Jakob Heide Jørgensen

Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle–Pock algorithm

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.

Ensuring convergence in total-variation-based reconstruction for accurate microcalcification imaging in breast X-ray CT

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 non-converged variables. For microcalcifications we find empirically a large correlation between nonzero gradient components and non-converged 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.

Convergence of iterative image reconstruction algorithms for Digital Breast Tomosynthesis

Most iterative image reconstruction algorithms are based on some form of optimization, such as minimization of a data-fidelity term plus an image regularizing penalty term. While achieving the solution of these optimization problems may not directly be clinically relevant, accurate optimization solutions can aid in iterative image reconstruction algorithm design. This issue is particularly acute for iterative image reconstruction in Digital Breast Tomosynthesis (DBT), where the corresponding data model IS particularly poorly conditioned. The impact of this poor conditioning is that iterative algorithms applied to this system can be slow to converge. Recent developments in first-order algorithms are now beginning to allow for accurate solutions to optimization problems of interest to tomographic imaging in general. In particular, we investigate an algorithm developed by Chambolle and Pock (2011 J. Math. Imag. Vol. 40, pgs 120-145) and apply it to iterative image reconstruction in DBT.

Sampling conditions for gradient-magnitude sparsity based image reconstruction algorithms

Image reconstruction from sparse-view data in 2D fan-beam CT is investigated by constrained, total-variation minimization. This optimization problem exploits possible sparsity in the gradient magnitude image (GMI). The investigation is performed in simulation under ideal, noiseless data conditions in order to reveal a possible link between GMI sparsity and the necessary number of projection views for reconstructing an accurate image. Results are shown for two, quite different phantoms of similar GMI sparsity.

Characterizing a discrete-to-discrete X-ray transform for iterative image reconstruction with limited angular-range scanning in CT

Iterative image reconstruction in computed tomography often employs a discrete-to-discrete (DD) linear data model, and many of the aspects of the image recovery relate directly to the properties of this linear model. While much is known about the properties of the continuous X-ray, the corresponding DD version can be more difficult to characterize due to non-standardization and wide variation in model parameters in the image expansion set and the integration model. For this work, we analyze in detail the DD model for fan-beam CT with a limited scanning range, namely less than 180 degrees plus the fan-angle. The analysis is performed by specifying the class of system matrices considered and computing their condition number. A scaling is observed that aids in relating the condition number for large system matrices to that of more easily analyzed small matrices.

A first-order primal-dual reconstruction algorithm for few-view SPECT

A sparsity-exploiting algorithm intended for few-view Single Photon Emission Computed Tomography (SPECT) reconstruction is proposed and characterized. The algorithm models the object as piecewise constant subject to a blurring operation. Monte Carlo simulations were performed to provide more projection data of a phantom with varying smoothness across the field of view. For all simulations, reconstructions were performed across a sweep of the two primary design parameters: the blurring parameter and the weighting of the total variation (TV) minimization term. Maximum-Likelihood Expectation Maximization (MLEM) reconstructions were performed to provide reference images. Spatial resolution, accuracy, and signal-to-noise ratio was calculated and compared for all reconstructions. In general, increased values of the blurring parameter and TV weighting parameters reduced noise and streaking artifacts, while decreasing spatial resolution. The reconstructed images demonstrate that the algorithm introduces low-frequency artifacts in some cases, but eliminates streak artifacts due to angular undersampling. Further, as the number of views was decreased from 60 to 9 the accuracy of images reconstructed using the proposed algorithm varied by less than 3%. Overall, the results demonstrate preliminary feasibility of a sparsity-exploiting reconstruction algorithm which may be beneficial for few-view SPECT.