Optimizing dual-energy CT technique for iodine-based contrast-to-noise ratio, a theoretical study

BACKGROUND

Dual-energy CT (DECT) systems provide valuable material-specific information by simultaneously acquiring two spectral measurements, resulting in superior image quality and contrast-to-noise ratio (CNR) while reducing radiation exposure and contrast agent usage. The selection of DECT scan parameters, including x-ray tube settings and fluence, is critical for the stability of the reconstruction process and hence the overall image quality.

PURPOSE

The goal of this study is to propose a systematic theoretical method for determining the optimal DECT parameters for minimal noise and maximum CNR in virtual monochromatic images (VMIs) for fixed subject size and total radiation dose.

METHODS

The noise propagation in the process of projection based material estimation from DECT measurements is analyzed. The main components of the study are the mean pixel variances for the sinogram and monochromatic image and the CNR, which were shown to depend on the Jacobian matrix of the sinograms-to-DECT measurements map. Analytic estimates for the mean sinogram and monochromatic image pixel variances and the CNR as functions of tube potentials, fluence, and VMI energy are derived, and then used in a virtual phantom experiment as an objective function for optimizing the tube settings and VMI energy to minimize the image noise and maximize the CNR.

RESULTS

It was shown that DECT measurements corresponding to kV settings that maximize the square of Jacobian determinant values over a domain of interest lead to improved stability of basis material reconstructions. Instances of non-uniqueness in DECT were addressed, focusing on scenarios where the Jacobian determinant becomes zero within the domain of interest despite significant spectral separation. The presence of non-uniqueness can lead to singular solutions during the inversion of sinograms-to-DECT measurements, underscoring the importance of considering uniqueness properties in parameter selection. Additionally, the optimal VMI energy and tube potentials for maximal CNR was determined. When the x-ray beam filter material was fixed at 2 mm 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 iodine CNR in the VMI at 53 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 set of expected line integrals leads 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.

Dual energy CT reconstruction using the constrained one step spectral image reconstruction algorithm

BACKGROUND

The constrained one-step spectral CT Image Reconstruction method (cOSSCIR) has been developed to estimate basis material maps directly from spectral CT data using a model of the polyenergetic x-ray transmissions and incorporating convex constraints into the inversion problem. This 'one-step' approach has been shown to stabilize the inversion in the case of photon-counting CT, and may provide similar benefits to dual-kV systems that utilize integrating detectors. Since the approach does not require the same rays be acquired for every spectral measurement, cOSSCIR can apply to dual energy protocols and systems used clinically, such as fast and slow kV switching systems and dual source scanning.

PURPOSE

The purpose of this study is to investigate the use of cOSSCIR applied to dual-kV data, using both registered and unregistered spectral acquisitions, specifically slow and fast kV switching imaging protocols. For this application, cOSSCIR is investigated using inverse crime simulations and dual-kV experiments. This study is the first demonstration of cOSSCIR on the dual-kV reconstruction problem.

METHODS

An integrating detector model was developed for the purpose of reconstructing dual-kV data, and an inverse crime study was used to validate the detector model within the cOSSCIR framework using a simulated pelvic phantom. Experiments were also used to evaluate cOSSCIR on the dual energy problem. Dual-kV data was obtained from a physical phantom containing analogs of adipose, bone, and liver tissues, with the aim of recovering the material coefficients in the bone and adipose basis material maps. cOSSCIR was applied to acquisitions where all rays performed both spectral measurements (registered) and fast and slow kV switching acquisitions (unregistered). cOSSCIR was also compared to two image-domain decomposition approaches, where image-domain methods are the conventional approach for decomposing unregistered spectral data.

RESULTS

Simulations demonstrate the application of cOSSCIR to the dual-kV inversion problem by successfully recovering the material basis maps on ideal data, while further showing that unregistered data presents a more challenging inversion problem. In our experimental reconstructions, the recovered basis material coefficient errors were found to be less than 6.5% in the bone, adipose, and liver regions for both registered and unregistered protocols. Similarly, the errors were less than 4% in the 50 keV virtual mono-energetic images, and the recovered material decomposition vectors nearly overlap their corresponding ground-truth vectors. Additionally, a preliminary two material decomposition study of iodine quantification recovered an average concentration of 9.2 mg/mL from a 10 mg/mL experimental iodine analog.

CONCLUSIONS

Using our integrating detector and spectral models, cOSCCIR is capable of accurately recovering material basis maps from dual-kV data for both registered and unregistered data. The material decomposition quantification compare favorably to the image domain approaches, and our results were not affected by the imaging protocol. Our results also suggest the extension of cOSSCIR to iodine quantification using two material decomposition.

Accurate reconstruction of 4D spectral-spatial images from sparse-view data in continuous-wave EPRI

In continuous-wave electron paramagnetic resonance imaging (CW EPRI), data are collected generally at densely sampled views sufficient for achieving accurate reconstruction of a four dimensional spectral-spatial (4DSS) image by use of the conventional filtered-backprojection (FBP) algorithm. It is desirable to minimize the scan time by collection of data only at sparsely sampled views, referred to as sparse-view data. Interest thus remains in investigation of algorithms for accurate reconstruction of 4DSS images from sparse-view data collected for potentially enabling fast data acquisition in CW EPRI. In this study, we investigate and demonstrate optimization-based algorithms for accurate reconstruction of 4DSS images from sparse-view data. Numerical studies using simulated and real sparse-view data acquired in CW EPRI are conducted that reveal, in terms of image visualization and physical-parameter estimation, the potential of the algorithms developed for yielding accurate 4DSS images from sparse-view data in CW EPRI. The algorithms developed may be exploited for enabling sparse-view scans with minimized scan time in CW EPRI for yielding 4DSS images of quality comparable to, or better than, that of the FBP reconstruction from data collected at densely sampled views.

Enhancing signal detectability in learning-based CT reconstruction with a model observer inspired loss function

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.

Simultaneous activity and attenuation estimation in TOF-PET with TV-constrained nonconvex optimization

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.

Simultaneous activity and attenuation estimation in TOF-PET with TV-constrained nonconvex optimization

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.

Accurate reconstruction of multiple basis images directly from dual-energy data in CT

OBJECTIVE

We develop optimization-based algorithms to accurately reconstruct multiple ( 2) basis images directly from dual-energy (DE) data in CT.

METHODS

In medical and industrial CT imaging, some basis materials such as bone, metals, and contrast agents of interest are confined often spatially within regions in the image. Exploiting this observation, we develop an optimization-based algorithm to reconstruct, directly from DE data, basis-region images from which multiple ( 2) basis images and virtual monochromatic images (VMIs) can be obtained over the entire image array.

RESULTS

We conduct experimental studies using simulated and real DE data in CT, and evaluate basis images and VMIs obtained in terms of visual inspection and quantitative metrics. The study results reveal that the algorithm developed can accurately and robustly reconstruct multiple ( 2) basis images directly from DE data.

CONCLUSIONS

The developed algorithm can yield accurate multiple ( 2) basis images, VMIs, and physical quantities of interest from DE data in CT.

SIGNIFICANCE

The work may provide insights into the development of practical procedures for reconstructing multiple basis images, VMIs, and physical quantities from DE data in applications. The work can be extended to reconstruct multiple basis images in multi-spectral CT or/and photon-counting CT.

Report on the AAPM deep-learning spectral CT Grand Challenge

BACKGROUND

This Special Report summarizes the 2022 AAPM Grand Challenge on Deep-Learning spectral Computed Tomography (DL-spectral CT) image reconstruction.

PURPOSE

The purpose of the challenge is to develop the most accurate image reconstruction algorithm possible for solving the inverse problem associated with a fast kilovolt switching dual-energy CT scan using a three tissue-map decomposition. Participants could choose to use a deep-learning (DL), iterative, or a hybrid approach.

METHODS

The challenge is based on a 2D breast CT simulation, where the simulated breast phantom consists of three tissue maps: adipose, fibroglandular, and calcification distributions. The phantom specification is stochastic so that multiple realizations can be generated for DL approaches. A dual-energy scan is simulated where the x-ray source potential of successive views alternates between 50 and 80 kilovolts (kV). A total of 512 views are generated, yielding 256 views for each source voltage. We generate 50 and 80 kV images by use of filtered back-projection (FBP) on negative logarithm processed transmission data. For participants who develop a DL approach, 1000 cases are available. Each case consists of the three 512 × 512 tissue maps, 50 and 80-kV transmission data sets and their corresponding FBP images. The goal of the DL network would then be to predict the material maps from either the transmission data, FBP images, or a combination of the two. For participants developing a physics-based approach, all of the required modeling parameters are made available: geometry, spectra, and tissue attenuation curves. The provided information also allows for hybrid approaches where physics is exploited as well as information about the scanned object derived from the 1000 training cases. Final testing is performed by computation of root-mean-square error (RMSE) for predictions on the tissue maps from 100 new cases.

RESULTS

Test phase submission were received from 18 research groups. Of the 18 submissions, 17 were results obtained with algorithms that involved DL. Only the second place finishing team developed a physics-based image reconstruction algorithm. 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 10 also achieved a high degree of accuracy; and as a result, this special report outlines the methodology developed by each of these groups.

CONCLUSIONS

The DL-spectral CT challenge successfully established a forum for developing image reconstruction algorithms that address an important inverse problem relevant for spectral CT.

Prototyping optimization-based image reconstructions from limited-angular-range data in dual-energy CT

Image reconstruction from data collected over full-angular range (FAR) in dual-energy CT (DECT) is well-studied. There exists interest in DECT with advanced scan configurations in which data are collected only over limited-angular ranges (LARs) for meeting unique workflow needs in certain practical imaging applications, and thus in the algorithm development for image reconstruction from such LAR data. The objective of the work is to investigate and prototype image reconstructions in DECT with LAR scans. We investigate and prototype optimization programs with various designs of constraints on the directional-total-variations (DTVs) of virtual monochromatic images and/or basis images, and derive the DTV algorithms to numerically solve the optimization programs for achieving accurate image reconstruction from data collected in a slew of different LAR scans. Using simulated and real data acquired with low- and high-kV spectra over LARs, we conduct quantitative studies to demonstrate and evaluate the optimization programs and their DTV algorithms developed. As the results of the numerical studies reveal, while the DTV algorithms yield images of visual quality and quantitative accuracy comparable to that of the existing algorithms from FAR data, the former reconstruct images with improved visualization, reduced artifacts, and also enhanced quantitative accuracy when applied to LAR data in DECT. Optimization-based, one-step algorithms, including the DTV algorithms demonstrated, can be developed for quantitative image reconstruction from spectral data collected over LARs of extents that are considerably smaller than the FAR in DECT. The theoretical and numerical results obtained can be exploited for prototyping designs of optimization-based reconstructions and LAR scans in DECT, and they may also yield insights into the development of reconstruction procedures in practical DECT applications. The approach and algorithms developed can naturally be applied to investigating image reconstruction from LAR data in multi-spectral and photon-counting CT.

Constrained one-step material decomposition reconstruction of head CT data from a silicon photon-counting prototype

BACKGROUND

Spectral CT material decomposition provides quantitative information but is challenged by the instability of the inversion into basis materials. We have previously proposed the constrained One-Step Spectral CT Image Reconstruction (cOSSCIR) algorithm to stabilize the material decomposition inversion by directly estimating basis material images from spectral CT data. cOSSCIR was previously investigated on phantom data.

PURPOSE

This study investigates the performance of cOSSCIR using head CT datasets acquired on a clinical photon-counting CT (PCCT) prototype. This is the first investigation of cOSSCIR for large-scale, anatomically complex, clinical PCCT data. The cOSSCIR decomposition is preceded by a spectrum estimation and nonlinear counts correction calibration step to address nonideal detector effects.

METHODS

Head CT data were acquired on an early prototype clinical PCCT system using an edge-on silicon detector with eight energy bins. Calibration data of a step wedge phantom were also acquired and used to train a spectral model to account for the source spectrum and detector spectral response, and also to train a nonlinear counts correction model to account for pulse pileup effects. The cOSSCIR algorithm optimized the bone and adipose basis images directly from the photon counts data, while placing a grouped total variation (TV) constraint on the basis images. For comparison, basis images were also reconstructed by a two-step projection-domain approach of Maximum Likelihood Estimation (MLE) for decomposing basis sinograms, followed by filtered backprojection (MLE + FBP) or a TV minimization algorithm (MLE + TVmin ) to reconstruct basis images. We hypothesize that the cOSSCIR approach will provide a more stable inversion into basis images compared to two-step approaches. To investigate this hypothesis, the noise standard deviation in bone and soft-tissue regions of interest (ROIs) in the reconstructed images were compared between cOSSCIR and the two-step methods for a range of regularization constraint settings.

RESULTS

cOSSCIR reduced the noise standard deviation in the basis images by a factor of two to six compared to that of MLE + TVmin , when both algorithms were constrained to produce images with the same TV. The cOSSCIR images demonstrated qualitatively improved spatial resolution and depiction of fine anatomical detail. The MLE + TVmin algorithm resulted in lower noise standard deviation than cOSSCIR for the virtual monoenergetic images (VMIs) at higher energy levels and constraint settings, while the cOSSCIR VMIs resulted in lower noise standard deviation at lower energy levels and overall higher qualitative spatial resolution. There were no statistically significant differences in the mean values within the bone region of images reconstructed by the studied algorithms. There were statistically significant differences in the mean values within the soft-tissue region of the reconstructed images, with cOSSCIR producing mean values closer to the expected values.

CONCLUSIONS

The cOSSCIR algorithm, combined with our previously proposed spectral model estimation and nonlinear counts correction method, successfully estimated bone and adipose basis images from high resolution, large-scale patient data from a clinical PCCT prototype. The cOSSCIR basis images were able to depict fine anatomical details with a factor of two to six reduction in noise standard deviation compared to that of the MLE + TVmin two-step approach.

4D-image reconstruction directly from limited-angular-range data in continuous-wave electron paramagnetic resonance imaging

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.

Spectral calibration of photon-counting detectors at high photon flux

BACKGROUND

Calibration of photon-counting detectors (PCDs) is necessary for quantitatively accurate spectral computed tomography (CT), but the calibration process can be complicated by nonlinear flux-dependent physical factors such as pulse pile-up.

PURPOSE

This work develops a method for spectral sensitivity calibration of a PCD-based spectral CT system that incorporates nonlinear flux dependence and can thus be employed at high photon flux.

METHODS

A calibration model for the spectral response and polynomial flux dependence is proposed, which incorporates prior x-ray source spectrum and PCD models and that has a small set of parameters for adjusting to the spectral CT system of interest. The model parameters are determined by fitting transmission data from a known object of known composition: a step-wedge phantom composed of different thicknesses of aluminum, a bone equivalent, and polymethyl methacrylate (PMMA), a soft-tissue equivalent. This fitting employs Tikhonov regularization, and the regularization strength and the polynomial order for the intensity modeling are determined by bias and variance analysis. The spectral calibration and nonlinear intensity correction is validated on transmission measurements through a third material, Teflon, at different x-ray photon flux levels.

RESULTS

The nonlinear intensity dependence is determined to be accurately accounted for with a third-order polynomial. The calibrated spectral CT model accurately predicts Teflon transmission to within 1% for flux levels up to 50% of the detector maximum.

CONCLUSIONS

The proposed PCD calibration method enables accurate physical modeling necessary for quantitative imaging in spectral CT. Furthermore, the model applies to high flux settings so that acquisition times will not be limited by restricting the spectral CT system to low flux levels.

Addressing CT metal artifacts using photon-counting detectors and one-step spectral CT image reconstruction

PURPOSE

The constrained one-step spectral CT image reconstruction (cOSSCIR) algorithm with a nonconvex alternating direction method of multipliers optimizer is proposed for addressing computed tomography (CT) metal artifacts caused by beam hardening, noise, and photon starvation. The quantitative performance of cOSSCIR is investigated through a series of photon-counting CT simulations.

METHODS

cOSSCIR directly estimates basis material maps from photon-counting data using a physics-based forward model that accounts for beam hardening. The cOSSCIR optimization framework places constraints on the basis maps, which we hypothesize will stabilize the decomposition and reduce streaks caused by noise and photon starvation. Another advantage of cOSSCIR is that the spectral data need not be registered, so that a ray can be used even if some energy window measurements are unavailable. Photon-counting CT acquisitions of a virtual pelvic phantom with low-contrast soft tissue texture and bilateral hip prostheses were simulated. Bone and water basis maps were estimated using the cOSSCIR algorithm and combined to form a virtual monoenergetic image for the evaluation of metal artifacts. The cOSSCIR images were compared to a "two-step" decomposition approach that first estimated basis sinograms using a maximum likelihood algorithm and then reconstructed basis maps using an iterative total variation constrained least-squares optimization (MLE+TV min $_{\text{min}}$ ). Images were also compared to a nonspectral TV min $_{\text{min}}$ reconstruction of the total number of counts detected for each ray with and without normalized metal artifact reduction (NMAR) applied. The simulated metal density was increased to investigate the effects of increasing photon starvation. The quantitative error and standard deviation in regions of the phantom were compared across the investigated algorithms. The ability of cOSSCIR to reproduce the soft-tissue texture, while reducing metal artifacts, was quantitatively evaluated.

RESULTS

Noiseless simulations demonstrated the convergence of the cOSSCIR and MLE+TV min $_{\text{min}}$ algorithms to the correct basis maps in the presence of beam-hardening effects. When noise was simulated, cOSSCIR demonstrated a quantitative error of -1 HU, compared to 2 HU error for the MLE+TV min $_{\text{min}}$ algorithm and -154 HU error for the nonspectral TV min $_{\text{min}}$ +NMAR algorithm. For the cOSSCIR algorithm, the standard deviation in the central iodine region of interest was 20 HU, compared to 299 HU for the MLE+TV min $_{\text{min}}$ algorithm, 41 HU for the MLE+TV min $_{\text{min}}$ +Mask algorithm that excluded rays through metal, and 55 HU for the nonspectral TV min $_{\text{min}}$ +NMAR algorithm. Increasing levels of photon starvation did not impact the bias or standard deviation of the cOSSCIR images. cOSSCIR was able to reproduce the soft-tissue texture when an appropriate regularization constraint value was selected.

CONCLUSIONS

By directly inverting photon-counting CT data into basis maps using an accurate physics-based forward model and a constrained optimization algorithm, cOSSCIR avoids metal artifacts due to beam hardening, noise, and photon starvation. The cOSSCIR algorithm demonstrated improved stability and accuracy compared to a two-step method of decomposition followed by reconstruction.

Report on the AAPM deep-learning sparse-view CT (DL-sparse-view CT) Grand Challenge

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. This article is protected by copyright. All rights reserved.

Image reconstruction from data over two orthogonal arcs of limited-angular ranges

PURPOSE

Computed tomography (CT) scanning over limited-angular ranges (LARs) is of practical interest in possible reduction of imaging dose and time and in design of non-standard scans. This work aims to investigate image reconstruction for two non-overlapping arcs of LARs, and to demonstrate that they may allow more accurate image reconstruction than may a single arc of LAR.

METHODS

We consider a configuration with two non-overlapping arcs of LARs α₁ and α₂ , whose centers are separated by 90°, and refer to it as a two-orthogonal-arc configuration. Data are generated from a chest phantom with two-orthogonal-arc configurations over total angular coverage α_τ = α₁ + α₂ ranging from 18° to 180°, and images are reconstructed subsequently by use of the directional-total-variation (DTV) algorithm. For comparison, we also consider image reconstruction for a single-arc configuration of angular range α_τ . Quantitative metrics such as the normalized root-mean-square-error (nRMSE) are used for evaluation of image reconstruction accuracy.

RESULTS

Visual inspection and quantitative analysis of images reconstructed reveal that a two-orthogonal-arc configuration generally yields more accurate image reconstruction than does its single-arc counterpart. As total angular range α_τ increases, the DTV algorithm yields image reconstruction with enhanced accuracy, as expected. Also, if α_τ remains constant, the two-orthogonal-arc configuration with α₁ = α₂ generally leads to image reconstruction more accurate than those of two-orthogonal-arc configurations with α₁ ≠ α₂ , as the nRMSE of the former can be lower than that of the latter for up to more than one order of magnitude.

CONCLUSIONS

Appropriately designed two-orthogonal-arc configurations may be exploited for improving image-reconstruction accuracy in CT imaging with reduced angular coverage. This study may yield insights into the design of innovative CT scans for lowering scan time and radiation dose, and/or for avoiding scan collision in, e.g., C-arm CT.

Dual-energy CT imaging with limited-angular-range data

In dual-energy computed tomography (DECT), low- and high-kVp data are collected often over a full-angular range (FAR) of 360^○. 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 ≤180^○by using a directional-total-variation (DTV) algorithm.Methods: Image reconstruction from LAR data is formulated as a convex optimization problem in which data-l₂is 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^○to 180^○, 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.

A signal detection model for quantifying overregularization in nonlinear image reconstruction

Many useful image quality metrics for evaluating linear image reconstruction techniques do not apply to or are difficult to interpret for nonlinear image reconstruction. The vast majority of metrics employed for evaluating nonlinear 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 overregularization 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. 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 nonlinear algorithm based on total variation constrained least-squares (TV-LSQ). The resulting images are studied as a function of three parameters: number of views acquired, total variation constraint value, and number of iterations. 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. Loss of signal detectability is seen to correlate well with the blocky or patchy appearance due to overregularization with TV-LSQ, and this trend runs counter to the image RMSE metric, which tends to favor the over-regularized images. 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 nonlinear image reconstruction is used.

Do CNNs Solve the CT Inverse 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

There is a significant discrepancy between the image obtained with the CNN and the image that generated the data. TVmin is able to accurately reconstruct the test images.

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.

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 and deep-learning to solve inverse problems in medical imaging.

Directional-TV algorithm for image reconstruction from limited-angular-range data

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.

Dual-energy CT imaging over non-overlapping, orthogonal arcs of limited-angular ranges

BACKGROUND

Interest exists in dual-energy computed tomography (DECT) imaging with scanning arcs of limited-angular ranges (LARs) for reducing scan time and radiation dose, and for enabling scan configurations of C-arm CT that can avoid possible collision between the rotating X-ray tube/detector and the imaged subject.

OBJECTIVE

In this work, we investigate image reconstruction for a type of configurations of practical DECT interest, referred to as the two-orthogonal-arc configuration, in which low- and high-kVp data are collected over two non-overlapping arcs of equal LAR α, ranging from 30° to 90°, separated by 90°. The configuration can readily be implemented, e.g., on CT with dual sources separated by 90° or with the slow-kVp-switching technique.

METHODS

The directional-total-variation (DTV) algorithm developed previously for image reconstruction in conventional, single-energy CT is tailored to enable image reconstruction in DECT with two-orthogonal-arc configurations.

RESULTS

Performing visual inspection and quantitative analysis of monochromatic images obtained and effective atomic numbers estimated, we observe that the monochromatic images of the DTV algorithm from LAR data are with substantially reduced LAR artifacts, which are observed otherwise in those of existing algorithms, and thus visually correlate reasonably well, in terms of metrics PCC and nMI, with their reference images obtained from full-angular-range data. In addition, effective atomic numbers estimated from LAR data of DECT with two-orthogonal-arc configurations are in reasonable agreement, with relative errors up to ∼  10%, with those estimated from full-angular-range data in DECT.

CONCLUSIONS

The results acquired in the work may yield insights into the design of LAR configurations of practical dual-energy application relevance in diagnostic CT or C-arm CT imaging.

Non-convex primal-dual algorithm for image reconstruction in spectral CT

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.

Optimization-based algorithm for solving the discrete x-ray transform with nonlinear partial volume effect

Purpose: Inverting the discrete x-ray transform (DXT) with the nonlinear partial volume (NLPV) effect, which we refer to as the NLPV DXT, remains of theoretical and practical interest. We propose an optimization-based algorithm for accurately and directly inverting the NLPV DXT. Methods: Formulating the inversion of the NLPV DXT as a nonconvex optimization program, we propose an iterative algorithm, referred to as the nonconvex primal-dual (NCPD) algorithm, to solve the problem. We obtain the NCPD algorithm by modifying a first-order primal-dual algorithm to address the nonconvex optimization. Subsequently, we perform quantitative studies to verify and characterize the NCPD algorithm. Results: In addition to proposing the NCPD algorithm, we perform numerical studies to verify that the NCPD algorithm can reach the devised numerically necessary convergence conditions and, under the study conditions considered, invert the NLPV DXT by yielding numerically accurate image reconstruction. Conclusion: We have developed and verified with numerical studies the NCPD algorithm for accurate inversion of the NLPV DXT. The study and results may yield insights into the effective compensation for the NLPV artifacts in CT imaging and into the algorithm development for nonconvex optimization programs in CT and other tomographic imaging technologies.

Optimization-Based Image Reconstruction From Low-Count, List-Mode TOF-PET Data

OBJECTIVE

We investigate an optimization-based approach to image reconstruction from list-mode data in digital time-of-flight (TOF) positron emission tomography (PET) imaging.

METHOD

In the study, the image to be reconstructed is designed as a solution to a convex, non-smooth optimization program, and a primal-dual algorithm is developed for image reconstruction by solving the optimization program. The algorithm is first applied to list-mode TOF-PET data of a typical count level from physical phantoms and a human subject. Subsequently, we explore the algorithm's potential for image reconstruction in low-dose and/or fast TOF-PET imaging of practical interest by applying the algorithm to list-mode TOF-PET data of different, low-count levels from the same physical phantoms and human subject.

RESULTS

Visual inspection and quantitative-metric analysis reveal that the optimization reconstruction approach investigated can yield images with enhanced spatial and contrast resolution, suppressed image noise, and increased axial volume coverage over the reference images obtained with a standard clinical reconstruction algorithm especially for low-dose TOF-PET data.

SIGNIFICANCE

The optimization-based reconstruction approach can be exploited for yielding insights into potential quality upper bound of reconstructed images in, and design of scanning protocols of, TOF-PET imaging of practical significance.

Imaging of fiber-like structures in digital breast tomosynthesis

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. Our 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.

Estimating the spectrum in computed tomography via Kullback-Leibler divergence constrained optimization

PURPOSE

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.

METHODS

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.

RESULTS

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.

CONCLUSIONS

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.

Reduction of Angularly-Varying-Data Truncation in C-Arm CBCT Imaging

C-arm cone-beam computed tomography (CBCT) has been used increasingly as an imaging tool for yielding 3D anatomical information about the subjects in surgical and interventional procedures. In the clinical applications, the limited field-of-view (FOV) of C-arm CBCT can lead to significant data truncation, resulting in image artifacts that can obscure low contrast tumor embedded within soft-tissue background, thus limiting the utility of C-arm CBCT. The truncation issue can become serious as most of the surgical and interventional procedures would involve devices and tubes that are placed outside the FOV of C-arm CBCT and thus can engender angularly-varying-data truncation. Existing methods may not be adequately applicable to dealing with the angularly-varying truncation. In this work, we seek to reduce truncation artifacts by tailoring optimization-based reconstruction directly from truncated data, without performing pre-reconstruction data compensation, collected from physical phantoms and human subjects. The reconstruction problem is formulated as a constrained optimization program in which a data-derivative-ℓ2-norm fidelity is included for effectively suppressing image artifacts caused by the angularly-varying-data truncation, and the generic Chambolle-Pock algorithm is tailored to solve the optimization program. The results of the study suggest that an appropriately designed optimization-based reconstruction can be exploited for yielding images with reduced artifacts caused by angularly-varying-data truncation.

Investigating simulation-based metrics for characterizing linear iterative reconstruction in digital breast tomosynthesis

PURPOSE

Simulation-based image quality metrics are adapted and investigated for characterizing the parameter dependences of linear iterative image reconstruction for DBT.

METHODS

Three metrics based on a 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 with resolution bar patterns at varying angles, and the ROI-HO metric is applied to two tasks relevant to DBT: lesion detection, modeled by use of a large, low-contrast signal, and microcalcification detection, modeled by use of a small, high-contrast signal. The RMSE metric trends are compared with visual assessment of the reconstructed bar-pattern phantom. The ROI-HO metric trends are compared with 3D reconstructed images from ACR phantom data acquired with a Hologic Selenia Dimensions DBT system.

RESULTS

Sensitivity of the image RMSE to mean pixel value is found to limit its applicability to the assessment of DBT image reconstruction. The 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 the reconstructed ACR phantom images appear to show a similar dependence with regularization strength.

CONCLUSIONS

From the limited studies presented it appears that image gradient RMSE trends correspond with visual assessment better than image RMSE for DBT image reconstruction. The ROI-HO metric for both detection tasks also appears to reflect visual trends in the ACR phantom reconstructions as a function of regularization strength. We point out, however, that the true utility of these metrics can only be assessed after amassing more data.

Algorithm-enabled partial-angular-scan configurations for dual-energy CT

PURPOSE

We seek to investigate an optimization-based one-step method for image reconstruction that explicitly compensates for nonlinear spectral response (i.e., the beam-hardening effect) in dual-energy CT, to investigate the feasibility of the one-step method for enabling two dual-energy partial-angular-scan configurations, referred to as the short- and half-scan configurations, on standard CT scanners without involving additional hardware, and to investigate the potential of the short- and half-scan configurations in reducing imaging dose and scan time in a single-kVp-switch full-scan configuration in which two full rotations are made for collection of dual-energy data.

METHODS

We use the one-step method to reconstruct images directly from dual-energy data through solving a nonconvex optimization program that specifies the images to be reconstructed in dual-energy CT. Dual-energy full-scan data are generated from numerical phantoms and collected from physical phantoms with the standard single-kVp-switch full-scan configuration, whereas dual-energy short- and half-scan data are extracted from the corresponding full-scan data. Besides visual inspection and profile-plot comparison, the reconstructed images are analyzed also in quantitative studies based upon tasks of linear-attenuation-coefficient and material-concentration estimation and of material differentiation.

RESULTS

Following the performance of a computer-simulation study to verify that the one-step method can reconstruct numerically accurately basis and monochromatic images of numerical phantoms, we reconstruct basis and monochromatic images by using the one-step method from real data of physical phantoms collected with the full-, short-, and half-scan configurations. Subjective inspection based upon visualization and profile-plot comparison reveals that monochromatic images, which are used often in practical applications, reconstructed from the full-, short-, and half-scan data are largely visually comparable except for some differences in texture details. Moreover, quantitative studies based upon tasks of linear-attenuation-coefficient and material-concentration estimation and of material differentiation indicate that the short- and half-scan configurations yield results in close agreement with the ground-truth information and that of the full-scan configuration.

CONCLUSIONS

The one-step method considered can compensate effectively for the nonlinear spectral response in full- and partial-angular-scan dual-energy CT. It can be exploited for enabling partial-angular-scan configurations on standard CT scanner without involving additional hardware. Visual inspection and quantitative studies reveal that, with the one-step method, partial-angular-scan configurations considered can perform at a level comparable to that of the full-scan configuration, thus suggesting the potential of the two partial-angular-scan configurations in reducing imaging dose and scan time in the standard single-kVp-switch full-scan CT in which two full rotations are performed. The work also yields insights into the investigation and design of other nonstandard scan configurations of potential practical significance in dual-energy CT.

A Convex Reconstruction Model for X-ray Tomographic Imaging with Uncertain Flat-fields

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.

Image reconstruction and scan configurations enabled by optimization-based algorithms in multispectral CT

Optimization-based algorithms for image reconstruction in multispectral (or photon-counting) computed tomography (MCT) remains a topic of active research. The challenge of optimization-based image reconstruction in MCT stems from the inherently non-linear data model that can lead to a non-convex optimization program for which no mathematically exact solver seems to exist for achieving globally optimal solutions. In this work, based upon a non-linear data model, we design a non-convex optimization program, derive its first-order-optimality conditions, and propose an algorithm to solve the program for image reconstruction in MCT. In addition to consideration of image reconstruction for the standard scan configuration, the emphasis is on investigating the algorithm's potential for enabling non-standard scan configurations with no or minimum hardware modification to existing CT systems, which has potential practical implications for lowered hardware cost, enhanced scanning flexibility, and reduced imaging dose/time in MCT. Numerical studies are carried out for verification of the algorithm and its implementation, and for a preliminary demonstration and characterization of the algorithm in reconstructing images and in enabling non-standard configurations with varying scanning angular range and/or x-ray illumination coverage in MCT.

A Spectral CT Method to Directly Estimate Basis Material Maps From Experimental Photon-Counting Data

The proposed spectral CT method solves the constrained one-step spectral CT reconstruction (cOSSCIR) optimization problem to estimate basis material maps while modeling the nonlinear X-ray detection process and enforcing convex constraints on the basis map images. In order to apply the optimization-based reconstruction approach to experimental data, the presented method empirically estimates the effective energy-window spectra using a calibration procedure. The amplitudes of the estimated spectra were further optimized as part of the reconstruction process to reduce ring artifacts. A validation approach was developed to select constraint parameters. The proposed spectral CT method was evaluated through simulations and experiments with a photon-counting detector. Basis material map images were successfully reconstructed using the presented empirical spectral modeling and cOSSCIR optimization approach. In simulations, the cOSSCIR approach accurately reconstructed the basis map images (<1% error). In experiments, the proposed method estimated the low-density polyethylene region of the basis maps with 0.5% error in the PMMA image and 4% error in the aluminum image. For the Teflon region, the experimental results demonstrated 8% and 31% error in the PMMA and aluminum basis material maps, respectively, compared with -24% and 126% error without estimation of the effective energy window spectra, with residual errors likely due to insufficient modeling of detector effects. The cOSSCIR algorithm estimated the material decomposition angle to within 1.3 degree error, where, for reference, the difference in angle between PMMA and muscle tissue is 2.1 degrees. The joint estimation of spectral-response scaling coefficients and basis material maps was found to reduce ring artifacts in both a phantom and tissue specimen. The presented validation procedure demonstrated feasibility for the automated determination of algorithm constraint parameters.

Optimization-based image reconstruction with artifact reduction in C-arm CBCT

We investigate an optimization-based reconstruction, with an emphasis on image-artifact reduction, from data collected in C-arm cone-beam computed tomography (CBCT) employed in image-guided interventional procedures. In the study, an image to be reconstructed is formulated as a solution to a convex optimization program in which a weighted data divergence is minimized subject to a constraint on the image total variation (TV); a data-derivative fidelity is introduced in the program specifically for effectively suppressing dominant, low-frequency data artifact caused by, e.g. data truncation; and the Chambolle-Pock (CP) algorithm is tailored to reconstruct an image through solving the program. Like any other reconstructions, the optimization-based reconstruction considered depends upon numerous parameters. We elucidate the parameters, illustrate their determination, and demonstrate their impact on the reconstruction. The optimization-based reconstruction, when applied to data collected from swine and patient subjects, yields images with visibly reduced artifacts in contrast to the reference reconstruction, and it also appears to exhibit a high degree of robustness against distinctively different anatomies of imaged subjects and scanning conditions of clinical significance. Knowledge and insights gained in the study may be exploited for aiding in the design of practical reconstructions of truly clinical-application utility.

Investigation of optimization-based reconstruction with an image-total-variation constraint in PET

Interest remains in reconstruction-algorithm research and development for possible improvement of image quality in current PET imaging and for enabling innovative PET systems to enhance existing, and facilitate new, preclinical and clinical applications. Optimization-based image reconstruction has been demonstrated in recent years of potential utility for CT imaging applications. In this work, we investigate tailoring the optimization-based techniques to image reconstruction for PET systems with standard and non-standard scan configurations. Specifically, given an image-total-variation (TV) constraint, we investigated how the selection of different data divergences and associated parameters impacts the optimization-based reconstruction of PET images. The reconstruction robustness was explored also with respect to different data conditions and activity up-takes of practical relevance. A study was conducted particularly for image reconstruction from data collected by use of a PET configuration with sparsely populated detectors. Overall, the study demonstrates the robustness of the TV-constrained, optimization-based reconstruction for considerably different data conditions in PET imaging, as well as its potential to enable PET configurations with reduced numbers of detectors. Insights gained in the study may be exploited for developing algorithms for PET-image reconstruction and for enabling PET-configuration design of practical usefulness in preclinical and clinical applications.

Reconstruction of 3D Regions-of-Interest from Data in Reduced Helical Cone-beam Scans

The suffciency conditions are derived for exact image reconstruction of a 3D ROI from projections acquired with a reduced helical scan over an angular range considerably smaller than that required by image reconstruction in, e.g., the conventional long object problem, for which the scanned angular range is often more than 2π. ROI reconstruction is investigated by a recently developed filtered-backprojection algorithm that can make use of data acquired with a reduced helical scan. Preliminary numerical studies demonstrate and validate the ROI reconstruction. This work may have significant practical implications because a reduced scan in CT often translates to reduced motion artifacts and reduced radiation dose delivered to the subject.

An algorithm for constrained one-step inversion of spectral CT data

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.

Artifact reduction in short-scan CBCT by use of optimization-based reconstruction

Increasing interest in optimization-based reconstruction in research on, and applications of, cone-beam computed tomography (CBCT) exists because it has been shown to have to potential to reduce artifacts observed in reconstructions obtained with the Feldkamp-Davis-Kress (FDK) algorithm (or its variants), which is used extensively for image reconstruction in current CBCT applications. In this work, we carried out a study on optimization-based reconstruction for possible reduction of artifacts in FDK reconstruction specifically from short-scan CBCT data. The investigation includes a set of optimization programs such as the image-total-variation (TV)-constrained data-divergency minimization, data-weighting matrices such as the Parker weighting matrix, and objects of practical interest for demonstrating and assessing the degree of artifact reduction. Results of investigative work reveal that appropriately designed optimization-based reconstruction, including the image-TV-constrained reconstruction, can reduce significant artifacts observed in FDK reconstruction in CBCT with a short-scan configuration.

Noise properties of CT images reconstructed by use of constrained total-variation, data-discrepancy minimization

PURPOSE

The authors develop and investigate iterative image reconstruction algorithms based on data-discrepancy minimization with a total-variation (TV) constraint. The various algorithms are derived with different data-discrepancy measures reflecting the maximum likelihood (ML) principle. Simulations demonstrate the iterative algorithms and the resulting image statistical properties for low-dose CT data acquired with sparse projection view angle sampling. Of particular interest is to quantify improvement of image statistical properties by use of the ML data fidelity term.

METHODS

An incremental algorithm framework is developed for this purpose. The instances of the incremental algorithms are derived for solving optimization problems including a data fidelity objective function combined with a constraint on the image TV. For the data fidelity term the authors, compare application of the maximum likelihood principle, in the form of weighted least-squares (WLSQ) and Poisson-likelihood (PL), with the use of unweighted least-squares (LSQ).

RESULTS

The incremental algorithms are applied to projection data generated by a simulation modeling the breast computed tomography (bCT) imaging application. The only source of data inconsistency in the bCT projections is due to noise, and a Poisson distribution is assumed for the transmitted x-ray photon intensity. In the simulations involving the incremental algorithms an ensemble of images, reconstructed from 1000 noise realizations of the x-ray transmission data, is used to estimate the image statistical properties. The WLSQ and PL incremental algorithms are seen to reduce image variance as compared to that of LSQ without sacrificing image bias. The difference is also seen at few iterations--short of numerical convergence of the corresponding optimization problems.

CONCLUSIONS

The proposed incremental algorithms prove effective and efficient for iterative image reconstruction in low-dose CT applications particularly with sparse-view projection data.

MOCCA: Mirrored Convex/Concave Optimization for Nonconvex Composite Functions

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 a 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.

Use of the Hotelling observer to optimize image reconstruction in digital breast tomosynthesis

We propose an implementation of the Hotelling observer that can be applied to the optimization of linear image reconstruction algorithms in digital breast tomosynthesis. The method is based on considering information within a specific region of interest, and it is applied to the optimization of algorithms for detectability of microcalcifications. Several linear algorithms are considered: simple back-projection, filtered back-projection, back-projection filtration, and [Formula: see text]-tomography. The optimized algorithms are then evaluated through the reconstruction of phantom data. The method appears robust across algorithms and parameters and leads to the generation of algorithm implementations which subjectively appear optimized for the task of interest.

Task-based optimization of dedicated breast CT via Hotelling observer metrics

PURPOSE

The purpose of this work is to develop and demonstrate a set of practical metrics for CT systems optimization. These metrics, based on the Hotelling observer (HO) figure of merit, are task-based. The authors therefore take the specific example of optimizing a dedicated breast CT system, including the reconstruction algorithm, for two relevant tasks, signal detection and Rayleigh discrimination.

METHODS

A dedicated breast CT system is simulated using specifications in the literature from an existing prototype. The authors optimize configuration and image reconstruction algorithm parameters for two tasks: the detection of simulated microcalcifications and the discrimination of two adjacent, high-contrast signals, known as the Rayleigh discrimination task. The effects on task performance of breast diameter, signal location, image grid size, projection view number, and reconstruction filter were all investigated. Two HO metrics were evaluated: the percentage of correct decisions in a two-alternative forced choice experiment (equivalent to area under the ROC curve or AUC), and the HO efficiency, defined as the squared ratio of HO signal-to-noise ratio (SNR) in the reconstructed image to HO SNR in the projection data.

RESULTS

The ease and efficiency of the HO metric computation allows a rapid high-resolution survey of many system parameters. Optimization of a range of system parameters using the HO results in images that subjectively appear optimal for the tasks investigated. Further, the results of assessment through the HO reproduce closely many existing results in the literature regarding the impact of parameter selection on image quality.

CONCLUSIONS

This study demonstrates the utility of a task-based approach to system design, evaluation, and optimization. The methodology presented is equally applicable to determining the impact of a wide range of factors, including patient parameters, system and acquisition design, and the reconstruction algorithm. The results demonstrate the versatility of the proposed HO formalism by not only generating a set of parameters that are optimal for a given task but also by qualitatively reproducing many existing results from the breast CT literature. Meanwhile, the implementation of the proposed methodology is straightforward and entirely simulation-based. This is an attractive feature for many system optimization problems, where the goal is to analyze the individual system components such as the image reconstruction algorithm. Final assessment of the system as a whole should be based also on real data studies.

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.

Algorithm-enabled exploration of image-quality potential of cone-beam CT in image-guided radiation therapy

Kilo-voltage (KV) cone-beam computed tomography (CBCT) unit mounted onto a linear accelerator treatment system, often referred to as on-board imager (OBI), plays an increasingly important role in image-guided radiation therapy. While the FDK algorithm is currently used for reconstructing images from clinical OBI data, optimization-based reconstruction has also been investigated for OBI CBCT. An optimization-based reconstruction involves numerous parameters, which can significantly impact reconstruction properties (or utility). The success of an optimization-based reconstruction for a particular class of practical applications thus relies strongly on appropriate selection of parameter values. In the work, we focus on tailoring the constrained-TV-minimization-based reconstruction, an optimization-based reconstruction previously shown of some potential for CBCT imaging conditions of practical interest, to OBI imaging through appropriate selection of parameter values. In particular, for given real data of phantoms and patient collected with OBI CBCT, we first devise utility metrics specific to OBI-quality-assurance tasks and then apply them to guiding the selection of parameter values in constrained-TV-minimization-based reconstruction. The study results show that the reconstructions are with improvement, relative to clinical FDK reconstruction, in both visualization and quantitative assessments in terms of the devised utility metrics.

Compressive sensing in medical imaging

The promise of compressive sensing, exploitation of compressibility to achieve high quality image reconstructions with less data, has attracted a great deal of attention in the medical imaging community. At the Compressed Sensing Incubator meeting held in April 2014 at OSA Headquarters in Washington, DC, presentations were given summarizing some of the research efforts ongoing in compressive sensing for x-ray computed tomography and magnetic resonance imaging systems. This article provides an expanded version of these presentations. Sparsity-exploiting reconstruction algorithms that have gained popularity in the medical imaging community are studied, and examples of clinical applications that could benefit from compressive sensing ideas are provided. The current and potential future impact of compressive sensing on the medical imaging field is discussed.

The effects of extending the spectral information acquired by a photon-counting detector for spectral CT

Photon-counting x-ray detectors with pulse-height analysis provide spectral information that may improve material decomposition and contrast-to-noise ratio (CNR) in CT images. The number of energy measurements that can be acquired simultaneously on a detector pixel is equal to the number of comparator channels. Some spectral CT designs have a limited number of comparator channels, due to the complexity of readout electronics. The spectral information could be extended by changing the comparator threshold levels over time, sub pixels, or view angle. However, acquiring more energy measurements than comparator channels increases the noise and/or dose, due to differences in noise correlations across energy measurements and decreased dose utilisation. This study experimentally quantified the effects of acquiring more energy measurements than comparator channels using a bench-top spectral CT system. An analytical and simulation study modeling an ideal detector investigated whether there was a net benefit for material decomposition or optimal energy weighting when acquiring more energy measurements than comparator channels. Experimental results demonstrated that in a two-threshold acquisition, acquiring the high-energy measurement independently from the low-energy measurement increased noise standard deviation in material-decomposition basis images by factors of 1.5-1.7 due to changes in covariance between energy measurements. CNR in energy-weighted images decreased by factors of 0.92-0.71. Noise standard deviation increased by an additional factor of [Formula: see text] due to reduced dose utilisation. The results demonstrated no benefit for two-material decomposition noise or energy-weighted CNR when acquiring more energy measurements than comparator channels. Understanding the noise penalty of acquiring more energy measurements than comparator channels is important for designing spectral detectors and for designing experiments and interpreting data from prototype systems with a limited number of comparator channels.

EMPIRICAL AVERAGE-CASE RELATION BETWEEN UNDERSAMPLING AND SPARSITY IN X-RAY CT

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 inspired 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 images 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 white noise.

Region of interest based Hotelling observer for computed tomography with comparison to alternative methods

We compare several approaches to estimation of Hotelling observer (HO) performance in x-ray computed tomography (CT). We consider the case where the signal of interest is small so that the reconstructed image can be restricted to a small region of interest (ROI) surrounding the signal. This reduces the dimensionality of the image covariance matrix so that direct computation of HO metrics within the ROI is feasible. We propose that this approach is directly applicable to systems optimization in CT; however, many alternative approaches exist, which make computation of HO performance tractable through a range of approximations, assumptions, or estimation strategies. Here, we compare several of these methods, including the use of Laguerre-Gauss channels, discrete Fourier domain computation of the HO (which assumes noise stationarity), and two approaches to HO estimation through samples of noisy images. Since our method computes HO performance exactly within an ROI, this allows us to investigate the validity of the assumptions inherent in various common approaches to HO estimation, such as the stationarity assumption in the case of the discrete Fourier transform domain method.

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.

Investigation of iterative image reconstruction in low-dose breast CT

There is interest in developing computed tomography (CT) dedicated to breast-cancer imaging. Because breast tissues are radiation-sensitive, the total radiation exposure in a breast-CT scan is kept low, often comparable to a typical two-view mammography exam, thus resulting in a challenging low-dose-data-reconstruction problem. In recent years, evidence has been found that suggests that iterative reconstruction may yield images of improved quality from low-dose data. In this work, based upon the constrained image total-variation minimization program and its numerical solver, i.e., the adaptive steepest descent-projection onto the convex set (ASD-POCS), we investigate and evaluate iterative image reconstructions from low-dose breast-CT data of patients, with a focus on identifying and determining key reconstruction parameters, devising surrogate utility metrics for characterizing reconstruction quality, and tailoring the program and ASD-POCS to the specific reconstruction task under consideration. The ASD-POCS reconstructions appear to outperform the corresponding clinical FDK reconstructions, in terms of subjective visualization and surrogate utility metrics.

Constrained TpV Minimization for Enhanced Exploitation of Gradient Sparsity: Application to CT Image Reconstruction

Exploiting sparsity in the image gradient magnitude has proved to be an effective means for reducing the sampling rate in the projection view angle in computed tomography (CT). Most of the image reconstruction algorithms, developed for this purpose, solve a nonsmooth convex optimization problem involving the image total variation (TV). The TV seminorm is the ℓ₁ norm of the image gradient magnitude, and reducing the ℓ₁ norm is known to encourage sparsity in its argument. Recently, there has been interest in employing nonconvex ℓ_p quasinorms with 0<p<1 for sparsity exploiting image reconstruction, which is potentially more effective than ℓ₁ because nonconvex ℓ_p is closer to ℓ₀-a direct measure of sparsity. This paper develops algorithms for constrained minimization of the total p-variation (TpV), ℓ_p of the image gradient. Use of the algorithms is illustrated in the context of breast CT-an imaging modality that is still in the research phase and for which constraints on X-ray dose are extremely tight. The TpV-based image reconstruction algorithms are demonstrated on computer simulated data for exploiting gradient magnitude sparsity to reduce the projection view angle sampling. The proposed algorithms are applied to projection data from a realistic breast CT simulation, where the total X-ray dose is equivalent to two-view digital mammography. Following the simulation survey, the algorithms are then demonstrated on a clinical breast CT data set.