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

Analysis of penalized likelihood image reconstruction for dynamic PET quantification

Quantification of tracer kinetics using dynamic positron emission tomography (PET) provides important information for understanding the physiological and biochemical processes in humans and animals. A common procedure is to reconstruct a sequence of dynamic images first, and then apply kinetic analysis to the time activity curve of a region of interest derived from the reconstructed images. Obviously, the choice of image reconstruction method and its parameters affect the accuracy of the time activity curve and hence the estimated kinetic parameters. This paper analyzes the effects of penalized likelihood image reconstruction on tracer kinetic parameter estimation. Approximate theoretical expressions are derived to study the bias, variance, and ensemble mean squared error of the estimated kinetic parameters. Computer simulations show that these formulae predict correctly the changes of these statistics as functions of the regularization parameter. It is found that the choice of the regularization parameter has a significant impact on kinetic parameter estimation, indicating proper selection of image reconstruction parameters is important for dynamic PET. A practical method has been developed to use the theoretical formulae to guide the selection of the regularization parameter in dynamic PET image reconstruction.

Quantitative Accuracy of Penalized-Likelihood Reconstruction for ROI Activity Estimation

Estimation of the tracer uptake in a region of interest (ROI) is an important task in emission tomography. ROI quantification is essential for measuring clinical factors such as tumor activity, growth rate, and the efficacy of therapeutic interventions. Accuracy of ROI quantification is significantly affected by image reconstruction algorithms. In penalized maximum-likelihood (PML) algorithm, the regularization parameter controls the resolution and noise tradeoff and, hence, affects ROI quantification. To obtain the optimum performance of ROI quantification, it is desirable to use a moderate regularization parameter to effectively suppress noise without introducing excessive bias. However, due to the non-linear and spatial-variant nature of PML reconstruction, choosing a proper regularization parameter is not an easy task. Our previous theoretical study [1] has shown that the bias-variance characteristic for ROI quantification task depends on the size and activity distribution of the ROI. In this work, we design physical phantom experiments to validate these predictions in a realistic situation. We found that the phantom data results match well the theoretical predictions. The good agreement between the phantom results and theoretical predictions shows that the theoretical expressions can be used to predict the accuracy of ROI activity quantification.

Iterative PET Image Reconstruction Using Translation Invariant Wavelet Transform

The present work describes a Bayesian maximum a posteriori (MAP) method using a statistical multiscale wavelet prior model. Rather than using the orthogonal discrete wavelet transform (DWT), this prior is built on the translation invariant wavelet transform (TIWT). The statistical modeling of wavelet coefficients relies on the generalized Gaussian distribution. Image reconstruction is performed in spatial domain with a fast block sequential iteration algorithm. We study theoretically the TIWT MAP method by analyzing the Hessian of the prior function to provide some insights on noise and resolution properties of image reconstruction. We adapt the key concept of local shift invariance and explore how the TIWT MAP algorithm behaves with different scales. It is also shown that larger support wavelet filters do not offer better performance in contrast recovery studies. These theoretical developments are confirmed through simulation studies. The results show that the proposed method is more attractive than other MAP methods using either the conventional Gibbs prior or the DWT-based wavelet prior.

A Vector Uniform Cramer-Rao Bound for SPECT System Design

In this paper, we present the use of modified uniform Cramer-Rao type bounds (MUCRB) for the design of single photon emission tomography (SPECT) systems. The MUCRB is the lowest attainable total variance using any estimator of an unknown vector parameter, whose mean gradient matrix satisfies a given constraint. Since the mean gradient is closely related to local impulse function, the MUCRB approach can be used to evaluate the fundamental tradeoffs between spatial resolution and variance that are achievable with a given SPECT system design. As a possible application, this approach allows one to compare different SPECT system designs based on the optimum average resolution-variance tradeoffs that can be achieved across multiple control-points inside a region-of-interest. The formulation of the MUCRB allows detailed modelling of physical aspects of practical SPECT systems and requests only a modest computation load. It can be used as an analytical performance index for comparing different SPECT system or aperture designs.

A survey of approaches for direct parametric image reconstruction in emission tomography

The quantitative data obtained by emission tomography are decoded using a number of techniques and methods in sequence to provide physiological information. Conventionally, the data are reconstructed to produce a series of static images. Then, pharmacokinetic modeling techniques are applied, and kinetic parameters that have physiological or functional significance are derived. Although it is possible to optimize each estimation step in this process, many simplifying assumptions have to be introduced to make the methods that are used practicable. Published research has shown that if the kinetic parameters are estimated directly from the measured data, the parametric images will have higher quality and lower mean-squared error than if this was done indirectly. This review highlights some aspects of the methods that have been proposed for such direct estimation of pharmacokinetic information from raw emission data.

Analysis of Resolution and Noise Properties of Nonquadratically Regularized Image Reconstruction Methods for PET

We present accurate and efficient methods for estimating the spatial resolution and noise properties of nonquadratically regularized image reconstruction for positron emission tomography (PET). It is well known that quadratic regularization tends to over-smooth sharp edges. Many types of edge-preserving nonquadratic penalties have been proposed to overcome this problem. However, there has been little research on the quantitative analysis of nonquadratic regularization due to its nonlinearity. In contrast, quadratically regularized estimators are approximately linear and are well understood in terms of resolution and variance properties. We derive new approximate expressions for the linearized local perturbation response (LLPR) and variance using the Taylor expansion with the remainder term. Although the expressions are implicit, we can use them to accurately predict resolution and variance for nonquadratic regularization where the conventional expressions based on the first-order Taylor truncation fail. They also motivate us to extend the use of a certainty-based modified penalty to nonquadratic regularization cases in order to achieve spatially uniform perturbation responses, analogous to uniform spatial resolution in quadratic regularization. Finally, we develop computationally efficient methods for predicting resolution and variance of nonquadratically regularized reconstruction and present simulations that illustrate the validity of these methods.

Effect of overlapping projections on reconstruction image quality in multipinhole SPECT

Multipinhole single photon emission computed tomography (SPECT) imaging has several advantages over single pinhole SPECT imaging, including an increased sensitivity and an improved sampling. However, the quest for a good design is challenging, due to the large number of design parameters. The effect of one of these, the amount of overlap in the projection images, on the reconstruction image quality, is examined in this paper. The evaluation of the quality is based on efficient approximations for the linearized local impulse response and the covariance in a voxel, and on the bias of the reconstruction of the noiseless projection data. Two methods are proposed that remove the overlap in the projection image by blocking certain projection rays with the use of extra shielding between the pinhole plate and the detector. Also two measures to quantify the amount of overlap are suggested. First, the approximate method, predicting the contrast-to-noise ratio (CNR), is validated using postsmoothed maximum likelihood expectation maximization (MLEM) reconstructions with an imposed target resolution. Second, designs with different amounts of overlap are evaluated to study the effect of multiplexing. In addition, the CNR of each pinhole design is also compared with that of the same design where overlap is removed. Third, the results are interpreted with the overlap quantification measures. Fourth, the two proposed overlap removal methods are compared. From the results we can conclude that, once the complete detector area has been used, the extra sensitivity due to multiplexing is only able to compensate for the loss of information, not to improve the CNR. Removing the overlap, however, improves the CNR. The gain is most prominent in the central field of view, though often at the cost of the CNR of some voxels at the edges, since after overlap removal very little information is left for their reconstruction. The reconstruction images provide insight in the multiplexing and truncation artifacts.

Single and multipinhole collimator design evaluation method for small animal SPECT

High-resolution functional imaging of small animals is often obtained by single pinhole SPECT with circular orbit acquisition. Multipinhole SPECT adds information due to its improved sampling, and can improve the trade-off between resolution and sensitivity. To evaluate different pinhole collimator designs an efficient method is needed that quantifies the reconstruction image quality. In this paper, we propose a fast, approximate method that examines the quality of individual voxels of a postsmoothed maximum likelihood expectation maximization (MLEM) reconstruction by studying their linearized local impulse response (LLIR) and (co)variance for a predefined target resolution. For validation, the contrast-to-noise ratios (CNRs) in some voxels of a homogeneous sphere and of a realistic rat brain software phantom were calculated for many single and multipinhole designs. A good agreement was observed between the CNRs obtained with the approximate method and those obtained with postsmoothed MLEM reconstructions of simulated noisy projections. This good agreement was quantified by a least squares fit through these results, which yielded a line with slope 1.02 (1.00 expected) and a y-intercept close to zero (0 expected). 95.4% of the validation points lie within three standard deviations from that line. Using the approximate method, the influence on the CNR of varying a parameter in realistic single and multipinhole designs was examined. The investigated parameters were the aperture diameter, the distance between the apertures and the axis-of-rotation, the focal distance, the acceptance angle, the position of the apertures, the focusing distance, and the number of pinholes. The results can generally be explained by the change in sensitivity, the amount of postsmoothing, and the amount of overlap in the projections. The method was applied to multipinhole designs with apertures focusing at a single point, but is also applicable to other designs.

Range Condition and ML-EM Checkerboard Artifacts

The expectation maximization (EM) algorithm for the maximum likelihood (ML) image reconstruction criterion generates severe checkerboard artifacts in the presence of noise. A classical remedy is to impose an a priori constraint for a penalized ML or maximum a posteriori probability solution. The penalty reduces the checkerboard artifacts and also introduces uncertainty because a priori information is usually unknown in clinic. Recent theoretical investigation reveals that the noise can be divided into two components: one is called null-space noise and the other is range-space noise. The null-space noise can be numerically estimated using filtered backprojection (FBP) algorithm. By the FBP algorithm, the null-space noise annihilates in the reconstruction while the range-space noise propagates into the reconstructed image. The aim of this work is to investigate the relation between the null-space noise and the checkerboard artifacts in the ML-EM reconstruction from noisy projection data. Our study suggests that removing the null-space noise from the projection data could improve the signal-to-noise ratio of the projection data and, therefore, reduce the checkerboard artifacts in the ML-EM reconstructed images. This study reveals an in-depth understanding of the different noise propagations in analytical and iterative image reconstructions, which may be useful to single photon emission computed tomography, where the noise has been a major factor for image degradation. The reduction of the ML-EM checkerboard artifacts by removing the null-space noise avoids the uncertainty of using a priori penalty.

Simulation-based evaluation of OSEM iterative reconstruction methods in dynamic brain PET studies

The reconstruction of dynamic PET data is usually performed using filtered backprojection algorithms (FBP). This method is fast, robust, linear and yields reliable quantitative results. However, the use of FBP for low count data, such as dynamic PET data, generally results in poor visual image quality, exhibiting high noise, disturbing streak artifacts and low contrast. These signal-to-noise ratio and contrast in the reconstructed images may alter the quantification of physiological indexes, such as the regional Binding Potential (BP) obtained from kinetic modeling. Iterative reconstruction methods are often presented as viable alternatives to FBP reconstruction. In this study, we investigated the characteristics of the UW-OSEM and the ANW-OSEM iterative reconstruction methods in the context of ligand-receptor PET studies with low counts. The assessment was conducted using replicates of simulated [18F]MPPF acquisitions. The quantitative accuracy obtained with the iterative and analytical methods was compared. The results show that analytical methods are more robust to the low count data than iterative methods, and therefore enable a better estimate of the regional activity values and binding potential. The positivity constraint in MLEM-based algorithms leads to overestimations of the activity in regions with low activity concentration, typically the cerebellum. This overestimation results in significant bias in BP estimates with iterative reconstruction methods. The bias is confirmed from the reconstruction of real PET data.

A priori selection of the regularization parameters in emission tomography by Fourier synthesis

This paper describes how the stability of the inverse problem underlying emission tomography can be measured and controlled in clinical settings. We show how the Lanczos approximation provides a way to regularize a certain class of iterative reconstruction algorithms through a given level of noise or resolution in the slices and for a given acquisition protocol. Moreover, we show how the same Lanczos approximation can be used to decide when the iterative reconstruction algorithm actually converges for a given machine precision. These ideas are illustrated by means of reconstructions of simulated and actual emission datasets.

A channelized Hotelling observer study of lesion detection in SPECT MAP reconstruction using anatomical priors

In emission tomography, anatomical side information, in the form of organ and lesion boundaries, derived from intra-patient coregistered CT or MR scans can be incorporated into the reconstruction. Our interest is in exploring the efficacy of such side information for lesion detectability. To assess detectability we used the SNR of a channelized Hotelling observer and a signal-known exactly/background-known exactly detection task. In simulation studies, we incorporated anatomical side information into a SPECT MAP (maximum a posteriori) reconstruction by smoothing within but not across organ or lesion boundaries. A non-anatomical prior was applied by uniform smoothing across the entire image. We investigated whether the use of anatomical priors with organ boundaries alone or with perfect lesion boundaries alone would change lesion detectability relative to the case of a prior with no anatomical information. Furthermore, we investigated whether any such detectability changes for the organ-boundary case would be a function of the distance of the lesion to the organ boundary. We also investigated whether any detectability changes for the lesion-boundary case would be a function of the degree of proximity, i.e. a difference in the radius of the true functional lesion and the radius of the anatomical lesion boundary. Our results showed almost no detectability difference with versus without organ boundaries at any lesion-to-organ boundary distance. Our results also showed no difference in lesion detectability with and without lesion boundaries, and no variation of lesion detectability with degree of proximity.

Noise propagation in linear and nonlinear inverse scattering

The propagation of noise from the data to the reconstructed speed of sound image by inverse scattering within the framework of the Lippmann-Schwinger integral equation of scattering is discussed. The inversion algorithm that was used consisted in minimizing a Tikhonov functional in the unknown speed of sound. The gradient of the objective functional was computed by the method of the adjoint fields. An analytical expression for the inverse scattering covariance matrix of the image noise was derived. It was shown that the covariance matrix in the linear x-ray computed tomography is a special case of the inverse scattering matrix derived in this paper. The matrix was also analyzed in the limit of the linearized Born approximation, and the results were found to be in qualitative agreement with those recently reported in the literature for Born inversion using filtered backpropagation algorithm. Finally, the applicability of the analysis reported here to the obstacle problem and the physical optics approximation was discussed.

Iterative image reconstruction using inverse Fourier rebinning for fully 3-D PET

We describe a fast forward and back projector pair based on inverse Fourier rebinning for use in iterative image reconstruction for fully 3-D positron emission tomography (PET). The projector pair is used as part of a factored system matrix that takes into account detector-pair response by using shift-variant sinogram blur kernels, thereby combining the computational advantages of Fourier rebinning with iterative reconstruction using accurate system models. The forward projector consists of a 2-D projector, which maps 3-D images into 2-D direct sinograms, followed by exact inverse rebinning which maps the 2-D into fully 3-D sinograms. The back projector is implemented as 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 (LORs). We compensate for two types of inaccuracies that arise in a cylindrical PET scanner when using inverse Fourier rebinning: 1) nonuniform radial sampling and 2) nonconstant oblique angles in the radial direction in a single oblique sinogram. We examine the effects of these corrections on sinogram accuracy and reconstructed image quality. We evaluate performance of the new projector pair for maximum a posteriori (MAP) reconstruction of simulated and in vivo data. The new projector results in only a small loss in resolution towards the edge of the field-of-view when compared to the fully 3-D geometric projector and requires an order of magnitude less computation.

Fast predictions of variance images for fan-beam transmission tomography with quadratic regularization

Accurate predictions of image variances can be useful for reconstruction algorithm analysis and for the design of regularization methods. Computing the predicted variance at every pixel using matrix-based approximations [1] is impractical. Even most recently adopted methods that are based on local discrete Fourier approximations are impractical since they would require a forward and backprojection and two fast Fourier transform (FFT) calculations for every pixel, particularly for shift-variant systems like fan-beam tomography. This paper describes new "analytical" approaches to predicting the approximate variance maps of 2-D images that are reconstructed by penalized-likelihood estimation with quadratic regularization in fan-beam geometries. The simplest of the proposed analytical approaches requires computation equivalent to one backprojection and some summations, so it is computationally practical even for the data sizes in X-ray computed tomography (CT). Simulation results show that it gives accurate predictions of the variance maps. The parallel-beam geometry is a simple special case of the fan-beam analysis. The analysis is also applicable to 2-D positron emission tomography (PET).

Optimization algorithms and weighting factors for analysis of dynamic PET studies

Positron emission tomography (PET) pharmacokinetic analysis involves fitting of measured PET data to a PET pharmacokinetic model. The fitted parameters may, however, suffer from bias or be unrealistic, especially in the case of noisy data. There are many optimization algorithms, each having different characteristics. The purpose of the present study was to evaluate (1) the performance of different optimization algorithms and (2) the effects of using incorrect weighting factors during optimization in terms of both accuracy and reproducibility of fitted PET pharmacokinetic parameters. In this study, the performance of commonly used optimization algorithms (i.e. interior-reflective Newton methods) and a simulated annealing (SA) method was evaluated. This SA algorithm, known as basin hopping, was modified for the present application. In addition, optimization was performed using various weighting factors. Algorithms and effects of using incorrect weighting factors were studied using both simulated and clinical time-activity curves (TACs). Input data, taken from [(15)O]H(2)O, [(11)C]flumazenil and [(11)C](R)-PK11195 studies, were used to simulate time-activity curves at various variance levels (0-15% COV). Clinical evaluation was based on studies with the same three tracers. SA was able to produce accurate results without the need for selecting appropriate starting values for (kinetic) parameters, in contrast to the interior-reflective Newton method. The latter gave biased results unless it was modified to allow for a range of starting values for the different parameters. For patient studies, where large variability is expected, both SA and the extended Newton method provided accurate results. Simulations and clinical assessment showed similar results for the evaluation of different weighting models in that small to intermediate mismatches between data variance and weighting factors did not significantly affect the outcome of the fits. Large errors were observed only when the mismatch between weighting model and data variance was large. It is concluded that selection of specific optimization algorithms and weighting factors can have a large effect on the accuracy and precision of PET pharmacokinetic analysis. Apart from carefully selecting appropriate algorithms and variance models, further improvement in accuracy might be obtained by using noise reducing strategies, such as wavelet filtering, provided that these methods do not introduce significant bias.

Penalized maximum-likelihood image reconstruction for lesion detection

Detecting cancerous lesions is one major application in emission tomography. In this paper, we study penalized maximum-likelihood image reconstruction for this important clinical task. Compared to analytical reconstruction methods, statistical approaches can improve the image quality by accurately modelling the photon detection process and measurement noise in imaging systems. To explore the full potential of penalized maximum-likelihood image reconstruction for lesion detection, we derived simplified theoretical expressions that allow fast evaluation of the detectability of a random lesion. The theoretical results are used to design the regularization parameters to improve lesion detectability. We conducted computer-based Monte Carlo simulations to compare the proposed penalty function, conventional penalty function, and a penalty function for isotropic point spread function. The lesion detectability is measured by a channelized Hotelling observer. The results show that the proposed penalty function outperforms the other penalty functions for lesion detection. The relative improvement is dependent on the size of the lesion. However, we found that the penalty function optimized for a 5 mm lesion still outperforms the other two penalty functions for detecting a 14 mm lesion. Therefore, it is feasible to use the penalty function designed for small lesions in image reconstruction, because detection of large lesions is relatively easy.

Iterative reconstruction techniques in emission computed tomography

In emission tomography statistically based iterative methods can improve image quality relative to analytic image reconstruction through more accurate physical and statistical modelling of high-energy photon production and detection processes. Continued exponential improvements in computing power, coupled with the development of fast algorithms, have made routine use of iterative techniques practical, resulting in their increasing popularity in both clinical and research environments. Here we review recent progress in developing statistically based iterative techniques for emission computed tomography. We describe the different formulations of the emission image reconstruction problem and their properties. We then describe the numerical algorithms that are used for optimizing these functions and illustrate their behaviour using small scale simulations.

Design and Feasibility Study of a Single Photon Emission Microscope System for Small Animal I-125 Imaging

This paper presents a design study of a single photon emission microscope (SPEM) system for small animal imaging using I-125 labelled radiotracers. This system is based on the use of a very-high resolution gamma camera coupled to a converging non-multiplexing multiple pinhole collimator. This enables one to "zoom" into a small local region inside the object to extract imaging information with a very high spatial resolution and a reasonable sensitivity for gamma rays emitted from this local region. The SPEM system also includes a pinhole (or multiple pinhole) gamma camera that has a full angular coverage of the entire object. The designed imaging spatial resolution for the SPEM system is between 250 μm to 500 μm FWHM.

Theoretical study of penalized-likelihood image reconstruction for region of interest quantification

Region of interest (ROI) quantification is an important task in emission tomography (e.g., positron emission tomography and single photon emission computed tomography). It is essential for exploring clinical factors such as tumor activity, growth rate, and the efficacy of therapeutic interventions. Statistical image reconstruction methods based on the penalized maximum-likelihood (PML) or maximum a posteriori principle have been developed for emission tomography to deal with the low signal-to-noise ratio of the emission data. Similar to the filter cut-off frequency in the filtered backprojection method, the regularization parameter in PML reconstruction controls the resolution and noise tradeoff and, hence, affects ROI quantification. In this paper, we theoretically analyze the performance of ROI quantification in PML reconstructions. Building on previous work, we derive simplified theoretical expressions for the bias, variance, and ensemble mean-squared-error (EMSE) of the estimated total activity in an ROI that is surrounded by a uniform background. When the mean and covariance matrix of the activity inside the ROI are known, the theoretical expressions are readily computable and allow for fast evaluation of image quality for ROI quantification with different regularization parameters. The optimum regularization parameter can then be selected to minimize the EMSE. Computer simulations are conducted for small ROIs with variable uniform uptake. The results show that the theoretical predictions match the Monte Carlo results reasonably well.

Estimation of variance in single-particle reconstruction using the bootstrap technique

Density maps of a molecule obtained by single-particle reconstruction from thousands of molecule projections exhibit strong changes in local definition and reproducibility, as a consequence of conformational variability of the molecule and non-stoichiometry of ligand binding. These changes complicate the interpretation of density maps in terms of molecular structure. A three-dimensional (3-D) variance map provides an effective tool to assess the structural definition in each volume element. In this work, the different contributions to the 3-D variance in a single-particle reconstruction are discussed, and an effective method for the estimation of the 3-D variance map is proposed, using a bootstrap technique of sampling. Computations with test data confirm the viability, computational efficiency, and accuracy of the method under conditions encountered in practical circumstances.

Mean and covariance properties of dynamic PET reconstructions from list-mode data

We derive computationally efficient methods for the estimation of the mean and variance properties of penalized likelihood dynamic positron emission tomography (PET) images. This allows us to predict the accuracy of reconstructed activity estimates and to compare reconstruction algorithms theoretically. We combine a bin-mode approach in which data is modeled as a collection of independent Poisson random variables at each spatiotemporal bin with the space-time separabilities in the imaging equation and penalties to derive rapidly computable analytic mean and variance approximations. We use these approximations to compare bias/variance properties of our dynamic PET image reconstruction algorithm with those of multiframe static PET reconstructions.

Analysis of observer performance in known-location tasks for tomographic image reconstruction

We consider the task of detecting a statistically varying signal of known location on a statistically varying background in a reconstructed tomographic image. We analyze the performance of linear observer models in this task. We show that, if one chooses a suitable reconstruction method, a broad family of linear observers can exactly achieve the optimal detection performance attainable with any combination of a linear observer and linear reconstructor. This conclusion encompasses several well-known observer models from the literature, including models with a frequency-selective channel mechanism and certain types of internal noise. Interestingly, the "optimal" reconstruction methods are unregularized and in some cases quite unconventional. These results suggest that, for the purposes of designing regularized reconstruction methods that optimize lesion detectability, known-location tasks are of limited use.

Neural network-based image reconstruction for positron emission tomography

Positron emission tomography (PET) is one of the key molecular imaging modalities in medicine and biology. Penalized iterative image reconstruction algorithms frequently used in PET are based on maximum-likelihood (ML) and maximum a posterior (MAP) estimation techniques. The ML algorithm produces noisy artifacts whereas the MAP algorithm eliminates noisy artifacts by utilizing availableprior information in the reconstruction process. The MAP-based algorithms fail to determine the density class in the reconstructed image and hence penalize the pixels irrespective of the density class and irrespective of the strength of interaction between the nearest neighbors. A Hebbian neural learning scheme is proposed to model the nature of interpixel interaction to reconstruct artifact-free edge preserving reconstruction. A key motivation of the proposed approach is to avoid oversmoothing across edges that is often the case with MAP algorithms. It is assumed that local correlation plays a significant role in PET image reconstruction, and proper modeling of correlation weight (which defines the strength of interpixel interaction) is essential to generate artifact-free reconstruction. The Hebbian learning-based approach modifies the interaction weight by adding a small correction that is proportional to the product of the input signal (neighborhood pixels) and output signal. Quantitative analysis shows that the Hebbian learning-based adaptive weight adjustment approach is capable of producing better reconstructed images compared with those reconstructed by conventional ML and MAP-based algorithms in PET image reconstruction.

Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio

Maximum-likelihood (ML) estimation is an established paradigm for the assessment of imaging system performance in nonlinear quantitation tasks. At high signal-to-noise ratio (SNR), ML estimates are asymptotically Gaussian-distributed, unbiased and efficient, thereby attaining the Cramer-Rao bound (CRB). Therefore, at high SNR the CRB is useful as a predictor of the variance of ML estimates and, consequently, as a basis for measures of estimation performance. At low SNR, however, the achievable parameter variances are often substantially larger than the CRB and the estimates are no longer Gaussian-distributed. These departures imply that inference about the estimates that is based on the CRB and the assumption of a normal distribution will not be valid. We have found previously that for some tasks these effects arise at noise levels considered clinically acceptable. We have derived the mathematical relationship between a new measure, chi2(pdf-ML), and the expected probability density of the ML estimates, and have justified the use of chi2(pdf-ML)-isocontours in parameter space to describe the ML estimates. We validated this approach by simulation experiments using spherical objects imaged with a Gaussian point spread function. The parameters, activity concentration and size, were estimated simultaneously by ML, and variances and covariances calculated over 1000 replications per condition from 3D image volumes and from 2D tomographic projections of the same object. At low SNR, where the CRB is no longer achievable, chi2(pdf-ML)-isocontours provide a robust prediction of the distribution of the ML estimates. At high SNR, the chi2(pdf-ML)-isocontours asymptotically approach the analogous chi2(pdf-F)-contours derived from the Fisher information matrix. The chi2(pdf-ML) model appears to be suitable for characterization of the influence of the noise level and characteristics, the task, and the object on the shape of the probability density of the ML estimates at low SNR. Furthermore, it provides unique insights into the causes of the variability of estimation performance.

Effect of errors in the system matrix on maximum a posteriori image reconstruction

Statistically based iterative image reconstruction methods have been developed for emission tomography. One important component in iterative image reconstruction is the system matrix, which defines the mapping from the image space to the data space. Several groups have demonstrated that an accurate system matrix can improve image quality in both single photon emission computed tomography (SPECT) and positron emission tomography (PET). While iterative methods are amenable to arbitrary and complicated system models, the true system response is never known exactly. In practice, one also has to sacrifice the accuracy of the system model because of limited computing and imaging resources. This paper analyses the effect of errors in the system matrix on iterative image reconstruction methods that are based on the maximum a posteriori principle. We derived an analytical expression for calculating artefacts in a reconstructed image that are caused by errors in the system matrix using the first-order Taylor series approximation. The theoretical expression is used to determine the required minimum accuracy of the system matrix in emission tomography. Computer simulations show that the theoretical results work reasonably well in low-noise situations.

Comparison between MAP and postprocessed ML for image reconstruction in emission tomography when anatomical knowledge is available

Previously, the noise characteristics obtained with penalized-likelihood reconstruction [or maximum a posteriori (MAP)] have been compared to those obtained with postsmoothed maximum-likelihood (ML) reconstruction, for emission tomography applications requiring uniform resolution. It was found that penalized-likelihood reconstruction was not superior to postsmoothed ML. In this paper, a similar comparison is made, but now for applications where the noise suppression is tuned with anatomical information. It is assumed that limited but exact anatomical information is available. Two methods were compared. In the first method, the anatomical information is incorporated in the prior of a MAP-algorithm and is, therefore, imposed during MAP-reconstruction. The second method starts from an unconstrained ML-reconstruction, and imposes the anatomical information in a postprocessing step. The theoretical analysis was verified with simulations: small lesions were inserted in two different objects, and noisy PET data were produced and reconstructed with both methods. The resulting images were analyzed with bias-noise curves, and by computing the detection performance of the nonprewhitening observer and a channelized Hotelling observer. Our analysis and simulations indicate that the postprocessing method is inferior, unless the noise correlations between neighboring pixels are taken into account. This can be done by applying a so-called prewhitening filter. However, because the prewhitening filter is shift variant and object dependent, it seems that MAP reconstruction is the more efficient method.

Fast LROC analysis of Bayesian reconstructed emission tomographic images using model observers

Lesion detection and localization is an important task in emission computed tomography. Detection and localization performance with signal location uncertainty may be summarized by a scalar figure of merit, the area under the localization receiver operating characteristic (LROC) curve, A(LROC). We consider model observers to compute A(LROC) for two-dimensional maximum a posteriori (MAP) reconstructions. Model observers may be used to rapidly prototype studies that use human observers. We address the case background-known-exactly (BKE) and signal known except for location. Our A(LROC) calculation makes use of theoretical expressions for the mean and covariance of the reconstruction and, unlike conventional methods that also use model observers, does not require computation of a large number of sample reconstructions. We validate the results of the procedure by comparison to A(LROC) obtained using a gold-standard Monte Carlo method employing a large set of reconstructed noise samples. Under reasonable simulation conditions, our theoretical calculation is about one to two orders of magnitude faster than the conventional Monte Carlo method.

A comparison of human and model observers in multislice LROC studies

Model and human observers have been compared in a series of localization receiver operating characteristic (LROC) studies involving single-slice and multislice image displays. The task was detection of Ga-avid lymphomas within single photon emission computed tomography (SPECT)-reconstructed transverse slices of a mathematical phantom, and the studies involved four reconstruction strategies: the filtered-backprojection (FBP) and ordered-subset expectation-maximization (OSEM) algorithms with two- and three-dimensional postreconstruction filtering. The human-observer data was drawn from studies performed by Wells et al. (2000), while multiclass versions of the nonprewhitening (NPW), channelized nonprewhitening (CNPW), and channelized Hotelling (CH) model observers, each capable of performing the tumor search task, were applied. The channelized observers were evaluated with multiple square-channel models and both with and without internal noise. For the multislice studies, two different capacities for integrating the slice information were also tested. The CH observer gave good quantitative agreement with the human data from both image-display studies when the internal-noise model was used. The CNPW observer performed similarly with the iterative strategies. Wells et al. had shown that human observers are imperfect integrators of multislice information, and this is characterized as increased internal noise with the model observers.

Efficient calculation of resolution and covariance for penalized-likelihood reconstruction in fully 3-D SPECT

Resolution and covariance predictors have been derived previously for penalized-likelihood estimators. These predictors can provide accurate approximations to the local resolution properties and covariance functions for tomographic systems given a good estimate of the mean measurements. Although these predictors may be evaluated iteratively, circulant approximations are often made for practical computation times. However, when numerous evaluations are made repeatedly (as in penalty design or calculation of variance images), these predictors still require large amounts of computing time. In Stayman and Fessler (2000), we discussed methods for precomputing a large portion of the predictor for shift-invariant system geometries. In this paper, we generalize the efficient procedure discussed in Stayman and Fessler (2000) to shift-variant single photon emission computed tomography (SPECT) systems. This generalization relies on a new attenuation approximation and several observations on the symmetries in SPECT systems. These new general procedures apply to both two-dimensional and fully three-dimensional (3-D) SPECT models, that may be either precomputed and stored, or written in procedural form. We demonstrate the high accuracy of the predictions based on these methods using a simulated anthropomorphic phantom and fully 3-D SPECT system. The evaluation of these predictors requires significantly less computation time than traditional prediction techniques, once the system geometry specific precomputations have been made.

Intensity-based image registration using robust correlation coefficients

The ordinary sample correlation coefficient is a popular similarity measure for aligning images from the same or similar modalities. However, this measure can be sensitive to the presence of "outlier" objects that appear in one image but not the other, such as surgical instruments, the patient table, etc., which can lead to biased registrations. This paper describes an intensity-based image registration technique that uses a robust correlation coefficient as a similarity measure. Relative to the ordinary sample correlation coefficient, the proposed similarity measure reduces the influence of outliers. We also compared the performance of the proposed method with the mutual information-based method. The robust correlation-based method should be useful for image registration in radiotherapy (KeV to MeV X-ray images) and image-guided surgery applications. We have investigated the properties of the proposed method by theoretical analysis, computer simulations, a phantom experiment, and with functional magnetic resonance imaging data.

Accurate estimation of the Fisher information matrix for the PET image reconstruction problem

The Fisher information matrix (FIM) plays a key role in the analysis and applications of statistical image reconstruction methods based on Poisson data models. The elements of the FIM are a function of the reciprocal of the mean values of sinogram elements. Conventional plug-in FIM estimation methods do not work well at low counts, where the FIM estimate is highly sensitive to the reciprocal mean estimates at individual detector pairs. A generalized error look-up table (GELT) method is developed to estimate the reciprocal of the mean of the sinogram data. This approach is also extended to randoms precorrected data. Based on these techniques, an accurate FIM estimate is obtained for both Poisson and randoms precorrected data. As an application, the new GELT method is used to improve resolution uniformity and achieve near-uniform image resolution in low count situations.

Propagation of errors from the sensitivity image in list mode reconstruction

List mode image reconstruction is attracting renewed attention. It eliminates the storage of empty sinogram bins. However, a single back projection of all LORs is still necessary for the pre-calculation of a sensitivity image. Since the detection sensitivity is dependent on the object attenuation and detector efficiency, it must be computed for each study. Exact computation of the sensitivity image can be a daunting task for modern scanners with huge numbers of LORs. Thus, some fast approximate calculation may be desirable. In this paper, we analyze the error propagation from the sensitivity image into the reconstructed image. The theoretical analysis is based on the fixed point condition of the list mode reconstruction. The nonnegativity constraint is modeled using the Kuhn-Tucker condition. With certain assumptions and the first-order Taylor series approximation, we derive a closed form expression for the error in the reconstructed image as a function of the error in the sensitivity image. The result shows that the error response is frequency-dependent and provides a simple expression for determining the required accuracy of the sensitivity image calculation. Computer simulations show that the theoretical results are in good agreement with the measured results.

Rapid calculation of detectability in Bayesian single photon emission computed tomography

We consider the calculation of lesion detectability using a mathematical model observer, the channelized Hotelling observer (CHO), in a signal-known-exactly/background-known-exactly detection task for single photon emission computed tomography (SPECT). We focus on SPECT images reconstructed with Bayesian maximum a posteriori methods. While model observers are designed to replace time-consuming studies using human observers, the calculation of CHO detectability is usually accomplished using a large number of sample images, which is still time consuming. We develop theoretical expressions for a measure of detectability, the signal-to-noise-ratio (SNR) of a CHO observer, that can be very rapidly evaluated. Key to our expressions are approximations to the reconstructed image covariance. In these approximations, we use methods developed in the PET literature, but modify them to reflect the different nature of attenuation and distance-dependent blur in SPECT. We validate our expressions with Monte Carlo methods. We show that reasonably accurate estimates of the SNR can be obtained at a computational expense equivalent to approximately two projection operations, and that evaluating SNR for subsequent lesion locations requires negligible additional computation.

A modified uniform Cramer-Rao bound for multiple pinhole aperture design

This paper presents a modified Uniform Cramer-Rao bound (UCRB) for studying estimator spatial resolution and variance tradeoffs. We proposed to use a resolution constraint that is imposed on mean gradient vectors of achieved estimators and derived the minimum achievable variance for any estimator satisfies this resolution constraint. This approach partially overcomes the limitations of the former UCRB approach based on a bias-gradient norm constraint. We applied this method in a feasibility study of using multiple pinhole apertures for small animal SPECT imaging applications. The SPECT system studied was based on an existing gamma camera. The achievable spatial resolution and variance tradeoffs for systems with different design parameters, such as number of pinholes and pinhole size, were studied.

Emission image reconstruction for randoms-precorrected PET allowing negative sinogram values

Most positron emission tomography (PET) emission scans are corrected for accidental coincidence (AC) events by real-time subtraction of delayed-window coincidences, leaving only the randoms-precorrected data available for image reconstruction. The real-time randoms precorrection compensates in mean for AC events but destroys the Poisson statistics. The exact log-likelihood for randoms-precorrected data is inconvenient, so practical approximations are needed for maximum likelihood or penalized-likelihood 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. With negative sinogram values, the log-likelihood functions can be nonconcave, complicating maximization; nevertheless, we develop monotonic algorithms for the new models by modifying the separable paraboloidal surrogates and the maximum-likelihood expectation-maximization (ML-EM) methods. These algorithms ascend to local maximizers of the objective function. Analysis and simulation results show that the new shifted Poisson (SP) model is nearly free of systematic bias yet keeps low variance. Despite its simpler implementation, the new SP performs comparably to the saddle-point model which has shown the best performance (as to systematic bias and variance) in randoms-precorrected PET emission reconstruction.

Analysis of lesion detectability in Bayesian emission reconstruction with nonstationary object variability

Bayesian methods based on the maximum a posteriori principle (also called penalized maximum-likelihood methods) have been developed to improve image quality in emission tomography. To explore the full potential of Bayesian reconstruction for lesion detection, we derive simplified theoretical expressions that allow fast evaluation of the detectability of a lesion in Bayesian reconstruction. This work is builded on the recent progress on the theoretical analysis of image properties of statistical reconstructions and the development of numerical observers. We explicitly model the nonstationary variation of the lesion and background without assuming that they are locally stationary. The results can be used to choose the optimum prior parameters for the maximum lesion detectability. The theoretical results are validated using Monte Carlo simulations. The comparisons show good agreement between the theoretical predictions and the Monte Carlo results. We also demonstrate that the lesion detectability can be reliably estimated using one noisy data set.

A unified noise analysis for iterative image estimation

Iterative image estimation methods have been widely used in emission tomography. Accurate estimation of the uncertainty of the reconstructed images is essential for quantitative applications. While both iteration-based noise analysis and fixed-point noise analysis have been developed, current iteration-based results are limited to only a few algorithms that have an explicit multiplicative update equation and some may not converge to the fixed-point result. This paper presents a theoretical noise analysis that is applicable to a wide range of preconditioned gradient-type algorithms. Under a certain condition, the proposed method does not require an explicit expression of the preconditioner. By deriving the fixed-point expression from the iteration-based result, we show that the proposed iteration-based noise analysis is consistent with fixed-point analysis. Examples in emission tomography and transmission tomography are shown. The results are validated using Monte Carlo simulations.

A Feasibility Study of Using Hybrid Collimation for Nuclear Environment

This paper presents a feasibility of a gamma ray imager using combined electronic and mechanical collimation methods. This detector is based on the use of a multiple pinhole collimator, a position sensitive scintillation detector with Anger logic readout. A pixelated semiconductor detector, located between the collimator and the scintillation detector, is used as a scattering detector. For gamma rays scattered in the first detector and then stopped in the second detector, an image can also be built up based on the joint probability of their passing through the collimator and falling into a broadened conical surface, defined by the detected Compton scattering event. Since these events have a much smaller angular uncertainty, they provide more information content per photon compared with using solely the mechanical or electronic collimation. Therefore, the overall image quality can be improved. This feasibility study adapted a theoretical approach, based on analysing the resolution-variance trade-off in images reconstructed using Maximum a priori (MAP) algorithm. The effect of factors such as the detector configuration, Doppler broadening and collimator configuration are studied. The results showed that the combined collimation leads to a significant improvement in image quality at energy range below 300keV. However, due to the mask penetration, the performance of such a detector configuration is worse than a standard Compton camera at above this energy.

Theoretical evaluation of the detectability of random lesions in Bayesian emission reconstruction

Detecting cancerous lesion is an important task in positron emission tomography (PET). Bayesian methods based on the maximum a posteriori principle (also called penalized maximum likelihood methods) have been developed to deal with the low signal to noise ratio in the emission data. Similar to the filter cut-off frequency in the filtered backprojection method, the prior parameters in Bayesian reconstruction control the resolution and noise trade-off and hence affect detectability of lesions in reconstructed images. Bayesian reconstructions are difficult to analyze because the resolution and noise properties are nonlinear and object-dependent. Most research has been based on Monte Carlo simulations, which are very time consuming. Building on the recent progress on the theoretical analysis of image properties of statistical reconstructions and the development of numerical observers, here we develop a theoretical approach for fast computation of lesion detectability in Bayesian reconstruction. The results can be used to choose the optimum hyperparameter for the maximum lesion detectability. New in this work is the use of theoretical expressions that explicitly model the statistical variation of the lesion and background without assuming that the object variation is (locally) stationary. The theoretical results are validated using Monte Carlo simulations. The comparisons show good agreement between the theoretical predications and the Monte Carlo results.

Experimental study of stochastic noise propagation in SPECT images reconstructed using the conjugate gradient algorithm

Thanks to an experimental study based on simulated and physical phantoms, the propagation of the stochastic noise in slices reconstructed using the conjugate gradient algorithm has been analysed versus iterations. After a first increase corresponding to the reconstruction of the signal, the noise stabilises before increasing linearly with iterations. The level of the plateau as well as the slope of the subsequent linear increase depends on the noise in the projection data.

Rapid Computation of LROC Figures of Merit Using Numerical Observers (for SPECT/PET Reconstruction)

The assessment of PET and SPECT image reconstructions by image quality metrics is typically time consuming, even if methods employing model observers and samples of reconstructions are used to replace human testing. We consider a detection task where the background is known exactly and the signal is known except for location. We develop theoretical formulae to rapidly evaluate two relevant figures of merit, the area under the LROC curve and the probability of correct localization. The formulae can accommodate different forms of model observer. The theory hinges on the fact that we are able to rapidly compute the mean and covariance of the reconstruction. For four forms of model observer, the theoretical expressions are validated by Monte Carlo studies for the case of MAP (maximum a posteriori) reconstruction. The theory method affords a 10(2) - 10(3) speedup relative to methods in which model observers are applied to sample reconstructions.

Scatter correction for positron emission mammography

In this paper we present a scatter correction method for a regularized list mode maximum likelihood reconstruction algorithm for the positron emission mammograph (PEM) that is being developed at our laboratory. The scatter events inside the object are modelled as additive Poisson random variables in the forward model of the reconstruction algorithm. The mean scatter sinogram is estimated using a Monte Carlo simulation program. With the assumption that the background activity is nearly uniform, the Monte Carlo scatter simulation only needs to run once for each PEM configuration. This saves computation time. The crystal scatters are modelled as a shift-invariant blurring in image domain because they are more localized. Thus, the useful information in the crystal scatters can be deconvolved in high-resolution reconstructions. The propagation of the noise from the estimated scatter sinogram into the reconstruction is analysed theoretically. The results provide an easy way to calculate the required number of events in the Monte Carlo scatter simulation for a given noise level in the image. The analysis is also applicable to other scatter estimation methods, provided that the covariance of the estimated scatter sinogram is available.

A non-parametric bootstrap approach for analysing the statistical properties of SPECT and PET images

Knowledge of the statistical properties of reconstructed single photon emission computed tomography (SPECT) and positron emission tomography (PET) images would be helpful for optimizing acquisition and image processing protocols. We describe a non-parametric bootstrap approach to accurately estimate the statistical properties of SPECT or PET images whatever the noise properties in the projections and the reconstruction algorithm. Using analytical simulations and real PET data, this method is shown to accurately predict the statistical properties, including the variance and covariance, of reconstructed pixel values for both linear (filtered backprojection) and non-linear (ordered subset expectation maximization) reconstruction algorithms.

Feasibility Study of Compton Scattering Enhanced Multiple Pinhole Imager for Nuclear Medicine

This paper presents a feasibility study of a Compton scattering enhanced (CSE) multiple pinhole imaging system for gamma rays with energy of 140keV or higher. This system consists of a multiple-pinhole collimator, a position sensitive scintillation detector as used in standard Gamma camera, and a Silicon pad detector array, inserted between the collimator and the scintillation detector. The problem of multiplexing, normally associated with multiple pinhole system, is reduced by using the extra information from the detected Compton scattering events. In order to compensate for the sensitivity loss, due to the low probability of detecting Compton scattered events, the proposed detector is designed to collect both Compton scattering and Non-Compton events. It has been shown that with properly selected pinhole spacing, the proposed detector design leads to an improved image quality.

A feasibility study of mutual information based setup error estimation for radiotherapy

We have investigated a fully automatic setup error estimation method that aligns DRRs (digitally reconstructed radiographs) from a three-dimensional planning computed tomography image onto two-dimensional radiographs that are acquired in a treatment room. We have chosen a MI (mutual information)-based image registration method, hoping for robustness to intensity differences between the DRRs and the radiographs. The MI-based estimator is fully automatic since it is based on the image intensity values without segmentation. Using 10 repeated scans of an anthropomorphic chest phantom in one position and two single scans in two different positions, we evaluated the performance of the proposed method and a correlation-based method against the setup error determined by fiducial marker-based method. The mean differences between the proposed method and the fiducial marker-based method were smaller than 1 mm for translational parameters and 0.8 degree for rotational parameters. The standard deviations of estimates from the proposed method due to detector noise were smaller than 0.3 mm and 0.07 degree for the translational parameters and rotational parameters, respectively.