted noise, TV regularization encourages a piecewise constant solution with sparse boundary regions. In this paper we propose an alternative ℓ₁ regularization approach for electron tomography based on higher order total variation (HOTV). Like TV, the HOTV approach promotes solutions with sparse boundary regions. In smooth regions however, the solution is not limited to piecewise constant behavior. We demonstrate that this allows for more accurate reconstruction of a broader class of images – even those for which TV was designed for – particularly when dealing with pragmatic tomographic sampling patterns and very fine image features. We develop results for an electron tomography data set as well as a phantom example, and we also make comparisons with discrete tomography approaches.