Toby Sanders

Toby Sanders, Anne Gelb, Rodrigo Platte et al.

Over the last decade or so, reconstruction methods using $\ell_1$ regularization, often categorized as compressed sensing (CS) algorithms, have significantly improved the capabilities of high fidelity imaging in electron tomography. The most popular $\ell_1$ 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 $\ell_1$ 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.

Toby Sanders, Rodrigo B. Platte

In the realm of signal and image denoising and reconstruction, $\ell_1$ regularization techniques have generated a great deal of attention with a multitude of variants. A key component for their success is that under certain assumptions, the solution of minimum $\ell_1$ norm is a good approximation to the solution of minimum $\ell_0$ norm. In this work, we demonstrate that this approximation can result in artifacts that are inconsistent with desired sparsity promoting $\ell_0$ properties, resulting in subpar results in {some} instances. With this as our motivation, we develop a multiscale higher order total variation (MHOTV) approach, which we show is related to the use of multiscale Daubechies wavelets. We also develop the tools necessary for MHOTV computations to be performed efficiently, via operator decomposition and alternatively converting the problem into Fourier space. The relationship of higher order regularization methods with wavelets, which we believe has generally gone unrecognized, is shown to hold in several numerical results, although notable improvements are seen with our approach over both wavelets and classical HOTV.

Toby Sanders

Popular methods for finding regularized solutions to inverse problems include sparsity promoting $\ell_1$ regularization techniques, one in particular which is the well known total variation (TV) regularization. More recently, several higher order (HO) methods similar to TV have been proposed, which we generally refer to as HOTV methods. In this letter, we investigate problem of the often debated selection of $λ$, the parameter used to carefully balance the interplay between data fitting and regularization terms. We theoretically argue for a scaling of the parameter that works for all orders for HOTV methods, based off of a single selection of the parameter for any one of the orders. We also provide numerical results which justify our theoretical findings.

Toby Sanders

Tomography is an imaging technique that works by reconstructing a scene from acquired data in the form of line integrals of the imaging domain. A fundamental underlying assumption in the reconstruction procedure is the precise alignment of the data values, i.e. the relationship between the data values and the paths of the lines of integration is accurately known. In many applications, e.g. electron and X-ray tomography, it is necessary to establish this relationship using software alignment techniques or image registration due to misalignment when rotating the physical specimen. Unfortunately, highly accurate software alignment is still a challenge to achieve in many cases, and improper alignment results in severe loss in the imaging resolution. In this article, we develop a new approach that considers the alignment problem through a completely different lens, as a problem of recovering phase shifts in Fourier domain {within} the reconstruction algorithm. The recovery of these phase shifts serves as the data alignment, which is done by calculating discrepancies between the misaligned data and the current reconstruction. In the development of the approach, we investigate proper selection of parameters, and we show that it is fairly flexible and surprisingly accurate. Finally, the analysis of our approach provides insight into why projection matching alignment by cross-correlation can be improved through low pass filtering, which we demonstrate. Our methods are validated in a wide range of examples and settings.