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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.

This work investigates conditions for quantitative image reconstruction in multispectral computed tomography (MSCT), which remains a topic of active research. In MSCT, one seeks to obtain from data the spatial distribution of linear attenuation coefficient, referred to as a virtual monochromatic image (VMI), at a given X-ray energy, within the subject imaged. As a VMI is decomposed often into a linear combination of basis images with known decomposition coefficients, the reconstruction of a VMI is thus tantamount to that of the basis images. An empirical, but highly effective, two-step data-domain-decomposition (DDD) method has been developed and used widely for quantitative image reconstruction in MSCT. In the two-step DDD method, step (1) estimates the so-called basis sinogram from data through solving a nonlinear transform, whereas step (2) reconstructs basis images from their basis sinograms estimated. Subsequently, a VMI can readily be obtained from the linear combination of basis images reconstructed. As step (2) involves the inversion of a straightforward linear system, step (1) is the key component of the DDD method in which a nonlinear system needs to be inverted for estimating the basis sinograms from data. In this work, we consider a {\it discrete} form of the nonlinear system in step (1), and then carry out theoretical and numerical analyses of conditions on the existence, uniqueness, and stability of a solution to the discrete nonlinear system for accurately estimating the discrete basis sinograms, leading to quantitative reconstruction of VMIs in MSCT., Comment: 27 pages, 12 figures

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

We propose an extended primal-dual algorithm framework for solving a general nonconvex optimization model. This work is motivated by image reconstruction problems in a class of nonlinear imaging, where the forward operator can be formulated as a nonlinear convex function with respect to the reconstructed image. Using the proposed framework, we put forward six specific iterative schemes, and present their detailed mathematical explanation. We also establish the relationship to existing algorithms. Moreover, under proper assumptions, we analyze the convergence of the schemes for the general model when the optimal dual variable regarding the nonlinear operator is non-vanishing. As a representative, the image reconstruction for spectral computed tomography is used to demonstrate the effectiveness of the proposed algorithm framework. By special properties of the concrete problem, we further prove the convergence of these customized schemes when the optimal dual variable regarding the nonlinear operator is vanishing. Finally, the numerical experiments show that the proposed algorithm has good performance on image reconstruction for various data with non-standard scanning configuration., Comment: 34 pages, 10 figures, 3 tables

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

Humans and animals are able to generalize or transfer information from previous experience so that they can behave appropriately in novel situations. What mechanisms--computations, representations, and neural systems--give rise to this remarkable ability? The members of this Generative Adversarial Collaboration (GAC) come from a range of academic backgrounds but are all interested in uncovering the mechanisms of generalization. We started out this GAC with the aim of arbitrating between two alternative conceptual accounts: (1) generalization stems from integration of multiple experiences into summary representations that reflect generalized knowledge, and (2) generalization is computed on-the-fly using separately stored individual memories. Across the course of this collaboration, we found that--despite using different terminology and techniques, and although some of our specific papers may provide evidence one way or the other--we in fact largely agree that both of these broad accounts (as well as several others) are likely valid. We believe that future research and theoretical synthesis across multiple lines of research is necessary to help determine the degree to which different candidate generalization mechanisms may operate simultaneously, operate on different scales, or be employed under distinct conditions. Here, as the first step, we introduce some of these candidate mechanisms and we discuss the issues currently hindering better synthesis of generalization research. Finally, we introduce some of our own research questions that have arisen over the course of this GAC, that we believe would benefit from future collaborative efforts.

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

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

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