Penalized Weighted Least-squares Research Articles

Evaluation of low-dose computed tomography reconstruction using spatial-radon domain total generalized variation regularization

The x-ray radiation dose in computed tomography (CT) examination has been a major concern for patients. Lowing the tube current and exposure time in data acquisition is a straightforward and cost-effective strategy to reduce the x-ray radiation dose. However, this will inevitably increase the noise fluctuations in measured projection data, and the corresponding CT image quality will be severely degraded if noise suppression is not performed during image reconstruction. To reconstruct high-quality low-dose CT image, we present a spatial-radon domain total generalized variation (SRDTGV) regularization for statistical iterative reconstruction based on penalized weighted least-squares (PWLS) principle, which is called PWLS-SRDTGV for simplicity. The presented PWLS-SRDTGV model can simultaneously reconstruct high-quality CT image in space domain and its corresponding projection in radon domain. An efficient split Bregman algorithm was applied to minimize the cost function of the proposed reconstruction model. Qualitative and quantitative studies were performed to evaluate the effectiveness of the PWLS-SRDTGV image reconstruction algorithm using a digital 3D XCAT phantom and an anthropomorphic torso phantom. The experimental results demonstrate that PWLS-SRDTGV algorithm achieves notable gains in noise reduction, streak artifact suppression, and edge preservation compared with competing reconstruction approaches.

Estimation of Multiresponse Multipredictor Nonparametric Regression Model Using Mixed Estimator

In data analysis using a nonparametric regression approach, we are often faced with the problem of analyzing a set of data that has mixed patterns, namely, some of the data have a certain pattern and the rest of the data have a different pattern. To handle this kind of datum, we propose the use of a mixed estimator. In this study, we theoretically discuss a developed estimation method for a nonparametric regression model with two or more response variables and predictor variables, and there is a correlation between the response variables using a mixed estimator. The model is called the multiresponse multipredictor nonparametric regression (MMNR) model. The mixed estimator used for estimating the MMNR model is a mixed estimator of smoothing spline and Fourier series that is suitable for analyzing data with patterns that partly change at certain subintervals, and some others that follow a recurring pattern in a certain trend. Since in the MMNR model there is a correlation between responses, a symmetric weight matrix is involved in the estimation process of the MMNR model. To estimate the MMNR model, we apply the reproducing kernel Hilbert space (RKHS) method to penalized weighted least square (PWLS) optimization for estimating the regression function of the MMNR model, which consists of a smoothing spline component and a Fourier series component. A simulation study to show the performance of proposed method is also given. The obtained results are estimations of the smoothing spline component, Fourier series component, MMNR model, weight matrix, and consistency of estimated regression function. In conclusion, the estimation of the MMNR model using the mixed estimator is a combination of smoothing spline component and Fourier series component estimators. It depends on smoothing and oscillation parameters, and it has linear in observation and consistent properties.

Multi-layer clustering-based residual sparsifying transform for low-dose CT image reconstruction.

The recently proposed sparsifying transform (ST) models incur low computational cost and have been applied to medical imaging. Meanwhile, deep models with nested network structure reveal great potential for learning features in different layers. In this study, we propose a network-structured ST learning approach for X-ray computed tomography (CT), which we refer to as multi-layer clustering-based residual sparsifying transform (MCST) learning. The proposed MCST scheme learns multiple different unitary transforms in each layer by dividing each layer's input into several classes. We apply the MCST model to low-dose CT (LDCT) reconstruction by deploying the learned MCST model into the regularizer in penalized weighted least squares (PWLS) reconstruction. The proposed MCST model combines a multi-layer sparse representation structure with multiple clusters for the features in each layer that are modeled by a rich collection of transforms. We train the MCST model in an unsupervised manner via a block coordinate descent (BCD) algorithm. Since our method is patch-based, the training can be performed with a limited set of images. For CT image reconstruction, we devise a novel algorithm called PWLS-MCST by integrating the pre-learned MCST signal model with PWLSoptimization. We conducted LDCT reconstruction experiments on XCAT phantom data, Numerical Mayo Clinical CT dataset and "LDCT image and projection dataset" (Clinical LDCT dataset). We trained the MCST model with two (or three) layers and with five clusters in each layer. The learned transforms in the same layer showed rich features while additional information is extracted from representation residuals. Our simulation results and clinical results demonstrate that PWLS-MCST achieves better image reconstruction quality than the conventional filtered back-projection (FBP) method and PWLS with edge-preserving (EP) regularizer. It also outperformed recent advanced methods like PWLS with a learned multi-layer residual sparsifying transform (MARS) prior and PWLS with a union of learned transforms (ULTRA), especially for displaying clear edges and preserving subtledetails. In this work, a multi-layer sparse signal model with a nested network structure is proposed. We refer this novel model as the MCST model that exploits multi-layer residual maps to sparsify the underlying image and clusters the inputs in each layer for accurate sparsification. We presented a new PWLS framework with a learned MCST regularizer for LDCT reconstruction. Experimental results show that the proposed PWLS-MCST provides clearer reconstructions than several baseline methods. The code for PWLS-MCST is released at https://github.com/Xikai97/PWLS-MCST.

Determining confidence interval and asymptotic distribution for parameters of multiresponse semiparametric regression model using smoothing spline estimator

The multiresponse semiparametric regression (MSR) model is a regression model with more than two response variables that are mutually correlated, and its regression function is composed of parametric and nonparametric components. The study objectives are propose a new method for estimating the MSR model using smoothing spline. Also, find the confidence interval (CI) of parameters and the distribution asymptotically of the model parameters estimator. Methods used in this study are reproducing kernel Hilbert space (RKHS) method and a developed penalized weighted least squares (PWLS), and apply pivotal quantity, central limit theorem, and theorems of Cramer-Wold and Slutsky. The results are an 100(1–α)% CI estimate and an asymptotic normal distribution for the parameters of the MSR model. In conclusion, the estimated MSR model is a combined components estimate of parametric and nonparametric which is linear to observation, and CIs of parameters depend on t distribution and estimator of parameters is asymptotically normally distributed. Future time, this study results can be used as theoretical bases to design standard growth charts of the toddlers which can then be used to assess the nutritional status of the toddlers.

Reproducing Kernel Hilbert Space Approach to Multiresponse Smoothing Spline Regression Function

In statistical analyses, especially those using a multiresponse regression model approach, a mathematical model that describes a functional relationship between more than one response variables and one or more predictor variables is often involved. The relationship between these variables is expressed by a regression function. In the multiresponse nonparametric regression (MNR) model that is part of the multiresponse regression model, estimating the regression function becomes the main problem, as there is a correlation between the responses such that it is necessary to include a symmetric weight matrix into a penalized weighted least square (PWLS) optimization during the estimation process. This is, of course, very complicated mathematically. In this study, to estimate the regression function of the MNR model, we developed a PWLS optimization method for the MNR model proposed by a previous researcher, and used a reproducing kernel Hilbert space (RKHS) approach based on a smoothing spline to obtain the solution to the developed PWLS optimization. Additionally, we determined the symmetric weight matrix and optimal smoothing parameter, and investigated the consistency of the regression function estimator. We provide an illustration of the effects of the smoothing parameters for the estimation results using simulation data. In the future, the theory generated from this study can be developed within the scope of statistical inference, especially for the purpose of testing hypotheses involving multiresponse nonparametric regression models and multiresponse semiparametric regression models, and can be used to estimate the nonparametric component of a multiresponse semiparametric regression model used to model Indonesian toddlers’ standard growth charts.

Multilayer residual sparsifying transform (MARS) model for low-dose CT image reconstruction.

Signal models based on sparse representations have received considerable attention in recent years. On the other hand, deep models consisting of a cascade of functional layers, commonly known as deep neural networks, have been highly successful for the task of object classification and have been recently introduced to image reconstruction. In this work, we develop a new image reconstruction approach based on a novel multilayer model learned in an unsupervised manner by combining both sparse representations and deep models. The proposed framework extends the classical sparsifying transform model for images to a Multilayer residual sparsifying transform (MARS) model, wherein the transform domain data are jointly sparsified over layers. We investigate the application of MARS models learned from limited regular-dose images for low-dose CT reconstruction using penalized weighted least squares (PWLS) optimization. We propose new formulations for multilayer transform learning and image reconstruction. We derive an efficient block coordinate descent algorithm to learn the transforms across layers, in an unsupervised manner from limited regular-dose images. The learned model is then incorporated into the low-dose image reconstruction phase. Low-dose CT experimental results with both the XCAT phantom and Mayo Clinic data show that the MARS model outperforms conventional methods such as filtered back-projection and PWLS methods based on the edge-preserving (EP) regularizer in terms of two numerical metrics (RMSE and SSIM) and noise suppression. Compared with the single-layer learned transform (ST) model, the MARS model performs better in maintaining some subtle details. This work presents a novel data-driven regularization framework for CT image reconstruction that exploits learned multilayer or cascaded residual sparsifying transforms. The image model is learned in an unsupervised manner from limited images. Our experimental results demonstrate the promising performance of the proposed multilayer scheme over single-layer learned sparsifying transforms. Learned MARS models also offer better image quality than typical nonadaptive PWLS methods.

An Analysis about Fourier Series Estimator in Nonparametric Regression for Longitudinal Data

Fourier series is a function that is often used Mathematically and Statistically especially for modeling. Here, Fourier series can be constructed as an estimator in nonparametric regression. Nonparametric regression is not only using cross section data, but also longitudinal data. Some of nonparametric regression estimators have been developed for longitudinal data case, such as kernel, and spline. In this study, we concentrate to develop an inference analysis that related to Fourier series estimator in nonparametric regression for longitudinal data. Nonparametric regression based on Fourier series is capable to model data relationship with fluctuation or oscillation pattern that represents with sine and cosine functions. For point estimation analysis, Penalized Weighted Least Square (PWLS) is used to determine an estimator for parameter vector in nonparametric regression. Different with previous studies, PWLS is used to get smooth estimator. The result is an estimator for nonparametric regression curve for longitudinal data based on Fourier series approach. In addition, this study also investigated the asymptotic properties of the nonparametric regression curve estimators using the Fourier series approach for longitudinal data, especially linearity and consistency. Some study cases based on previous research and a new study case is given to make sure that Fourier series estimator in nonparametric regression has good performance in longitudinal data modeling. This study is important in order to develop further inferences Statistics, such as interval estimation and test hypothesis that related nonparametric regression with Fourier series estimator for longitudinal data.

Iterative reconstruction for low-dose cerebral perfusion computed tomography using prior image induced diffusion tensor

Cerebral perfusion computed tomography (CPCT) can depict the functional status of cerebral circulation at the tissue level; hence, it has been increasingly used to diagnose patients with cerebrovascular disease. However, there is a significant concern that CPCT scanning protocol could expose patients to excessive radiation doses. Although reducing the x-ray tube current when acquiring CPCT projection data is an effective method for reducing radiation dose, this technique usually results in degraded image quality. To enhance the image quality of low-dose CPCT, we present a prior image induced diffusion tensor (PIDT) for statistical iterative reconstruction, based on the penalized weighted least-squares (PWLS) criterion, which we referred to as PWLS-PIDT, for simplicity. Specifically, PIDT utilizes the geometric features of pre-contrast scanned high-quality CT image as a structure prior for PWLS reconstruction; therefore, the low-dose CPCT images are enhanced while preserving important features in the target image. An effective alternating minimization algorithm is developed to solve the associated objective function in the PWLS-PIDT reconstruction. We conduct qualitative and quantitative studies to evaluate the PWLS-PIDT reconstruction with a digital brain perfusion phantom and patient data. With this method, the noise in the reconstructed CPCT images is more substantially reduced than that of other competing methods, without sacrificing structural details significantly. Furthermore, the CPCT sequential images reconstructed via the PWLS-PIDT method can derive more accurate hemodynamic parameter maps than those of other competing methods.

Mixed Spline Smoothing and Kernel Estimator in Biresponse Nonparametric Regression

Mixed estimators in nonparametric regression have been developed in models with one response. The biresponse cases with different patterns among predictor variables that tend to be mixed estimators are often encountered. Therefore, in this article, we propose a biresponse nonparametric regression model with mixed spline smoothing and kernel estimators. This mixed estimator is suitable for modeling biresponse data with several patterns (response vs. predictors) that tend to change at certain subintervals such as the spline smoothing pattern, and other patterns that tend to be random are commonly modeled using kernel regression. The mixed estimator is obtained through two-stage estimation, i.e., penalized weighted least square (PWLS) and weighted least square (WLS). Furthermore, the proposed biresponse modeling with mixed estimators is validated using simulation data. This estimator is also applied to the percentage of the poor population and human development index data. The results show that the proposed model can be appropriately implemented and gives satisfactory results.

Nonparametric Regression Model for Longitudinal Data with Mixed Truncated Spline and Fourier Series

Existing literature in nonparametric regression has established a model that only applies one estimator to all predictors. This study is aimed at developing a mixed truncated spline and Fourier series model in nonparametric regression for longitudinal data. The mixed estimator is obtained by solving the two-stage estimation, consisting of a penalized weighted least square (PWLS) and weighted least square (WLS) optimization. To demonstrate the performance of the proposed method, simulation and real data are provided. The results of the simulated data and case study show a consistent finding.

Statistical image-based material decomposition for triple-energy computed tomography using total variation regularization.

Triple-energy computed tomography (TECT) can obtain x-ray attenuation measurements at three energy spectra, thereby allowing identification of different material compositions with same or very similar attenuation coefficients. This ability is known as material decomposition, which can decompose TECT images into different basis material image. However, the basis material image would be severely degraded when material decomposition is directly performed on the noisy TECT measurements using a matrix inversion method. To achieve high quality basis material image, we present a statistical image-based material decomposition method for TECT, which uses the penalized weighted least-squares (PWLS) criteria with total variation (TV) regularization (PWLS-TV). The weighted least-squares term involves the noise statistical properties of the material decomposition process, and the TV regularization penalizes differences between local neighboring pixels in a decomposed image, thereby contributing to improving the quality of the basis material image. Subsequently, an alternating optimization method is used to minimize the objective function. The performance of PWLS-TV is quantitatively evaluated using digital and mouse thorax phantoms. The experimental results show that PWLS-TV material decomposition method can greatly improve the quality of decomposed basis material image compared to the quality of images obtained using the competing methods in terms of suppressing noise and preserving edge and fine structure details. The PWLS-TV method can simultaneously perform noise reduction and material decomposition in one iterative step, and it results in a considerable improvement of basis material image quality.

Cone-beam CT for imaging of the head/brain: Development and assessment of scanner prototype and reconstruction algorithms.

Our aim was to develop a high-quality, mobile cone-beam computed tomography (CBCT) scanner for point-of-care detection and monitoring of low-contrast, soft-tissue abnormalities in the head/brain, such as acute intracranial hemorrhage (ICH). This work presents an integrated framework of hardware and algorithmic advances for improving soft-tissue contrast resolution and evaluation of its technical performance with human subjects. Four configurations of a CBCT scanner prototype were designed and implemented to investigate key aspects of hardware (including system geometry, antiscatter grid, bowtie filter) and technique protocols. An integrated software pipeline (c.f., a serial cascade of algorithms) was developed for artifact correction (image lag, glare, beam hardening and x-ray scatter), motion compensation, and three-dimensional image (3D) reconstruction [penalized weighted least squares (PWLS), with a hardware-specific statistical noise model]. The PWLS method was extended in this work to accommodate multiple, independently moving regions with different resolution (to address both motion compensation and image truncation). Imaging performance was evaluated quantitatively and qualitatively with 41 human subjects in the neurosciences critical care unit (NCCU) at our institution. The progression of four scanner configurations exhibited systematic improvement in the quality of raw data by variations in system geometry (source-detector distance), antiscatter grid, and bowtie filter. Quantitative assessment of CBCT images in 41 subjects demonstrated: ~70% reduction in image nonuniformity with artifact correction methods (lag, glare, beam hardening, and scatter); ~40% reduction in motion-induced streak artifacts via the multi-motion compensation method; and ~15% improvement in soft-tissue contrast-to-noise ratio (CNR) for PWLS compared to filtered backprojection (FBP) at matched resolution. Each of these components was important to improve contrast resolution for point-of-care cranial imaging. This work presents the first application of a high-quality, point-of-care CBCT system for imaging of the head/ brain in a neurological critical care setting. Hardware configuration iterations and an integrated software pipeline for artifacts correction and PWLS reconstruction mitigated artifacts and noise to achieve image quality that could be valuable for point-of-care detection and monitoring of a variety of intracranial abnormalities, including ICH and hydrocephalus.

Low-Dose CT Image Reconstruction With a Deep Learning Prior

In low-dose computed tomography (LDCT), a penalized weighted least squares (PWLS) approach that incorporates the Poisson statistics of X-ray photons can significantly reduce excessive quantum noise. To improve the quality of LDCT images, prior information such as the total variation, Markov random field, and nonlocal mean, can be imposed onto the target image. However, this information may be limited to reflect the characteristics of the target images, thereby resulting in unexpected side effects (e.g. blurry images). In this paper, we propose a PWLS method combined with a deep learning prior, which is learned from standard dose CT (SDCT) images. The proposed model learns a noise reduction function that maps an LDCT image to its corresponding SDCT images and estimates the prior distribution using a Pearson χ 2 divergence. The model can be converted to the least squares generative adversarial network with an added PWLS objective, where the optimal generator acts as the noise reduction function. We reformulate the proposed model as a constrained optimization problem and solve it using the alternating optimization (AO) algorithm. Clinical SDCT and simulated LDCT scans of ten patients were used to show the validity of the proposed method. Results show that the proposed method outperforms other PWLS methods, by imposing priors such as the total variation and the nonlocal mean.

DECT-MULTRA: Dual-Energy CT Image Decomposition With Learned Mixed Material Models and Efficient Clustering.

Dual-energy computed tomography (DECT) imaging plays an important role in advanced imaging applications due to its material decomposition capability. Image-domain decomposition operates directly on CT images using linear matrix inversion, but the decomposed material images can be severely degraded by noise and artifacts. This paper proposes a new method dubbed DECT-MULTRA for image-domain DECT material decomposition that combines conventional penalized weighted-least squares (PWLS) estimation with regularization based on a mixed union of learned transforms (MULTRA) model. Our proposed approach pre-learns a union of common-material sparsifying transforms from patches extracted from all the basis materials, and a union of cross-material sparsifying transforms from multi-material patches. The common-material transforms capture the common properties among different material images, while the cross-material transforms capture the cross-dependencies. The proposed PWLS formulation is optimized efficiently by alternating between an image update step and a sparse coding and clustering step, with both of these steps having closed-form solutions. The effectiveness of our method is validated with both XCAT phantom and clinical head data. The results demonstrate that our proposed method provides superior material image quality and decomposition accuracy compared to other competing methods.

Sparse-view statistical image reconstruction with improved total variation regularization for X-ray micro-CT imaging

Sparse-view x-ray micro computed tomography (micro-CT) reconstruction algorithms via total variation (TV) optimize the data without introducing notable noise and artifacts, resulting in significant scanning time reduction while maintaining image quality. However, due to the piecewise constant assumption for the image, a conventional TV minimization often suffers from patchy artifacts in reconstructed images. Moreover, for lack of directional gradient in TV some directional information are lost. To obviate these drawbacks, in this study we develop a penalized weighted least-square (PWLS) strategy for micro-CT sparse-view image reconstruction by incorporating an adaptive weighted total variation in combination with an adaptive weighted diagonal total variation (AwTV+AwDTV) penalty term. The AwTV considers the vertical and horizontal gradients while the AwDTV uses the diagonal gradients. The associated weights which are defined based on the anisotropic edge properties of an image, are expressed as an exponential function and can be adaptively adjusted by the amount of the difference between voxel intensities to preserve the edge details. To evaluate the presented (AwTV+AwDTV)-PWLS algorithm, both qualitative and quantitative studies were performed by computer simulations and micro-CT data experiments. The Shepp-Logan phantom for computer simulation and the micro-CT water phantom and a rat skull for micro-CT experiments are employed to perform image reconstruction. To evaluate the performance of AwTV+AwDTV algorithm, we compared it with TV and AwTV reconstruction algorithms. The simulation results show that the presented (AwTV+AwDTV)-PWLS algorithm can achieve the lowest RMSE and highest PSNR, SSIM and MTF for different number of projections as compared to the AwTV and conventional TV algorithms. The micro-CT data results confirmed the superiority of the proposed (AwTV+AwDTV) method to the AwTV and TV methods for different number of projections.

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.

Known-component 3D image reconstruction for improved intraoperative imaging in spine surgery: A clinical pilot study.

Intraoperative imaging plays an increased role in support of surgical guidance and quality assurance for interventional approaches. However, image quality sufficient to detect complications and provide quantitative assessment of the surgical product is often confounded by image noise and artifacts. In this work, we translated a three-dimensional model-based image reconstruction (referred to as "Known-Component Reconstruction," KC-Recon) for the first time to clinical studies with the aim of resolving both limitations. KC-Recon builds upon a penalized weighted least-squares (PWLS) method by incorporating models of surgical instrumentation ("known components") within a joint image registration-reconstruction process to improve image quality. Under IRB approval, a clinical pilot study was conducted with 17 spine surgery patients imaged under informed consent using the O-arm cone-beam CT system (Medtronic, Littleton MA) before and after spinal instrumentation. Volumetric images were generated for each patient using KC-Recon in comparison to conventional filtered backprojection (FBP). Imaging performance prior to instrumentation ("preinstrumentation") was evaluated in terms of soft-tissue contrast-to-noise ratio (CNR) and spatial resolution. The quality of images obtained after the instrumentation ("postinstrumentation") was assessed by quantifying the magnitude of metal artifacts (blooming and streaks) arising from pedicle screws. The potential low-dose advantages of the algorithm were tested by simulating low-dose data (down to one-tenth of the dose of standard protocols) from images acquired at normal dose. Preinstrumentation images (at normal clinical dose and matched resolution) exhibited an average 24.0% increase in soft-tissue CNR with KC-Recon compared to FBP (N=16, P=0.02), improving visualization of paraspinal muscles, major vessels, and other soft-tissues about the spine and abdomen. For a total of 72 screws in postinstrumentation images, KC-Recon yielded a significant reduction in metal artifacts: 66.3% reduction in overestimation of screw shaft width due to blooming (P<0.0001) and reduction in streaks at the screw tip (65.8% increase in attenuation accuracy, P<0.0001), enabling clearer depiction of the screw within the pedicle and vertebral body for an assessment of breach. Depending on the imaging task, dose reduction up to an order of magnitude appeared feasible while maintaining soft-tissue visibility and metal artifact reduction. KC-Recon offers a promising means to improve visualization in the presence of surgical instrumentation and reduce patient dose in image-guided procedures. The improved soft-tissue visibility could facilitate the use of cone-beam CT to soft-tissue surgeries, and the ability to precisely quantify and visualize instrument placement could provide a valuable check against complications in the operating room (cf., postoperative CT).

The Estimation Function Approach Smoothing Spline Regression Analysis for Longitudinal Data

It is given the paired data following a nonparametric regression model for longitudinal data. The regression curve is approached by the smoothing spline function. The smoothing spline is a function that is able to map the data well and has a small error variance. This current study aims to obtained from completing PWLS (Penalized Weighted Least Square) optimaziton. Besides, the GCV method is used to select the smoothing parameter.

Convergence criterion for MBIR based on the local noise-power spectrum: Theory and implementation in a framework for accelerated 3D image reconstruction with a morphological pyramid.

Model-based iterative reconstruction (MBIR) offers improved noise-resolution tradeoffs and artifact reduction in cone-beam CT compared to analytical reconstruction, but carries increased computational burden. An important consideration in minimizing computation time is reliable selection of the stopping criterion to perform the minimum number of iterations required to obtain the desired image quality. Most MBIR methods rely on a fixed number of iterations or relative metrics on image or cost-function evolution, and it would be desirable to use metrics that are more representative of the underlying image properties. A second front for reduction of computation time is the use of acceleration techniques (e.g. subsets or momentum). However, most of these techniques do not strictly guarantee convergence of the resulting MBIR method. A data-dependent analytical model of noise-power spectrum (NPS) for penalized weighted least squares (PWLS) reconstruction is proposed as an absolute metric of image properties for the fully converged volume. Distance to convergence is estimated as the root mean squared error (RMSE) between the estimated NPS and an NPS measured on a uniform region of interest (ROI) in the evolving volume. Iterations are stopped when the RMSE falls below a threshold directly related with the properties of the target image. Further acceleration was achieved by combining the spectral stopping criterion with a morphological pyramid (mPyr) in which the minimization of the PWLS cost-function is divided in a cascade of stages. The algorithm parameters (voxel size in this work) change between stages to achieve faster evolution in early stages, and a final stage with the target parameters to guarantee convergence. Transition between stages is governed by the spectral stopping criterion. The approach was evaluated on simulated CBCT data of a realistic digital abdomen phantom. Accuracy of the NPS model and evolution with time of the distance from the measured NPS was assessed in two ROIs. Performance of the spectrally-driven mPyr architecture was compared to a conventional, single stage, PWLS, and to two mPyr designs running a fixed number of iterations. The spectrally-driven mPyr achieved faster convergence, with 40% lower RMSE than the single stage PWLS, and between 10% and 20% RMSE reduction compared to other mPyr designs. The proposed spectral stopping criterion proved to be a suitable choice for a stopping rule, and, in particular, to govern mPyr stage transition.

Structure tensor total variation for CBCT reconstruction

The total variation (TV) regularization has been widely used in statistically iterative cone-beam computed tomography (CBCT) reconstruction, showing ability to preserve object edges. However, the TV regularization can also produce staircase effect and tend to over-smooth the reconstructed images due to its piecewise constant assumption. In this study, we proposed to use the structure tensor total variation (STV) that penalizes the eigenvalues of the structure tensor for CBCT reconstruction. The STV penalty extends the TV penalty, with many important properties maintained such as convexity and rotation and translation invariance. The STV penalty utilizes gradient information more effectively and has a stronger ability to capture local image structural variation. The objective function was constructed with the penalized weighted least-square (PWLS) strategy and the gradient descent (GD) method was used to optimize the objective function. Besides, we investigated whether the norms involved in the STV penalty affected the reconstruction performance and found that the l1-norm gave the better performance than the l2-norm and l∞-norm. We also examined performance of the STV penalties constructed using different kernel functions and found that the STV with the Gaussian kernel had the best performance, and the STVs with Uniform, Logistic, and Sigmoid kernels had similar performance to each other. We evaluated our reconstruction method with the STV penalty on computer simulated phantoms and physical phantoms. The results demonstrated that STV led to better reconstruction performance than TV, both visually and quantitatively. For the Catphan 600 physical phantom, the STV1 penalty was 175% and 623% better than the low-dose FDK and the high-dose FDK, and 14% better than the TV penalty at the matched noise level, according to the average contrast-to-noise ratio (CNR); while for the Compressed Sensing simulation phantom, the peak signal to noise ratio (PSNR) of reconstructed results using STV1, STV2, and STV∞ were 40.67 dB, 38.72 dB, and 37.40 dB, respectively, all being significantly better than 36.84 dB using TV.