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It is important to predict and understand how scanner design characteristics will affect image quality relevant to clinical tasks when developing a new system. It is a complex task to understand the interplay among multiple system parameters such as time-of-flight (TOF) resolution, sensitivity, intrinsic spatial resolution, and depth-of-interaction (DOI) resolution. This can be achieved by Monte Carlo (MC) simulation. Here we perform MC simulation using SimSET to simulate brain-dedicated positron emission tomography (PET) scanners with varying system parameters such as DOI resolution, intrinsic spatial resolution and TOF resolution. We report preliminary results from simulating various phantoms. The results show that DOI information greatly improves image resolution particularly in the peripheral regions, reducing parallax errors even without point spread function modeling, that smaller detector elements improve image resolution, and that TOF information lowers image noise. The MC simulation analysis provides useful results for optimizing detector geometry and hardware for high resolution neuro-PET scanner design.

Statistical PET image reconstruction methods are often accelerated by the use of a subset of available projections at each iteration. It is known that many subset algorithms, such as ordered subset expectation maximisation, will not converge to a single solution but to a limit cycle. Reconstruction methods exist to relax the update step sizes of subset algorithms to obtain convergence, however, this introduces additional parameters that may result in extended reconstruction times. Another approach is to gradually decrease the number of subsets to reduce the effect of the limit cycle at later iterations, but the optimal iteration numbers for these reductions may be data dependent. We propose an automatic method to increase subset sizes so a reconstruction can take advantage of the acceleration provided by small subset sizes during early iterations, while at later iterations reducing the effects of the limit cycle behaviour providing estimates closer to the maximum a posteriori solution. At each iteration, two image updates are computed from a common estimate using two disjoint subsets. The divergence of the two update vectors is measured and, if too great, subset sizes are increased in future iterations. We show results for both sinogram and list mode data using various subset selection methodologies.

In dual-modality systems, using anatomical priors has been shown to improve image quality and quantification in emission tomography. However, alignment between the functional and anatomical images is crucial. In this study, we propose two algorithms for solving misalignment issues. Both approaches are based on a recently published joint motion estimation and image reconstruction method. The first approach deforms the anatomical image to align it with the functional one while the second approach deforms both images to align them with the measured data. Our current implementation uses alternates between image reconstruction and alignment estimation. To evaluate the potential of these approaches, we have chosen Parallel Level Sets (PLS) as a representative anatomical penalty since it has shown promising results in literature, incorporating a spatially-variant penalty strength to achieve uniform local contrast and fast convergence rate. The performance evaluation was achieved by using simulated non-TOF data generated with an XCAT phantom in the thorax region. We used the attenuation image in the anatomical prior. The results demonstrated that both methods are able to estimate the misalignment and deform the anatomical image accordingly when a proper workflow for the alternating optimization is applied. However, the performance of the first approach depends highly on the workflow of the alternating process. In contrast, the second approach shows the ability to converge to the correct alignment faster than the first approach does, independent of the workflow. Our results indicate that it is possible to align functional and anatomical information, enabling the use of anatomical priors in practice.

This study explores the use of a spatially-variant penalty strength, proposed initially for quadratic penalties, in penalized image reconstruction using anatomical information. We have used the recently proposed Parallel Level Sets (PLS) anatomical prior as it has shown promising results in the literature. It was incorporated into the previously proposed preconditioned algorithm (L-BFGS-B-PC) for achieving both good image quality and fast convergence rate. A 2-dimensional (2D) disc phantom with a hot spot at the center and a 3D XCAT thorax phantom with lesions inserted in different slices are used to study how surrounding activity and lesion location affect both the visual appearance and quantitative consistency, respectively. Anatomical information is provided and assumed to be well-aligned with the corresponding activity images. For the XCAT phantom, the inserted lesions are either present or absent in the anatomical images to investigate the influence of the anatomical penalty. The reconstructed images for both phantoms with and without applying the spatially-variant penalty strength are compared. Preliminary results demonstrate that applying the spatially-variant penalization with an anatomical prior can reduce the dependence of local contrast on background activity and lesion location. Further work to explore the potential benefit in clinical imaging is warranted.

Projection data acquired from a positron emission tomography (PET) scanner consist of true, scattered and random events. Scattered events can cause severe artifacts and quantitation errors in reconstructed PET images unless corrected for properly. A scatter correction algorithm is required to predict scattered events from the measurement. Scatter correction requires estimation of both single scatter and multiple scatter profiles. Usually, single scatter profiles are calculated by modelbased simulation and multiple scatter profiles are estimated by a kernel-based convolution method. However, design of the convolution kernels for multiple scatter estimation is sophisticated and requires fine parameter tuning. In this work, we adopt deep learning techniques for scatter estimation. We propose two convolutional neural networks. The first network estimates multiple scatter profiles from single scatter profiles, replacing the kernel-based convolution method. The second network is designed to predict the total scatter profiles (including single and multiple scatters) directly from the input of emission and attenuation sinograms. Initial results from both networks show a promise with the potential for more accurate and faster scatter correction for PET.

A penalized-likelihood (PL) image reconstruction algorithm using the relative difference penalty (RDP), called a PL-RDP algorithm, has recently been developed for clinical PET and has previously demonstrated improved Iesion quantitation accuracy and Iesion detectability over the current clinical standard PET image reconstruction algorithm, ordered subsets expectation maximization (OSEM). Patient size is one of the main factors affecting PET image quality, but the effects of patient size on PL-RDP have not been investigated. The purpose of this study was to evaluate the impact of patient size on image quality for PL-RDP compared to OSEM. We inserted simulated liver Sesions with varying sizes and contrasts into clinical data. To evaluate Iesion detectability, we applied a channelized Hotelling observer and calculated the signal-to-noise ratio (SNR). To evaluate Iesion quantitation accuracy, we compared contrast recovery coefficient (CRC) at matched image roughness, which is a surrogate image noise measure. The Iesion detectability and the quantitation accuracy in a group of smaller-sized patients with body mass index (BMI) 26 kg/m2. The detection performance of TOF PL-RDP was better than that of TOF OSEM for 20 mm Iesions with contrast $\geq 0.5$ in the smaller-sized patient group; and for 10 mm Iesion with contrast $\geq$ 1.25, 15 mm Iesions with contrast $\geq 0.5$ and 20 mm Iesions with contrast $\geq$ 0.25 in the larger-sized patient group. The detectability for non-TOF PL-RDP was better than that for non- TOF OSEM for 20 mm Iesions with contrast $\geq 0.5$ in the Iarger- sized patient group. The quantitation performance of non-TOF PL-RDP was better than that of non-TOF OSEM for 20 mm Iesions with contrast $\geq 0.5$, 15 mm Iesions with contrast $\geq 1$, and 10 mm Iesions with contrast $\geq 4$ in the smaller-sized patient group; and for 20 mm Iesions with contrast $\geq 0.5$, 15 mm Iesions with contrast $\geq 0.5$, and 10 mm Iesions with contrast $\geq 4$ in the larger-sized patient group. In addition, the improvement of PL- RDP in detection over OSEM for the larger-sized patients was larger than that for the smaller-sized patients for 15 mm Iesion with contrast $\geq 0.75$ and 20 mm Iesions with contrast $\geq 0.5$ in the TOF case; and the improvement of PL-RDP in quantitation accuracy over OSEM for the larger-sized patients was larger than that for the smaller-sized patients for 20 mm Sesions with contrast $\geq 0.5$ in the non-TOF case.

We previously proposed a fast maximum a posteriori (MAP) algorithm, limited-memory Broyden-Fletcher-Goldfarb-Shanno with boundary constrains (LBFGS-B-PC), combining LBFGS-B with diagonal preconditioning. Previous results have shown in simulations that it converges using around 40 projections independent of many factors. The aim of this study is to improve the algorithm further by using a better initial image and a modified preconditioner that is less sensitive to noise and data scale. By initializing the algorithm with the best initial image (one full iteration of OSEM with 35 subsets), ROI values can converge almost twice as fast for the same computation time. Moreover, the new preconditioner makes the performance more consistent between high and low count data sets. In addition, we have found a means to choose the stopping criteria to reach a desired level of quantitative accuracy in the reconstructed image. Based on the results with patient data, the optimized LBFGS-B-PC shows promise for clinical imaging.

Attenuation correction is critical to accurate PET quantitation. Accurate and robust attenuation correction remains challenging in hybrid PET/MR because it is difficult to segment bones, internal air and lungs accurately in MR images and MR often suffers from artifacts due to metal implants and signal shading. Joint estimation (JE) of activity and attenuation from time-of-flight (TOF) PET data has recently gained substantial interest because the JE approach shows a promise to address the challenges in MR based attenuation correction. We have previously implemented a JE algorithm with MR-based priors and demonstrated its feasibility using FDG PET/MR clinical data. However, JE algorithms have rarely been evaluated on non-FDG tracers which often have spatially more specific uptake patterns with little activity in background tissue, and hence may pose challenges to JE. In this study, we evaluate the JE algorithm using non-FDG PET/MR clinical data. We demonstrate that the JE algorithm improves the accuracy and robustness of MR-based attenuation correction not only for FDG as previously shown but also for non-FDG tracers such as 68Ga-DOTATOC and Fluoride. In addition, we demonstrate the critical role of TOF information in JE algorithms.

Attenuation correction is critical to accurate PET quantitation. However, it is challenging to extract accurate attenuation from MR data because of distinct physics of MR and PET. The goal of this study is to achieve robust and accurate attenuation correction in PET/MR. We combine an MR-segmentation based approach and a joint estimation approach synergistically by using an MR-segmentation based attenuation map as an MR-based prior for joint estimation. We evaluate the joint estimation algorithm with MR-based priors using time-of-flight (TOF) PET/MR clinical data. It is demonstrated that the joint estimation method using MR-based priors can recover the attenuation of metal implants, internal air cavities and bones in a robust way.

The detector response due to inter-crystal scatter, penetration and non-colinearity in PET can be modeled using blurring point spread functions (PSFs) in sinogram or image space. Incorporating PSFs into image reconstruction is expected to provide better image quality because of accurate system modeling. It has been widely observed that PSF-based reconstruction produces better spatial resolution with more pronounced overshoot/undershoot near edges and yields lower variance and more correlated image noise than non-PSF reconstruction. These findings have been mostly empirically made and there has not been much of rigorous explanation about the reason for the behaviors. In this study, we investigate the underlying mathematical structure of PSF-based and non-PSF image reconstruction and attempt to provide a mathematical explanation about the distinct behaviors they show. The local impulse response and the covariance of PSF-based and non-PSF penalized-likelihood image reconstruction are analyzed in continuous space to obtain useful insights. From the continuous-space analysis, we address why PSF-based reconstruction produces pronounced overshoot/undershoot near edges and what is the limitation of non-PSF reconstruction in achievable image resolution. In addition, it is analytically shown that PSF-based reconstruction results in spatially more correlated noise than non-PSF reconstruction at matched ensemble pixel variance. Finally, it is demonstrated that PSF-based reconstruction not only produces a lower pixel variance than non-PSF reconstruction at matched spatial resolution but also nearly achieves the minimum pixel variance for given spatial resolution.

Attenuation correction is critical to positron emission tomography (PET) quantitation. Errors in an attenuation map propagate into PET image reconstruction, subsequently affecting PET quantitation. It is important to understand quantitatively how the errors in the attenuation map impact on PET reconstruction in order to avoid misinterpretation of PET images in the presence of such errors and to set a specification on the accuracy of attenuation maps. In this study we present a computationally efficient method to predict the perturbation in the reconstructed PET image due to a perturbation in the attenuation map for time-of-flight (TOF) PET, which is known to be more robust to modeling inaccuracy than non-TOF PET. Building on existing methods that focus on non-TOF PET, we derive an analytical expression to predict the propagated errors for TOF PET by using a stationary point analysis applied to penalized-likelihood image reconstruction. We calculate the analytical expression efficiently in Fourier space without additional image reconstruction or matrix inversion. In addition, we demonstrate the propagated errors in PET quantitation are proportional to TOF timing resolution. The analytical method enables one to quickly examine the effects of inaccuracies in an attenuation map on PET quantitation. Its applications include analysis of MR-based attenuation correction in PET/MR, where MR-based attenuation maps often have inaccuracies in bones, lungs and implants.

Most image reconstruction methods have parameters for users to determine: for example, an iteration number and post-reconstruction filter parameters in OSEM, or a regularization parameter in penalized-likelihood (PL). To optimize such reconstruction parameters, one needs to quantitatively understand the relationship among those parameters and image quality. However, image quality is a function of not only the reconstruction parameters but also patient, scanner and imaging protocol. A major advantage of PL over OSEM is the availability of a computationally efficient algebraic procedure to predict resolution and noise properties as a function of the regularization parameter for a given sinogram data set while taking into account the dependence on patient, scanner and protocol. But the procedure, which is based on discrete-space matrix and vector operations, despite its usefulness, lacks intuitive insights such as can be obtained from studying continuous-space Radon transforms. Here, by continuous-space analysis of PL with quadratic (or Gaussian) penalties, we derive approximate yet insightful closed-form expressions for functional relationships, which turn out to be power laws, among the regularization parameter and such image quality measures as resolution, variance and spatial correlation. The expressions we derive provide intuitive insights into how a regularization parameter affects the image quality measures. As a by-product, we develop an understanding of why the ensemble voxel variance in PL is a function of the regularization parameter only and is, somewhat surprisingly, independent of other factors including patient size, scan time and dose.

Computational cost is a major challenge for 3D PET image reconstruction. Traditional ordered subsets based methods, such as OS-EM, accelerate the reconstruction process but in general are not convergent. In recent years, accelerated and convergent algorithms have been developed for PET image reconstruction. In this work, we study the numerical convergence speeds for three types of ordered subsets based convergent algorithms for penalized likelihood image reconstruction: BSREM, COSEM, and OS-MAP. The study of the computational cost for these algorithms is important in guiding numerical algorithm selection in clinical applications of penalized likelihood image reconstruction. A special case of the q-generalized Gaussian Markov random field (q-GGMRF) was used as the prior and a separable surrogate function for the prior was derived for COSEM and OS-MAP. The experiments were performed with simulated data obtained from a resolution phantom and the NCAT phantom as well as with clinical data containing inserted lesions. We investigated the local convergence in lesion activity recovery and the global convergence in the objective function as a function of iteration number. The experiments indicate that BSREM achieves faster convergence with a good selection of relaxation parameters compared to COSEM. OS-MAP's convergence speed with a good rule for gradually reducing the number of subsets to one was comparable to that of BSREM.

Partial volume errors, caused by finite image resolution, that is, blurring in reconstructed images, prevent accurate quantitation of lesion uptake in oncological applications of PET. However, partial volume correction (PVC) is a challenging problem because image resolution is affected in a nontrivial way by a number of factors including patient, imaging protocol, background activity, and reconstruction parameters such as iteration numbers in OSEM and smoothing parameters in penalized-likelihood (PL). One of the advantages of PL over OSEM is that one can systematically identify key factors affecting image resolution and exploit them for PVC. Indeed, image resolution can be efficiently and accurately predicted for PL particularly with quadratic penalties. However, quadratic penalties do not necessarily offer clinically acceptable visual image quality comparable to OSEM. Here we focus on nonquadratic relative difference penalties, which we have previously found provide clinically acceptable visual image quality while achieving better quantitation accuracy than OSEM. It is straightforward to apply the methods presented here to conventional nonquadratic edge-preserving penalties. Our approach to PVC is to correct for partial volume errors through correcting for contrast recovery errors. First, we analytically identify key factors affecting contrast recovery and tabulate the functional dependence of contrast recovery coefficients (CRCs) on those factors empirically from a simulation study. Then we use the precomputed CRC table for PVC given an individual clinical data set. A main contrast between our approach and existing PVC methods using CRC or recovery coefficients is that thanks to PL we can analytically identify crucial factors affecting partial volume errors and correct for their effects.

Joint reconstruction of activity and attenuation maps from emission data only, while being a long-standing problem in emission tomography, has been gaining recent interests because of its application to MR-based attenuation correction in PET/MR scanners where CT images are not available. Furthermore, recent studies showed that TOF (time-of-flight) information can substantially reduce, or completely remove in theory, crosstalk artifacts, which had been one of the hurdles preventing joint reconstruction techniques from being used clinically. Nonetheless, estimating both activity and attenuation from TOF emission data is a computationally challenging nonconvex optimization problem with high-dimensional data size. Therefore, as a tool for investigating and optimizing the joint estimation techniques for clinical use, we need numerical algorithms that are derived in a principled way and guaranteed to converge to a solution. Here, in a PL (penalized-likelihood) framework, we present a block alternating MM (minorization maximization) algorithm, which is provably globally convergent although the PL objective function is nonconcave. By using linear parameterization of attenuation maps, the algorithm applies to a variety of scenarios, depending on the type of and the degree of prior information, in a unified way. In addition, we provide a proof of the uniqueness of solutions to joint estimation problems for a continuous-space TOF PET system whose TOF kernels do not need to be Gaussian.

Clinical widespread use of edge-preserving penalized-likelihood (PL) methods has been hindered by the properties of the resulting images such as blocky background noise textures, piecewise-constant appearances of organs and relative noise strengths in high and low activity regions despite their potential for improved lesion quantitation over OSEM and quadratically penalized PL. Here, we investigate the use of the convex relative difference penalty first introduced by Nuyts et at. (TNS '02) for improved quantitation over OSEM in whole-body clinical PET imaging while maintaining visual image properties similar to OSEM and therefore clinical acceptability. We perform data-independent axial smoothing modulation based on the system sensitivity profile in order to avoid excessively smooth bed-position overlap regions. We also perform data-independent transaxial smoothing modulation to avoid oversmoothing the central portions of the field-of-view that occur with the use of a constant smoothing parameter. The resulting overall smoothing modulation profile allows for improved resolution uniformity in regions with high sensitivity and improved noise uniformity between regions of low and high sensitivity. We evaluate our approach in multiple clinical datasets with lesions inserted into representative locations with time-of-flight (TOF) and non-TOF reconstructions. Such "hybrid" datasets combine clinically realistic image backgrounds with known lesion activities. We demonstrate that using the relative difference penalty with proper smoothing modulation, superior quantitation over early-stopped and post-filtered OS EM can be achieved while maintaining clinically acceptable image quality. Furthermore, the approach lends itself to theoretical contrast recovery prediction and bias correction for improved contrast recovery consistency across lesions and further improvements in quantitation.

We investigate noise strength and correlation length tradeoffs for non time-of-flight (TOF) and TOF penalized likelihood image reconstruction as functions of object size and imaging system timing resolution. We show that for non-TOF imaging, noise correlation lengths have a strong dependence on object size. When smoothing parameters are varied to match correlation lengths among different sized objects, noise levels differ greatly. Since TOF-PET is a better conditioned reconstruction problem compared to the non-TOF case and mostly benefits larger objects, we investigate whether TOF is able to reduce the object size dependency of noise strength and correlation lengths. Through analytic expressions and Monte Carlo simulations, we show that as the timing resolution improves, noise correlation lengths in larger objects are shortened and it becomes possible to reduce their dependence on object size. Noise texture across the body is therefore more uniform with TOF-PET whole body imaging compared to non-TOF imaging.

Accurate system modeling is essential for improved quantitation and lesion detection. Many investigators have made efforts to accurately model detector blurring using point spread functions (PSFs) in sinogram space and to incorporate them into image reconstruction for accurate quantitation. It has been observed that incorporating detector PSF into reconstruction leads to improved contrast recovery and resolution with reduced noise but introduces edge artifacts. It is not straightforward to investigate the impact of PSF kernels on image qualities because of lack of a tool to quantitatively analyze nonlinearly object-dependent OSEM. Accordingly, there have been few methods to reduce edge artifacts in a systematically object-independent way. Our goal is to analyze edge artifacts as well as contrast recovery, resolution and image noise in image reconstruction using various PSF models including full, under-modeled and no PSF kernels, and to provide a systematic solution to reduce edge artifacts without loss of contrast recovery. We focus on penalized likelihood reconstruction with quadratic regularization. Building on previous work, we derive analytical expressions for local impulse response and covariance where a PSF model mismatch exists so that one can analytically predict image qualities, such as contrast recovery, noise and edge artifacts, as a function of regularization parameters and reconstruction PSF kernels. Using the analytical tools, we show that there exists a trade-off between contrast recovery (or resolution), image noise and edge artifacts and that one can control the trade-off by tuning regularization parameters and reconstruction PSF kernels.

In many PET scanners, there exist gaps between detector blocks and the data corresponding to lines of response associated with these gaps are missing. The missing gap data must be estimated prior to image reconstruction in cases where the sinogram data are rebinned into lower dimensional formats or when filtered backprojection (FBP) is used. Both bilinear interpolation and filters enforcing data consistency conditions have been used for non-TOF PET data. Here we develop a method for gap filling for TOF data by iteratively applying a bow-tie filter in the Fourier domain. We use a Fourier rebinning method (FORET-2D) to rebin the 3D TOF PET data to 2D non-TOF data and then apply bow-tie filtering to each of the stacked 2D sinograms. The estimated gap data are extracted from the 3D TOF sinograms computed by inverse rebinning from the bow-tie filtered 2D non-TOF data. The procedure can be repeated iteratively to further refine the gap estimates. Preliminary results indicate that the new method provides improved estimates of the gap data compared to bilinear interpolation and reduces artifacts due to interpolation error.

Time-of-flight (TOF) PET scanners provide the potential for significantly improved signal-to-noise ratio (SNR) and lesion detectability in clinical PET. Therefore, it is likely that TOF will become the standard for clinical whole body PET in the near future. However, fully 3D TOF PET image reconstruction is a challenging task due to the huge data size. One solution to this problem is to rebin TOF data into a lower dimensional format. We have recently developed Fourier rebinning methods for mapping TOF data into non-TOF formats and achieved substantial SNR advantages over sinograms acquired without TOF information. However, such mappings for rebinning into non-TOF formats are not unique and optimization of rebinning methods has not been widely investigated. In this paper we address the question of optimal rebinning in order to make full use of TOF information and consequently to maximize image quality. We focus on FORET-3D, which rebins 3D TOF data into 3D non-TOF sinogram formats without requiring a Fourier transform in the axial direction. We optimize the weighting for FORET-3D using a uniformly minimum variance unbiased (UMVU) estimator under reasonable approximations. We show that the rebinned data with optimal weights are a sufficient statistic for the unknown image, implying that any information loss due to rebinning is as a result only of the approximations used in developing the optimal weighting. We demonstrate using simulated and real phantom TOF data that the optimal rebinning method achieves significant variance reduction and better contrast recovery compared to other rebinning weightings.

We investigate fast iterative image reconstruction methods for fully 3D multispectral optical bioluminescence tomography where inhomogeneous optical properties are modeled using the finite element method (FEM). We compare two approaches to incorporating the forward model into the solution of the inverse problem. In the direct calculation approach we compute the full forward model by repeated solution of the PEM problem, once for each potential source location. In the on-the-fly approach, we do not explicitly solve for the full forward model. Instead, the solution to the forward problem is included implicitly in the formulation of the inverse problem, and the FEM is solved once at each iteration for the current image estimate. In both approaches we use reordered Cholesky factorization and substitution (CFS) to solve the FEM problem. We compare the computation cost of these two approaches, concluding that the on-the-fly approach can lead to substantial reductions in cost when combined with a rapidly converging iterative algorithm

This paper examines the spatial resolution properties of non-quadratically regularized image reconstruction methods for positron emission tomography (PET). It is well known that quadratic regularization tends to over-smooth edges. Many types of edge-preserving nonquadratic penalties have been proposed to overcome this problem. However, there has been little research on quantitative analysis of nonquadratic regularization due to its nonlinear nature, whereas quadratic regularization is fairly well-understood in terms of its resolution and variance properties. We analyze the spatial resolution property of nonquadratic regularization by the local perturbation response. Our analysis shows that one can design a non-quadratic penalty to yield improved shift-invariant spatial resolution properties by using the certainty-based method based on the assumption of localized perturbations and locally uniform backgrounds. We also derive a useful scaling property of nonquadratic regularization. Simulations of 2D PET scans illustrate the validity of our analysis.

We describe a hybrid approach to iterative PET image reconstruction in which we combine three algorithms: preconditioned conjugate gradient (PCG), ordered subsets separable paraboloidal surrogate (OSSPS), and emission reconstruction incremental optimization transfer (ERIOT). These algorithms exhibit quite different convergence behavior, e.g. the initial convergence of OSSPS is fast but it soon enters a limit cycle. Conversely, initial convergence of PCG is slow compared to OSSPS and ERIOT but its asymptotic behavior is the fastest. The hybrid approach estimates convergence behavior for each method by fitting an exponential to the objective function as a function of iteration number and switches between the algorithms to optimize the convergence behavior throughout the iterations. This hybrid approach is compared to each of the component algorithms in application to simulated PET data demonstrating a reduction of at least 50% in iterations required for effective convergence compared to use of PCG alone.

We describe a fast forward and back projector pair based on inverse Fourier rebinning for use in iterative fully 3D image reconstruction. The fast projector pair is used as part of a factored system matrix that models detector-pair response with a shift variant sinogram blur kernel. In this way we are able to combine the computational advantages of Fourier rebinning with iterative reconstruction using accurate system models. The forward projector uses the exact inverse rebinning equation to map stacked 2D transaxial projections into a full 3D sinogram. The back projector is simply the transpose of the forward projector and differs from the true exact rebinning operator in the sense that it does not require reprojection to compute missing lines of response (LOR). The inverse rebinning projector differs in performance by less than 1% in RMS error from the fully 3D geometric projector computed using solid angle calculations of voxel-wise sensitivity along each LOR. The cost of forward and back projection for the microPET Focus220 small animal scanner using the inverse rebinning projector is 5 to 15 times less than that for the fully 3D geometric projector, depending on the reconstructed field of view and voxel size. We compare performance between the inverse rebinning and 3D geometric projectors using simulated point and phantom data and in vivo FDG mouse data.

Quantifying PET tracer uptake in a region of interest (ROI) is an important task in a variety of applications including brain imaging, myocardial imaging, and tumor activity assessment. Many post-reconstruction correction methods that compensate for spatial resolution or partial volume effects have been proposed for ROI quantification. Here our goal is to optimize the image reconstruction methods themselves for this task through the use of spatially variant regularization. We investigate and analyze a selective regularization strategy where reduced smoothing is imposed across the boundary of a pre-specified ROI which can be drawn, for example, from a coregistered CT image. Preliminary simulation results show that this strategy leads to better bias/variance trade-offs than spatially uniform regularization. However, the penalty function for selective smoothing is space-variant and therefore it is not straightforward to predict the bias and variance of ROI uptake estimators in a computationally efficient way without expensive Monte Carlo simulation. Here we develop a computationally efficient method to predict bias and variance through use of the Sherman-Morrison-Woodbury matrix identity and local Fourier approximations. Simulation results show that the prediction is reasonably accurate.

We provide approximate expressions for the covariance matrix of kinetic parameter estimators based on time activity curve (TAC) reconstructions when TACs are modeled as a linear combination of temporal basis functions such as B-splines. The approximations are useful tools for assessing and optimizing the basis functions for TACs and the temporal bins for data in terms of computation and efficiency. In this paper we analyze a 1D temporal problem for simplicity, and we consider a scenario where TACs are reconstructed by penalized-likelihood (PL) estimation incorporating temporal regularization, and kinetic parameters are obtained by maximum likelihood (ML) estimation. We derive approximate formulas for the covariance of the kinetic parameter estimators using 1) the mean and variance approximations for PL estimators in (Fessler, 1996) and 2) Cramer-Rao bounds. The approximations apply to list-mode data as well as bin-mode data.

Many conventional PET emission scans are corrected for accidental coincidence (AC) events, or randoms, by real-time subtraction of delayed-window coincidences, leaving only the randoms-precorrected data available for image reconstruction. The real-time precorrection compensates in mean for AC events but destroys Poisson statistics. Since the exact log-likelihood for randoms-precorrected data is inconvenient to maximize, practical approximations are desirable for statistical image reconstruction. Conventional approximations involve setting negative sinogram values to zero, which can induce positive systematic biases, particularly for scans with low counts per ray. We propose new likelihood approximations that allow negative sinogram values without requiring zero-thresholding. We also develop monotonic algorithms for the new models by using "optimization transfer" principles. Simulation results show that our new model, SP/sup -/, is free of systematic bias yet keeps low variance. Despite its simpler implementation, the new model performs comparably to the saddle-point (SD) model which has previously shown the best performance (as to systematic bias and variance) in randoms-precorrected PET emission reconstruction.