First-order convex feasibility algorithms for x-ray CT

Optimization-based region-of-interest reconstruction for X-ray computed tomography based on total variation and data derivative

Region-of-interest (ROI) and interior reconstructions for computed tomography (CT) have drawn much attention and can be of practical value for potential applications in reducing radiation dose and hardware cost. The conventional wisdom is that the exact reconstruction of an interior ROI is very difficult to be obtained by only using data associated with lines through the ROI. In this study, we propose and investigate optimization-based methods for ROI and interior reconstructions based on total variation (TV) and data derivative. Objective functions are built by the image TV term plus the data finite difference term. Different data terms in the forms of L1-norm, L2-norm, and Kullback-Leibler divergence are incorporated and investigated in the optimizations. Efficient algorithms are developed using the proximal alternating direction method of multipliers (ADMM) for each program. All sub-problems of ADMM are solved by using closed-form solutions with high efficiency. The customized optimizations and algorithms based on the TV and derivative-based data terms can serve as a powerful tool for interior reconstructions. Simulations and real-data experiments indicate that the proposed methods can be of practical value for CT imaging applications.

X-ray CT image reconstruction from few-views via total generalized p-variation minimization

Total variation (TV)-based CT image reconstruction, employing the image gradient sparsity, has shown to be experimentally capable of reducing the X-ray sampling rate and removing the unwanted artifacts, yet may cause unfavorable over-smoothing and staircase effects by the piecewise constant assumption. In this paper, we present a total generalized p-variation (TGpV) regularization model to adaptively preserve the edge information while avoiding the staircase effect. The new model is solved by splitting variables with an efficient alternating minimization scheme. With the utilization of generalized p-shrinkage mappings and partial Fourier transform, all the subproblems have closed solutions. The proposed method shows excellent properties of edge preserving as well as the smoothness features by the consideration of high order derivatives. Experimental results indicate that the proposed method could avoid the mentioned effects and reconstruct more accurately than both the TV and TGV minimization algorithms when applied to a few-view problem.

Fast alternating projection methods for constrained tomographic reconstruction

The alternating projection algorithms are easy to implement and effective for large-scale complex optimization problems, such as constrained reconstruction of X-ray computed tomography (CT). A typical method is to use projection onto convex sets (POCS) for data fidelity, nonnegative constraints combined with total variation (TV) minimization (so called TV-POCS) for sparse-view CT reconstruction. However, this type of method relies on empirically selected parameters for satisfactory reconstruction and is generally slow and lack of convergence analysis. In this work, we use a convex feasibility set approach to address the problems associated with TV-POCS and propose a framework using full sequential alternating projections or POCS (FS-POCS) to find the solution in the intersection of convex constraints of bounded TV function, bounded data fidelity error and non-negativity. The rationale behind FS-POCS is that the mathematically optimal solution of the constrained objective function may not be the physically optimal solution. The breakdown of constrained reconstruction into an intersection of several feasible sets can lead to faster convergence and better quantification of reconstruction parameters in a physical meaningful way than that in an empirical way of trial-and-error. In addition, for large-scale optimization problems, first order methods are usually used. Not only is the condition for convergence of gradient-based methods derived, but also a primal-dual hybrid gradient (PDHG) method is used for fast convergence of bounded TV. The newly proposed FS-POCS is evaluated and compared with TV-POCS and another convex feasibility projection method (CPTV) using both digital phantom and pseudo-real CT data to show its superior performance on reconstruction speed, image quality and quantification.

Optimization-based image reconstruction in computed tomography by alternating direction method with ordered subsets

Nowadays, diversities of task-specific applications for computed tomography (CT) have already proposed multiple challenges for algorithm design of image reconstructions. Consequently, efficient algorithm design tool is necessary to be established. A fast and efficient algorithm design framework for CT image reconstruction, which is based on alternating direction method (ADM) with ordered subsets (OS), is proposed, termed as OS-ADM. The general ideas of ADM and OS have been abstractly introduced and then they are combined for solving convex optimizations in CT image reconstruction. Standard procedures are concluded for algorithm design which contain 1) model mapping, 2) sub-problem dividing and 3) solving, 4) OS level setting and 5) algorithm evaluation. Typical reconstruction problems are modeled as convex optimizations, including (non-negative) least-square, constrained L1 minimization, constrained total variation (TV) minimization and TV minimizations with different data fidelity terms. Efficient working algorithms for these problems are derived with detailed derivations by the proposed framework. In addition, both simulations and real CT projections are tested to verify the performances of two TV-based algorithms. Experimental investigations indicate that these algorithms are of the state-of-the-art performances. The algorithm instances show that the proposed OS-ADM framework is promising for practical applications.

Constrained Total Generalized p-Variation Minimization for Few-View X-Ray Computed Tomography Image Reconstruction

Total generalized variation (TGV)-based computed tomography (CT) image reconstruction, which utilizes high-order image derivatives, is superior to total variation-based methods in terms of the preservation of edge information and the suppression of unfavorable staircase effects. However, conventional TGV regularization employs l1-based form, which is not the most direct method for maximizing sparsity prior. In this study, we propose a total generalized p-variation (TGpV) regularization model to improve the sparsity exploitation of TGV and offer efficient solutions to few-view CT image reconstruction problems. To solve the nonconvex optimization problem of the TGpV minimization model, we then present an efficient iterative algorithm based on the alternating minimization of augmented Lagrangian function. All of the resulting subproblems decoupled by variable splitting admit explicit solutions by applying alternating minimization method and generalized p-shrinkage mapping. In addition, approximate solutions that can be easily performed and quickly calculated through fast Fourier transform are derived using the proximal point method to reduce the cost of inner subproblems. The accuracy and efficiency of the simulated and real data are qualitatively and quantitatively evaluated to validate the efficiency and feasibility of the proposed method. Overall, the proposed method exhibits reasonable performance and outperforms the original TGV-based method when applied to few-view problems.

3D pulse EPR imaging from sparse-view projections via constrained, total variation minimization

Tumors and tumor portions with low oxygen concentrations (pO2) have been shown to be resistant to radiation therapy. As such, radiation therapy efficacy may be enhanced if delivered radiation dose is tailored based on the spatial distribution of pO2 within the tumor. A technique for accurate imaging of tumor oxygenation is critically important to guide radiation treatment that accounts for the effects of local pO2. Electron paramagnetic resonance imaging (EPRI) has been considered one of the leading methods for quantitatively imaging pO2 within tumors in vivo. However, current EPRI techniques require relatively long imaging times. Reducing the number of projection scan considerably reduce the imaging time. Conventional image reconstruction algorithms, such as filtered back projection (FBP), may produce severe artifacts in images reconstructed from sparse-view projections. This can lower the utility of these reconstructed images. In this work, an optimization based image reconstruction algorithm using constrained, total variation (TV) minimization, subject to data consistency, is developed and evaluated. The algorithm was evaluated using simulated phantom, physical phantom and pre-clinical EPRI data. The TV algorithm is compared with FBP using subjective and objective metrics. The results demonstrate the merits of the proposed reconstruction algorithm.

How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray computed tomography

We introduce phase-diagram analysis, a standard tool in compressed sensing (CS), to the X-ray computed tomography (CT) community as a systematic method for determining how few projections suffice for accurate sparsity-regularized reconstruction. In CS, a phase diagram is a convenient way to study and express certain theoretical relations between sparsity and sufficient sampling. We adapt phase-diagram analysis for empirical use in X-ray CT for which the same theoretical results do not hold. We demonstrate in three case studies the potential of phase-diagram analysis for providing quantitative answers to questions of undersampling. First, we demonstrate that there are cases where X-ray CT empirically performs comparably with a near-optimal CS strategy, namely taking measurements with Gaussian sensing matrices. Second, we show that, in contrast to what might have been anticipated, taking randomized CT measurements does not lead to improved performance compared with standard structured sampling patterns. Finally, we show preliminary results of how well phase-diagram analysis can predict the sufficient number of projections for accurately reconstructing a large-scale image of a given sparsity by means of total-variation regularization.

Combining ordered subsets and momentum for accelerated X-ray CT image reconstruction

Statistical X-ray computed tomography (CT) reconstruction can improve image quality from reduced dose scans, but requires very long computation time. Ordered subsets (OS) methods have been widely used for research in X-ray CT statistical image reconstruction (and are used in clinical PET and SPECT reconstruction). In particular, OS methods based on separable quadratic surrogates (OS-SQS) are massively parallelizable and are well suited to modern computing architectures, but the number of iterations required for convergence should be reduced for better practical use. This paper introduces OS-SQS-momentum algorithms that combine Nesterov's momentum techniques with OS-SQS methods, greatly improving convergence speed in early iterations. If the number of subsets is too large, the OS-SQS-momentum methods can be unstable, so we propose diminishing step sizes that stabilize the method while preserving the very fast convergence behavior. Experiments with simulated and real 3D CT scan data illustrate the performance of the proposed algorithms.

Analysis of iterative region-of-interest image reconstruction for x-ray computed tomography

One of the challenges for iterative image reconstruction (IIR) is that such algorithms solve an imaging model implicitly, requiring a complete representation of the scanned subject within the viewing domain of the scanner. This requirement can place a prohibitively high computational burden for IIR applied to x-ray computed tomography (CT), especially when high-resolution tomographic volumes are required. In this work, we aim to develop an IIR algorithm for direct region-of-interest (ROI) image reconstruction. The proposed class of IIR algorithms is based on an optimization problem that incorporates a data fidelity term, which compares a derivative of the estimated data with the available projection data. In order to characterize this optimization problem, we apply it to computer-simulated two-dimensional fan-beam CT data, using both ideal noiseless data and realistic data containing a level of noise comparable to that of the breast CT application. The proposed method is demonstrated for both complete field-of-view and ROI imaging. To demonstrate the potential utility of the proposed ROI imaging method, it is applied to actual CT scanner data.