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The primal–dual optimization algorithm developed in Chambolle and Pock (CP) (2011 J. Math. Imag. Vis. 40 1–26) is applied to various convex optimization problems of interest in computed tomography (CT) image reconstruction. This algorithm allows for rapid prototyping of optimization problems for the purpose of designing iterative image reconstruction algorithms for CT. The primal–dual algorithm is briefly summarized in this paper, and its potential for prototyping is demonstrated by explicitly deriving CP algorithm instances for many optimization problems relevant to CT. An example application modeling breast CT with low-intensity x-ray illumination is presented.

Total Variation (TV) regularization is a powerful technique for image reconstruction tasks such as denoising, in-painting, and deblurring, because of its ability to produce sharp edges in the images. In this talk we discuss the use of TV regularization for tomographic imaging, where we compute a 2D or 3D reconstruction from noisy projections. We demonstrate that for a small signal-to-noise ratio, this new approach allows us to compute better (i.e., more reliable) reconstructions than those obtained by classical methods. This is possible due to the use of the TV reconstruction model, which incorporates our prior information about the solution and thus compensates for the loss of accuracy in the data. A consequence is that smaller data acquisition times can be used, thus reducing a patients exposure to X-rays in medical scanning and speeding up non-destructive measurements in materials science.

Breast X-ray CT imaging is being considered in screening as an extension to mammography. As a large fraction of the population will be exposed to radiation, low·dose imaging is essential. Iterative image reconstruction based on solving an optimization problem, such as Total-Variation minimization, shows potential for reconstruction from sparse-view data. For iterative methods it is important to ensure convergence to an accurate solution, since important diagnostic image features, such as presence of microcalcifications indicating breast cancer, may not be visible in a non-converged reconstruction, and this can have clinical significance. To prevent excessively long computational times, which is a practical concern for the large image arrays in CT, it is desirable to keep the number of iterations low, while still ensuring a sufficiently accurate reconstruction for the specific imaging task. This motivates the study of accurate convergence criteria for iterative image reconstruction. In simulation studies with a realistic breast phantom with microcalcifications we investigate the issue of ensuring sufficiently converged solution for reliable reconstruction. Our results show that it can be challenging to ensure a sufficiently accurate microcalcification reconstruction, when using standard convergence criteria. In particular, the gray level of the small microcalcifications may not have converged long after the background tissue is reconstructed uniformly. We propose the use of the individual objective function gradient components to better monitor possible regions of nonconverged variables. For microcalcifications we find empirically a large correlation between nonzero gradient components and nonconverged variables, which occur precisely within the microcalcifications. This supports our claim that gradient components can be used to ensure convergence to a sufficiently accurate reconstruction.

A realistic computer-simulation of a breast computed tomography (CT) system and subject is constructed. The model is used to investigate the optimal number of views for the scan given a fixed total X-ray fluence. The reconstruction algorithm is based on accurate solution to a constrained, TVminimization problem, which has received much interest recently for sparse-view CT data.