Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-ray CT

Selecting projection views based on error equidistribution for computed tomography

BACKGROUND

Nonuniform sampling is a useful technique to optimize the acquisition of projections with a limited budget. Existing methods for selecting important projection views have limitations, such as relying on blueprint images or excessive computing resources.

METHODS

We aim to develop a simple nonuniform sampling method for selecting informative projection views suitable for practical CT applications. The proposed algorithm is inspired by two key observations: projection errors contain angle-specific information, and adding views around error peaks effectively reduces errors and improves reconstruction. Given a budget and an initial view set, the proposed method involves: estimating projection errors based on current set of projection views, adding more projection views based on error equidistribution to smooth out errors, and final image reconstruction based on the new set of projection views. This process can be recursive, and the initial view can be obtained uniformly or from a prior for greater efficiency.

RESULTS

Comparison with popular view selection algorithms using simulated and real data demonstrates consistently superior performance in identifying optimal views and generating high-quality reconstructions. Notably, the new algorithm performs well in both PSNR and SSIM metrics while being computationally efficient, enhancing its practicality for CT optimization.

CONCLUSIONS

A projection view selection algorithm based on error equidistribution is proposed, offering superior reconstruction quality and efficiency over existing methods. It is ready for real CT applications to optimize dose utilization.

TMAA-net: tensor-domain multi-planal anti-aliasing network for sparse-view CT image reconstruction

Objective.Among various deep-network-based sparse-view CT image reconstruction studies, the sinogram upscaling network has been predominantly employed to synthesize additional view information. However, the performance of the sinogram-based network is limited in terms of removing aliasing streak artifacts and recovering low-contrast small structures. In this study, we used a view-by-view back-projection (VVBP) tensor-domain network to overcome such limitations of the sinogram-based approaches.Approach.The proposed method offers advantages of addressing the aliasing artifacts directly in the 3D tensor domain over the 2D sinogram. In the tensor-domain network, the multi-planal anti-aliasing modules were used to remove artifacts within the coronal and sagittal tensor planes. In addition, the data-fidelity-based refinement module was also implemented to successively process output images of the tensor network to recover image sharpness and textures.Main result.The proposed method showed outperformance in terms of removing aliasing artifacts and recovering low-contrast details compared to other state-of-the-art sinogram-based networks. The performance was validated for both numerical and clinical projection data in a circular fan-beam CT configuration.Significance.We observed that view-by-view aliasing artifacts in sparse-view CT exhibit distinct patterns within the tensor planes, making them effectively removable in high-dimensional representations. Additionally, we demonstrated that the co-domain characteristics of tensor space processing offer higher generalization performance for aliasing artifact removal compared to conventional sinogram-domain processing.

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.

Tiny changes in tomographic system matrices can cause large changes in reconstruction quality

This paper studies the impact of tiny changes in region-of-interest (ROI) tomography system matrices on the variance of the reconstructed ROI. In small-scale and medium-scale examples, the variance in the reconstructed ROI was estimated for different system matrices. The results revealed a striking and counterintuitive phenomenon: a tiny change in the system matrix can dramatically affect the variance of the ROI estimate. In one of our examples, a decrease of 0.1% in one element out of hundreds of thousands of the system matrix resulted in a systematic reduction of the variance inside the ROI, and by a factor of 5 to 10 for some pixels. Our results agree with a recently proven theorem about the ability of additional measurements to reduce the variance in ROI tomography.

Static coded illumination strategies for low-dose x-ray material decomposition

Static coded aperture x-ray tomography was introduced recently where a static illumination pattern is used to interrogate an object with a low radiation dose, from which an accurate 3D reconstruction of the object can be attained computationally. Rather than continuously switching the pattern of illumination with each view angle, as traditionally done, static code computed tomography (CT) places a single pattern for all views. The advantages are many, including the feasibility of practical implementation. This paper generalizes this powerful framework to develop single-scan dual-energy coded aperture spectral tomography that enables material characterization at a significantly reduced exposure level. Two sensing strategies are explored: rapid kV switching with a single-static block/unblock coded aperture, and coded apertures with non-uniform thickness. Both systems rely on coded illumination with a plurality of x-ray spectra created by kV switching or 3D coded apertures. The structured x-ray illumination is projected through the objects of interest and measured with standard x-ray energy integrating detectors. Then, based on the tensor representation of projection data, we develop an algorithm to estimate a full set of synthesized measurements that can be used with standard reconstruction algorithms to accurately recover the object in each energy channel. Simulation and experimental results demonstrate the effectiveness of the proposed cost-effective solution to attain material characterization in low-dose dual-energy CT.

Total variation combining nonlocal means filtration for image reconstruction in X-ray computed tomography

BACKGROUND

Image reconstruction for realistic medical images under incomplete observation is still one of the core tasks for computed tomography (CT). However, the stair-case artifacts of Total variation (TV) based ones have restricted the usage of the reconstructed images.

OBJECTIVE

This work aims to propose and test an accurate and efficient algorithm to improve reconstruction quality under the idea of synergy between local and nonlocal regularizations.

METHODS

The total variation combining the nonlocal means filtration is proposed and the alternating direction method of multipliers is utilized to develop an efficient algorithm. The first order approximation of linear expansion at intermediate point is applied to overcome the computation of the huge CT system matrix.

RESULTS

The proposed method improves root mean squared error by 25.6% compared to the recent block-matching sparsity regularization (BMSR) on simulation dataset of 19 views. The structure similarities of image of the new method is higher than 0.95, while that of BMSR is about 0.92. Moreover, on real rabbit dataset of 20 views, the peak signal-to-noise ratio (PSNR) of the new method is 36.84, while using other methods PSNR are lower than 35.81.

CONCLUSIONS

The proposed method shows advantages on noise suppression and detail preservations over the competing algorithms used in CT image reconstruction.

Experimental demonstration and optimization of X-ray StaticCodeCT

As the use of X-ray computed tomography (CT) grows in medical diagnosis, so does the concern for the harm a radiation dose can cause and the biological risks it represents. StaticCodeCT is a new low-dose imaging architecture that uses a single-static coded aperture (CA) in a CT gantry. It exploits the highly correlated data in the projection domain to estimate the unobserved measurements on the detector. We previously analyzed the StaticCodeCT system by emulating the effect of the coded mask on experimental CT data. In contrast, this manuscript presents test-bed reconstructions using an experimental cone-beam X-ray CT system with a CA holder. We analyzed the reconstruction quality using three different techniques to manufacture the CAs: metal additive manufacturing, cold casting, and ceramic additive manufacturing. Furthermore, we propose an optimization method to design the CA pattern based on the algorithm developed for the measurement estimation. The obtained results point to the possibility of the real deployment of StaticCodeCT systems in practice.

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.

StaticCodeCT: single coded aperture tensorial X-ray CT

Coded aperture X-ray CT (CAXCT) is a new low-dose imaging technology that promises far-reaching benefits in industrial and clinical applications. It places various coded apertures (CA) at a time in front of the X-ray source to partially block the radiation. The ill-posed inverse reconstruction problem is then solved using l1-norm-based iterative reconstruction methods. Unfortunately, to attain high-quality reconstructions, the CA patterns must change in concert with the view-angles making the implementation impractical. This paper proposes a simple yet radically different approach to CAXCT, which is coined StaticCodeCT, that uses a single-static CA in the CT gantry, thus making the imaging system amenable for practical implementations. Rather than using conventional compressed sensing algorithms for recovery, we introduce a new reconstruction framework for StaticCodeCT. Namely, we synthesize the missing measurements using low-rank tensor completion principles that exploit the multi-dimensional data correlation and low-rank nature of a 3-way tensor formed by stacking the 2D coded CT projections. Then, we use the FDK algorithm to recover the 3D object. Computational experiments using experimental projection measurements exhibit up to 10% gains in the normalized root mean square distance of the reconstruction using the proposed method compared with those attained by alternative low-dose systems.

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.

A Novel Low-Dose Dual-Energy Imaging Method for a Fast-Rotating Gantry-Type CT Scanner

CT scan by use of a beam-filter placed between the x-ray source and the patient allows a single-scan low-dose dual-energy imaging with a minimal hardware modification to the existing CT systems. We have earlier demonstrated the feasibility of such imaging method with a multi-slit beam-filter reciprocating along the direction perpendicular to the CT rotation axis in a cone-beam CT system. However, such method would face mechanical challenges when the beam-filter is supposed to cooperate with a fast-rotating gantry in a diagnostic CT system. In this work, we propose a new scanning method and associated image reconstruction algorithm that can overcome these challenges. We propose to slide a beam-filter that has multi-slit structure with its slits being at a slanted angle with the CT gantry rotation axis during a scan. A streaky pattern would show up in the sinogram domain as a result. Using a notch filter in the Fourier domain of the sinogram, we removed the streaks and reconstructed an image by use of the filtered-backprojection algorithm. The remaining image artifacts were suppressed by applying l₀ norm based smoothing. Using this image as a prior, we have reconstructed low- and high-energy CT images in the iterative reconstruction framework. An image-based material decomposition then followed. We conducted a simulation study to test its feasibility using the XCAT phantom and also an experimental study using the Catphan phantom, a head phantom, an iodine-solution phantom, and a monkey in anesthesia, and showed its successful performance in image reconstruction and in material decomposition.

Noise suppression-guided image filtering for low-SNR CT reconstruction

In practical computed tomography (CT) applications, projections with low signal-to-noise ratio (SNR) are often encountered due to the reduction of radiation dose or device limitations. In these situations, classical reconstruction algorithms, like simultaneous algebraic reconstruction technique (SART), cannot reconstruct high-quality CT images. Block-matching and 3D filtering (BM3D)-based iterative reconstruction algorithm (POCS-BM3D) has remarkable effect in dealing with CT reconstruction from noisy projections. However, BM3D may restrain noise with excessive loss of details in the case of low-SNR CT reconstruction. In order to achieve a preferable trade-off between noise suppression and edge preservation, we introduce guided image filtering (GIF) into low-SNR CT reconstruction, and propose noise suppression-guided image filtering reconstruction (NSGIFR) algorithm. In each iteration of NSGIFR, the output image of SART reserves more details and is used as input image of GIF, while the image denoised by BM3D serves as guidance image of GIF. Experimental results indicate that the proposed algorithm displays outstanding performance on preserving structures and suppressing noise for low-SNR CT reconstruction. NSGIFR can achieve more superior image quality than SART, POCS-TV and POCS-BM3D in terms of visual effect and quantitative analysis. Graphical abstract Block-matching and 3D filtering (BM3D)-based iterative reconstruction algorithm (POCS-BM3D) has remarkable effect in dealing with CT reconstruction from noisy projections. However, BM3D may restrain noise with excessive loss of details in the case of low-SNR CT reconstruction. In order to achieve a preferable trade-off between noise suppression and edge preservation, we introduce guided image filtering (GIF) into low-SNR CT reconstruction, and propose noise suppression-guided image filtering reconstruction (NSGIFR) algorithm.

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.

Retrospective correction of intensity inhomogeneity with sparsity constraints in transform-domain: Application to brain MRI

An effective retrospective correction method is introduced in this paper for intensity inhomogeneity which is an inherent artifact in MR images. Intensity inhomogeneity problem is formulated as the decomposition of acquired image into true image and bias field which are expected to have sparse approximation in suitable transform domains based on their known properties. Piecewise constant nature of the true image lends itself to have a sparse approximation in framelet domain. While spatially smooth property of the bias field supports a sparse representation in Fourier domain. The algorithm attains optimal results by seeking the sparsest solutions for the unknown variables in the search space through L₁ norm minimization. The objective function associated with defined problem is convex and is efficiently solved by the linearized alternating direction method. Thus, the method estimates the optimal true image and bias field simultaneously in an L₁ norm minimization framework by promoting sparsity of the solutions in suitable transform domains. Furthermore, the methodology doesn't require any preprocessing, any predefined specifications or parametric models that are critically controlled by user-defined parameters. The qualitative and quantitative validation of the proposed methodology in simulated and real human brain MR images demonstrates the efficacy and superiority in performance compared to some of the distinguished algorithms for intensity inhomogeneity correction.

Compressive spectral X-ray tomography based on spatial and spectral coded illumination

Spectral computed tomography (CT) relies on the spectral dependence of X-ray attenuation coefficients to separate projection measurements into more than two energy bins. Such data can be used to unveil tomographic material characterization - key in national security and medical imaging. This paper explores a radical departure from conventional methods used in spectral imaging. It relies on K-edge coded apertures to create spatially and spectrally coded, lower-dose, X-ray bundles that interrogate specific voxels of the object. The new approach referred to as compressive spectral X-ray imaging (CSXI) uses low-cost standard X-ray integrating detectors and acquires compressive measurements, which enable the reconstruction of energy binned images from fewer measurements. Various spectral and spatial coding strategies for structured illumination are explored. Subsampling in CSXI is accomplished by either view angle spectral subsampling, spatial subsampling enabled by block-unblock coded apertures placed at the source or detector side, or both. The careful design of subsampling strategies, spectral filters, coded apertures, and their placement, are shown to be critical for the quality of tomographic image reconstruction. The forward imaging model of CSXI, which is a non-linear ill-posed problem, is analyzed and a multi-stage algorithm is developed to address the estimation of the energy binned sinograms from the integrating detector measurements. Then, an Alternating Direction Method of Multipliers (ADMM) is used to solve a joint sparse and low-rank optimization problem for reconstruction that exploits the structure of the spectral X-ray data cube.

Source and coded aperture joint optimization for compressive X-ray tomosynthesis

Compressive X-ray tomosynthesis is an emerging technique to reconstruct three-dimensional (3D) objects from two-dimensional projection measurements generated by a set of spatially distributed X-ray sources, where coded apertures are used in front of each source to modulate a set of X-rays to interrogate an object with a reduced radiation dose without loss of image reconstruction quality. The reconstruction performance in compressive tomosynthesis is influenced by several factors including the locations of the X-ray sources, their incident angles, and the coded apertures that determine the structured illumination patterns. This paper develops a source and coded aperture joint optimization (SCO) approach to improve the image reconstruction performance of compressive X-ray tomosynthesis. Based on compressive sensing theory, the synergy among the source pattern, source orientation, and the coded apertures is utilized to minimize the coherence of the sensing matrix of the imaging system. In concert with a gradient-based optimization algorithm, regularization methods are used to reduce the convergence error and achieve uniform sensing of the object under inspection. Compared to the optimization of either the source orientation, or the coded aperture individually, the proposed method effectively increases the degree of optimization freedom, and thus achieves considerable improvement in the 3D imaging reconstruction accuracy.

Efficient solving algorithm for determining the exact sampling condition of limited-angle computed tomography reconstruction

Total variation (TV) regularization-based iterative reconstruction algorithms have an impressive potential to solve limited-angle computed tomography with insufficient sampling projections. The analysis of exact reconstruction sampling conditions for a TV-minimization reconstruction model can determine the minimum number of scanning angle and minimize the scanning range. However, the large-scale matrix operations caused by increased testing phantom size are the computation bottleneck in determining the exact reconstruction sampling conditions in practice. When the size of the testing phantom increases to a certain scale, it is very difficult to analyze quantitatively the exact reconstruction sampling condition using existing methods. In this paper, we propose a fast and efficient algorithm to determine the exact reconstruction sampling condition for large phantoms. Specifically, the sampling condition of a TV minimization model is modeled as a convex optimization problem, which is derived from the sufficient and necessary condition of solution uniqueness for the L1 minimization model. An effective alternating direction minimization algorithm is developed to optimize the objective function by alternatively solving two sub-problems split from the convex problem. The Cholesky decomposition method is used in solving the first sub-problem to reduce computational complexity. Experimental results show that the proposed method can efficiently solve the verification problem of the accurate reconstruction sampling condition. Furthermore, we obtain the lower bounds of scanning angle range for the exact reconstruction of a specific phantom with the larger size.

Iterative image reconstruction for sparse-view CT via total variation regularization and dictionary learning

Recently, low-dose computed tomography (CT) has become highly desirable due to the increasing attention paid to the potential risks of excessive radiation of the regular dose CT. However, ensuring image quality while reducing the radiation dose in the low-dose CT imaging is a major challenge. Compared to classical filtered back-projection (FBP) algorithms, statistical iterative reconstruction (SIR) methods for modeling measurement statistics and imaging geometry can significantly reduce the radiation dose, while maintaining the image quality in a variety of CT applications. To facilitate low-dose CT imaging, we in this study proposed an improved statistical iterative reconstruction scheme based on the penalized weighted least squares (PWLS) standard combined with total variation (TV) minimization and sparse dictionary learning (DL), which is named as a method of PWLS-TV-DL. To evaluate this PWLS-TV-DL method, we performed experiments on digital phantoms and physical phantoms, and analyzed the results in terms of image quality and calculation. The results show that the proposed method is better than the comparison methods, which indicates the potential of applying this PWLS-TV-DL method to reconstruct low-dose CT images.

An autoencoder based formulation for compressed sensing reconstruction

This work proposes a new formulation for image reconstruction based on the autoencoder framework. The work follows the adaptive approach used in prior dictionary and transform learning based reconstruction techniques. Existing autoencoder based reconstructions are non-adaptive; they are trained on a separate training set and applied on another. In this work, the autoencoder is learnt from the patches of the image it is reconstructing. Experimental studies on MRI reconstruction shows that the proposed method outperforms state-of-the-art methods in dictionary learning, transform learning and (non-adaptive) autoencoder based approaches.

Fast optimization of coded apertures in X-ray computed tomography

Coded aperture X-ray computed tomography (CAXCT) is a novel X-ray imaging system capable of reconstructing high quality images from a reduced set of measurements. Coded apertures are placed in front of the X-ray source in CAXCT so as to obtain patterned projections onto a detector array. Then, compressive sensing (CS) reconstruction algorithms are used to reconstruct the linear attenuation coefficients. The coded aperture is an important factor that influences the point spread function (PSF), which in turn determines the capability to sample the linear attenuation coefficients of the object. A coded aperture optimization approach was recently proposed based on the coherence of the system matrix; however, this algorithm is memory intensive and it is not able to optimize the coded apertures for large image sizes required in many applications. This paper introduces a significantly more efficient approach for coded aperture optimization that reduces the memory requirements and the execution time by orders of magnitude. The features are defined as the inner product of the vectors representing the geometric paths of the X-rays with the sparse basis representation of the object; therefore, the algorithm aims to find a subset of features that minimizes the information loss compared to the complete set of projections. This subset corresponds to the unblocking elements in the optimized coded apertures. The proposed approach solves the memory and runtime limitations of the previously proposed algorithm and provides a significant gain in the reconstruction image quality compared to that attained by random coded apertures in both simulated datasets and real datasets.

Limited-view CT Reconstruction Based on Autoencoder-like Generative Adversarial Networks with Joint Loss

Limiting the scan views of X-ray computed tomography (CT) can make radiation dose reduced efficiently and consequently weaken the damage of ionizing radiation. However, it will degrade the reconstructed CT images. In this paper, we proposed to predict the missing projections and improve the reconstructed CT images by constructing an autoencoder-like generative adversarial network (GAN) with joint loss function. In the generator network, we train an autoencoder-like convolutional neural network (CNN) to generate the missing projections given a sinogram of the limited-view CT projections. For the discriminator network, a CNN is used to classify an input sinogram as real or synthetic one. To produce more realistic images, the joint loss function which includes not only reconstruction loss, but the adversarial loss is employed. While reconstruction loss can capture the overall structure of the missing projections, the latter can pick a particular mode from the distribution and make the results much sharper. After the missing projections have been estimated, we reconstruct the CT images from the completed projections by utilizing conventional filtered back-projection (FBP) method. The experiments prove the capability of our method to achieve a considerable improvement in limited-view CT reconstruction.

Motion guided Spatiotemporal Sparsity for high quality 4D-CBCT reconstruction

Conventional cone-beam computed tomography is often deteriorated by respiratory motion blur, which negatively affects target delineation. On the other side, the four dimensional cone-beam computed tomography (4D-CBCT) can be considered to describe tumor and organ motion. But for current on-board CBCT imaging system, the slow rotation speed limits the projection number at each phase, and the associated reconstructions are contaminated by noise and streak artifacts using the conventional algorithm. To address the problem, we propose a novel framework to reconstruct 4D-CBCT from the under-sampled measurements—Motion guided Spatiotemporal Sparsity (MgSS). In this algorithm, we try to divide the CBCT images at each phase into cubes (3D blocks) and track the cubes with estimated motion field vectors through phase, then apply regional spatiotemporal sparsity on the tracked cubes. Specifically, we recast the tracked cubes into four-dimensional matrix, and use the higher order singular value decomposition (HOSVD) technique to analyze the regional spatiotemporal sparsity. Subsequently, the blocky spatiotemporal sparsity is incorporated into a cost function for the image reconstruction. The phantom simulation and real patient data are used to evaluate this algorithm. Results show that the MgSS algorithm achieved improved 4D-CBCT image quality with less noise and artifacts compared to the conventional algorithms.

Sampling limits for electron tomography with sparsity-exploiting reconstructions

Electron tomography (ET) has become a standard technique for 3D characterization of materials at the nano-scale. Traditional reconstruction algorithms such as weighted back projection suffer from disruptive artifacts with insufficient projections. Popularized by compressed sensing, sparsity-exploiting algorithms have been applied to experimental ET data and show promise for improving reconstruction quality or reducing the total beam dose applied to a specimen. Nevertheless, theoretical bounds for these methods have been less explored in the context of ET applications. Here, we perform numerical simulations to investigate performance of ℓ₁-norm and total-variation (TV) minimization under various imaging conditions. From 36,100 different simulated structures, our results show specimens with more complex structures generally require more projections for exact reconstruction. However, once sufficient data is acquired, dividing the beam dose over more projections provides no improvements-analogous to the traditional dose-fraction theorem. Moreover, a limited tilt range of ±75° or less can result in distorting artifacts in sparsity-exploiting reconstructions. The influence of optimization parameters on reconstructions is also discussed.

Coded aperture optimization in compressive X-ray tomography: a gradient descent approach

Coded aperture X-ray computed tomography (CT) has the potential to revolutionize X-ray tomography systems in medical imaging and air and rail transit security - both areas of global importance. It allows either a reduced set of measurements in X-ray CT without degradation in image reconstruction, or measure multiplexed X-rays to simplify the sensing geometry. Measurement reduction is of particular interest in medical imaging to reduce radiation, and airport security often imposes practical constraints leading to limited angle geometries. Coded aperture compressive X-ray CT places a coded aperture pattern in front of the X-ray source in order to obtain patterned projections onto a detector. Compressive sensing (CS) reconstruction algorithms are then used to recover the image. To date, the coded illumination patterns used in conventional CT systems have been random. This paper addresses the code optimization problem for general tomography imaging based on the point spread function (PSF) of the system, which is used as a measure of the sensing matrix quality which connects to the restricted isometry property (RIP) and coherence of the sensing matrix. The methods presented are general, simple to use, and can be easily extended to other imaging systems. Simulations are presented where the peak signal to noise ratios (PSNR) of the reconstructed images using optimized coded apertures exhibit significant gain over those attained by random coded apertures. Additionally, results using real X-ray tomography projections are presented.

Superiorized algorithm for reconstruction of CT images from sparse-view and limited-angle polyenergetic data

Recent work in CT image reconstruction has seen increasing interest in the use of total variation (TV) and related penalties to regularize problems involving reconstruction from undersampled or incomplete data. Superiorization is a recently proposed heuristic which provides an automatic procedure to 'superiorize' an iterative image reconstruction algorithm with respect to a chosen objective function, such as TV. Under certain conditions, the superiorized algorithm is guaranteed to find a solution that is as satisfactory as any found by the original algorithm with respect to satisfying the constraints of the problem; this solution is also expected to be superior with respect to the chosen objective. Most work on superiorization has used reconstruction algorithms which assume a linear measurement model, which in the case of CT corresponds to data generated from a monoenergetic x-ray beam. Many CT systems generate x-rays from a polyenergetic spectrum, however, in which the measured data represent an integral of object attenuation over all energies in the spectrum. This inconsistency with the linear model produces the well-known beam hardening artifacts, which impair analysis of CT images. In this work we superiorize an iterative algorithm for reconstruction from polyenergetic data, using both TV and an anisotropic TV (ATV) penalty. We apply the superiorized algorithm in numerical phantom experiments modeling both sparse-view and limited-angle scenarios. In our experiments, the superiorized algorithm successfully finds solutions which are as constraints-compatible as those found by the original algorithm, with significantly reduced TV and ATV values. The superiorized algorithm thus produces images with greatly reduced sparse-view and limited angle artifacts, which are also largely free of the beam hardening artifacts that would be present if a superiorized version of a monoenergetic algorithm were used.

Discriminative feature representation: an effective postprocessing solution to low dose CT imaging

This paper proposes a concise and effective approach termed discriminative feature representation (DFR) for low dose computerized tomography (LDCT) image processing, which is currently a challenging problem in medical imaging field. This DFR method assumes LDCT images as the superposition of desirable high dose CT (HDCT) 3D features and undesirable noise-artifact 3D features (the combined term of noise and artifact features induced by low dose scan protocols), and the decomposed HDCT features are used to provide the processed LDCT images with higher quality. The target HDCT features are solved via the DFR algorithm using a featured dictionary composed by atoms representing HDCT features and noise-artifact features. In this study, the featured dictionary is efficiently built using physical phantom images collected from the same CT scanner as the target clinical LDCT images to process. The proposed DFR method also has good robustness in parameter setting for different CT scanner types. This DFR method can be directly applied to process DICOM formatted LDCT images, and has good applicability to current CT systems. Comparative experiments with abdomen LDCT data validate the good performance of the proposed approach.

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

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

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.

Volumetric CT with sparse detector arrays (and application to Si-strip photon counters)

Novel x-ray medical imaging sensors, such as photon counting detectors (PCDs) and large area CCD and CMOS cameras can involve irregular and/or sparse sampling of the detector plane. Application of such detectors to CT involves undersampling that is markedly different from the commonly considered case of sparse angular sampling. This work investigates volumetric sampling in CT systems incorporating sparsely sampled detectors with axial and helical scan orbits and evaluates performance of model-based image reconstruction (MBIR) with spatially varying regularization in mitigating artifacts due to sparse detector sampling. Volumetric metrics of sampling density and uniformity were introduced. Penalized-likelihood MBIR with a spatially varying penalty that homogenized resolution by accounting for variations in local sampling density (i.e. detector gaps) was evaluated. The proposed methodology was tested in simulations and on an imaging bench based on a Si-strip PCD (total area 5 cm  ×  25 cm) consisting of an arrangement of line sensors separated by gaps of up to 2.5 mm. The bench was equipped with translation/rotation stages allowing a variety of scanning trajectories, ranging from a simple axial acquisition to helical scans with variable pitch. Statistical (spherical clutter) and anthropomorphic (hand) phantoms were considered. Image quality was compared to that obtained with a conventional uniform penalty in terms of structural similarity index (SSIM), image uniformity, spatial resolution, contrast, and noise. Scan trajectories with intermediate helical width (~10 mm longitudinal distance per 360° rotation) demonstrated optimal tradeoff between the average sampling density and the homogeneity of sampling throughout the volume. For a scan trajectory with 10.8 mm helical width, the spatially varying penalty resulted in significant visual reduction of sampling artifacts, confirmed by a 10% reduction in minimum SSIM (from 0.88 to 0.8) and a 40% reduction in the dispersion of SSIM in the volume compared to the constant penalty (both penalties applied at optimal regularization strength). Images of the spherical clutter and wrist phantoms confirmed the advantages of the spatially varying penalty, showing a 25% improvement in image uniformity and 1.8  ×  higher CNR (at matched spatial resolution) compared to the constant penalty. The studies elucidate the relationship between sampling in the detector plane, acquisition orbit, sampling of the reconstructed volume, and the resulting image quality. They also demonstrate the benefit of spatially varying regularization in MBIR for scenarios with irregular sampling patterns. Such findings are important and integral to the incorporation of a sparsely sampled Si-strip PCD in CT imaging.

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

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

Estimation of noise properties for TV-regularized image reconstruction in computed tomography

A method for predicting the image covariance resulting from total-variation-penalized iterative image reconstruction (TV-penalized IIR) is presented and demonstrated in a variety of contexts. The method is validated against the sample covariance from statistical noise realizations for a small image using a variety of comparison metrics. Potential applications for the covariance approximation include investigation of image properties such as object- and signal-dependence of noise, and noise stationarity. These applications are demonstrated, along with the construction of image pixel variance maps for two-dimensional 128 × 128 pixel images. Methods for extending the proposed covariance approximation to larger images and improving computational efficiency are discussed. Future work will apply the developed methodology to the construction of task-based image quality metrics such as the Hotelling observer detectability for TV-based IIR.

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.

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.

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.

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.

Effects of sparse sampling schemes on image quality in low-dose CT

PURPOSE

Various scanning methods and image reconstruction algorithms are actively investigated for low-dose computed tomography (CT) that can potentially reduce a health-risk related to radiation dose. Particularly, compressive-sensing (CS) based algorithms have been successfully developed for reconstructing images from sparsely sampled data. Although these algorithms have shown promises in low-dose CT, it has not been studied how sparse sampling schemes affect image quality in CS-based image reconstruction. In this work, the authors present several sparse-sampling schemes for low-dose CT, quantitatively analyze their data property, and compare effects of the sampling schemes on the image quality.

METHODS

Data properties of several sampling schemes are analyzed with respect to the CS-based image reconstruction using two measures: sampling density and data incoherence. The authors present five different sparse sampling schemes, and simulated those schemes to achieve a targeted dose reduction. Dose reduction factors of about 75% and 87.5%, compared to a conventional scan, were tested. A fully sampled circular cone-beam CT data set was used as a reference, and sparse sampling has been realized numerically based on the CBCT data.

RESULTS

It is found that both sampling density and data incoherence affect the image quality in the CS-based reconstruction. Among the sampling schemes the authors investigated, the sparse-view, many-view undersampling (MVUS)-fine, and MVUS-moving cases have shown promising results. These sampling schemes produced images with similar image quality compared to the reference image and their structure similarity index values were higher than 0.92 in the mouse head scan with 75% dose reduction.

CONCLUSIONS

The authors found that in CS-based image reconstructions both sampling density and data incoherence affect the image quality, and suggest that a sampling scheme should be devised and optimized by use of these indicators. With this strategic approach, one can acquire optimally sampled sparse data so that the CS-based algorithms can best perform in terms of image quality.

Adaptive bacteria colony picking in unstructured environments using intensity histogram and unascertained LS-SVM classifier

Features analysis is an important task which can significantly affect the performance of automatic bacteria colony picking. Unstructured environments also affect the automatic colony screening. This paper presents a novel approach for adaptive colony segmentation in unstructured environments by treating the detected peaks of intensity histograms as a morphological feature of images. In order to avoid disturbing peaks, an entropy based mean shift filter is introduced to smooth images as a preprocessing step. The relevance and importance of these features can be determined in an improved support vector machine classifier using unascertained least square estimation. Experimental results show that the proposed unascertained least square support vector machine (ULSSVM) has better recognition accuracy than the other state-of-the-art techniques, and its training process takes less time than most of the traditional approaches presented in this paper.

Sparse-view x-ray CT reconstruction via total generalized variation regularization

Sparse-view CT reconstruction algorithms via total variation (TV) optimize the data iteratively on the basis of a noise- and artifact-reducing model, resulting in significant radiation dose reduction while maintaining image quality. However, the piecewise constant assumption of TV minimization often leads to the appearance of noticeable patchy artifacts in reconstructed images. To obviate this drawback, we present a penalized weighted least-squares (PWLS) scheme to retain the image quality by incorporating the new concept of total generalized variation (TGV) regularization. We refer to the proposed scheme as 'PWLS-TGV' for simplicity. Specifically, TGV regularization utilizes higher order derivatives of the objective image, and the weighted least-squares term considers data-dependent variance estimation, which fully contribute to improving the image quality with sparse-view projection measurement. Subsequently, an alternating optimization algorithm was adopted to minimize the associative objective function. To evaluate the PWLS-TGV method, both qualitative and quantitative studies were conducted by using digital and physical phantoms. Experimental results show that the present PWLS-TGV method can achieve images with several noticeable gains over the original TV-based method in terms of accuracy and resolution properties.

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.

First-order convex feasibility algorithms for x-ray CT

PURPOSE

Iterative image reconstruction (IIR) algorithms in computed tomography (CT) are based on algorithms for solving a particular optimization problem. Design of the IIR algorithm, therefore, is aided by knowledge of the solution to the optimization problem on which it is based. Often times, however, it is impractical to achieve accurate solution to the optimization of interest, which complicates design of IIR algorithms. This issue is particularly acute for CT with a limited angular-range scan, which leads to poorly conditioned system matrices and difficult to solve optimization problems. In this paper, we develop IIR algorithms which solve a certain type of optimization called convex feasibility. The convex feasibility approach can provide alternatives to unconstrained optimization approaches and at the same time allow for rapidly convergent algorithms for their solution-thereby facilitating the IIR algorithm design process.

METHODS

An accelerated version of the Chambolle-Pock (CP) algorithm is adapted to various convex feasibility problems of potential interest to IIR in CT. One of the proposed problems is seen to be equivalent to least-squares minimization, and two other problems provide alternatives to penalized, least-squares minimization.

RESULTS

The accelerated CP algorithms are demonstrated on a simulation of circular fan-beam CT with a limited scanning arc of 144°. The CP algorithms are seen in the empirical results to converge to the solution of their respective convex feasibility problems.

CONCLUSIONS

Formulation of convex feasibility problems can provide a useful alternative to unconstrained optimization when designing IIR algorithms for CT. The approach is amenable to recent methods for accelerating first-order algorithms which may be particularly useful for CT with limited angular-range scanning. The present paper demonstrates the methodology, and future work will illustrate its utility in actual CT application.