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

Adaptive-weighted high order TV algorithm for sparse-view CT reconstruction

BACKGROUND

With the development of low-dose computed tomography (CT), incomplete data reconstruction has been widely concerned. The total variation (TV) minimization algorithm can accurately reconstruct images from sparse or noisy data.

PURPOSE

However, the traditional TV algorithm ignores the direction of structures in images, leading to the loss of edge information and block artifacts when the object is not piecewise constant. Since the anisotropic information can facilitate preserving the edge and detail information in images, we aim to improve the TV algorithm in terms of reconstruction accuracy via this approach.

METHODS

In this paper, we propose an adaptive-weighted high order total variation (awHOTV) algorithm. We construct the second order TV-norm using the second order gradient, adapt the anisotropic edge property between neighboring image pixels, adjust the local image-intensity gradient to keep edge information, and design the corresponding Chambolle-Pock (CP) solving algorithm. Implementing the proposed algorithm, comprehensive studies are conducted in the ideal projection data experiment where the Structural similarity (SSIM), Root Mean Square Error (RMSE), Contrast to noise ratio (CNR), and modulation transform function (MTF) curves are utilized to evaluate the quality of reconstructed images in statism, structure, spatial resolution, and contrast, respectively. In the noisy data experiment, we further use the noise power spectrum (NPS) curve to evaluate the reconstructed images and compare it with other three algorithms.

RESULTS

We use the 2D slice in the XCAT phantom, 2D slice in TCIA Challenge data and FORBILD phantom as simulation phantoms and use real bird data for real verification. The results show that, compared with the traditional TV and FBP algorithms, the awHOTV has better performance in terms of RMSE, SSIM, and Pearson correlation coefficient (PCC) under the projected data with different sparsity. In addition, the awHOTV algorithm is robust against the noise of different intensities.

CONCLUSIONS

The proposed awHOTV method can reconstruct the images with high accuracy under sparse or noisy data. The awHOTV solves the strip artifacts caused by sparse data in the FBP method. Compared with the TV method, the awHOTV can effectively suppress block artifacts and has good detail protection ability.

A feasibility study on deep-neural-network-based dose-neutral dual-energy digital breast tomosynthesis

BACKGROUND

Diagnostic performance based on x-ray breast imaging is subject to breast density. Although digital breast tomosynthesis (DBT) is reported to outperform conventional mammography in denser breasts, mass detection and malignancy characterization are often considered challenging yet.

PURPOSE

As an improved diagnostic solution to the dense breast cases, we propose a dual-energy DBT imaging technique that enables breast compositional imaging at comparable scanning time and patient dose compared to the conventional single-energy DBT.

METHODS

The proposed dual-energy DBT acquires projection data by alternating two different energy spectra. Then, we synthesize unmeasured projection data using a deep neural network that exploits the measured projection data and adjacent projection data obtained under the other x-ray energy spectrum. For material decomposition, we estimate partial path lengths of an x-ray through water, lipid, and protein from the measured and the synthesized projection data with the object thickness information. After material decomposition in the projection domain, we reconstruct material-selective DBT images. The deep neural network is trained with the numerical breast phantoms. A pork meat phantom is scanned with a prototype dual-energy DBT system to demonstrate the feasibility of the proposed imaging method.

RESULTS

The developed deep neural network successfully synthesized missing projections. Material-selective images reconstructed from the synthesized data present comparable compositional contrast of the cancerous masses compared with those from the fully measured data.

CONCLUSIONS

The proposed dual-energy DBT scheme is expected to substantially contribute to enhancing mass malignancy detection accuracy particularly in dense breasts.

RISING: A new framework for model-based few-view CT image reconstruction with deep learning

Medical image reconstruction from low-dose tomographic data is an active research field, recently revolutionized by the advent of deep learning. In fact, deep learning typically yields superior results than classical optimization approaches, but unstable results have been reported both numerically and theoretically in the literature. This paper proposes RISING, a new framework for sparse-view tomographic image reconstruction combining an early-stopped Rapid Iterative Solver with a subsequent Iteration Network-based Gaining step. In our two-step approach, the first phase executes very few iterations of a regularized model-based algorithm, whereas the second step completes the missing iterations by means of a convolutional neural network. The proposed method is ground-truth free; it exploits the computational speed and flexibility of a data-driven approach, but it also imposes sparsity constraints to the solution as in the model-based setting. Experiments performed both on a digitally created and on a real abdomen data set confirm the numerical and visual accuracy of the reconstructed RISING images in short computational times. These features make the framework promising to be used on real systems for clinical exams.

A fast tomosynthesis method for printed circuit boards based on a multiple multi-resolution reconstruction algorithm

Digital tomosynthesis (DTS) technology has attracted much attention in the field of nondestructive testing of printed circuit boards (PCB) due to its high resolution and suitability to thin slab objects. However, the traditional DTS iterative algorithm is computationally demanding, and its real-time processing of high-resolution and large volume reconstruction is infeasible. To address this issue, we in this study propose a multiple multi-resolution algorithm, including two multi-resolution strategies: volume domain multi-resolution and projection domain multi-resolution. The first multi-resolution scheme employs a LeNet-based classification network to divide the roughly reconstructed low-resolution volume into two sub-volumes namely, (1) the region of interest (ROI) with welding layers that necessitates high-resolution reconstruction, and (2) the remaining volume with unimportant information which can be reconstructed in low-resolution. When X-rays in adjacent projection angles pass through many identical voxels, information redundancy is prevalent between the adjacent image projections. Therefore, the second multi-resolution scheme divides the projections into non-overlapping subsets, using only one subset for each iteration. The proposed algorithm is evaluated using both the simulated and real image data. The results demonstrate that the proposed algorithm is approximately 6.5 times faster than the full-resolution DTS iterative reconstruction algorithm without compromising image reconstruction quality.

Low-Light Enhancement Using a Plug-and-Play Retinex Model With Shrinkage Mapping for Illumination Estimation

Low-light photography conditions degrade image quality. This study proposes a novel Retinex-based low-light enhancement method to correctly decompose an input image into reflectance and illumination. Subsequently, we can improve the viewing experience by adjusting the illumination using intensity and contrast enhancement. Because image decomposition is a highly ill-posed problem, constraints must be properly imposed on the optimization framework. To meet the criteria of ideal Retinex decomposition, we design a nonconvex L_p norm and apply shrinkage mapping to the illumination layer. In addition, edge-preserving filters are introduced using the plug-and-play technique to improve illumination. Pixel-wise weights based on variance and image gradients are adopted to suppress noise and preserve details in the reflectance layer. We choose the alternating direction method of multipliers (ADMM) to solve the problem efficiently. Experimental results on several challenging low-light datasets show that our proposed method can more effectively enhance image brightness as compared with state-of-the-art methods. In addition to subjective observations, the proposed method also achieved competitive performance in objective image quality assessments.

Smoothed-NUV Priors for Imaging

Variations of L1 -regularization including, in particular, total variation regularization, have hugely improved computational imaging. However, sharper edges and fewer staircase artifacts can be achieved with convex-concave regularizers. We present a new class of such regularizers using normal priors with unknown variance (NUV), which include smoothed versions of the logarithm function and smoothed versions of Lp norms with p ≤ 1 . All NUV priors allow variational representations that lead to efficient algorithms for image reconstruction by iterative reweighted descent. A preferred such algorithm is iterative reweighted coordinate descent, which has no parameters (in particular, no step size to control) and is empirically robust and efficient. The proposed priors and algorithms are demonstrated with applications to tomography. We also note that the proposed priors come with built-in edge detection, which is demonstrated by an application to image segmentation.

Optimization of a customized simultaneous algebraic reconstruction technique algorithm for phase-contrast breast computed tomography

Objective. To introduce the optimization of a customized GPU-based simultaneous algebraic reconstruction technique (cSART) in the field of phase-contrast breast computed tomography (bCT). The presented algorithm features a 3D bilateral regularization filter that can be tuned to yield optimal performance for clinical image visualization and tissues segmentation. Approach. Acquisitions of a dedicated test object and a breast specimen were performed at Elettra, the Italian synchrotron radiation (SR) facility (Trieste, Italy) using a large area CdTe single-photon counting detector. Tomographic images were obtained at 5 mGy of mean glandular dose, with a 32 keV monochromatic x-ray beam in the free-space propagation mode. Three independent algorithms parameters were optimized by using contrast-to-noise ratio (CNR), spatial resolution, and noise texture metrics. The results obtained with the cSART algorithm were compared with conventional SART and filtered back projection (FBP) reconstructions. Image segmentation was performed both with gray scale-based and supervised machine-learning approaches. Main results. Compared to conventional FBP reconstructions, results indicate that the proposed algorithm can yield images with a higher CNR (by 35% or more), retaining a high spatial resolution while preserving their textural properties. Alternatively, at the cost of an increased image ‘patchiness’, the cSART can be tuned to achieve a high-quality tissue segmentation, suggesting the possibility of performing an accurate glandularity estimation potentially of use in the realization of realistic 3D breast models starting from low radiation dose images. Significance. The study indicates that dedicated iterative reconstruction techniques could provide significant advantages in phase-contrast bCT imaging. The proposed algorithm offers great flexibility in terms of image reconstruction optimization, either toward diagnostic evaluation or image segmentation.

A Green Prospective for Learned Post-Processing in Sparse-View Tomographic Reconstruction

Deep Learning is developing interesting tools that are of great interest for inverse imaging applications. In this work, we consider a medical imaging reconstruction task from subsampled measurements, which is an active research field where Convolutional Neural Networks have already revealed their great potential. However, the commonly used architectures are very deep and, hence, prone to overfitting and unfeasible for clinical usages. Inspired by the ideas of the green AI literature, we propose a shallow neural network to perform efficient Learned Post-Processing on images roughly reconstructed by the filtered backprojection algorithm. The results show that the proposed inexpensive network computes images of comparable (or even higher) quality in about one-fourth of time and is more robust than the widely used and very deep ResUNet for tomographic reconstructions from sparse-view protocols.

A Model-Based Optimization Framework for Iterative Digital Breast Tomosynthesis Image Reconstruction

Digital Breast Tomosynthesis is an X-ray imaging technique that allows a volumetric reconstruction of the breast, from a small number of low-dose two-dimensional projections. Although it is already used in the clinical setting, enhancing the quality of the recovered images is still a subject of research. The aim of this paper was to propose and compare, in a general optimization framework, three slightly different models and corresponding accurate iterative algorithms for Digital Breast Tomosynthesis image reconstruction, characterized by a convergent behavior. The suggested model-based implementations are specifically aligned to Digital Breast Tomosynthesis clinical requirements and take advantage of a Total Variation regularizer. We also tune a fully-automatic strategy to set a proper regularization parameter. We assess our proposals on real data, acquired from a breast accreditation phantom and a clinical case. The results confirm the effectiveness of the presented framework in reconstructing breast volumes, with particular focus on the masses and microcalcifications, in few iterations and in enhancing the image quality in a prolonged execution.

A TV-minimization image-reconstruction algorithm without system matrix

PURPOSE

Total Variation (TV) minimization algorithm is a classical compressed sensing (CS) based iterative image reconstruction algorithm that can accurately reconstruct images from sparse-view projections in computed tomography (CT). However, the system matrix used in the algorithm is often too large to be stored in computer memory. The purpose of this study is to investigate a new TV algorithm based on image rotation and without system matrix to avoid the memory requirement of system matrix.

METHODS

Without loss of generality, a rotation-based adaptive steepest descent-projection onto convex sets (R-ASD-POCS) algorithm is proposed and tested to solve the TV model in parallel beam CT. Specifically, simulation experiments are performed via the Shepp-Logan, FORBILD and real CT image phantoms are used to verify the inverse-crime capability of the algorithm and evaluate the sparse reconstruction capability and the noise suppression performance of the algorithm.

RESULTS

Experimental results show that the algorithm can achieve inverse-crime, accurate sparse reconstruction and thus accurately reconstruct images from noisy projections. Compared with the classical ASD-POCS algorithm, the new algorithm may yield the similar image reconstruction accuracy without use of the huge system matrix, which saves the computational memory space significantly. Additionally, the results also show that R-ASD-POCS algorithm is faster than ASD-POCS.

CONCLUSIONS

The proposed new algorithm can effectively solve the problem of using huge memory in large scale and iterative image reconstruction. Integrating with ASD-POCS frame, this no-system-matrix based scheme may be readily extended and applied to any iterative image reconstructions.

Low-dose spectral CT reconstruction based on image-gradient L0-norm and adaptive spectral PICCS

The photon-counting detector based spectral computed tomography (CT) is promising for lesion detection, tissue characterization, and material decomposition. However, the lower signal-to-noise ratio within multi-energy projection dataset can result in poorly reconstructed image quality. Recently, as prior information, a high-quality spectral mean image was introduced into the prior image constrained compressed sensing (PICCS) framework to suppress noise, leading to spectral PICCS (SPICCS). In the original SPICCS model, the image gradient L₁-norm is employed, and it can cause blurred edge structures in the reconstructed images. Encouraged by the advantages in edge preservation and finer structure recovering, the image gradient L₀-norm was incorporated into the PICCS model. Furthermore, due to the difference of energy spectrum in different channels, a weighting factor is introduced and adaptively adjusted for different channel-wise images, leading to an L₀-norm based adaptive SPICCS (L₀-ASPICCS) algorithm for low-dose spectral CT reconstruction. The split-Bregman method is employed to minimize the objective function. Extensive numerical simulations and physical phantom experiments are performed to evaluate the proposed method. By comparing with the state-of-the-art algorithms, such as the simultaneous algebraic reconstruction technique, total variation minimization, and SPICCS, the advantages of our proposed method are demonstrated in terms of both qualitative and quantitative evaluation results.

Ordered subsets Non-Local means constrained reconstruction for sparse view cone beam CT system

Sparse-view sampling scans reduce the patient's radiation dose by reducing the total exposure duration. CT reconstructions under such scan mode are often accompanied by severe artifacts due to the high ill-posedness of the problem. In this paper, we use a Non-Local means kernel as a regularization constraint to reconstruct image volumes from sparse-angle sampled cone-beam CT scans. To overcome the huge computational cost of the 3D reconstruction, we propose a sequential update scheme relying on ordered subsets in the image domain. It is shown through experiments on simulated and real data and comparisons with other methods that the proposed approach is robust enough to deal with the number of views reduced up to 1/10. When coupled with a CUDA parallel computing technique, the computation speed of the iterative reconstruction is greatly improved.

A new in-line X-ray phase-contrast computed tomography reconstruction algorithm based on adaptive-weighted anisotropic TpV regularization for insufficient data

In-line X-ray phase-contrast computed tomography (IL-PCCT) is a valuable tool for revealing the internal detailed structures in weakly absorbing objects (e.g. biological soft tissues), and has a great potential to become clinically applicable. However, the long scanning time for IL-PCCT will result in a high radiation dose to biological samples, and thus impede the wider use of IL-PCCT in clinical and biomedical imaging. To alleviate this problem, a new iterative CT reconstruction algorithm is presented that aims to decrease the radiation dose by reducing the projection views, while maintaining the high quality of reconstructed images. The proposed algorithm combines the adaptive-weighted anisotropic total p-variation (AwaTpV, 0 < p < 1) regularization technique with projection onto convex sets (POCS) strategy. Noteworthy, the AwaTpV regularization term not only contains the horizontal and vertical image gradients but also adds the diagonal image gradients in order to enforce the directional continuity in the gradient domain. To evaluate the effectiveness and ability of the proposed algorithm, experiments with a numerical phantom and synchrotron IL-PCCT were performed, respectively. The results demonstrated that the proposed algorithm had the ability to significantly reduce the artefacts caused by insufficient data and effectively preserved the edge details under noise-free and noisy conditions, and thus could be used as an effective approach to decrease the radiation dose for IL-PCCT.

Image gradient L0-norm based PICCS for swinging multi-source CT reconstruction

Dynamic computed tomography (CT) is usually employed to image motion objects, such as beating heart, coronary artery and cerebral perfusion, etc. Recently, to further improve the temporal resolution for aperiodic industrial process imaging, the swinging multi-source CT (SMCT) systems and the corresponding swinging multi-source prior image constrained compressed sensing (SM-PICCS) method were developed. Since the SM-PICCS uses the L1-norm of image gradient, the edge structures in the reconstructed images are blurred and motion artifacts are still present. Inspired by the advantages in terms of image edge preservation and fine structure recovering, the L0-norm of image gradient is incorporated into the prior image constrained compressed sensing, leading to an L0-PICCS Algorithm 1Table 1The parameters of L0-PICCS (δ1,δ2,λ1*,λ2*) for numerical simulation.Sourceswδ1(10-2)δ2(10-2)λ1*(10-2)λ2*(10-8)Noise-free510522.001.525522.001.55035002.00471014.33332.00500025522.00500050222.005000Noise51062002.505002554502.501.55054502.901.571027.385.91.5810000258.285.91.5850050522.001.5. The experimental results confirm that the L0-PICCS outperforms the SM-PICCS in both visual inspection and quantitative analysis.

Infrared Image Super-Resolution Reconstruction Based on Quaternion Fractional Order Total Variation with Lp Quasinorm

Owing to the limitations of the imaging principle as well as the properties of imaging systems, infrared images often have some drawbacks, including low resolution, a lack of detail, and indistinct edges. Therefore, it is essential to improve infrared image quality. Considering the information of neighbors, a description of sparse edges, and by avoiding staircase artifacts, a new super-resolution reconstruction (SRR) method is proposed for infrared images, which is based on fractional order total variation (FTV) with quaternion total variation and the L p quasinorm. Our proposed method improves the sparsity exploitation of FTV, and efficiently preserves image structures. Furthermore, we adopt the plug-and-play alternating direction method of multipliers (ADMM) and the fast Fourier transform (FFT) theory for the proposed method to improve the efficiency and robustness of our algorithm; in addition, an accelerated step is adopted. Our experimental results show that the proposed method leads to excellent performances in terms of an objective evaluation and the subjective visual effect.

X-ray CT image reconstruction from few-views via total generalized p-variation minimization

Total variation (TV)-based CT image reconstruction, employing the image gradient sparsity, has shown to be experimentally capable of reducing the X-ray sampling rate and removing the unwanted artifacts, yet may cause unfavorable over-smoothing and staircase effects by the piecewise constant assumption. In this paper, we present a total generalized p-variation (TGpV) regularization model to adaptively preserve the edge information while avoiding the staircase effect. The new model is solved by splitting variables with an efficient alternating minimization scheme. With the utilization of generalized p-shrinkage mappings and partial Fourier transform, all the subproblems have closed solutions. The proposed method shows excellent properties of edge preserving as well as the smoothness features by the consideration of high order derivatives. Experimental results indicate that the proposed method could avoid the mentioned effects and reconstruct more accurately than both the TV and TGV minimization algorithms when applied to a few-view problem.

Block matching sparsity regularization-based image reconstruction for incomplete projection data in computed tomography

In medical imaging many conventional regularization methods, such as total variation or total generalized variation, impose strong prior assumptions which can only account for very limited classes of images. A more reasonable sparse representation frame for images is still badly needed. Visually understandable images contain meaningful patterns, and combinations or collections of these patterns can be utilized to form some sparse and redundant representations which promise to facilitate image reconstructions. In this work, we propose and study block matching sparsity regularization (BMSR) and devise an optimization program using BMSR for computed tomography (CT) image reconstruction for an incomplete projection set. The program is built as a constrained optimization, minimizing the L1-norm of the coefficients of the image in the transformed domain subject to data observation and positivity of the image itself. To solve the program efficiently, a practical method based on the proximal point algorithm is developed and analyzed. In order to accelerate the convergence rate, a practical strategy for tuning the BMSR parameter is proposed and applied. The experimental results for various settings, including real CT scanning, have verified the proposed reconstruction method showing promising capabilities over conventional regularization.

A weighted difference of L1 and L2 on the gradient minimization based on alternating direction method for circular computed tomography

Iterative reconstruction algorithms for computed tomography (CT) through total variation (TV) regularization can provide accurate and stable reconstruction results. TV minimization is the L1-norm of gradient-magnitude images and can be regarded as a convex relaxation method to replace the L0 norm. In this study, a fast and efficient algorithm, which is named a weighted difference of L1 and L2 (L1 - αL2) on the gradient minimization, was proposed and investigated. The new algorithm provides a better description of sparsity for the optimization-based algorithms than TV minimization algorithms. The alternating direction method is an efficient method to solve the proposed model, which is utilized in this study. Both simulations and real CT projections were tested to verify the performances of the proposed algorithm. In the simulation experiments, the reconstructions from the proposed method provided better image quality than TV minimization algorithms with only 7 views in 180 degrees, which is also computationally faster. Meanwhile, the new algorithm enabled to achieve the final solution with less iteration numbers.

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.

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

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

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.

Constrained Total Generalized p-Variation Minimization for Few-View X-Ray Computed Tomography Image Reconstruction

Total generalized variation (TGV)-based computed tomography (CT) image reconstruction, which utilizes high-order image derivatives, is superior to total variation-based methods in terms of the preservation of edge information and the suppression of unfavorable staircase effects. However, conventional TGV regularization employs l1-based form, which is not the most direct method for maximizing sparsity prior. In this study, we propose a total generalized p-variation (TGpV) regularization model to improve the sparsity exploitation of TGV and offer efficient solutions to few-view CT image reconstruction problems. To solve the nonconvex optimization problem of the TGpV minimization model, we then present an efficient iterative algorithm based on the alternating minimization of augmented Lagrangian function. All of the resulting subproblems decoupled by variable splitting admit explicit solutions by applying alternating minimization method and generalized p-shrinkage mapping. In addition, approximate solutions that can be easily performed and quickly calculated through fast Fourier transform are derived using the proximal point method to reduce the cost of inner subproblems. The accuracy and efficiency of the simulated and real data are qualitatively and quantitatively evaluated to validate the efficiency and feasibility of the proposed method. Overall, the proposed method exhibits reasonable performance and outperforms the original TGV-based method when applied to few-view problems.

MOCCA: Mirrored Convex/Concave Optimization for Nonconvex Composite Functions

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

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.

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.

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.