Proximal ADMM for multi-channel image reconstruction in spectral X-ray CT

Robust PCA with Lw,∗ and L2,1 Norms: A Novel Method for Low-Quality Retinal Image Enhancement

Nonmydriatic retinal fundus images often suffer from quality issues and artifacts due to ocular or systemic comorbidities, leading to potential inaccuracies in clinical diagnoses. In recent times, deep learning methods have been widely employed to improve retinal image quality. However, these methods often require large datasets and lack robustness in clinical settings. Conversely, the inherent stability and adaptability of traditional unsupervised learning methods, coupled with their reduced reliance on extensive data, render them more suitable for real-world clinical applications, particularly in the limited data context of high noise levels or a significant presence of artifacts. However, existing unsupervised learning methods encounter challenges such as sensitivity to noise and outliers, reliance on assumptions like cluster shapes, and difficulties with scalability and interpretability, particularly when utilized for retinal image enhancement. To tackle these challenges, we propose a novel robust PCA (RPCA) method with low-rank sparse decomposition that also integrates affine transformations τi, weighted nuclear norm, and the L2,1 norms, aiming to overcome existing method limitations and to achieve image quality improvement unseen by these methods. We employ the weighted nuclear norm (Lw,∗) to assign weights to singular values to each retinal images and utilize the L2,1 norm to eliminate correlated samples and outliers in the retinal images. Moreover, τi is employed to enhance retinal image alignment, making the new method more robust to variations, outliers, noise, and image blurring. The Alternating Direction Method of Multipliers (ADMM) method is used to optimally determine parameters, including τi, by solving an optimization problem. Each parameter is addressed separately, harnessing the benefits of ADMM. Our method introduces a novel parameter update approach and significantly improves retinal image quality, detecting cataracts, and diabetic retinopathy. Simulation results confirm our method's superiority over existing state-of-the-art methods across various datasets.

Direct Multi-Material Reconstruction via Iterative Proximal Adaptive Descent for Spectral CT Imaging

Spectral computed tomography (spectral CT) is a promising medical imaging technology because of its ability to provide information on material characterization and quantification. However, with an increasing number of basis materials, the nonlinearity of measurements causes difficulty in decomposition. In addition, noise amplification and beam hardening further reduce image quality. Thus, improving the accuracy of material decomposition while suppressing noise is pivotal for spectral CT imaging. This paper proposes a one-step multi-material reconstruction model as well as an iterative proximal adaptive decent method. In this approach, a proximal step and a descent step with adaptive step size are designed under the forward-backward splitting framework. The convergence analysis of the algorithm is further discussed according to the convexity of the optimization objective function. For simulation experiments with different noise levels, the peak signal-to-noise ratio (PSNR) obtained by the proposed method increases approximately 23 dB, 14 dB, and 4 dB compared to those of other algorithms. Magnified areas of thorax data further demonstrated that the proposed method has a better ability to preserve details in tissues, bones, and lungs. Numerical experiments verify that the proposed method efficiently reconstructed the material maps, and reduced noise and beam hardening artifacts compared with the state-of-the-art methods.

Bicolor K-edge spectral photon-counting CT imaging for the diagnosis of thoracic endoleaks: A dynamic phantom study

PURPOSE

The purpose of this study was to investigate the feasibility of identifying and characterizing the three most common types of endoleaks within a thoracic aorta aneurysm model using bicolor K-edge imaging with a spectral photon-counting computing tomography (SPCCT) system in combination with a biphasic contrast agent injection.

MATERIALS AND METHODS

Three types of thoracic endoleaks (type 1, 2 and 3) were created in a dynamic anthropomorphic thoracic aorta phantom. Protocol consisted in an injection of an iodinated contrast material followed 80 seconds after an injection of a gadolinium-based contrast agent (GBCA). The phantom was scanned using a clinical prototype SPCCT during bicolor phase imaging consisting in an early distribution of GBCA and a late distribution of iodine. Conventional and spectral images were reconstructed for differentiating between the contrast agents and measuring their respective attenuation values and concentrations inside and outside the stent graft.

RESULTS

Conventional images failed to provide specific dynamic imaging contrast agents in the aneurysmal sac and outside the stent graft while spectral images differentiated their specific distribution. In type 1 and 3 thoracic endoleaks, GBCA concentration was measured outside the stent graft at 6.1 ± 3.7 (standard deviation [SD]) mg/mL and 6.0 ± 4.0 (SD) mg/mL, respectively, in favor of an early blood flow. In type 2 thoracic endoleak, iodine was measured outside the stent graft at 24.3 ± 5.5 (SD) mg/mL in favor of a late blood flow in the aneurysmal sac.

CONCLUSION

Bicolor K-edge imaging enabled SPCCT allows a bicolor characterization of thoracic aorta endoleaks in a single acquisition in combination with a biphasic contrast agent injection.

Distributed and scalable optimization for robust proton treatment planning

BACKGROUND

The importance of robust proton treatment planning to mitigate the impact of uncertainty is well understood. However, its computational cost grows with the number of uncertainty scenarios, prolonging the treatment planning process.

PURPOSE

We developed a fast and scalable distributed optimization platform that parallelizes the robust proton treatment plan computation over the uncertainty scenarios.

METHODS

We modeled the robust proton treatment planning problem as a weighted least-squares problem. To solve it, we employed an optimization technique called the alternating direction method of multipliers with Barzilai-Borwein step size (ADMM-BB). We reformulated the problem in such a way as to split the main problem into smaller subproblems, one for each proton therapy uncertainty scenario. The subproblems can be solved in parallel, allowing the computational load to be distributed across multiple processors (e.g., CPU threads/cores). We evaluated ADMM-BB on four head-and-neck proton therapy patients, each with 13 scenarios accounting for 3 mm setup and 3.5% range uncertainties. We then compared the performance of ADMM-BB with projected gradient descent (PGD) applied to the same problem.

RESULTS

For each patient, ADMM-BB generated a robust proton treatment plan that satisfied all clinical criteria with comparable or better dosimetric quality than the plan generated by PGD. However, ADMM-BB's total runtime averaged about 6 to 7 times faster. This speedup increased with the number of scenarios.

CONCLUSIONS

ADMM-BB is a powerful distributed optimization method that leverages parallel processing platforms, such as multicore CPUs, GPUs, and cloud servers, to accelerate the computationally intensive work of robust proton treatment planning. This results in (1) a shorter treatment planning process and (2) the ability to consider more uncertainty scenarios, which improves plan quality.

Deep learning-based virtual noncalcium imaging in multiple myeloma using dual-energy CT

BACKGROUND

Dual-energy CT with virtual noncalcium (VNCa) images allows the evaluation of focal intramedullary bone marrow involvement in patients with multiple myeloma. However, current commercial VNCa techniques suffer from excessive image noise and artifacts due to material decomposition used in synthesizing VNCa images.

OBJECTIVES

In this work, we aim to improve VNCa image quality for the assessment of focal multiple myeloma, using an Artificial intelligence based Generalizable Algorithm for mulTi-Energy CT (AGATE) method.

MATERIALS AND METHODS

AGATE method used a custom dual-task convolutional neural network (CNN) that concurrently carries out material classification and quantification. The material classification task provided an auxiliary regularization to the material quantification task. CNN parameters were optimized using custom loss functions that involved cross-entropy, physics-informed constraints, structural redundancy in spectral and material images, and texture information in spectral images. For training data, CT phantoms (diameters 30 to 45 cm) with tissue-mimicking inserts were scanned on a third generation dual-source CT system. Scans were performed at routine dose and half of the routine dose. Small image patches (i.e., 40 × 40 pixels) of tissue-mimicking inserts with known basis material densities were extracted for training samples. Numerically simulated insert materials with various shapes increased diversity of training samples. Generalizability of AGATE was evaluated using CT images from phantoms and patients. In phantoms, material decomposition accuracy was estimated using mean-absolute-percent-error (MAPE), using physical inserts that were not used during the training. Noise power spectrum (NPS) and modulation transfer function (MTF) were compared across phantom sizes and radiation dose levels. Five patients with multiple myeloma underwent dual-energy CT, with VNCa images generated using a commercial method and AGATE. Two fellowship-trained musculoskeletal radiologists reviewed the VNCa images (commercial and AGATE) side-by-side using a dual-monitor display, blinded to VNCa type, rating the image quality for focal multiple myeloma lesion visualization using a 5-level Likert comparison scale (-2 = worse visualization and diagnostic confidence, -1 = worse visualization but equivalent diagnostic confidence, 0 = equivalent visualization and diagnostic confidence, 1 = improved visualization but equivalent diagnostic confidence, 2 = improved visualization and diagnostic confidence). A post hoc assignment of comparison ratings was performed to rank AGATE images in comparison to commercial ones.

RESULTS

AGATE demonstrated consistent material quantification accuracy across phantom sizes and radiation dose levels, with MAPE ranging from 0.7% to 4.4% across all testing materials. Compared to commercial VNCa images, the AGATE-synthesized VNCa images yielded considerably lower image noise (50-77% noise reduction) without compromising noise texture or spatial resolution across different phantom sizes and two radiation doses. AGATE VNCa images had markedly reduced area under NPS curves and maintained NPS peak frequency (0.7 lp/cm to 1.0 lp/cm), with similar MTF curves (50% MTF at 3.0 lp/cm). In patients, AGATE demonstrated reduced image noise and artifacts with improved delineation of focal multiple myeloma lesions (all readers comparison scores indicating improved overall diagnostic image quality [scores 1 or 2]).

CONCLUSIONS

AGATE demonstrated reduced noise and artifacts in VNCa images and ability to improve visualization of bone marrow lesions for assessing multiple myeloma.

Dual Alternating Direction Method of Multipliers for Inverse Imaging

Inverse imaging covers a wide range of imaging applications, including super-resolution, deblurring, and compressive sensing. We propose a novel scheme to solve such problems by combining duality and the alternating direction method of multipliers (ADMM). In addition to a conventional ADMM process, we introduce a second one that solves the dual problem to find the estimated nontrivial lower bound of the objective function, and the related iteration results are used in turn to guide the primal iterations. We call this D-ADMM, and show that it converges to the global minimum when the regularization function is convex and the optimization problem has at least one optimizer. Furthermore, we show how the scheme can give rise to two specific algorithms, called D-ADMM-L2 and D-ADMM-TV, by having different regularization functions. We compare D-ADMM-TV with other methods on image super-resolution and demonstrate comparable or occasionally slightly better quality results. This paves the way of incorporating advanced operators and strategies designed for basic ADMM into the D-ADMM method as well to further improve the performances of those methods.

Ultra-Low-Dose Spectral CT Based on a Multi-level Wavelet Convolutional Neural Network

Spectral computed tomography (CT) based on a photon-counting detector (PCD) is a promising technique with the potential to improve lesion detection, tissue characterization, and material decomposition. PCD-based scanners have several technical issues including operation in the step-and-scan mode and long data acquisition time. One straightforward solution to these issues is to reduce the number of projection views. However, if the projection data are under-sampled or noisy, it would be challenging to produce a correct solution without precise prior information. Recently, deep-learning approaches have demonstrated impressive performance for under-sampled CT reconstruction. In this work, the authors present a multilevel wavelet convolutional neural network (MWCNN) to address the limitations of PCD-based scanners. Data properties of the proposed method in under-sampled spectral CT are analyzed with respect to the proposed deep-running-network-based image reconstruction using two measures: sampling density and data incoherence. This work presents the proposed method and four different methods to restore sparse sampling. We investigate and compare these methods through a simulation and real experiments. In addition, data properties are quantitatively analyzed and compared for the effect of sparse sampling on the image quality. Our results indicate that both sampling density and data incoherence affect the image quality in the studied methods. Among the different methods, the proposed MWCNN shows promising results. Our method shows the highest performance in terms of various evaluation parameters such as the structural similarity, root mean square error, and resolution. Based on the results of imaging and quantitative evaluation, this study confirms that the proposed deep-running network structure shows excellent image reconstruction in sparse-view PCD-based CT. These results demonstrate the feasibility of sparse-view PCD-based CT using the MWCNN. The advantage of sparse view CT is that it can significantly reduce the radiation dose and obtain images with several energy bands by fusing PCDs. These results indicate that the MWCNN possesses great potential for sparse-view PCD-based CT.

Multi-segment spectral reconstruction via zero-value set prior

Most scanning schemes of multi-energy computed tomography (MECT) require multiple sets of full-scan measurements under different x-ray spectra, which limits the application of MECT with incomplete scan. To handle this problem, a flexible MECT scanning strategy is proposed in this paper, which divides one half scan into three curves. Also, a novel MECT reconstruction algorithm is developed to relax the requirement of data acquisition of MECT. For MECT, gradient images of CT images at different energies ideally share the same position of zero-value set (Pos-OS) for the same object. Based on this observation, the characteristics of limited-angle artifacts is first explored, and it is found that the limited-angle artifacts in the image domain are closely related to the angle trajectory of the scan. Inspired by this discovery, the Pos-OS of the gradient image from the fusion CT image is extracted, and it is incorporated as prior knowledge into the TV minimization model in the form of equality constraints. The alternating direction method is exploited to solve the improved optimization model iteratively. Based on this, the proposed algorithm is derived to eliminate the limited angle artifacts in the image domain.The experimental results show that the proposed method achieves higher reconstruction quality under the designed scanning configuration than other methods in the literature.

Regularization by denoising sub-sampled Newton method for spectral CT multi-material decomposition

Spectral Computed Tomography (CT) is an emerging technology that enables us to estimate the concentration of basis materials within a scanned object by exploiting different photon energy spectra. In this work, we aim at efficiently solving a model-based maximum-a-posterior problem to reconstruct multi-materials images with application to spectral CT. In particular, we propose to solve a regularized optimization problem based on a plug-in image-denoising function using a randomized second order method. By approximating the Newton step using a sketching of the Hessian of the likelihood function, it is possible to reduce the complexity while retaining the complex prior structure given by the data-driven regularizer. We exploit a non-uniform block sub-sampling of the Hessian with inexact but efficient conjugate gradient updates that require only Jacobian-vector products for denoising term. Finally, we show numerical and experimental results for spectral CT materials decomposition. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 1'.

Photon-counting x-ray detectors for CT

The introduction of photon-counting detectors is expected to be the next major breakthrough in clinical x-ray computed tomography (CT). During the last decade, there has been considerable research activity in the field of photon-counting CT, in terms of both hardware development and theoretical understanding of the factors affecting image quality. In this article, we review the recent progress in this field with the intent of highlighting the relationship between detector design considerations and the resulting image quality. We discuss detector design choices such as converter material, pixel size, and readout electronics design, and then elucidate their impact on detector performance in terms of dose efficiency, spatial resolution, and energy resolution. Furthermore, we give an overview of data processing, reconstruction methods and metrics of imaging performance; outline clinical applications; and discuss potential future developments.

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.

Multi-energy CT reconstruction using tensor nonlocal similarity and spatial sparsity regularization

Background

Multi-energy computed tomography (MECT) based on a photon-counting detector is an emerging imaging modality that collects projections at several energy bins with a single scan. However, the limited number of photons collected into the divided, narrow energy bins results in high quantum noise levels in reconstructed images. This study aims to improve MECT image quality by minimizing noise levels while retaining image details.

Methods

A novel MECT reconstruction method was proposed by exploiting the nonlocal tensor similarity among interchannel images and spatial sparsity in single-channel images. Similar patches were initially extracted from the interchannel images in spectral and spatial domains, then stacked into a new three-order tensor. Intrinsic tensor sparsity regularization that combined the Tuker and canonical polyadic (CP) low-rank decomposition techniques were applied to exploit the nonlocal similarity of the formulated tensor. Spatial sparsity in single-channel images was modeled by total variation (TV) regularization that utilizes the compressibility of gradient image. A new MECT reconstruction model was established by simultaneously incorporating the intrinsic tensor sparsity and TV regularizations. The iterative alternating minimization method was utilized to solve the reconstruction model based on a flexible framework.

Results

The proposed method was applied to the digital phantom and real mouse data to assess its feasibility and reliability. The reconstruction and decomposition results in the mouse data were encouraging and demonstrated the ability of the proposed method in noise suppression while preserving image details, not observed with other methods. Imaging data from the digital phantom illustrated this method as achieving the best intuitive reconstruction and decomposition results among all compared methods. They reduced the root mean square error (RMSE) by 89.75%, 50.75%, and 36.54% on the reconstructed images compared with analytic, TV-based, and tensor-based methods, respectively. This phenomenon was also observed with decomposition results, where the RMSE was also reduced by 97.96%, 67.74%, 72.05%, respectively.

Conclusions

In this study, we proposed a reconstruction method for photon counting detector-based MECT, using the intrinsic tensor sparsity and TV regularizations. Improvements in noise suppression and detail preservation in the digital phantom and real mouse data were validated by the qualitative and quantitative evaluations on the reconstruction and decomposition results, verifying the potential of the proposed method in MECT reconstruction.

Multi-energy CT imaging for large patients using dual-source photon-counting detector CT

Multi-energy CT imaging of large patients with conventional dual-energy (DE)-CT using an energy-integrating-detector (EID) is challenging due to photon starvation-induced image artifacts, especially in lower tube potential (80-100 kV) images. Here, we performed phantom experiments to investigate the performance of DECT for morbidly obese patients, using an iodine and water material decomposition task as an example, on an emulated dual-source (DS)-photon-counting-detector (PCD)-CT, and compared its performance with a clinical DS-EID-CT. An abdominal CT phantom with iodine inserts of different concentrations was wrapped with tissue-equivalent gel layers to emulate a large patient (50 cm lateral size). The phantom was scanned on a research whole-body single-source (SS)-PCD-CT (140 kV tube potential), a DS-PCD-CT (100/Sn140 kV; Sn140 indicates 140 kV with Sn filter), and a clinical DS-EID-CT (100/Sn140 kV) with the same radiation dose. Phantom scans were repeated five times on each system. The DS-PCD-CT acquisition was emulated by scanning twice on the SS-PCD-CT using different tube potentials. The multi-energy CT images acquired on each system were then reconstructed, and iodine- and water-specific images were generated using material decomposition. The root-mean-square-error (RMSE) between true and measured iodine concentrations were calculated for each system and compared. The images acquired on the DS-EID-CT showed severe artifacts, including ringing, reduced uniformity, and photon starvation artifacts, especially for low-energy images. These were largely reduced in DS-PCD-CT images. The CT number difference that was measured using regions-of-interest across field-of-view were reduced from 20.3 ± 0.9 (DS-EID-CT) to 2.5 ± 0.4 HU on DS-PCD-CT, showing improved image uniformity using DS-PCD-CT. Iodine RMSE was reduced from 3.42 ± 0.03 mg ml^-1 (SS-PCD-CT) and 2.90 ± 0.03 mg ml^-1 (DS-EID-CT) to 2.39 ± 0.05 mg ml^-1 using DS-PCD-CT. DS-PCD-CT out-performed a clinical DS-EID-CT for iodine and water-based material decomposition on phantom emulating obese patients by reducing image artifacts and improving iodine quantification (RMSE reduced by 20%). With DS-PCD-CT, multi-energy CT can be performed on large patients that cannot be accommodated with current DECT.

Reconstruction method for DECT with one half-scan plus a second limited-angle scan using prior knowledge of complementary support set (Pri-CSS)

Dual-energy computed tomography (DECT) has capability to improve material differentiation, but most scanning schemes require two sets of full-scan measurements at different x-ray spectra, limiting its application to imaging system with incomplete scan. In this study, using one half-scan and a second limited-angle scan, we propose a DECT reconstruction method by exploiting the consistent information of gradient images at high- and low-energy spectra, which relaxes the requirement of data acquisition of DECT. Based on the theory of sampling condition analysis, the complementary support set of gradient images plays an important role in image reconstruction because it constitutes the sufficient and necessary condition for accurate CT reconstruction. For DECT, the gradient images of high- and low-energy CT images ideally share the same complementary support set for the same object. Inspired by this idea, we extract the prior knowledge of complementary support set (Pri-CSS) from the gradient image of the first half-scan CT image to promote the second limited-angle CT reconstruction. Pri-CSS will be incorporated into total variation regularization model in the form of constrains. Alternative direction method is applied to iteratively solve the modified optimization model, thereby deriving the proposed algorithm to recover low-energy CT image from limited-angle measurements. The qualitative and quantitative experiments on digital and real data are performed to validate the proposed method. The results show that the proposed method outperforms its counterparts and achieve high reconstruction quality for the designed scanning configuration.

Comparison of data-acquisition designs for single-shot edge-illumination X-ray phase-contrast tomography

Edge-illumination X-ray phase-contrast tomography (EIXPCT) is an emerging technique that enables practical phase-contrast imaging with laboratory-based X-ray sources. A joint reconstruction method was proposed for reconstructing EIXPCT images, enabling novel flexible data-acquisition designs. However, only limited efforts have been devoted to optimizing data-acquisition designs for use with the joint reconstruction method. In this study, several promising designs are introduced, such as the constant aperture position (CAP) strategy and the alternating aperture position (AAP) strategy covering different angular ranges. In computer-simulation studies, these designs are analyzed and compared. Experimental data are employed to test the designs in real-world applications. All candidate designs are also compared for their implementation complexity. The tradeoff between data-acquisition time and image quality is discussed.

Multi-energy computed tomography reconstruction using a nonlocal spectral similarity model

Multi-energy computed tomography (MECT) is able to acquire simultaneous multi-energy measurements from one scan. In addition, it allows material differentiation and quantification effectively. However, due to the limited energy bin width, the number of photons detected in an energy-specific channel is smaller than that in traditional CT, which results in image quality degradation. To address this issue, in this work, we develop a statistical iterative reconstruction algorithm to acquire high-quality MECT images and high-accuracy material-specific images. Specifically, this algorithm fully incorporates redundant self-similarities within nonlocal regions in the MECT image at one bin and rich spectral similarities among MECT images at all bins. For simplicity, the presented algorithm is referred to as 'MECT-NSS'. Moreover, an efficient optimization algorithm is developed to solve the MECT-NSS objective function. Then, a comprehensive evaluation of parameter selection for the MECT-NSS algorithm is conducted. In the experiment, the datasets include images from three phantoms and one patient to validate and evaluate the MECT-NSS reconstruction performance. The qualitative and quantitative results demonstrate that the presented MECT-NSS can successfully yield better MECT image quality and more accurate material estimation than the competing algorithms.

Comparison of five one-step reconstruction algorithms for spectral CT

Over the last decade, dual-energy CT scanners have gone from prototypes to clinically available machines, and spectral photon counting CT scanners are following. They require a specific reconstruction process, consisting of two steps: material decomposition and tomographic reconstruction. Image-based methods perform reconstruction, then decomposition, while projection-based methods perform decomposition first, and then reconstruction. As an alternative, 'one-step inversion' methods have been proposed, which perform decomposition and reconstruction simultaneously. Unfortunately, one-step methods are typically slower than their two-step counterparts, and in most CT applications, reconstruction time is critical. This paper therefore proposes to compare the convergence speeds of five one-step algorithms. We adapted all these algorithms to solve the same problem: spectral photon-counting CT reconstruction from five energy bins, using a three materials decomposition basis and spatial regularization. The paper compares a Bayesian method which uses non-linear conjugate gradient for minimization (Cai et al 2013 Med. Phys. 40 111916-31), three methods based on quadratic surrogates (Long and Fessler 2014 IEEE Trans. Med. Imaging 33 1614-26, Weidinger et al 2016 Int. J. Biomed. Imaging 2016 1-15, Mechlem et al 2018 IEEE Trans. Med. Imaging 37 68-80), and a primal-dual method based on MOCCA, a modified Chambolle-Pock algorithm (Barber et al 2016 Phys. Med. Biol. 61 3784). Some of these methods have been accelerated by using μ-preconditioning, i.e. by performing all internal computations not with the actual materials the object is made of, but with carefully chosen linear combinations of those. In this paper, we also evaluated the impact of three different μ-preconditioners on convergence speed. Our experiments on simulated data revealed vast differences in the number of iterations required to reach a common image quality objective: Mechlem et al (2018 IEEE Trans. Med. Imaging 37 68-80) needed ten iterations, Cai et al (2013 Med. Phys. 40 111916-31), Long and Fessler (2014 IEEE Trans. Med. Imaging 33 1614-26) and Weidinger et al (2016 Int. J. Biomed. Imaging 2016 1-15) several hundreds, and Barber et al (2016 Phys. Med. Biol. 61 3784) several thousands. We also sum up other practical aspects, like memory footprint and the need to tune extra parameters.

The evolution of image reconstruction for CT-from filtered back projection to artificial intelligence

Abstract

The first CT scanners in the early 1970s already used iterative reconstruction algorithms; however, lack of computational power prevented their clinical use. In fact, it took until 2009 for the first iterative reconstruction algorithms to come commercially available and replace conventional filtered back projection. Since then, this technique has caused a true hype in the field of radiology. Within a few years, all major CT vendors introduced iterative reconstruction algorithms for clinical routine, which evolved rapidly into increasingly advanced reconstruction algorithms. The complexity of algorithms ranges from hybrid-, model-based to fully iterative algorithms. As a result, the number of scientific publications on this topic has skyrocketed over the last decade. But what exactly has this technology brought us so far? And what can we expect from future hardware as well as software developments, such as photon-counting CT and artificial intelligence? This paper will try answer those questions by taking a concise look at the overall evolution of CT image reconstruction and its clinical implementations. Subsequently, we will give a prospect towards future developments in this domain.

Key Points

• Advanced CT reconstruction methods are indispensable in the current clinical setting.

• IR is essential for photon-counting CT, phase-contrast CT, and dark-field CT.

• Artificial intelligence will potentially further increase the performance of reconstruction methods.

Iterative reconstruction for photon-counting CT using prior image constrained total generalized variation

In this paper, we present an iterative reconstruction for photon-counting CT using prior image constrained total generalized variation (PICTGV). This work aims to exploit structural correlation in the energy domain to reduce image noise in photon-counting CT with narrow energy bins. This is motived by the fact that the similarity between high-quality full-spectrum image and target image is an important prior knowledge for photon-counting CT reconstruction. The PICTGV method is implemented using a splitting-based fast iterative shrinkage-threshold algorithm (FISTA). Evaluations conducted with simulated and real photon-counting CT data demonstrate that PICTGV method outperforms the existing prior image constrained compressed sensing (PICCS) method in terms of noise reduction, artifact suppression and resolution preservation. In the simulated head data study, the average relative root mean squared error is reduced from 2.3% in PICCS method to 1.2% in PICTGV method, and the average universal quality index increases from 0.67 in PICCS method to 0.76 in PICTGV method. The results show that the present PICTGV method improves the performance of the PICCS method for photon-counting CT reconstruction with narrow energy bins.

Multicolour imaging with spectral photon-counting CT: a phantom study

Background

To evaluate the feasibility of multicolour quantitative imaging with spectral photon-counting computed tomography (SPCCT) of different mixed contrast agents.

Methods

Phantoms containing eleven tubes with mixtures of varying proportions of two contrast agents (i.e. two selected from gadolinium, iodine or gold nanoparticles) were prepared so that the attenuation of each tube was about 280 HU. Scans were acquired at 120 kVp and 100 mAs using a five-bin preclinical SPCCT prototype, generating conventional, water, iodine, gadolinium and gold images. The correlation between prepared and measured concentrations was assessed using linear regression. The cross-contamination was measured for each material as the root mean square error (RMSE) of its concentration in the other material images, where no signal was expected. The contrast-to-noise ratio (CNR) relative to a phosphate buffered saline tube was calculated for each contrast agent.

Results

The solutions had similar attenuations (279 ± 10 HU, mean ± standard deviation) and could not be differentiated on conventional images. However, a distinction was observed in the material images within the same samples, and the measured and prepared concentrations were strongly correlated (R² ≥ 0.97, 0.81 ≤ slope ≤ 0.95, -0.68 ≤ offset ≤ 0.89 mg/mL). Cross-contamination in the iodine images for the mixture of gold and gadolinium contrast agents (RMSE = 0.34 mg/mL) was observed. CNR for 1 mg/mL of contrast agent was better for the mixture of iodine and gadolinium (CNR_iodine = 3.20, CNR_gadolinium = 2.80) than gold and gadolinium (CNR_gadolinium = 1.67, CNR_gold = 1.37).

Conclusions

SPCCT enables multicolour quantitative imaging. As a result, it should be possible to perform imaging of multiple uptake phases of a given tissue/organ within a single scan by injecting different contrast agents sequentially.

Joint Statistical Iterative Material Image Reconstruction for Spectral Computed Tomography Using a Semi-Empirical Forward Model

By acquiring tomographic measurements with several distinct photon energy spectra, spectral computed tomography (spectral CT) is able to provide additional material-specific information compared with conventional CT. This information enables the generation of material selective images, which have found various applications in medical imaging. However, material decomposition typically leads to noise amplification and a degradation of the signal-to-noise ratio. This is still a fundamental problem of spectral CT, especially for low-dose medical applications. Inspired by the success for low-dose conventional CT, several statistical iterative reconstruction algorithms for spectral CT have been developed. These algorithms typically rely on detailed knowledge about the spectrum and the detector response. Obtaining this knowledge is often difficult in practice, especially if photon counting detectors are used to acquire the energy specific information. In this paper, a new algorithm for joint statistical iterative material image reconstruction is presented. It relies on a semi-empirical forward model which is tuned by calibration measurements. This strategy allows to model spatially varying properties of the imaging system without requiring detailed prior knowledge of the system parameters. We employ an efficient optimization algorithm based on separable surrogate functions to accelerate convergence and reduce the reconstruction time. Numerical as well as real experiments show that our new algorithm leads to reduced statistical bias and improved image quality compared with projection-based material decomposition followed by analytical or iterative image reconstruction.

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.

A comparison of linear interpolation models for iterative CT reconstruction

PURPOSE

Recent reports indicate that model-based iterative reconstruction methods may improve image quality in computed tomography (CT). One difficulty with these methods is the number of options available to implement them, including the selection of the forward projection model and the penalty term. Currently, the literature is fairly scarce in terms of guidance regarding this selection step, whereas these options impact image quality. Here, the authors investigate the merits of three forward projection models that rely on linear interpolation: the distance-driven method, Joseph's method, and the bilinear method. The authors' selection is motivated by three factors: (1) in CT, linear interpolation is often seen as a suitable trade-off between discretization errors and computational cost, (2) the first two methods are popular with manufacturers, and (3) the third method enables assessing the importance of a key assumption in the other methods.

METHODS

One approach to evaluate forward projection models is to inspect their effect on discretized images, as well as the effect of their transpose on data sets, but significance of such studies is unclear since the matrix and its transpose are always jointly used in iterative reconstruction. Another approach is to investigate the models in the context they are used, i.e., together with statistical weights and a penalty term. Unfortunately, this approach requires the selection of a preferred objective function and does not provide clear information on features that are intrinsic to the model. The authors adopted the following two-stage methodology. First, the authors analyze images that progressively include components of the singular value decomposition of the model in a reconstructed image without statistical weights and penalty term. Next, the authors examine the impact of weights and penalty on observed differences.

RESULTS

Image quality metrics were investigated for 16 different fan-beam imaging scenarios that enabled probing various aspects of all models. The metrics include a surrogate for computational cost, as well as bias, noise, and an estimation task, all at matched resolution. The analysis revealed fundamental differences in terms of both bias and noise. Task-based assessment appears to be required to appreciate the differences in noise; the estimation task the authors selected showed that these differences balance out to yield similar performance. Some scenarios highlighted merits for the distance-driven method in terms of bias but with an increase in computational cost. Three combinations of statistical weights and penalty term showed that the observed differences remain the same, but strong edge-preserving penalty can dramatically reduce the magnitude of these differences.

CONCLUSIONS

In many scenarios, Joseph's method seems to offer an interesting compromise between cost and computational effort. The distance-driven method offers the possibility to reduce bias but with an increase in computational cost. The bilinear method indicated that a key assumption in the other two methods is highly robust. Last, strong edge-preserving penalty can act as a compensator for insufficiencies in the forward projection model, bringing all models to similar levels in the most challenging imaging scenarios. Also, the authors find that their evaluation methodology helps appreciating how model, statistical weights, and penalty term interplay together.

Superiorization-based multi-energy CT image reconstruction

The recently-developed superiorization approach is efficient and robust for solving various constrained optimization problems. This methodology can be applied to multi-energy CT image reconstruction with the regularization in terms of the prior rank, intensity and sparsity model (PRISM). In this paper, we propose a superiorized version of the simultaneous algebraic reconstruction technique (SART) based on the PRISM model. Then, we compare the proposed superiorized algorithm with the Split-Bregman algorithm in numerical experiments. The results show that both the Superiorized-SART and the Split-Bregman algorithms generate good results with weak noise and reduced artefacts.

Dose-efficient ultrahigh-resolution scan mode using a photon counting detector computed tomography system

An ultrahigh-resolution (UHR) data collection mode was enabled on a whole-body, research photon counting detector (PCD) computed tomography system. In this mode, 64 rows of [Formula: see text] detector pixels were used, which corresponded to a pixel size of [Formula: see text] at the isocenter. Spatial resolution and image noise were quantitatively assessed for the UHR PCD scan mode, as well as for a commercially available UHR scan mode that uses an energy-integrating detector (EID) and a set of comb filters to decrease the effective detector size. Images of an anthropomorphic lung phantom, cadaveric swine lung, swine heart specimen, and cadaveric human temporal bone were qualitatively assessed. Nearly equivalent spatial resolution was demonstrated by the modulation transfer function measurements: 15.3 and [Formula: see text] spatial frequencies were achieved at 10% and 2% modulation, respectively, for the PCD system and 14.2 and [Formula: see text] for the EID system. Noise was 29% lower in the PCD UHR images compared to the EID UHR images, representing a potential dose savings of 50% for equivalent image noise. PCD UHR images from the anthropomorphic phantom and cadaveric specimens showed clear delineation of small structures.

Iterative reconstruction for CT perfusion with a prior-image induced hybrid nonlocal means regularization: Phantom studies

PURPOSE

In computed tomography perfusion (CTP) imaging, an initial phase CT acquired with a high-dose protocol can be used to improve the image quality of later phase CT acquired with a low-dose protocol. For dynamic regions, signals in the later low-dose CT may not be completely recovered if the initial CT heavily regularizes the iterative reconstruction process. The authors propose a hybrid nonlocal means (hNLM) regularization model for iterative reconstruction of low-dose CTP to overcome the limitation of the conventional prior-image induced penalty.

METHODS

The hybrid penalty was constructed by combining the NLM of the initial phase high-dose CT in the stationary region and later phase low-dose CT in the dynamic region. The stationary and dynamic regions were determined by the similarity between the initial high-dose scan and later low-dose scan. The similarity was defined as a Gaussian kernel-based distance between the patch-window of the same pixel in the two scans, and its measurement was then used to weigh the influence of the initial high-dose CT. For regions with high similarity (e.g., stationary region), initial high-dose CT played a dominant role for regularizing the solution. For regions with low similarity (e.g., dynamic region), the regularization relied on a low-dose scan itself. This new hNLM penalty was incorporated into the penalized weighted least-squares (PWLS) for CTP reconstruction. Digital and physical phantom studies were performed to evaluate the PWLS-hNLM algorithm.

RESULTS

Both phantom studies showed that the PWLS-hNLM algorithm is superior to the conventional prior-image induced penalty term without considering the signal changes within the dynamic region. In the dynamic region of the Catphan phantom, the reconstruction error measured by root mean square error was reduced by 42.9% in PWLS-hNLM reconstructed image.

CONCLUSIONS

The PWLS-hNLM algorithm can effectively use the initial high-dose CT to reconstruct low-dose CTP in the stationary region while reducing its influence in the dynamic region.

Accelerated fast iterative shrinkage thresholding algorithms for sparsity-regularized cone-beam CT image reconstruction

PURPOSE

The development of iterative image reconstruction algorithms for cone-beam computed tomography (CBCT) remains an active and important research area. Even with hardware acceleration, the overwhelming majority of the available 3D iterative algorithms that implement nonsmooth regularizers remain computationally burdensome and have not been translated for routine use in time-sensitive applications such as image-guided radiation therapy (IGRT). In this work, two variants of the fast iterative shrinkage thresholding algorithm (FISTA) are proposed and investigated for accelerated iterative image reconstruction in CBCT.

METHODS

Algorithm acceleration was achieved by replacing the original gradient-descent step in the FISTAs by a subproblem that is solved by use of the ordered subset simultaneous algebraic reconstruction technique (OS-SART). Due to the preconditioning matrix adopted in the OS-SART method, two new weighted proximal problems were introduced and corresponding fast gradient projection-type algorithms were developed for solving them. We also provided efficient numerical implementations of the proposed algorithms that exploit the massive data parallelism of multiple graphics processing units.

RESULTS

The improved rates of convergence of the proposed algorithms were quantified in computer-simulation studies and by use of clinical projection data corresponding to an IGRT study. The accelerated FISTAs were shown to possess dramatically improved convergence properties as compared to the standard FISTAs. For example, the number of iterations to achieve a specified reconstruction error could be reduced by an order of magnitude. Volumetric images reconstructed from clinical data were produced in under 4 min.

CONCLUSIONS

The FISTA achieves a quadratic convergence rate and can therefore potentially reduce the number of iterations required to produce an image of a specified image quality as compared to first-order methods. We have proposed and investigated accelerated FISTAs for use with two nonsmooth penalty functions that will lead to further reductions in image reconstruction times while preserving image quality. Moreover, with the help of a mixed sparsity-regularization, better preservation of soft-tissue structures can be potentially obtained. The algorithms were systematically evaluated by use of computer-simulated and clinical data sets.

Penalized weighted least-squares approach for multienergy computed tomography image reconstruction via structure tensor total variation regularization

Multienergy computed tomography (MECT) allows identifying and differentiating different materials through simultaneous capture of multiple sets of energy-selective data belonging to specific energy windows. However, because sufficient photon counts are not available in each energy window compared with that in the whole energy window, the MECT images reconstructed by the analytical approach often suffer from poor signal-to-noise and strong streak artifacts. To address the particular challenge, this work presents a penalized weighted least-squares (PWLS) scheme by incorporating the new concept of structure tensor total variation (STV) regularization, which is henceforth referred to as 'PWLS-STV' for simplicity. Specifically, the STV regularization is derived by penalizing higher-order derivatives of the desired MECT images. Thus it could provide more robust measures of image variation, which can eliminate the patchy artifacts often observed in total variation (TV) regularization. Subsequently, an alternating optimization algorithm was adopted to minimize the objective function. Extensive experiments with a digital XCAT phantom and meat specimen clearly demonstrate that the present PWLS-STV algorithm can achieve more gains than the existing TV-based algorithms and the conventional filtered backpeojection (FBP) algorithm in terms of both quantitative and visual quality evaluations.

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.

Line Integral Alternating Minimization Algorithm for Dual-Energy X-Ray CT Image Reconstruction

We propose a new algorithm, called line integral alternating minimization (LIAM), for dual-energy X-ray CT image reconstruction. Instead of obtaining component images by minimizing the discrepancy between the data and the mean estimates, LIAM allows for a tunable discrepancy between the basis material projections and the basis sinograms. A parameter is introduced that controls the size of this discrepancy, and with this parameter the new algorithm can continuously go from a two-step approach to the joint estimation approach. LIAM alternates between iteratively updating the line integrals of the component images and reconstruction of the component images using an image iterative deblurring algorithm. An edge-preserving penalty function can be incorporated in the iterative deblurring step to decrease the roughness in component images. Images from both simulated and experimentally acquired sinograms from a clinical scanner were reconstructed by LIAM while varying the regularization parameters to identify good choices. The results from the dual-energy alternating minimization algorithm applied to the same data were used for comparison. Using a small fraction of the computation time of dual-energy alternating minimization, LIAM achieves better accuracy of the component images in the presence of Poisson noise for simulated data reconstruction and achieves the same level of accuracy for real data reconstruction.

Model-based iterative reconstruction for flat-panel cone-beam CT with focal spot blur, detector blur, and correlated noise

While model-based reconstruction methods have been successfully applied to flat-panel cone-beam CT (FP-CBCT) systems, typical implementations ignore both spatial correlations in the projection data as well as system blurs due to the detector and focal spot in the x-ray source. In this work, we develop a forward model for flat-panel-based systems that includes blur and noise correlation associated with finite focal spot size and an indirect detector (e.g. scintillator). This forward model is used to develop a staged reconstruction framework where projection data are deconvolved and log-transformed, followed by a generalized least-squares reconstruction that utilizes a non-diagonal statistical weighting to account for the correlation that arises from the acquisition and data processing chain. We investigate the performance of this novel reconstruction approach in both simulated data and in CBCT test-bench data. In comparison to traditional filtered backprojection and model-based methods that ignore noise correlation, the proposed approach yields a superior noise-resolution tradeoff. For example, for a system with 0.34 mm FWHM scintillator blur and 0.70 FWHM focal spot blur, using the correlated noise model instead of an uncorrelated noise model increased resolution by 42% (with variance matched at 6.9  ×  10(-8) mm(-2)). While this advantage holds across a wide range of systems with differing blur characteristics, the improvements are greatest for systems where source blur is larger than detector blur.

Sparsity-constrained three-dimensional image reconstruction for C-arm angiography

X-ray C-arm is an important imaging tool in interventional radiology, road-mapping and radiation therapy because it provides accurate descriptions of vascular anatomy and therapeutic end point. In common interventional radiology, the C-arm scanner produces a set of two-dimensional (2D) X-ray projection data obtained with a detector by rotating the scanner gantry around the patient. Unlike conventional fluoroscopic imaging, three-dimensional (3D) C-arm computed tomography (CT) provides more accurate cross-sectional images, which are helpful for therapy planning, guidance and evaluation in interventional radiology. However, 3D vascular imaging using the conventional C-arm fluoroscopy encounters some geometry challenges. Inspired by the theory of compressed sensing, we developed an image reconstruction algorithm for conventional angiography C-arm scanners. The main challenge in this image reconstruction problem is the projection data limitations. We consider a small number of views acquired from a short rotation orbit with offset scan geometry. The proposed method, called sparsity-constrained angiography (SCAN), is developed using the alternating direction method of multipliers, and the results obtained from simulated and real data are encouraging. SCAN algorithm provides a framework to generate 3D vascular images using the conventional C-arm scanners in lower cost than conventional 3D imaging scanners.

Joint reconstruction of multi-channel, spectral CT data via constrained total nuclear variation minimization

We explore the use of the recently proposed 'total nuclear variation' (TVN) as a regularizer for reconstructing multi-channel, spectral CT images. This convex penalty is a natural extension of the total variation (TV) to vector-valued images and has the advantage of encouraging common edge locations and a shared gradient direction among image channels. We show how it can be incorporated into a general, data-constrained reconstruction framework and derive update equations based on the first-order, primal-dual algorithm of Chambolle and Pock. Early simulation studies based on the numerical XCAT phantom indicate that the inter-channel coupling introduced by the TVN leads to better preservation of image features at high levels of regularization, compared to independent, channel-by-channel TV reconstructions.