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An optimization-based image reconstruction algorithm is developed for contrast enhanced digital breast tomosynthesis (DBT) using dual-energy scanning. The algorithm minimizes directional total variation (TV) with a data discrepancy and non-negativity constraints. Iodinated contrast agent (ICA) imaging is performed by reconstructing images from dual-energy DBT data followed by weighted subtraction. Physical DBT data is acquired with a Siemens Mammomat scanner of a structured breast phantom with ICA inserts. Results are shown for both directional TV minimization and filtered back-projection for reference. It is seen that directional TV is able to substantially reduce depth blur for the ICA objects., Comment: Proceedings paper for accepted contribution to the 8th International Conference on Image Formation in X-Ray Computed Tomography (https://www.ct-meeting.org)

Deep neural networks used for reconstructing sparse-view CT data are typically trained by minimizing a pixel-wise mean-squared error or similar loss function over a set of training images. However, networks trained with such pixel-wise losses are prone to wipe out small, low-contrast features that are critical for screening and diagnosis. To remedy this issue, we introduce a novel training loss inspired by the model observer framework to enhance the detectability of weak signals in the reconstructions. We evaluate our approach on the reconstruction of synthetic sparse-view breast CT data, and demonstrate an improvement in signal detectability with the proposed loss.

Purpose: This study proposes a systematic method for determining the optimal x-ray tube settings/energy windows and fluence for minimal noise and maximum CNR in material density images obtained from DECT scans by fixing the subject size and the total radiation dose. Methods: The noise propagation in the process of sinogram and image reconstruction from DECT measurements is analyzed. Analytic estimates for the sinogram and monochromatic image pixel variances and the CNR as functions of tube potentials, fluence, and virtual monochromatic image (VMI) energy are derived, and then used in a phantom experiment as an objective function for optimizing the tube settings to minimize the image noise and maximize the CNR. Results: A non-trivial example that shows the existence of singular solutions to the inversion of sinograms-to-DECT measurements map was presented. Additionally, the optimal VMI energy for maximal CNR was determined. The optimal energy VMI was found to be the least noisy monochromatic image synthesized from the iodine and water density images, and it was shown that using more general weights in combining the two images linearly does not improve image quality. When the x-ray beam filter material was fixed at 2mm of Aluminum and the photon fluence for low and high kV scans were considered equal, the tube potential pair of 60/120 kV led to the maximal CNR in the VMI formed at energy 55 KeV. Conclusions: Optimizing DECT scan parameters to maximize the CNR can be done in a systematic way. Also choosing the parameters that maximize the Jacobian determinant over the sinogram domain would lead to more stable reconstructions due to the reduced amplification of the measurement noise. Since the values of the Jacobian determinant depend strongly on the imaging task, careful consideration of all of the relevant factors is needed when implementing the proposed framework.

Objective: We investigate and develop optimization-based algorithms for accurate reconstruction of four-dimensional (4D)-spectral-spatial (SS) images directly from data collected over limited angular ranges (LARs) in continuous-wave (CW) electron paramagnetic resonance imaging (EPRI). Methods: Basing on a discrete-to-discrete data model devised in CW EPRI employing the Zeeman-modulation (ZM) scheme for data acquisition, we first formulate the image reconstruction problem as a convex, constrained optimization program that includes a data fidelity term and also constraints on the individual directional total variations (DTVs) of the 4D-SS image. Subsequently, we develop a primal-dual-based DTV algorithm, simply referred to as the DTV algorithm, to solve the constrained optimization program for achieving image reconstruction from data collected in LAR scans in CW-ZM EPRI. Results: We evaluate the DTV algorithm in simulated- and real-data studies for a variety of LAR scans of interest in CW-ZM EPRI, and visual and quantitative results of the studies reveal that 4D-SS images can be reconstructed directly from LAR data, which are visually and quantitatively comparable to those obtained from data acquired in the standard, full-angular-range (FAR) scan in CW-ZM EPRI. Conclusion: An optimization-based DTV algorithm is developed for accurately reconstructing 4D-SS images directly from LAR data in CW-ZM EPRI. Future work includes the development and application of the optimization-based DTV algorithm for reconstructions of 4D-SS images from FAR and LAR data acquired in CW EPRI employing schemes other than the ZM scheme. Significance: The DTV algorithm developed may be exploited potentially for enabling and optimizing CW EPRI with minimized imaging time and artifacts by acquiring data in LAR scans., Comment: Submitted to Journal of Magnetic Resonance

An alternating direction method of multipliers (ADMM) framework is developed for nonsmooth biconvex optimization for inverse problems in imaging. In particular, the simultaneous estimation of activity and attenuation (SAA) problem in time-of-flight positron emission tomography (TOF-PET) has such a structure when maximum likelihood estimation (MLE) is employed. The ADMM framework is applied to MLE for SAA in TOF-PET, resulting in the ADMM-SAA algorithm. This algorithm is extended by imposing total variation (TV) constraints on both the activity and attenuation map, resulting in the ADMM-TVSAA algorithm. The performance of this algorithm is illustrated using the penalized maximum likelihood activity and attenuation estimation (P-MLAA) algorithm as a reference. Additional results on step-size tuning and on the use of unconstrained ADMM-SAA are presented in the previous arXiv submission: arXiv:2303.17042v1., Comment: Manuscript accepted at IEEE transactions on medical imaging. This version contains the appendix for the ADMM-TVSAA pseudocode

Quantitative image reconstruction in dual-energy computed tomography (CT) remains a topic of active research. We read with interest ``DIRECT-Net: A unified mutual-domain material decomposition network for quantitative dual-energy CT imaging,'' which appears in the 2022 February Issue of Med Phys. In the paper the authors propose a deep-learning (DL) method, referred to as the Direct-Net method, to address the problem of quantitative image reconstruction directly from data in full-scan dual-energy CT (DECT). We comment on the study and conclusion in the paper. The Reply to this comment appears under Communications on medphys.org: https://www.medphys.org/Communications/Reply-Pan_Response-Su.pdf In order to have context for the Reply, we provide the full text of our Comments.

This Special Report summarizes the 2022 AAPM Grand Challenge on Deep-Learning spectral Computed Tomography (DL-spectral CT) image reconstruction. The purpose of the challenge is to develop the most accurate image reconstruction algorithm for solving the inverse problem associated with a fast kVp switching dual-energy CT scan using a three tissue-map decomposition. Participants could choose to use deep-learning (DL), iterative, or a hybrid approach. Test phase submission were received from 18 research groups. Both the winning and second place teams had highly accurate results where the RMSE was nearly zero to single floating point precision. Results from the top ten also achieved a high degree of accuracy; and as a result this special report outlines the methodology developed by each of these groups. The DL-spectral CT challenge successfully established a forum for developing image reconstruction algorithms based on deep-learning that address an important inverse problem relevant for spectral CT., Comment: To be submitted as a Special Report to Medical Physics

The generalized minimal residual (GMRES) algorithm is applied to image reconstruction using linear computed tomography (CT) models. The GMRES algorithm iteratively solves square, non-symmetric linear systems and it has practical application to CT when using unmatched back-projector/projector pairs and when applying preconditioning. The GMRES algorithm is demonstrated on a 3D CT image reconstruction problem where it is seen that use of unmatched projection matrices does not prevent convergence, while using an unmatched pair in the related conjugate gradients for least-squares (CGLS) algorithm leads to divergent iteration. Implementation of preconditioning using GMRES is also demonstrated., Comment: Accepted to the 2022 CT Meeting

Purpose: The purpose of the challenge is to find the deep-learning technique for sparse-view CT image reconstruction that can yield the minimum RMSE under ideal conditions, thereby addressing the question of whether or not deep learning can solve inverse problems in imaging. Methods: The challenge set-up involves a 2D breast CT simulation, where the simulated breast phantom has random fibro-glandular structure and high-contrast specks. The phantom allows for arbitrarily large training sets to be generated with perfectly known truth. The training set consists of 4000 cases where each case consists of the truth image, 128-view sinogram data, and the corresponding 128-view filtered back-projection (FBP) image. The networks are trained to predict the truth image from either the sinogram or FBP data. Geometry information is not provided. The participating algorithms are tested on a data set consisting of 100 new cases. Results: About 60 groups participated in the challenge at the validation phase, and 25 groups submitted test-phase results along with reports on their deep-learning methodology. The winning team improved reconstruction accuracy by two orders of magnitude over our previous CNN-based study on a similar test problem. Conclusions: The DL-sparse-view challenge provides a unique opportunity to examine the state-of-the-art in deep-learning techniques for solving the sparse-view CT inverse problem., Comment: To appear in Medical Physics

In dual-energy computed tomography (DECT), low- and high- kVp data are collected often over a full-angular range (FAR) of $360^\circ$. While there exists strong interest in DECT with low- and high-kVp data acquired over limited-angular ranges (LARs), there remains little investigation of image reconstruction in DECT with LAR data. Objective: We investigate image reconstruction with minimized LAR artifacts from low- and high-kVp data over LARs of $\le 180^\circ$ by using a directional-total-variation (DTV) algorithm. Methods: Image reconstruction from LAR data is formulated as a convex optimization problem in which data-$\ell_2$ is minimized with constraints on image's DTVs along orthogonal axes. We then achieve image reconstruction by applying the DTV algorithm to solve the optimization problem. We conduct numerical studies from data generated over arcs of LARs, ranging from $14^\circ$ to $180^\circ$, and perform visual inspection and quantitative analysis of images reconstructed. Results: Monochromatic images of interest obtained with the DTV algorithm from LAR data show substantially reduced artifacts that are observed often in images obtained with existing algorithms. The improved image quality also leads to accurate estimation of physical quantities of interest, such as effective atomic number and iodine-contrast concentration. Conclusion: Our study reveals that from LAR data of low- and high-kVp, monochromatic images can be obtained that are visually, and physical quantities can be estimated that are quantitatively, comparable to those obtained in FAR DECT. Significance: As LAR DECT is of high practical application interest, the results acquired in the work may engender insights into the design of DECT with LAR scanning configurations of practical application significance., Comment: 23 pages, 18 figures, to be published in Physics in Medicine & Biology

Investigation of image reconstruction from data collected over a limited angular range in X-ray CT remains a topic of active research because it may yield insight into the development of imaging workflow of practical significance. This reconstruction problem is well-known to be challenging, however, because it is highly ill-conditioned. In the work, we investigate optimization-based image reconstruction from data acquired over a limited-angular range that is considerably smaller than the angular range in short-scan CT. We first formulate the reconstruction problem as a convex optimization program with directional total-variation (TV) constraints applied to the image, and then develop an iterative algorithm, referred to as the directional-TV (DTV) algorithm for image reconstruction through solving the optimization program. We use the DTV algorithm to reconstruct images from data collected over a variety of limited-angular ranges for breast and bar phantoms of clinical- and industrial-application relevance. The study demonstrates that the DTV algorithm accurately recovers the phantoms from data generated over a significantly reduced angular range, and that it considerably diminishes artifacts observed otherwise in reconstructions of existing algorithms. We have also obtained empirical conditions on minimal angular ranges sufficient for numerically accurate image reconstruction with the DTV algorithm.

The work seeks to develop an algorithm for image reconstruction by directly inverting the non-linear data model in spectral CT. Using the non-linear data model, we formulate the image-reconstruction problem as a non-convex optimization program, and develop a non-convex primal-dual (NCPD) algorithm to solve the program. We devise multiple convergence conditions and perform verification studies numerically to demonstrate that the NCPD algorithm can solve the non-convex optimization program and under appropriate data condition, can invert the non-linear data model. Using the NCPD algorithm, we then reconstruct monochromatic images from simulated and real data of numerical and physical phantoms acquired with a standard, full-scan dual-energy configuration. The result of the reconstruction studies shows that the NCPD algorithm can correct accurately for the non-linear beam-hardening effect. Furthermore, we apply the NCPD algorithm to simulated and real data of the numerical and physical phantoms collected with non-standard, short-scan dual-energy configurations, and obtain monochromatic images comparable to those of the standard, full-scan study, thus revealing the potential of the NCPD algorithm for enabling non-standard scanning configurations in spectral CT, where the existing indirect methods are limited., Comment: 16 pages, 12 figures, submitted to Computerized Medical Imaging and Graphics

Purpose: Many useful image quality metrics for evaluating linear image reconstruction techniques do not apply to or are difficult to interpret for non-linear image reconstruction. The vast majority of metrics employed for evaluating non-linear image reconstruction are based on some form of global image fidelity, such as image root mean square error (RMSE). Use of such metrics can lead to over-regularization in the sense that they can favor removal of subtle details in the image. To address this shortcoming, we develop an image quality metric based on signal detection that serves as a surrogate to the qualitative loss of fine image details. Methods: The metric is demonstrated in the context of a breast CT simulation, where different equal-dose configurations are considered. The configurations differ in the number of projections acquired. Image reconstruction is performed with a non-linear algorithm based on total variation constrained least-squares (TV-LSQ). The images are evaluated visually, with image RMSE, and with the proposed signal-detection based metric. The latter uses a small signal, and computes detectability in the sinogram and in the reconstructed image. Loss of signal detectability through the image reconstruction process is taken as a quantitative measure of loss of fine details in the image. Results: Loss of signal detectability is seen to correlate well with the blocky or patchy appearance due to over-regularization with TV-LSQ, and this trend runs counter to the image RMSE metric, which tends to favor the over-regularized images. Conclusions: The proposed signal detection based metric provides an image quality assessment that is complimentary to that of image RMSE. Using the two metrics in concert may yield a useful prescription for determining CT algorithm and configuration parameters when non-linear image reconstruction is used., Comment: 25 pages, 10 figures. Submitted to Medical Physics

The alternating direction method of multipliers (ADMM) algorithm is a powerful and flexible tool for complex optimization problems of the form $\min\{f(x)+g(y) : Ax+By=c\}$. ADMM exhibits robust empirical performance across a range of challenging settings including nonsmoothness and nonconvexity of the objective functions $f$ and $g$, and provides a simple and natural approach to the inverse problem of image reconstruction for computed tomography (CT) imaging. From the theoretical point of view, existing results for convergence in the nonconvex setting generally assume smoothness in at least one of the component functions in the objective. In this work, our new theoretical results provide convergence guarantees under a restricted strong convexity assumption without requiring smoothness or differentiability, while still allowing differentiable terms to be treated approximately if needed. We validate these theoretical results empirically, with a simulated example where both $f$ and $g$ are nondifferentiable -- and thus outside the scope of existing theory -- as well as a simulated CT image reconstruction problem.

Objective: This work examines the claim made in the literature that the inverse problem associated with image reconstruction in sparse-view computed tomography (CT) can be solved with a convolutional neural network (CNN). Methods: Training and testing image/data pairs are generated in a dedicated breast CT simulation for sparse-view sampling, using two different object models. The trained CNN is tested to see if images can be accurately recovered from their corresponding sparse-view data. For reference, the same sparse-view CT data is reconstructed by the use of constrained total-variation (TV) minimization (TVmin), which exploits sparsity in the gradient magnitude image (GMI). Results: Using sparse-view data from images either in the training or testing set, there is a significant discrepancy between the image obtained with the CNN and the image that generated the data. For the same simulated scanning conditions, TVmin is able to accurately reconstruct the test image. Conclusion: We find that the sparse-view CT inverse problem cannot be solved for the particular published CNN-based methodology that we chose and the particular object model that we tested. Furthermore, this negative result is obtained for conditions where TVmin is able to recover the test images. Significance: The inability of the CNN to solve the inverse problem associated with sparse-view CT, for the specific conditions of the presented simulation, draws into question similar unsupported claims being made for the use of CNNs to solve inverse problems in medical imaging., Comment: 12 pages, 11 figures. Will be submitted to IEEE Transactions on Biomedical Engineering

We study the problem of spectrum estimation from transmission data of a known phantom. The goal is to reconstruct an x-ray spectrum that can accurately model the x-ray transmission curves and reflects a realistic shape of the typical energy spectra of the CT system. To this end, spectrum estimation is posed as an optimization problem with x-ray spectrum as unknown variables, and a Kullback-Leibler (KL) divergence constraint is employed to incorporate prior knowledge of the spectrum and enhance numerical stability of the estimation process. The formulated constrained optimization problem is convex and can be solved efficiently by use of the exponentiated-gradient (EG) algorithm. We demonstrate the effectiveness of the proposed approach on the simulated and experimental data. The comparison to the expectation-maximization (EM) method is also discussed. In simulations, the proposed algorithm is seen to yield x-ray spectra that closely match the ground truth and represent the attenuation process of x-ray photons in materials, both included and not included in the estimation process. In experiments, the calculated transmission curve is in good agreement with the measured transmission curve, and the estimated spectra exhibits physically realistic looking shapes. The results further show the comparable performance between the proposed optimization-based approach and EM. In conclusion, our formulation of a constrained optimization provides an interpretable and flexible framework for spectrum estimation. Moreover, a KL-divergence constraint can include a prior spectrum and appears to capture important features of x-ray spectrum, allowing accurate and robust estimation of x-ray spectrum in CT imaging., Comment: 17 pages, 9 figures

Fiber-like features are an important aspect of breast imaging. Vessels and ducts are present in all breast images, and spiculations radiating from a mass can indicate malignancy. Accordingly, fiber objects are one of the three types of signals used in the American College of Radiology digital mammography (ACR-DM) accreditation phantom. This work focuses on the image properties of fiber-like structures in digital breast tomosynthesis (DBT) and how image reconstruction can affect their appearance. The impact of DBT image reconstruction algorithm and regularization strength on the conspicuity of fiber-like signals of various orientations is investigated in simulation. A metric is developed to characterize this orientation dependence and allow for quantitative comparison of algorithms and associated parameters in the context of imaging fiber signals. The imaging properties of fibers, characterized in simulation, are then demonstrated in detail with physical DBT data of the ACR-DM phantom. The characterization of imaging of fiber signals is used to explain features of an actual clinical DBT case. For the algorithms investigated, at low regularization setting, the results show a striking variation in conspicuity as a function of orientation in the viewing plane. In particular, the conspicuity of fibers nearly aligned with the plane of the X-ray source trajectory is decreased relative to more obliquely oriented fibers. Increasing regularization strength mitigates this orientation dependence at the cost of increasing depth blur of these structures., Comment: This version has been updated with changes made during the peer-review process. The manuscript has now been published in the SPIE Journal of Medical Imaging

There are a number of clinically relevant tasks in digital breast tomosynthesis (DBT) involving the detection and visual assessment of fiber-like structures such as Cooper's ligaments, blood vessels, and spiculated lesions. Such structures can exhibit orientation dependent variations in conspicuity. This study demonstrates the presence of in-plane orientation-dependent signal conspicuity for fiber-like signals in DBT and shows how reconstruction algorithm design can mitigate this phenomenon. We uncover a tradeoff between minimizing orientation-dependence and preserving depth resolution that is dictated by the regularization strength employed in reconstruction., Comment: 4 pages, 3 figures, submitted to the 60th Annual Meeting and Exhibition of AAPM

A potential application for spectral computed tomography (CT) with multi-energy-window photon-counting detectors is quantitative medical imaging with K-edge contrast agents. Image reconstruction for spectral CT with such contrast agents necessitates expression of the X-ray linear attenuation map in at least three expansion functions, for example, bone/water/K-edge-material or photo-electric- process/Compton-process/K-edge-material. The use of three expansion functions can result in slow convergence for iterative image reconstruction (IIR) algorithms applied to spectral CT. We propose a block-diagonal step-preconditioner for use with a primal-dual iterative image reconstruction framework that we have been developing for spectral CT. We demonstrate the advantage of the new step-preconditioner on a sensitive spectral CT simulation where the test object has low concentration of Gadolinium (Gd) contrast agent and the X-ray attenuation map is represented by three materials - PMMA, a soft-tissue equivalent, Aluminum, a bone equivalent, and Gd., Comment: Extend version of a manuscript submitted to the The Fifth International Conference on Image Formation in X-Ray Computed Tomography

Classical methods for X-ray computed tomography are based on the assumption that the X-ray source intensity is known, but in practice, the intensity is measured and hence uncertain. Under normal operating conditions, when the exposure time is sufficiently high, this kind of uncertainty typically has a negligible effect on the reconstruction quality. However, in time- or dose-limited applications such as dynamic CT, this uncertainty may cause severe and systematic artifacts known as ring artifacts. By carefully modeling the measurement process and by taking uncertainties into account, we derive a new convex model that leads to improved reconstructions despite poor quality measurements. We demonstrate the effectiveness of the methodology based on simulated and real data sets., Comment: Accepted at IEEE Transactions on Computational Imaging

Simulation-based image quality metrics are adapted and investigated for characterizing the parameter dependences of linear iterative image reconstruction for DBT. Three metrics based on 2D DBT simulation are investigated: (1) a root-mean-square-error (RMSE) between the test phantom and reconstructed image, (2) a gradient RMSE where the comparison is made after taking a spatial gradient of both image and phantom, and (3) a region-of-interest (ROI) Hotelling observer (HO) for signal-known-exactly/background-known-exactly (SKE/BKE) and signal-known-exactly/background-known-statistically (SKE/BKS) detection tasks. Two simulation studies are performed using the aforementioned metrics, varying voxel aspect ratio and regularization strength for two types of Tikhonov regularized least-squares optimization. The RMSE metrics are applied to a 2D test phantom and the ROI-HO metric is applied to two tasks relevant to DBT: large, low contrast lesion detection and small, high contrast microcalcification detection. The RMSE metric trends are compared with visual assessment of the reconstructed test phantom. The ROI-HO metric trends are compared with 3D reconstructed images from ACR phantom data acquired with a Hologic Selenia Dimensions DBT system. Sensitivity of image RMSE to mean pixel value is found to limit its applicability to the assessment of DBT image reconstruction. Image gradient RMSE is insensitive to mean pixel value and appears to track better with subjective visualization of the reconstructed bar-pattern phantom. The ROI-HO metric shows an increasing trend with regularization strength for both forms of Tikhonov-regularized least-squares; however, this metric saturates at intermediate regularization strength indicating a point of diminishing returns for signal detection. Visualization with reconstructed ACR phantom images appears to show a similar dependence with regularization strength., Comment: The manuscript has been submitted to Medical Physics

This article is intended to supplement our 2015 paper in Medical Physics titled "Noise properties of CT images reconstructed by use of constrained total-variation, data-discrepancy minimization", in which ordered subsets methods were employed to perform total-variation constrained data-discrepancy minimization for image reconstruction in X-ray computed tomography. Here we provide details regarding implementation of the ordered subsets algorithms and suggestions for selection of algorithm parameters. Detailed pseudo-code is included for every algorithm implemented in the original manuscript., Comment: 8 pages

We develop a primal-dual algorithm that allows for one-step inversion of spectral CT transmission photon counts data to a basis map decomposition. The algorithm allows for image constraints to be enforced on the basis maps during the inversion. The derivation of the algorithm makes use of a local upper bounding quadratic approximation to generate descent steps for non-convex spectral CT data discrepancy terms, combined with a new convex-concave optimization algorithm. Convergence of the algorithm is demonstrated on simulated spectral CT data. Simulations with noise and anthropomorphic phantoms show examples of how to employ the constrained one-step algorithm for spectral CT data., Comment: Submitted to Physics in Medicine and Biology

Many optimization problems arising in high-dimensional statistics decompose naturally into a sum of several terms, where the individual terms are relatively simple but the composite objective function can only be optimized with iterative algorithms. In this paper, we are interested in optimization problems of the form F(Kx) + G(x), where K is a fixed linear transformation, while F and G are functions that may be nonconvex and/or nondifferentiable. In particular, if either of the terms are nonconvex, existing alternating minimization techniques may fail to converge; other types of existing approaches may instead be unable to handle nondifferentiability. We propose the MOCCA (mirrored convex/concave) algorithm, a primal/dual optimization approach that takes local convex approximation to each term at every iteration. Inspired by optimization problems arising in computed tomography (CT) imaging, this algorithm can handle a range of nonconvex composite optimization problems, and offers theoretical guarantees for convergence when the overall problem is approximately convex (that is, any concavity in one term is balanced out by convexity in the other term). Empirical results show fast convergence for several structured signal recovery problems.

We introduce phase-diagram analysis, a standard tool in compressed sensing, to the X-ray CT community as a systematic method for determining how few projections suffice for accurate sparsity-regularized reconstruction. In compressed sensing 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 comparable with an optimal compressed sensing 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 to 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., Comment: 24 pages, 13 figures

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. To validate that the algorithm closely approximates the true object in the noiseless case, projection data were generated from an object assuming this model and using the system matrix. Monte Carlo simulations were performed to provide more realistic data of a phantom with varying smoothness across the field of view. Reconstructions were performed across a sweep of two primary design parameters. The results demonstrate that the algorithm recovers the object in a noiseless simulation case. While the algorithm assumes a specific blurring model, the results suggest that the algorithm may provide high reconstruction accuracy even when the object does not match the assumed blurring model. Generally, increased values of the blurring parameter and TV weighting parameters reduced noise and streaking artifacts, while decreasing spatial resolution. As the number of views 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., Comment: 38 pages, 25 figures, submitted to journal

In x-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such reconstruction methods is motivated by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, nor any relation between sparsity and a sufficient number of measurements for recovery, i.e., perfect reconstruction from noise-free data. We consider reconstruction through 1-norm minimization, as proposed in CS, from data obtained using a standard CT fan-beam sampling pattern. In empirical simulation studies we establish quantitatively a relation between the image sparsity and the sufficient number of measurements for recovery within image classes motivated by tomographic applications. We show empirically that the specific relation depends on the image class and in many cases exhibits a sharp phase transition as seen in CS, i.e. same-sparsity image require the same number of projections for recovery. Finally we demonstrate that the relation holds independently of image size and is robust to small amounts of additive Gaussian noise., Comment: 15 pages, 8 figures. Submitted to journal

Iterative image reconstruction (IIR) algorithms in Computed Tomography (CT) are based on algorithms for solving a particular optimization problem. Design of the IIR algorithm, therefore, is aided by knowledge of the solution to the optimization problem on which it is based. Often times, however, it is impractical to achieve accurate solution to the optimization of interest, which complicates design of IIR algorithms. This issue is particularly acute for CT with a limited angular-range scan, which leads to poorly conditioned system matrices and difficult to solve optimization problems. In this article, we develop IIR algorithms which solve a certain type of optimization called convex feasibility. The convex feasibility approach can provide alternatives to unconstrained optimization approaches and at the same time allow for efficient algorithms for their solution -- thereby facilitating the IIR algorithm design process. An accelerated version of the Chambolle-Pock (CP) algorithm is adapted to various convex feasibility problems of potential interest to IIR in CT. One of the proposed problems is seen to be equivalent to least-squares minimization, and two other problems provide alternatives to penalized, least-squares minimization., Comment: Revised version to appear March 2013 in Medical Physics. Version 1 has an error in line 5 of pseudocodes in Figs. 2 and 9 (now 12). This has been corrected in Version 2

The primal-dual optimization algorithm developed in Chambolle and Pock (CP), 2011 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 the article, 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., Comment: Resubmitted to Physics in Medicine and Biology. Text has been modified according to referee comments, and typos in the equations have been corrected

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 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 compare different convergence criteria 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., Comment: 5 pages, 4 figures, extended version of conference paper for 2011 IEEE Nuclear Science Symposium and Medical Imaging Conference

Iterative image reconstruction (IIR) with sparsity-exploiting methods, such as total variation (TV) minimization, investigated in compressive sensing (CS) claim potentially large reductions in sampling requirements. Quantifying this claim for computed tomography (CT) is non-trivial, because both full sampling in the discrete-to-discrete imaging model and the reduction in sampling admitted by sparsity-exploiting methods are ill-defined. The present article proposes definitions of full sampling by introducing four sufficient-sampling conditions (SSCs). The SSCs are based on the condition number of the system matrix of a linear imaging model and address invertibility and stability. In the example application of breast CT, the SSCs are used as reference points of full sampling for quantifying the undersampling admitted by reconstruction through TV-minimization. In numerical simulations, factors affecting admissible undersampling are studied. Differences between few-view and few-detector bin reconstruction as well as a relation between object sparsity and admitted undersampling are quantified., Comment: Revised version that was submitted to IEEE Transactions on Medical Imaging on 8/16/2012

Total-variation (TV)-based Computed Tomography (CT) image reconstruction has shown experimentally to be capable of producing accurate reconstructions from sparse-view data. In particular TV-based reconstruction is very well suited for images with piecewise nearly constant regions. Computationally, however, TV-based reconstruction is much more demanding, especially for 3D imaging, and the reconstruction from clinical data sets is far from being close to real-time. This is undesirable from a clinical perspective, and thus there is an incentive to accelerate the solution of the underlying optimization problem. The TV reconstruction can in principle be found by any optimization method, but in practice the large-scale systems arising in CT image reconstruction preclude the use of memory-demanding methods such as Newton's method. The simple gradient method has much lower memory requirements, but exhibits slow convergence. In the present work we consider the use of two accelerated gradient-based methods, GPBB and UPN, for reducing the number of gradient method iterations needed to achieve a high-accuracy TV solution in CT image reconstruction. The former incorporates several heuristics from the optimization literature such as Barzilai-Borwein (BB) step size selection and nonmonotone line search. The latter uses a cleverly chosen sequence of auxiliary points to achieve a better convergence rate. The methods are memory efficient and equipped with a stopping criterion to ensure that the TV reconstruction has indeed been found. An implementation of the methods (in C with interface to Matlab) is available for download from http://www2.imm.dtu.dk/~pch/TVReg/. We compare the proposed methods with the standard gradient method, applied to a 3D test problem with synthetic few-view data. We find experimentally that for realistic parameters the proposed methods significantly outperform the gradient method., Comment: 4 pages, 2 figures

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, TV-minimization problem, which has received much interest recently for sparse-view CT data., Comment: accepted to the 11th International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine 2011

This work is concerned with applying iterative image reconstruction, based on constrained total-variation minimization, to low-intensity X-ray CT systems that have a high sampling rate. Such systems pose a challenge for iterative image reconstruction, because a very fine image grid is needed to realize the resolution inherent in such scanners. These image arrays lead to under-determined imaging models whose inversion is unstable and can result in undesirable artifacts and noise patterns. There are many possibilities to stabilize the imaging model, and this work proposes a method which may have an advantage in terms of algorithm efficiency. The proposed method introduces additional constraints in the optimization problem; these constraints set to zero high spatial frequency components which are beyond the sensing capability of the detector. The method is demonstrated with an actual CT data set and compared with another method based on projection up-sampling., Comment: This manuscript appears in the proceedings of the 2010 IEEE medical imaging conference

Purpose: We develop an iterative image-reconstruction algorithm for application to low-intensity computed tomography (CT) projection data, which is based on constrained, total-variation (TV) minimization. The algorithm design focuses on recovering structure on length scales comparable to a detector-bin width. Method: Recovering the resolution on the scale of a detector bin, requires that pixel size be much smaller than the bin width. The resulting image array contains many more pixels than data, and this undersampling is overcome with a combination of Fourier upsampling of each projection and the use of constrained, TV-minimization, as suggested by compressive sensing. The presented pseudo-code for solving constrained, TV-minimization is designed to yield an accurate solution to this optimization problem within 100 iterations. Results: The proposed image-reconstruction algorithm is applied to a low-intensity scan of a rabbit with a thin wire, to test resolution. The proposed algorithm is compared with filtered back-projection (FBP). Conclusion: The algorithm may have some advantage over FBP in that the resulting noise-level is lowered at equivalent contrast levels of the wire., Comment: This article has been submitted to "Medical Physics" on 9/13/2010

In practical applications of tomographic imaging, there are often challenges for image reconstruction due to under-sampling and insufficient data. In computed tomography (CT), for example, image reconstruction from few views would enable rapid scanning with a reduced x-ray dose delivered to the patient. Limited-angle problems are also of practical significance in CT. In this work, we develop and investigate an iterative image reconstruction algorithm based on the minimization of the image total variation (TV) that applies to divergent-beam CT. Numerical demonstrations of our TV algorithm are performed with various insufficient data problems in fan-beam CT. The TV algorithm can be generalized to cone-beam CT as well as other tomographic imaging modalities., Comment: appeared in Journal of X-Ray Science and Technology

PURPOSE: We develop a practical, iterative algorithm for image-reconstruction in under-sampled tomographic systems, such as digital breast tomosynthesis (DBT). METHOD: The algorithm controls image regularity by minimizing the image total $p$-variation (TpV), a function that reduces to the total variation when $p=1.0$ or the image roughness when $p=2.0$. Constraints on the image, such as image positivity and estimated projection-data tolerance, are enforced by projection onto convex sets (POCS). The fact that the tomographic system is under-sampled translates to the mathematical property that many widely varied resultant volumes may correspond to a given data tolerance. Thus the application of image regularity serves two purposes: (1) reduction of the number of resultant volumes out of those allowed by fixing the data tolerance, finding the minimum image TpV for fixed data tolerance, and (2) traditional regularization, sacrificing data fidelity for higher image regularity. The present algorithm allows for this dual role of image regularity in under-sampled tomography. RESULTS: The proposed image-reconstruction algorithm is applied to three clinical DBT data sets. The DBT cases include one with microcalcifications and two with masses. CONCLUSION: Results indicate that there may be a substantial advantage in using the present image-reconstruction algorithm for microcalcification imaging., Comment: Submitted to Medical Physics