Approximations of noise covariance in multi-slice helical CT scans: impact on lung nodule size estimation

Experimental measurement of local noise power spectrum (NPS) in photon counting detector-CT (PCD-CT) using a single data acquisition

BACKGROUND

Accurate noise power spectra (NPS) measurement in clinical X-ray CT exams is challenging due to the need for repeated scans, which expose patients to high radiation risks. A reliable method for single CT acquisition NPS estimation is thus highly desirable.

PURPOSE

To develop a method for estimating local NPS from a single photon counting detector-CT (PCD-CT) acquisition.

METHODS

A novel nearly statistical bias-free estimator was constructed from the raw counts data of PCD-CT scan to estimate the variance of sinogram projection data. An analytical algorithm is employed to reconstruct point-wise covariancecov ( x i , x j ) $\text{cov}({\bf x}_i,{\bf x}_j)$ between any two image pixel/voxel locationsx i ${\bf x}_i$ andx j ${\bf x_j}$ . A Fourier transform is applied to obtain the desired point-wise NPS for any chosen locationx i ${\bf x}_i$ . The method was validated using experimental data acquired from a benchtop PCD-CT system with various physical phantoms, and the results were compared with the conventional local NPS measurement method using repeated scans and statistical ensemble averaging.

RESULTS

The experimental results demonstrate that (1) the proposed method can achieve pointwise/local NPS measurement for a region of interest (ROI) located at any chosen position, accurately characterizing the NPS with spatial structures resulting from image content heterogeneity; (2) the local NPS measured using the proposed method show a higher precision in the measured NPS compared to the conventional measurement method; (3) spatial averaging of the local NPS yields the conventional NPS for a given local ROI.

CONCLUSION

A new method was developed to enable local NPS from a single PCD-CT acquisition.

Noise power spectrum (NPS) in computed tomography: Enabling local NPS measurement without stationarity and ergodicity assumptions

BACKGROUND

Conventional methods for estimating the noise power spectrum (NPS) often necessitate multiple computed tomography (CT) data acquisitions and are required to satisfy stringent stationarity and ergodicity conditions, which prove challenging in CT imaging systems.

PURPOSE

The aim was to revisit the conventional NPS estimation method, leading to a new framework that estimates local NPS without relying on stationarity or ergodicity, thus facilitating experimental NPS estimations.

METHODS

The scientific foundation of the conventional CT NPS measurement method, based on the Wiener-Khintchine theorem, was reexamined, emphasizing the critical conditions of stationarity and ergodicity. This work proposes an alternative framework, characterized by its independence from stationarity and ergodicity, and its ability to facilitate local NPS estimations. A spatial average of local NPS over a Region of Interest (ROI) yields the conventional NPS for that ROI. The connections and differences between the proposed alternative method and the conventional method are discussed. Experimental studies were conducted to validate the new method.

RESULTS

(1) The NPS estimated using the conventional method was demonstrated to correspond to the spatial average of pointwise NPS from the proposed NPS estimation framework. (2) The NPS estimated over an ROI with the conventional method was shown to be the sum of the NPS estimated from the proposed method and a contribution from measurement uncertainty. (3) Local NPS estimations from the proposed method in this work elucidate the impact of surrounding image content on local NPS variations.

CONCLUSION

The NPS estimation method proposed in this work allows for the estimation of local NPS without relying on stationarity and ergodicity conditions, offering local NPS estimations with significantly improved precision.

Performance of a deep learning-based CT image denoising method: Generalizability over dose, reconstruction kernel and slice thickness

PURPOSE

Deep learning (DL) is rapidly finding applications in low-dose CT image denoising. While having the potential to improve image quality (IQ) over the filtered back projection method (FBP) and produce images quickly, performance generalizability of the data-driven DL methods is not fully understood yet. The main purpose of this work is to investigate the performance generalizability of a low-dose CT image denoising neural network in data acquired under different scan conditions, particularly relating to these three parameters: reconstruction kernel, slice thickness and dose (noise) level. A secondary goal is to identify any underlying data property associated with the CT scan settings that might help predict the generalizability of the denoising network.

METHODS

We select the residual encoder-decoder convolutional neural network (REDCNN) as an example of a low-dose CT image denoising technique in this work. To study how the network generalizes on the three imaging parameters, we grouped the CT volumes in the Low-Dose Grand Challenge (LDGC) data into three pairs of training datasets according to their imaging parameters, changing only one parameter in each pair. We trained REDCNN with them to obtain six denoising models. We test each denoising model on datasets of matching and mismatching parameters with respect to its training sets regarding dose, reconstruction kernel and slice thickness, respectively, to evaluate the denoising performance changes. Denoising performances are evaluated on patient scans, simulated phantom scans and physical phantom scans using IQ metrics including mean squared error (MSE), contrast-dependent modulation transfer function (MTF), pixel-level noise power spectrum (pNPS) and low-contrast lesion detectability (LCD).

RESULTS

REDCNN had larger MSE when the testing data was different from the training data in reconstruction kernel, but no significant MSE difference when varying slice thickness in the testing data. REDCNN trained with quarter-dose data had slightly worse MSE in denoising higher-dose images than that trained with mixed-dose data (17-80%). The MTF tests showed that REDCNN trained with the two reconstruction kernels and slice thicknesses yielded images of similar image resolution. However, REDCNN trained with mixed-dose data preserved the low-contrast resolution better compared to REDCNN trained with quarter-dose data. In the pNPS test, it was found that REDCNN trained with smooth-kernel data could not remove high-frequency noise in the test data of sharp kernel, possibly because the lack of high-frequency noise in the smooth-kernel data limited the ability of the trained model in removing high-frequency noise. Finally, in the LCD test, REDCNN improved the lesion detectability over the original FBP images regardless of whether the training and testing data had matching reconstruction kernels.

CONCLUSIONS

REDCNN is observed to be poorly generalizable between reconstruction kernels, more robust in denoising data of arbitrary dose levels when trained with mixed-dose data, and not highly sensitive to slice thickness. It is known that reconstruction kernel affects the in-plane pNPS shape of a CT image whereas slice thickness and dose level do not, so it is possible that the generalizability performance of this CT image denoising network highly correlates to the pNPS similarity between the testing and training data. This article is protected by copyright. All rights reserved.

Unified Supervised-Unsupervised (SUPER) Learning for X-Ray CT Image Reconstruction

Traditional model-based image reconstruction (MBIR) methods combine forward and noise models with simple object priors. Recent machine learning methods for image reconstruction typically involve supervised learning or unsupervised learning, both of which have their advantages and disadvantages. In this work, we propose a unified supervised-unsupervised (SUPER) learning framework for X-ray computed tomography (CT) image reconstruction. The proposed learning formulation combines both unsupervised learning-based priors (or even simple analytical priors) together with (supervised) deep network-based priors in a unified MBIR framework based on a fixed point iteration analysis. The proposed training algorithm is also an approximate scheme for a bilevel supervised training optimization problem, wherein the network-based regularizer in the lower-level MBIR problem is optimized using an upper-level reconstruction loss. The training problem is optimized by alternating between updating the network weights and iteratively updating the reconstructions based on those weights. We demonstrate the learned SUPER models' efficacy for low-dose CT image reconstruction, for which we use the NIH AAPM Mayo Clinic Low Dose CT Grand Challenge dataset for training and testing. In our experiments, we studied different combinations of supervised deep network priors and unsupervised learning-based or analytical priors. Both numerical and visual results show the superiority of the proposed unified SUPER methods over standalone supervised learning-based methods, iterative MBIR methods, and variations of SUPER obtained via ablation studies. We also show that the proposed algorithm converges rapidly in practice.

Accurate Image Domain Noise Insertion in CT Images

Tools to simulate lower dose, noisy computed tomography (CT) images from existing data enable protocol optimization by quantifying the trade-off between patient dose and image quality. Many studies have developed and validated noise insertion techniques; however, most of these tools operate on proprietary projection data which can be difficult to access and can be time consuming when a large number of realizations is needed. In response, this work aims to develop and validate an image domain approach to accurately insert CT noise and simulate low dose scans. In this framework, information from the image is utilized to estimate the variance map and local noise power spectra (NPS). Normally distributed noise is filtered within small patches in the image domain using the inverse Fourier transform of the square root of the estimated local NPS to generate noise with the appropriate spatial correlation. The patches are overlapped and element-wise multiplied by the standard deviation map to produce locally varying, spatially correlated noise. The resulting noise image is scaled based on the relationship between the initial and desired dose and added to the original image. The results demonstrate excellent agreement between traditional projection domain methods and the proposed method, both for simulated and real data sets. This new framework is not intended to replace projection domain methods; rather, it fills a gap in CT noise simulation tools and is an accurate alternative when projection domain methods are not practical, for example, in large scale repeatability or detectability studies.

A data-efficient method for local noise power spectrum (NPS) estimation in FDK-reconstructed 3D cone-beam CT

PURPOSE

For computed tomography (CT) systems in which noise is nonstationary, a local noise power spectrum (NPS) is often needed to characterize its noise property. We have previously developed a data-efficient radial NPS method to estimate the two-dimensional (2D) local NPS for filtered back projection (FBP)-reconstructed fan-beam CT utilizing the polar separability of CT NPS. In this work, we extend this method to estimate three-dimensional (3D) local NPS for feldkamp-davis-kress (FDK)-reconstructed cone-beam CT (CBCT) volumes.

METHODS

Starting from the 2D polar separability, we analyze the CBCT geometry and FDK image reconstruction process to derive the 3D expression of the polar separability for CBCT local NPS. With the polar separability, the 3D local NPS of CBCT can be decomposed into a 2D radial NPS shape function and a one-dimensional (1D) angular amplitude function with certain geometrical transforms. The 2D radial NPS shape function is a global function characterizing the noise correlation structure, while the 1D angular amplitude function is a local function reflecting the varying local noise amplitudes. The 3D radial local NPS method is constructed from the polar separability. We evaluate the accuracy of the 3D radial local NPS method using simulated and real CBCT data by comparing the radial local NPS estimates to a reference local NPS in terms of normalized mean squared error (NMSE) and a task-based performance metric (lesion detectability).

RESULTS

In both simulated and physical CBCT examples, a very small NMSE (<5%) was achieved by the radial local NPS method from as few as two scans, while for the traditional local NPS method, about 20 scans were needed to reach this accuracy. The results also showed that the detectability-based system performances computed using the local NPS estimated with the NPS method developed in this work from two scans closely reflected the actual system performance.

CONCLUSIONS

The polar separability greatly reduces the data dimensionality of the 3D CBCT local NPS. The radial local NPS method developed based on this property is shown to be capable of estimating the 3D local NPS from only two CBCT scans with acceptable accuracy. The minimum data requirement indicates the potential utility of local NPS in CBCT applications even for clinical situations.

Assessment of CT Image Quality Using a Bayesian Framework

In computed tomography, there is a tradeoff between the quality of the reconstructed image and the radiation dose received by the patient. In order to find an appropriate compromise between the image quality of the reconstructed images and the radiation dose, it is important to have reliable methods for evaluating the quality of the reconstructed images. A successful family of methods for the assessment of image quality is task-based image quality assessment, which often involves the use of model observers, and which assesses the quality of the image reconstruction by deriving a figure of merit. Here, we present a Bayesian framework that can be used in task-based image quality assessment. Our framework is applicable to binary classification problems with normally distributed observations, and we make the additional assumption that the covariance matrix is the same in both image classes. We choose a particular non-informative prior for the parameters of our model, which allows us to derive an expression for the Bayes factor for the binary classification problem which to the best of our knowledge is novel. We introduce a novel model observer based on this Bayes factor. Further, we have developed a methodology for estimating the posterior distribution of the figure of merit for this type of classification problem. Compared with classical statistical approaches, our Bayesian approach has the advantage that it provides a full characterization of the uncertainty of the figure of merit. Our choice of prior allows us to design a simple Monte Carlo algorithm to efficiently sample the posterior of the figure of merit of the ideal observer, in contrast to common Bayesian procedures which rely on computationally expensive Markov chain Monte Carlo sampling. We have shown that for training samples of sufficient size, our estimated credible intervals for the figure of merit have coverage probabilities close to their credibility, so that our approach can reasonably be used within a classical statistical framework as well.

Exact confidence intervals for channelized Hotelling observer performance in image quality studies

Task-based assessments of image quality constitute a rigorous, principled approach to the evaluation of imaging system performance. To conduct such assessments, it has been recognized that mathematical model observers are very useful, particularly for purposes of imaging system development and optimization. One type of model observer that has been widely applied in the medical imaging community is the channelized Hotelling observer (CHO), which is well-suited to known-location discrimination tasks. In the present work, we address the need for reliable confidence interval estimators of CHO performance. Specifically, we show that the bias associated with point estimates of CHO performance can be overcome by using confidence intervals proposed by Reiser for the Mahalanobis distance. In addition, we find that these intervals are well-defined with theoretically-exact coverage probabilities, which is a new result not proved by Reiser. The confidence intervals are tested with Monte Carlo simulation and demonstrated with two examples comparing X-ray CT reconstruction strategies. Moreover, commonly-used training/testing approaches are discussed and compared to the exact confidence intervals. MATLAB software implementing the estimators described in this work is publicly available at http://code.google.com/p/iqmodelo/.

Volume estimation of low-contrast lesions with CT: a comparison of performances from a phantom study, simulations and theoretical analysis

Measurements of lung nodule volume with multi-detector computed tomography (MDCT) have been shown to be more accurate and precise compared to conventional lower dimensional measurements. Quantifying the size of lesions is potentially more difficult when the object-to-background contrast is low as with lesions in the liver. Physical phantom and simulation studies are often utilized to analyze the bias and variance of lesion size estimates because a ground truth or reference standard can be established. In addition, it may also be useful to derive theoretical bounds as another way of characterizing lesion sizing methods. The goal of this work was to study the performance of a MDCT system for a lesion volume estimation task with object-to-background contrast less than 50 HU, and to understand the relation among performances obtained from phantom study, simulation and theoretical analysis. We performed both phantom and simulation studies, and analyzed the bias and variance of volume measurements estimated by a matched-filter-based estimator. We further corroborated results with a theoretical analysis to estimate the achievable performance bound, which was the Cramer-Rao's lower bound (CRLB) of minimum variance for the size estimates. Results showed that estimates of non-attached solid small lesion volumes with object-to-background contrast of 31-46 HU can be accurate and precise, with less than 10.8% in percent bias and 4.8% in standard deviation of percent error (SPE), in standard dose scans. These results are consistent with theoretical (CRLB), computational (simulation) and empirical phantom bounds. The difference between the bounds is rather small (for SPE less than 1.9%) indicating that the theoretical- and simulation-based performance bounds can be good surrogates for physical phantom studies.

Task-based detectability in CT image reconstruction by filtered backprojection and penalized likelihood estimation

PURPOSE

Nonstationarity is an important aspect of imaging performance in CT and cone-beam CT (CBCT), especially for systems employing iterative reconstruction. This work presents a theoretical framework for both filtered-backprojection (FBP) and penalized-likelihood (PL) reconstruction that includes explicit descriptions of nonstationary noise, spatial resolution, and task-based detectability index. Potential utility of the model was demonstrated in the optimal selection of regularization parameters in PL reconstruction.

METHODS

Analytical models for local modulation transfer function (MTF) and noise-power spectrum (NPS) were investigated for both FBP and PL reconstruction, including explicit dependence on the object and spatial location. For FBP, a cascaded systems analysis framework was adapted to account for nonstationarity by separately calculating fluence and system gains for each ray passing through any given voxel. For PL, the point-spread function and covariance were derived using the implicit function theorem and first-order Taylor expansion according to Fessler ["Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): Applications to tomography," IEEE Trans. Image Process. 5(3), 493-506 (1996)]. Detectability index was calculated for a variety of simple tasks. The model for PL was used in selecting the regularization strength parameter to optimize task-based performance, with both a constant and a spatially varying regularization map.

RESULTS

Theoretical models of FBP and PL were validated in 2D simulated fan-beam data and found to yield accurate predictions of local MTF and NPS as a function of the object and the spatial location. The NPS for both FBP and PL exhibit similar anisotropic nature depending on the pathlength (and therefore, the object and spatial location within the object) traversed by each ray, with the PL NPS experiencing greater smoothing along directions with higher noise. The MTF of FBP is isotropic and independent of location to a first order approximation, whereas the MTF of PL is anisotropic in a manner complementary to the NPS. Task-based detectability demonstrates dependence on the task, object, spatial location, and smoothing parameters. A spatially varying regularization "map" designed from locally optimal regularization can improve overall detectability beyond that achievable with the commonly used constant regularization parameter.

CONCLUSIONS

Analytical models for task-based FBP and PL reconstruction are predictive of nonstationary noise and resolution characteristics, providing a valuable framework for understanding and optimizing system performance in CT and CBCT.

New Theoretical Results on Channelized Hotelling Observer Performance Estimation with Known Difference of Class Means

Task-based assessments of image quality constitute a rigorous, principled approach to the evaluation of imaging system performance. To conduct such assessments, it has been recognized that mathematical model observers are very useful, particularly for purposes of imaging system development and optimization. One type of model observer that has been widely applied in the medical imaging community is the channelized Hotelling observer (CHO). Since estimates of CHO performance typically include statistical variability, it is important to control and limit this variability to maximize the statistical power of image-quality studies. In a previous paper, we demonstrated that by including prior knowledge of the image class means, a large decrease in the bias and variance of CHO performance estimates can be realized. The purpose of the present work is to present refinements and extensions of the estimation theory given in our previous paper, which was limited to point estimation with equal numbers of images from each class. Specifically, we present and characterize minimum-variance unbiased point estimators for observer signal-to-noise ratio (SNR) that allow for unequal numbers of lesion-absent and lesion-present images. Building on this SNR point estimation theory, we then show that confidence intervals with exactly-known coverage probabilities can be constructed for commonly-used CHO performance measures. Moreover, we propose simple, approximate confidence intervals for CHO performance, and we show that they are well-behaved in most scenarios of interest.

On Efficient Assessment of Image-Quality Metrics Based on Linear Model Observers

This paper is motivated by the problem of image-quality assessment using model observers for the purpose of development and optimization of medical imaging systems. Specifically, we present a study regarding the estimation of the receiver operating characteristic (ROC) curve for the observer and associated summary measures. This study evaluates the statistical advantage that may be gained in ROC estimates of observer performance by assuming that the difference of the class means for the observer ratings is known. Such knowledge is frequently available in image-quality studies employing known-location lesion detection tasks together with linear model observers. The study is carried out by introducing parametric point and confidence interval estimators that incorporate a known difference of class means. An evaluation of the new estimators for the area under the ROC curve establishes that a large reduction in statistical variability can be achieved through incorporation of knowledge of the difference of class means. Namely, the mean 95% AUC confidence interval length can be as much as seven times smaller in some cases. We also examine how knowledge of the difference of class means can be advantageously used to compare the areas under two correlated ROC curves, and observe similar gains.