Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography

Variance Estimation for Myocardial Blood Flow by Dynamic PET

The estimation of myocardial blood flow (MBF) by (13)N-ammonia or (82)Rb dynamic PET typically relies on an empirically determined generalized Renkin-Crone equation to relate the kinetic parameter K1 to MBF. Because the Renkin-Crone equation defines MBF as an implicit function of K1, the MBF variance cannot be determined using standard error propagation techniques. To overcome this limitation, we derived novel analytical approximations that provide first- and second-order estimates of MBF variance in terms of the mean and variance of K1 and the Renkin-Crone parameters. The accuracy of the analytical expressions was validated by comparison with Monte Carlo simulations, and MBF variance was evaluated in clinical (82)Rb dynamic PET scans. For both (82)Rb and (13)N-ammonia, good agreement was observed between both (first- and second-order) analytical variance expressions and Monte Carlo simulations, with moderately better agreement for second-order estimates. The contribution of the Renkin-Crone relation to overall MBF uncertainty was found to be as high as 68% for (82)Rb and 35% for (13)N-ammonia. For clinical (82)Rb PET data, the conventional practice of neglecting the statistical uncertainty in the Renkin-Crone parameters resulted in underestimation of the coefficient of variation of global MBF and coronary flow reserve by 14-49%. Knowledge of MBF variance is essential for assessing the precision and reliability of MBF estimates. The form and statistical uncertainty in the empirical Renkin-Crone relation can make substantial contributions to the variance of MBF. The novel analytical variance expressions derived in this work enable direct estimation of MBF variance which includes this previously neglected contribution.

Precision and accuracy of clinical quantification of myocardial blood flow by dynamic PET: A technical perspective

A number of exciting advances in PET/CT technology and improvements in methodology have recently converged to enhance the feasibility of routine clinical quantification of myocardial blood flow and flow reserve. Recent promising clinical results are pointing toward an important role for myocardial blood flow in the care of patients. Absolute blood flow quantification can be a powerful clinical tool, but its utility will depend on maintaining precision and accuracy in the face of numerous potential sources of methodological errors. Here we review recent data and highlight the impact of PET instrumentation, image reconstruction, and quantification methods, and we emphasize (82)Rb cardiac PET which currently has the widest clinical application. It will be apparent that more data are needed, particularly in relation to newer PET technologies, as well as clinical standardization of PET protocols and methods. We provide recommendations for the methodological factors considered here. At present, myocardial flow reserve appears to be remarkably robust to various methodological errors; however, with greater attention to and more detailed understanding of these sources of error, the clinical benefits of stress-only blood flow measurement may eventually be more fully realized.

Evaluation of Fisher Information Matrix-Based Methods for Fast Assessment of Image Quality in Pinhole SPECT

The accurate determination of the local impulse response and the covariance in voxels from penalized maximum likelihood reconstructed images requires performing reconstructions from many noise realizations of the projection data. As this is usually a very time-consuming process, efficient analytical approximations based on the Fisher information matrix (FIM) have been extensively used in PET and SPECT to estimate these quantities. For 3D imaging, however, additional approximations need to be made to the FIM in order to speed up the calculations. The most common approach is to use the local shift-invariant (LSI) approximation of the FIM, but this assumes specific conditions which are not always necessarily valid. In this paper we take a single-pinhole SPECT system and compare the accuracy of the LSI approximation against two other methods that have been more recently put forward: the non-uniform object-space pixelation (NUOP) and the subsampled FIM. These methods do not assume such restrictive conditions while still increasing the speed of the calculations considerably. Our results indicate that in pinhole SPECT the NUOP and subsampled FIM approaches could be more reliable than the LSI approximation, especially when a high accuracy is required.

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.

Development and evaluation of convergent and accelerated penalized SPECT image reconstruction methods for improved dose-volume histogram estimation in radiopharmaceutical therapy

PURPOSE

Three-dimensional (3D) dosimetry has the potential to provide better prediction of response of normal tissues and tumors and is based on 3D estimates of the activity distribution in the patient obtained from emission tomography. Dose-volume histograms (DVHs) are an important summary measure of 3D dosimetry and a widely used tool for treatment planning in radiation therapy. Accurate estimates of the radioactivity distribution in space and time are desirable for accurate 3D dosimetry. The purpose of this work was to develop and demonstrate the potential of penalized SPECT image reconstruction methods to improve DVHs estimates obtained from 3D dosimetry methods.

METHODS

The authors developed penalized image reconstruction methods, using maximum a posteriori (MAP) formalism, which intrinsically incorporate regularization in order to control noise and, unlike linear filters, are designed to retain sharp edges. Two priors were studied: one is a 3D hyperbolic prior, termed single-time MAP (STMAP), and the second is a 4D hyperbolic prior, termed cross-time MAP (CTMAP), using both the spatial and temporal information to control noise. The CTMAP method assumed perfect registration between the estimated activity distributions and projection datasets from the different time points. Accelerated and convergent algorithms were derived and implemented. A modified NURBS-based cardiac-torso phantom with a multicompartment kidney model and organ activities and parameters derived from clinical studies were used in a Monte Carlo simulation study to evaluate the methods. Cumulative dose-rate volume histograms (CDRVHs) and cumulative DVHs (CDVHs) obtained from the phantom and from SPECT images reconstructed with both the penalized algorithms and OS-EM were calculated and compared both qualitatively and quantitatively. The STMAP method was applied to patient data and CDRVHs obtained with STMAP and OS-EM were compared qualitatively.

RESULTS

The results showed that the penalized algorithms substantially improved the CDRVH and CDVH estimates for large organs such as the liver compared to optimally postfiltered OS-EM. For example, the mean squared errors (MSEs) of the CDRVHs for the liver at 5 h postinjection obtained with CTMAP and STMAP were about 15% and 17%, respectively, of the MSEs obtained with optimally filtered OS-EM. For the CDVH estimates, the MSEs obtained with CTMAP and STMAP were about 16% and 19%, respectively, of the MSEs from OS-EM. For the kidneys and renal cortices, larger residual errors were observed for all algorithms, likely due to partial volume effects. The STMAP method showed promising qualitative results when applied to patient data.

CONCLUSIONS

Penalized image reconstruction methods were developed and evaluated through a simulation study. The study showed that the MAP algorithms substantially improved CDVH estimates for large organs such as the liver compared to optimally postfiltered OS-EM reconstructions. For small organs with fine structural detail such as the kidneys, a large residual error was observed for both MAP algorithms and OS-EM. While CTMAP provided marginally better MSEs than STMAP, given the extra effort needed to handle misregistration of images at different time points in the algorithm and the potential impact of residual misregistration, 3D regularization methods, such as that used in STMAP, appear to be a more practical choice.

Noise properties and task-based evaluation of diffraction-enhanced imaging

Diffraction-enhanced imaging (DEI) is an emerging x-ray imaging method that simultaneously yields x-ray attenuation and refraction images and holds great promise for soft-tissue imaging. The DEI has been mainly studied using synchrotron sources, but efforts have been made to transition the technology to more practical implementations using conventional x-ray sources. The main technical challenge of this transition lies in the relatively lower x-ray flux obtained from conventional sources, leading to photon-limited data contaminated by Poisson noise. Several issues that must be understood in order to design and optimize DEI imaging systems with respect to noise performance are addressed. Specifically, we: (a) develop equations describing the noise properties of DEI images, (b) derive the conditions under which the DEI algorithm is statistically optimal,

Estimation of noise-free variance to measure heterogeneity

Variance is a statistical parameter used to characterize heterogeneity or variability in data sets. However, measurements commonly include noise, as random errors superimposed to the actual value, which may substantially increase the variance compared to a noise-free data set. Our aim was to develop and validate a method to estimate noise-free spatial heterogeneity of pulmonary perfusion using dynamic positron emission tomography (PET) scans. On theoretical grounds, we demonstrate a linear relationship between the total variance of a data set derived from averages of n multiple measurements, and the reciprocal of n. Using multiple measurements with varying n yields estimates of the linear relationship including the noise-free variance as the constant parameter. In PET images, n is proportional to the number of registered decay events, and the variance of the image is typically normalized by the square of its mean value yielding a coefficient of variation squared (CV²). The method was evaluated with a Jaszczak phantom as reference spatial heterogeneity (CV_r²) for comparison with our estimate of noise-free or ‘true’ heterogeneity (CV_t²). We found that CV_t² was only 5.4% higher than CV_r². Additional evaluations were conducted on 38 PET scans of pulmonary perfusion using ¹³NN-saline injection. The mean CV_t² was 0.10 (range: 0.03–0.30), while the mean CV² including noise was 0.24 (range: 0.10–0.59). CV_t² was in average 41.5% of the CV² measured including noise (range: 17.8–71.2%). The reproducibility of CV_t² was evaluated using three repeated PET scans from five subjects. Individual CV_t² were within 16% of each subject's mean and paired t-tests revealed no difference among the results from the three consecutive PET scans. In conclusion, our method provides reliable noise-free estimates of CV_t² in PET scans, and may be useful for similar statistical problems in experimental data.

Task-driven image acquisition and reconstruction in cone-beam CT

This work introduces a task-driven imaging framework that incorporates a mathematical definition of the imaging task, a model of the imaging system, and a patient-specific anatomical model to prospectively design image acquisition and reconstruction techniques to optimize task performance. The framework is applied to joint optimization of tube current modulation, view-dependent reconstruction kernel, and orbital tilt in cone-beam CT. The system model considers a cone-beam CT system incorporating a flat-panel detector and 3D filtered backprojection and accurately describes the spatially varying noise and resolution over a wide range of imaging parameters in the presence of a realistic anatomical model. Task-based detectability index (d') is incorporated as the objective function in a task-driven optimization of image acquisition and reconstruction techniques. The orbital tilt was optimized through an exhaustive search across tilt angles ranging ± 30°. For each tilt angle, the view-dependent tube current and reconstruction kernel (i.e. the modulation profiles) that maximized detectability were identified via an alternating optimization. The task-driven approach was compared with conventional unmodulated and automatic exposure control (AEC) strategies for a variety of imaging tasks and anthropomorphic phantoms. The task-driven strategy outperformed the unmodulated and AEC cases for all tasks. For example, d' for a sphere detection task in a head phantom was improved by 30% compared to the unmodulated case by using smoother kernels for noisy views and distributing mAs across less noisy views (at fixed total mAs) in a manner that was beneficial to task performance. Similarly for detection of a line-pair pattern, the task-driven approach increased d' by 80% compared to no modulation by means of view-dependent mA and kernel selection that yields modulation transfer function and noise-power spectrum optimal to the task. Optimization of orbital tilt identified the tilt angle that reduced quantum noise in the region of the stimulus by avoiding highly attenuating anatomical structures. The task-driven imaging framework offers a potentially valuable paradigm for prospective definition of acquisition and reconstruction protocols that improve task performance without increase in dose.

Regularization designs for uniform spatial resolution and noise properties in statistical image reconstruction for 3-D X-ray CT

Statistical image reconstruction methods for X-ray computed tomography (CT) provide improved spatial resolution and noise properties over conventional filtered back-projection (FBP) reconstruction, along with other potential advantages such as reduced patient dose and artifacts. Conventional regularized image reconstruction leads to spatially variant spatial resolution and noise characteristics because of interactions between the system models and the regularization. Previous regularization design methods aiming to solve such issues mostly rely on circulant approximations of the Fisher information matrix that are very inaccurate for undersampled geometries like short-scan cone-beam CT. This paper extends the regularization method proposed in to 3-D cone-beam CT by introducing a hypothetical scanning geometry that helps address the sampling properties. The proposed regularization designs were compared with the original method in with both phantom simulation and clinical reconstruction in 3-D axial X-ray CT. The proposed regularization methods yield improved spatial resolution or noise uniformity in statistical image reconstruction for short-scan axial cone-beam CT.

Task-based measures of image quality and their relation to radiation dose and patient risk

The theory of task-based assessment of image quality is reviewed in the context of imaging with ionizing radiation, and objective figures of merit (FOMs) for image quality are summarized. The variation of the FOMs with the task, the observer and especially with the mean number of photons recorded in the image is discussed. Then various standard methods for specifying radiation dose are reviewed and related to the mean number of photons in the image and hence to image quality. Current knowledge of the relation between local radiation dose and the risk of various adverse effects is summarized, and some graphical depictions of the tradeoffs between image quality and risk are introduced. Then various dose-reduction strategies are discussed in terms of their effect on task-based measures of image quality.

Alternating direction method of multiplier for tomography with nonlocal regularizers

The ordered subset expectation maximization (OSEM) algorithm approximates the gradient of a likelihood function using a subset of projections instead of using all projections so that fast image reconstruction is possible for emission and transmission tomography such as SPECT, PET, and CT. However, OSEM does not significantly accelerate reconstruction with computationally expensive regularizers such as patch-based nonlocal (NL) regularizers, because the regularizer gradient is evaluated for every subset. We propose to use variable splitting to separate the likelihood term and the regularizer term for penalized emission tomographic image reconstruction problem and to optimize it using the alternating direction method of multiplier (ADMM). We also propose a fast algorithm to optimize the ADMM parameter based on convergence rate analysis. This new scheme enables more sub-iterations related to the likelihood term. We evaluated our ADMM for 3-D SPECT image reconstruction with a patch-based NL regularizer that uses the Fair potential function. Our proposed ADMM improved the speed of convergence substantially compared to other existing methods such as gradient descent, EM, and OSEM using De Pierro's approach, and the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm.

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.

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

The development of spectral X-ray computed tomography (CT) using binned photon-counting detectors has received great attention in recent years and has enabled selective imaging of contrast agents loaded with K-edge materials. A practical issue in implementing this technique is the mitigation of the high-noise levels often present in material-decomposed sinogram data. In this work, the spectral X-ray CT reconstruction problem is formulated within a multi-channel (MC) framework in which statistical correlations between the decomposed material sinograms can be exploited to improve image quality. Specifically, a MC penalized weighted least squares (PWLS) estimator is formulated in which the data fidelity term is weighted by the MC covariance matrix and sparsity-promoting penalties are employed. This allows the use of any number of basis materials and is therefore applicable to photon-counting systems and K-edge imaging. To overcome numerical challenges associated with use of the full covariance matrix as a data fidelity weight, a proximal variant of the alternating direction method of multipliers is employed to minimize the MC PWLS objective function. Computer-simulation and experimental phantom studies are conducted to quantitatively evaluate the proposed reconstruction method.

Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT

The development of spectral computed tomography (CT) using binned photon-counting detectors has garnered great interest in recent years and has enabled selective imaging of K-edge materials. A practical challenge in CT image reconstruction of K-edge materials is the mitigation of image artifacts that arise from reduced-view and/or noisy decomposed sinogram data. In this note, we describe and investigate sparsity-regularized penalized weighted least squares-based image reconstruction algorithms for reconstructing K-edge images from few-view decomposed K-edge sinogram data. To exploit the inherent sparseness of typical K-edge images, we investigate use of a total variation (TV) penalty and a weighted sum of a TV penalty and an ℓ1-norm with a wavelet sparsifying transform. Computer-simulation and experimental phantom studies are conducted to quantitatively demonstrate the effectiveness of the proposed reconstruction algorithms.

Efficient determination of the uncertainty for the optimization of SPECT system design: a subsampled fisher information matrix

System designs in single photon emission tomography (SPECT) can be evaluated based on the fundamental trade-off between bias and variance that can be achieved in the reconstruction of emission tomograms. This trade off can be derived analytically using the Cramer-Rao type bounds, which imply the calculation and the inversion of the Fisher information matrix (FIM). The inverse of the FIM expresses the uncertainty associated to the tomogram, enabling the comparison of system designs. However, computing, storing and inverting the FIM is not practical with 3-D imaging systems. In order to tackle the problem of the computational load in calculating the inverse of the FIM, a method based on the calculation of the local impulse response and the variance, in a single point, from a single row of the FIM, has been previously proposed for system design. However this approximation (circulant approximation) does not capture the global interdependence between the variables in shift-variant systems such as SPECT, and cannot account e.g., for data truncation or missing data. Our new formulation relies on subsampling the FIM. The FIM is calculated over a subset of voxels arranged in a grid that covers the whole volume. Every element of the FIM at the grid points is calculated exactly, accounting for the acquisition geometry and for the object. This new formulation reduces the computational complexity in estimating the uncertainty, but nevertheless accounts for the global interdependence between the variables, enabling the exploration of design spaces hindered by the circulant approximation. The graphics processing unit accelerated implementation of the algorithm reduces further the computation times, making the algorithm a good candidate for real-time optimization of adaptive imaging systems. This paper describes the subsampled FIM formulation and implementation details. The advantages and limitations of the new approximation are explored, in comparison with the circulant approximation, in the context of design optimization of a parallel-hole collimator SPECT system and of an adaptive imaging system (similar to the commercially available D-SPECT).

Quality and precision of parametric images created from PET sinogram data by direct reconstruction: proof of concept

We have previously implemented the direct reconstruction of dense kinetic model parameter images ("parametric images") from sinogram data, and compared it to conventional image domain kinetic parameter estimation methods . Although it has been shown that the direct reconstruction algorithm estimates the kinetic model parameters with lower root mean squared error than the conventional image domain techniques, some theoretical obstacles remain. These obstacles include the difficulty of evaluating the accuracy and precision of the estimated parameters. In image domain techniques, the reconstructed time activity curve (TAC) and the model predicted TAC are compared, and the goodness-of-fit is evaluated as a measure of the accuracy and precision of the estimated parameters. This approach cannot be applied to the direct reconstruction technique as there are no reconstructed TACs. In this paper, we propose ways of evaluating the precision and goodness-of-fit of the kinetic model parameters estimated by the direct reconstruction algorithm. Specifically, precision of the estimates requires the calculation of variance images for each parameter, and goodness-of-fit is addressed by reconstructing the difference between the measured and the fitted sinograms. We demonstrate that backprojecting the difference from sinogram space to image space creates error images that can be examined for goodness-of-fit and model selection purposes. The presence of nonrandom structures in the error images may indicate an inadequacy of the kinetic model that has been incorporated into the direct reconstruction algorithm. We introduce three types of goodness-of-fit images. We propose and demonstrate a number-of-runs image as a means of quantifying the adequacy or deficiency of the model. We further propose and demonstrate images of the F statistic and the change in the Akaike Information Criterion as devices for identifying the statistical advantage of one model over another at each voxel. As direct reconstruction to parametric images proliferates, it will be essential for imagers to adopt methods such as those proposed herein to assess the accuracy and precision of their parametric images.

Regularization design in penalized maximum-likelihood image reconstruction for lesion detection in 3D PET

Detecting cancerous lesions is a major clinical application in emission tomography. In previous work, we have studied penalized maximum-likelihood (PML) image reconstruction for the detection task and proposed a method to design a shift-invariant quadratic penalty function to maximize detectability of a lesion at a known location in a two dimensional image. Here we extend the regularization design to maximize detectability of lesions at unknown locations in fully 3D PET. We used a multiview channelized Hotelling observer (mvCHO) to assess the lesion detectability in 3D images to mimic the condition where a human observer examines three orthogonal views of a 3D image for lesion detection. We derived simplified theoretical expressions that allow fast prediction of the detectability of a 3D lesion. The theoretical results were used to design the regularization in PML reconstruction to improve lesion detectability. We conducted computer-based Monte Carlo simulations to compare the optimized penalty with the conventional penalty for detecting lesions of various sizes. Only true coincidence events were simulated. Lesion detectability was also assessed by two human observers, whose performances agree well with that of the mvCHO. Both the numerical observer and human observer results showed a statistically significant improvement in lesion detection by using the proposed penalty function compared to using the conventional penalty function.

Non-local means denoising of dynamic PET images

Objective

Dynamic positron emission tomography (PET), which reveals information about both the spatial distribution and temporal kinetics of a radiotracer, enables quantitative interpretation of PET data. Model-based interpretation of dynamic PET images by means of parametric fitting, however, is often a challenging task due to high levels of noise, thus necessitating a denoising step. The objective of this paper is to develop and characterize a denoising framework for dynamic PET based on non-local means (NLM).

Theory

NLM denoising computes weighted averages of voxel intensities assigning larger weights to voxels that are similar to a given voxel in terms of their local neighborhoods or patches. We introduce three key modifications to tailor the original NLM framework to dynamic PET. Firstly, we derive similarities from less noisy later time points in a typical PET acquisition to denoise the entire time series. Secondly, we use spatiotemporal patches for robust similarity computation. Finally, we use a spatially varying smoothing parameter based on a local variance approximation over each spatiotemporal patch.

Methods

To assess the performance of our denoising technique, we performed a realistic simulation on a dynamic digital phantom based on the Digimouse atlas. For experimental validation, we denoised PET images from a mouse study and a hepatocellular carcinoma patient study. We compared the performance of NLM denoising with four other denoising approaches – Gaussian filtering, PCA, HYPR, and conventional NLM based on spatial patches.

Results

The simulation study revealed significant improvement in bias-variance performance achieved using our NLM technique relative to all the other methods. The experimental data analysis revealed that our technique leads to clear improvement in contrast-to-noise ratio in Patlak parametric images generated from denoised preclinical and clinical dynamic images, indicating its ability to preserve image contrast and high intensity details while lowering the background noise variance.

Quantitative statistical methods for image quality assessment

Quantitative measures of image quality and reliability are critical for both qualitative interpretation and quantitative analysis of medical images. While, in theory, it is possible to analyze reconstructed images by means of Monte Carlo simulations using a large number of noise realizations, the associated computational burden makes this approach impractical. Additionally, this approach is less meaningful in clinical scenarios, where multiple noise realizations are generally unavailable. The practical alternative is to compute closed-form analytical expressions for image quality measures. The objective of this paper is to review statistical analysis techniques that enable us to compute two key metrics: resolution (determined from the local impulse response) and covariance. The underlying methods include fixed-point approaches, which compute these metrics at a fixed point (the unique and stable solution) independent of the iterative algorithm employed, and iteration-based approaches, which yield results that are dependent on the algorithm, initialization, and number of iterations. We also explore extensions of some of these methods to a range of special contexts, including dynamic and motion-compensated image reconstruction. While most of the discussed techniques were developed for emission tomography, the general methods are extensible to other imaging modalities as well. In addition to enabling image characterization, these analysis techniques allow us to control and enhance imaging system performance. We review practical applications where performance improvement is achieved by applying these ideas to the contexts of both hardware (optimizing scanner design) and image reconstruction (designing regularization functions that produce uniform resolution or maximize task-specific figures of merit).

Model observers in medical imaging research

Model observers play an important role in the optimization and assessment of imaging devices. In this review paper, we first discuss the basic concepts of model observers, which include the mathematical foundations and psychophysical considerations in designing both optimal observers for optimizing imaging systems and anthropomorphic observers for modeling human observers. Second, we survey a few state-of-the-art computational techniques for estimating model observers and the principles of implementing these techniques. Finally, we review a few applications of model observers in medical imaging research.

A macroquantification approach for region-of-interest assessment in emission tomography

In this article, we propose a quantification methodology for estimating the statistical parameters of the activity inside regions of interest (ROIs). Macroquantification implies a rearrangement of the emission projection data into macroprojections and a redefinition of the system matrix based either on an image reconstruction involving iterative ROI-wise regularization or on an ROI uniformity assumption. The technique allows a very fast computation of the ROI activities and covariance matrix in the least squares sense using a low-dimensional model of the tomographic problem. The macroquantification approach is evaluated through Monte Carlo simulations using a numerical thorax phantom, without taking into account the measurement artifacts and assuming a perfect a priori ROI definition. Various tumor ROI configurations and count rates are considered to reflect clinical situations. The results show that our technique yields low-bias ROI estimations that turn out to be more accurate than classical estimates relying on pixel summation. Macroquantification also provides an approximation for the ROI variance that describes the effective variance obtained through the simulations fairly well. The technique is then validated using single photon emission computed tomography (SPECT) data from a physical phantom composed of cylinders filled with different Tc concentrations for the task of ROI comparison. Here again, the study shows excellent agreement between the measured and predicted values of the ROI variance resulting in efficient estimations of ROI ratios and highly accurate ROI comparisons. In its simplest formulation, macroquantification has a short computation time, making it an ideal technique for quantitative ROI assessment that is compatible with a wide range of routine clinical applications.

Magnetic resonance-guided positron emission tomography image reconstruction

The resolution of positron emission tomography (PET) images is limited by the physics of positron-electron annihilation and instrumentation for photon coincidence detection. Model-based methods that incorporate accurate physical and statistical models have produced significant improvements in reconstructed image quality when compared with filtered backprojection reconstruction methods. However, it has often been suggested that by incorporating anatomical information, the resolution and noise properties of PET images could be further improved, leading to better quantitation or lesion detection. With the recent development of combined MR-PET scanners, we can now collect intrinsically coregistered magnetic resonance images. It is therefore possible to routinely make use of anatomical information in PET reconstruction, provided appropriate methods are available. In this article, we review research efforts over the past 20 years to develop these methods. We discuss approaches based on the use of both Markov random field priors and joint information or entropy measures. The general framework for these methods is described, and their performance and longer-term potential and limitations are discussed.

Least-squares dual characterization for ROI assessment in emission tomography

Our aim is to describe an original method for estimating the statistical properties of regions of interest (ROIs) in emission tomography. Drawn upon the works of Louis on the approximate inverse, we propose a dual formulation of the ROI estimation problem to derive the ROI activity and variance directly from the measured data without any image reconstruction. The method requires the definition of an ROI characteristic function that can be extracted from a co-registered morphological image. This characteristic function can be smoothed to optimize the resolution-variance tradeoff. An iterative procedure is detailed for the solution of the dual problem in the least-squares sense (least-squares dual (LSD) characterization), and a linear extrapolation scheme is described to compensate for sampling partial volume effect and reduce the estimation bias (LSD-ex). LSD and LSD-ex are compared with classical ROI estimation using pixel summation after image reconstruction and with Huesman's method. For this comparison, we used Monte Carlo simulations (GATE simulation tool) of 2D PET data of a Hoffman brain phantom containing three small uniform high-contrast ROIs and a large non-uniform low-contrast ROI. Our results show that the performances of LSD characterization are at least as good as those of the classical methods in terms of root mean square (RMS) error. For the three small tumor regions, LSD-ex allows a reduction in the estimation bias by up to 14%, resulting in a reduction in the RMS error of up to 8.5%, compared with the optimal classical estimation. For the large non-specific region, LSD using appropriate smoothing could intuitively and efficiently handle the resolution-variance tradeoff.

Parameter optimization of relaxed Ordered Subsets Pre-computed Back Projection (BP) based Penalized-Likelihood (OS-PPL) reconstruction in limited-angle X-ray tomography

This paper presents a two-step strategy to provide a quality-predictable image reconstruction. A Pre-computed Back Projection based Penalized-Likelihood (PPL) method is proposed in the strategy to generate consistent image quality. To solve PPL efficiently, relaxed Ordered Subsets (OS) is applied. A training sets based evaluation is performed to quantify the effect of the undetermined parameters in OS, which lets the results as consistent as possible with the theoretical one.

SPECT system optimization against a discrete parameter space

In this paper, we present an analytical approach for optimizing the design of a static SPECT system or optimizing the sampling strategy with a variable/adaptive SPECT imaging hardware against an arbitrarily given set of system parameters. This approach has three key aspects. First, it is designed to operate over a discretized system parameter space. Second, we have introduced an artificial concept of virtual detector as the basic building block of an imaging system. With a SPECT system described as a collection of the virtual detectors, one can convert the task of system optimization into a process of finding the optimum imaging time distribution (ITD) across all virtual detectors. Thirdly, the optimization problem (finding the optimum ITD) could be solved with a block-iterative approach or other nonlinear optimization algorithms. In essence, the resultant optimum ITD could provide a quantitative measure of the relative importance (or effectiveness) of the virtual detectors and help to identify the system configuration or sampling strategy that leads to an optimum imaging performance. Although we are using SPECT imaging as a platform to demonstrate the system optimization strategy, this development also provides a useful framework for system optimization problems in other modalities, such as positron emission tomography and x-ray computed tomography (Moore et al (2009 IEEE Nucl. Sci. Symp. Conf. Rec. pp 4154-7), Freed et al (2008 Med. Phys. 35 1912-25)).

Modeling and Control of Nonstationary Noise Characteristics in Filtered-Backprojection and Penalized Likelihood Image Reconstruction

Purpose

Nonstationarity of CT noise presents a major challenge to the assessment of image quality. This work presents models for imaging performance in both filtered backprojection (FBP) and penalized likelihood (PL) reconstruction that describe not only the dependence on the imaging chain but also the dependence on the object as well as the nonstationary characteristics of the signal and noise. The work furthermore demonstrates the ability to impart control over the imaging process by adjusting reconstruction parameters to exploit nonstationarity in a manner advantageous to a particular imaging task.

Methods

A cascaded systems analysis model was used to model the local noise-power spectrum (NPS) and modulation transfer function (MTF) for FBP reconstruction, with locality achieved by separate calculation of fluence and system gain for each view as a function of detector location. The covariance and impulse response function for PL reconstruction (quadratic penalty) were computed using the implicit function theorem and Taylor expansion. Detectability index was calculated under the assumption of local stationarity to show the variation in task-dependent image quality throughout the image for simple and complex, heterogeneous objects. Control of noise magnitude and correlation was achieved by applying a spatially varying roughness penalty in PL reconstruction in a manner that improved overall detectability.

Results

The models provide a foundation for task-based imaging performance assessment in FBP and PL image reconstruction. For both FBP and PL, noise is anisotropic and varies in a manner dependent on the path length of each view traversing the object. The anisotropy in turn affects task performance, where detectability is enhanced or diminished depending on the frequency content of the task relative to that of the NPS. Spatial variation of the roughness penalty can be exploited to control noise magnitude and correlation (and hence detectability).

Conclusions

Nonstationarity of image noise is a significant effect that can be modeled in both FBP and PL image reconstruction. Prevalent spatial-frequency-dependent metrics of spatial resolution and noise can be analyzed under assumptions of local stationarity, providing a means to analyze imaging performance as a function of location throughout the image. Knowledgeable selection of a spatially-varying roughness penalty in PL can potentially improve local noise and spatial resolution in a manner tuned to a particular imaging task.

Statistical properties of amplitude and decay parameter estimators for fluorescence lifetime imaging

We analyze the statistical properties of the maximum likelihood estimator, least squares estimator, and Pearson's χ(2)-based and Neyman's χ(2)-based estimators for the estimation of decay constants and amplitudes for fluorescence lifetime imaging. Our analysis is based on the linearization of the gradient of the objective functions around true parameters. The analysis shows that only the maximum likelihood estimator based on the Poisson likelihood function yields unbiased and efficient estimation. All other estimators yield either biased or inefficient estimations. We validate our analysis by using simulations.

Noise properties of motion-compensated tomographic image reconstruction methods

Motion-compensated image reconstruction (MCIR) methods incorporate motion models to improve image quality in the presence of motion. MCIR methods differ in terms of how they use motion information and they have been well studied separately. However, there have been less theoretical comparisions of different MCIR methods. This paper compares the theoretical noise properties of three popular MCIR methods assuming known nonrigid motion. We show the relationship among three MCIR methods-motion-compensated temporal regularization (MTR), the parametric motion model (PMM), and post-reconstruction motion correction (PMC)-for penalized weighted least square cases. These analyses show that PMM and MTR are matrix-weighted sums of all registered image frames, while PMC is a scalar-weighted sum. We further investigate the noise properties of MCIR methods with Poisson models and quadratic regularizers by deriving accurate and fast variance prediction formulas using an "analytical approach." These theoretical noise analyses show that the variances of PMM and MTR are lower than or comparable to the variance of PMC due to the statistical weighting. These analyses also facilitate comparisons of the noise properties of different MCIR methods, including the effects of different quadratic regularizers, the influence of the motion through its Jacobian determinant, and the effect of assuming that total activity is preserved. Two-dimensional positron emission tomography simulations demonstrate the theoretical results.

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.

Data analysis in emission tomography using emission-count posteriors

A novel approach to the analysis of emission tomography data using the posterior probability of the number of emissions per voxel (emission count) conditioned on acquired tomographic data is explored. The posterior is derived from the prior and the Poisson likelihood of the emission-count data by marginalizing voxel activities. Based on emission-count posteriors, examples of Bayesian analysis including estimation and classification tasks in emission tomography are provided. The application of the method to computer simulations of 2D tomography is demonstrated. In particular, the minimum-mean-square-error point estimator of the emission count is demonstrated. The process of finding this estimator can be considered as a tomographic image reconstruction technique since the estimates of the number of emissions per voxel divided by voxel sensitivities and acquisition time are the estimates of the voxel activities. As an example of a classification task, a hypothesis stating that some region of interest (ROI) emitted at least or at most r-times the number of events in some other ROI is tested. The ROIs are specified by the user. The analysis described in this work provides new quantitative statistical measures that can be used in decision making in diagnostic imaging using emission tomography.

Computation of variance in compartment model parameter estimates from dynamic PET data

PURPOSE

This paper investigates the validity of the analytical framework for variance ("analytical variance") in kinetic parameter and macroparameter estimations. Analytical variance is compared against the variance obtained from Monte Carlo simulations ("MC variance") for two different compartment models at different noise levels.

METHODS

Kinetic parameters for one-tissue (1T) and two-tissue (2T) compartment models are used to generate time-activity curves (TAC). Gaussian noise is added to the noiseless TAC to generate noise realizations for each noise level. The kinetic parameters are then estimated by minimizing the weighted squared error between the noisy TAC and the model output. Standard deviation is computed statistically from the estimated parameters and computed analytically using the framework at each noise level. The ratio of standard deviation to true parameter value obtained from Monte Carlo simulations and analytical computations is compared.

RESULTS

Difference between the analytical and MC variance increases with the level of noise and complexity of the compartment model. The standard deviation of the analytical variance also increases with the noise-level and model complexity. The difference between the analytical and MC variance is less than 3% for 1T compartment model and less than 10% for 2T compartment model at all noise levels. In addition, the standard deviation in the analytical variance is less than 15% for 1T and 2T compartment models at all noise levels.

CONCLUSIONS

These results indicate that the proposed framework for the variance in the kinetic parameter estimations can be used for 1T and 2T compartment models even in the existence of high noise.

Spatial resolution properties of motion-compensated tomographic image reconstruction methods

Many motion-compensated image reconstruction (MCIR) methods have been proposed to correct for subject motion in medical imaging. MCIR methods incorporate motion models to improve image quality by reducing motion artifacts and noise. This paper analyzes the spatial resolution properties of MCIR methods and shows that nonrigid local motion can lead to nonuniform and anisotropic spatial resolution for conventional quadratic regularizers. This undesirable property is akin to the known effects of interactions between heteroscedastic log-likelihoods (e.g., Poisson likelihood) and quadratic regularizers. This effect may lead to quantification errors in small or narrow structures (such as small lesions or rings) of reconstructed images. This paper proposes novel spatial regularization design methods for three different MCIR methods that account for known nonrigid motion. We develop MCIR regularization designs that provide approximately uniform and isotropic spatial resolution and that match a user-specified target spatial resolution. Two-dimensional PET simulations demonstrate the performance and benefits of the proposed spatial regularization design methods.

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.

Parametric imaging with Bayesian priors: a validation study with (11)C-Altropane PET

It has been suggested that Bayesian estimation methods may be used to improve the signal-to-noise ratio of parametric images. However, there is little experience with the method and some of the underlying assumptions and performance properties of Bayesian estimation remain to be investigated. We used a sample population of 54 subjects, studied previously with (11)C-Altropane, to empirically evaluate the assumptions, performance and some practical issues in forming parametric images. By using normality tests, we showed that the underpinning normality assumptions of data and parametric distribution apply to more than 80% of voxels. The standard deviation of the binding potential can be reduced 30-50% using Bayesian estimation, without introducing substantial bias. The sample size required to form the a priori information was found to be modest; as little as ten subjects may be sufficient and the choice of specific subjects has little effect on Bayesian estimation. A realistic simulation study showed that detection of localized differences in parametric images, e.g. by statistical parametric mapping (SPM), could be made more reliable and/or conducted with smaller sample size using Bayesian estimation. In conclusion, Bayesian estimation can improve the SNR of parametric images and better detect localized changes in cohorts of subjects.

Noise propagation for iterative penalized-likelihood image reconstruction based on Fisher information

Iterative reconstruction algorithms have been widely used in PET and SPECT emission tomography. Accurate modeling of photon noise propagation is crucial for quantitative tomography applications. Iteration-based noise propagation methods have been developed for only a few algorithms that have explicit multiplicative update equations. And there are discrepancies between the iteration-based methods and Fessler's fixed-point method because of improper approximations. In this paper, we present a unified theoretical prediction of noise propagation for any penalized expectation maximization (EM) algorithm where the EM approach incorporates a penalty term. The proposed method does not require an explicit update equation. The update equation is assumed to be implicitly defined by a differential equation of a surrogate function. We derive the expressions using the implicit function theorem, Taylor series and the chain rule from vector calculus. We also derive the fixed-point expressions when iterative algorithms converge and show the consistency between the proposed method and the fixed-point method. These expressions are solely defined in terms of the partial derivatives of the surrogate function and the Fisher information matrices. We also apply the theoretical noise predictions for iterative reconstruction algorithms in emission tomography. Finally, we validate the theoretical predictions for MAP-EM and OSEM algorithms using Monte Carlo simulations with Jaszczak-like and XCAT phantoms, respectively.

Adaptive imaging for lesion detection using a zoom-in PET system

Positron emission tomography (PET) has become a leading modality in molecular imaging. Demands for further improvements in spatial resolution and sensitivity remain high with growing number of applications. In this paper we present a novel PET system design that integrates a high-resolution depth-of-interaction (DOI) detector into an existing PET system to obtain higher-resolution and higher-sensitivity images in a target region around the face of the high-resolution detector. A unique feature of the proposed PET system is that the high-resolution detector can be adaptively positioned based on the detectability or quantitative accuracy of a feature of interest. This paper focuses on the signal-known-exactly, background-known-exactly (SKE-BKE) detection task. We perform theoretical analysis of lesion detectability using computer observers, and then develop methods that can efficiently calculate the optimal position of the high-resolution detector that maximizes the lesion detectability. We simulated incorporation of a high-resolution DOI detector into the microPET II scanner. Quantitative results verified that the new system has better performance than the microPET II scanner in terms of spatial resolution and lesion detectability, and that the optimal position for lesion detection can be reliably predicted by the proposed method.

Propagation of Blood Function Errors to the Estimates of Kinetic Parameters with Dynamic PET

Dynamic PET, in contrast to static PET, can identify temporal variations in the radiotracer concentration. Mathematical modeling of the tissue of interest in dynamic PET can be simplified using compartment models as a linear system where the time activity curve of a specific tissue is the convolution of the tracer concentration in the plasma and the impulse response of the tissue containing kinetic parameters. Since the arterial sampling of blood to acquire the value of tracer concentration is invasive, blind methods to estimate both blood input function and kinetic parameters have recently drawn attention. Several methods have been developed, but the effect of accuracy of the estimated blood function on the estimation of the kinetic parameters is not studied. In this paper, we present a method to compute the error in the kinetic parameter estimates caused by the error in the blood input function. Computer simulations show that analytical expressions we derive are sufficiently close to results obtained from numerical methods. Our findings are important to observe the effect of the blood function on kinetic parameter estimation, but also useful to evaluate various blind methods and observe the dependence of kinetic parameter estimates to certain parts of the blood function.

Comparison between parallel hole and rotating slat collimation: analytical noise propagation models

We have previously proposed a method to compare tomographic systems. It is assumed that each system acquires a tomographic scan of a certain tracer distribution in the same acquisition time. From this scan, each system is forced to reconstruct an image with a predefined spatial resolution. The system that can perform this task with the "most favorable" noise propagation is considered as the best system. The variance on pixel values or region-of-interest (ROI) values is used to assess the noise in the reconstructed image. In this paper, we extend this idea to compare the performance of parallel hole (PH) and rotating slat (RS) collimations. Two different analytical approaches were used to analyze the variance of the reconstructed pixel/ROI values. The first method is based on the filtered-backprojection (FBP) theory, and was applied to the central point of a uniform symmetrical phantom. It yields analytical expressions for the optimal collimator aperture and the corresponding variance of the reconstructed pixel values, but it can only be applied to highly symmetrical configurations. The second method is based on approximations for the Fisher information matrix. It provides numerical results, and it is more general and can be applied to nonsymmetrical objects and shift-variant tomographic systems. The collimations were compared for both planar imaging and volume imaging. The main results are as follows. 1) For cases where both methods are valid, they are in excellent agreement. 2a) The optimal collimator aperture varies linearly with the target resolution. 2b) For a fixed target resolution, the optimal collimator aperture depends on the collimator type and the imaging mode (planar or volume). 2c) The optimal aperture of PH is a factor of √2 larger than that of RS. 3a) The relative performance of the two collimators is determined by both the object size and the object-to-detector distance. 3b) Pixel variance and variances of ROIs with varying sizes yield very similar relative performance for RS versus PH.

Dynamic single photon emission computed tomography--basic principles and cardiac applications

The very nature of nuclear medicine, the visual representation of injected radiopharmaceuticals, implies imaging of dynamic processes such as the uptake and wash-out of radiotracers from body organs. For years, nuclear medicine has been touted as the modality of choice for evaluating function in health and disease. This evaluation is greatly enhanced using single photon emission computed tomography (SPECT), which permits three-dimensional (3D) visualization of tracer distributions in the body. However, to fully realize the potential of the technique requires the imaging of in vivo dynamic processes of flow and metabolism. Tissue motion and deformation must also be addressed. Absolute quantification of these dynamic processes in the body has the potential to improve diagnosis. This paper presents a review of advancements toward the realization of the potential of dynamic SPECT imaging and a brief history of the development of the instrumentation. A major portion of the paper is devoted to the review of special data processing methods that have been developed for extracting kinetics from dynamic cardiac SPECT data acquired using rotating detector heads that move as radiopharmaceuticals exchange between biological compartments. Recent developments in multi-resolution spatiotemporal methods enable one to estimate kinetic parameters of compartment models of dynamic processes using data acquired from a single camera head with slow gantry rotation. The estimation of kinetic parameters directly from projection measurements improves bias and variance over the conventional method of first reconstructing 3D dynamic images, generating time-activity curves from selected regions of interest and then estimating the kinetic parameters from the generated time-activity curves. Although the potential applications of SPECT for imaging dynamic processes have not been fully realized in the clinic, it is hoped that this review illuminates the potential of SPECT for dynamic imaging, especially in light of new developments that enable measurement of dynamic processes directly from projection measurements.

Band-restricted estimation of noise variance in filtered backprojection reconstructions using repeated scans

We introduce a new estimator for noise variance in tomographic images reconstructed using algorithms of the filtered backprojection type. The new estimator operates on data acquired from repeated scans of the object under examination, is unbiased, and is shown to have significantly lower variance than the conventional unbiased estimator for many scenarios of practical interest. We provide an extensive theoretical analysis of this estimator, highlighting the circumstances under which it is most effective. This analysis includes both general and specific data-correlation patterns. Moreover, we have applied our estimator to real X-ray computed tomography data and present preliminary results that support the theory and provide experimental evidence of the new estimator's efficacy.

Noise and signal properties in PSF-based fully 3D PET image reconstruction: an experimental evaluation

The addition of accurate system modeling in PET image reconstruction results in images with distinct noise texture and characteristics. In particular, the incorporation of point spread functions (PSF) into the system model has been shown to visually reduce image noise, but the noise properties have not been thoroughly studied. This work offers a systematic evaluation of noise and signal properties in different combinations of reconstruction methods and parameters. We evaluate two fully 3D PET reconstruction algorithms: (1) OSEM with exact scanner line of response modeled (OSEM+LOR), (2) OSEM with line of response and a measured point spread function incorporated (OSEM+LOR+PSF), in combination with the effects of four post-reconstruction filtering parameters and 1-10 iterations, representing a range of clinically acceptable settings. We used a modified NEMA image quality (IQ) phantom, which was filled with 68Ge and consisted of six hot spheres of different sizes with a target/background ratio of 4:1. The phantom was scanned 50 times in 3D mode on a clinical system to provide independent noise realizations. Data were reconstructed with OSEM+LOR and OSEM+LOR+PSF using different reconstruction parameters, and our implementations of the algorithms match the vendor's product algorithms. With access to multiple realizations, background noise characteristics were quantified with four metrics. Image roughness and the standard deviation image measured the pixel-to-pixel variation; background variability and ensemble noise quantified the region-to-region variation. Image roughness is the image noise perceived when viewing an individual image. At matched iterations, the addition of PSF leads to images with less noise defined as image roughness (reduced by 35% for unfiltered data) and as the standard deviation image, while it has no effect on background variability or ensemble noise. In terms of signal to noise performance, PSF-based reconstruction has a 7% improvement in contrast recovery at matched ensemble noise levels and 20% improvement of quantitation SNR in unfiltered data. In addition, the relations between different metrics are studied. A linear correlation is observed between background variability and ensemble noise for all different combinations of reconstruction methods and parameters, suggesting that background variability is a reasonable surrogate for ensemble noise when multiple realizations of scans are not available.

Statistical sinogram restoration in dual-energy CT for PET attenuation correction

Dual-energy (DE) X-ray computed tomography (CT) has been found useful in various applications. In medical imaging, one promising application is using low-dose DECT for attenuation correction in positron emission tomography (PET). Existing approaches to sinogram material decomposition ignore noise characteristics and are based on logarithmic transforms, producing noisy component sinogram estimates for low-dose DECT. In this paper, we propose two novel sinogram restoration methods based on statistical models: penalized weighted least square (PWLS) and penalized likelihood (PL), yielding less noisy component sinogram estimates for low-dose DECT than classical methods. The proposed methods consequently provide more precise attenuation correction of the PET emission images than do previous methods for sinogram material decomposition with DECT. We report simulations that compare the proposed techniques and existing approaches.

High-resolution adaptive PET imaging

While the performance of small animal PET systems has been improved impressively in terms of spatial resolution and sensitivity, demands for further improvements remain high with growing number of applications. Here we propose a novel PET system design that integrates a high-resolution detector into an existing PET system to obtain higher-resolution images in a target region. The high-resolution detector will be adaptively positioned based on the detectability or quantitative accuracy of a feature of interest. The proposed system will be particularly effective for studying human cancers using animal models where tumors are often grown near the skin surface and therefore permit close contact with the high resolution detector. It will also be useful for the high-resolution brain imaging in rodents. In this paper, we present the theoretical analysis and Monte Carlo simulation studies of the performance of the proposed system.

Theoretical analysis and simulation study of a high-resolution zoom-in PET system

We study a novel PET system that integrates a high-resolution zoom-in detector into an existing PET scanner to provide higher resolution and sensitivity in a target region. In contrast to a full-ring PET insert, the proposed system is designed to focus on the target region close to the face of the high-resolution detector. The proposed design is easier to implement than a full-ring insert and provides flexibility for adaptive PET imaging. We developed a maximum a posteriori (MAP) image reconstruction method for the proposed system. Theoretical analysis of the resolution and noise properties of the MAP reconstruction is performed. We show that the proposed PET system offers better performance in terms of resolution-noise tradeoff and lesion detectability. The results are validated using computer simulations.

Estimation of channelized hotelling observer performance with known class means or known difference of class means

This paper concerns task-based image quality assessment for the task of discriminating between two classes of images. We address the problem of estimating two widely-used detection performance measures, SNR and AUC, from a finite number of images, assuming that the class discrimination is performed with a channelized Hotelling observer. In particular, we investigate the advantage that can be gained when either 1) the means of the signal-absent and signal-present classes are both known, or 2) when the difference of class means is known. For these two scenarios, we propose uniformly minimum variance unbiased estimators of SNR(2), derive the corresponding sampling distributions and provide variance expressions. In addition, we demonstrate how the bias and variance for the related AUC estimators may be calculated numerically by using the sampling distributions for the SNR(2) estimators. We find that for both SNR(2) and AUC, the new estimators have significantly lower bias and mean-square error than the traditional estimator, which assumes that the class means, and their difference, are unknown.