Globally convergent algorithms for maximum a posteriori transmission tomography

Prospects for quantitative computed tomography imaging in the presence of foreign metal bodies using statistical image reconstruction

X-ray computed tomography (CT) images of patients bearing metal intracavitary applicators or other metal foreign objects exhibit severe artifacts including streaks and aliasing. We have systematically evaluated via computer simulations the impact of scattered radiation, the polyenergetic spectrum, and measurement noise on the performance of three reconstruction algorithms: conventional filtered backprojection (FBP), deterministic iterative deblurring, and a new iterative algorithm, alternating minimization (AM), based on a CT detector model that includes noise, scatter, and polyenergetic spectra. Contrary to the dominant view of the literature, FBP streaking artifacts are due mostly to mismatches between FBP's simplified model of CT detector response and the physical process of signal acquisition. Artifacts on AM images are significantly mitigated as this algorithm substantially reduces detector-model mismatches. However, metal artifacts are reduced to acceptable levels only when prior knowledge of the metal object in the patient, including its pose, shape, and attenuation map, are used to constrain AM's iterations. AM image reconstruction, in combination with object-constrained CT to estimate the pose of metal objects in the patient, is a promising approach for effectively mitigating metal artifacts and making quantitative estimation of tissue attenuation coefficients a clinical possibility.

Modeling the axial extension of a transmission line source within iterative reconstruction via multiple transmission sources

Reconstruction algorithms for transmission tomography have generally assumed that the photons reaching a particular detector bin at a particular angle originate from a single point source. In this paper, we highlight several cases of extended transmission sources, in which it may be useful to approach the estimation of attenuation coefficients as a problem involving multiple transmission point sources. Examined in detail is the case of a fixed transmission line source with a fan-beam collimator. This geometry can result in attenuation images that have significant axial blur. Herein it is also shown, empirically, that extended transmission sources can result in biased estimates of the average attenuation, and an explanation is proposed. The finite axial resolution of the transmission line source configuration is modeled within iterative reconstruction using an expectation-maximization algorithm that was previously derived for estimating attenuation coefficients from single photon emission computed tomography (SPECT) emission data. The same algorithm is applicable to both problems because both can be thought of as involving multiple transmission sources. It is shown that modeling axial blur within reconstruction removes the bias in the average estimated attenuation and substantially improves the axial resolution of attenuation images.

An interior point iterative maximum-likelihood reconstruction algorithm incorporating upper and lower bounds with application to SPECT transmission imaging

The algorithm we consider here is a block-iterative (or ordered subset) version of the interior point algorithm for transmission reconstruction. Our algorithm is an interior point method because each vector of the iterative sequence [x(k)], k = 0, 1, 2, ... satisfies the constraints a(j) < x(j)k < b(j), j = 1, ..., J. Because it is a block-iterative algorithm that reconstructs the transmission attenuation map and places constraints above and below the pixel values of the reconstructed image, we call it the BITAB method. Computer simulations using the three-dimensional mathematical cardiac and torso phantom, reveal that the BITAB algorithm in conjunction with reasonably selected prior upper and lower bounds has the potential to improve the accuracy of the reconstructed attenuation coefficients from truncated fan beam transmission projections. By suitably selecting the bounds, it is possible to restrict the over estimation of coefficients outside the fully sampled region, that results from reconstructing truncated fan beam projections with iterative transmission algorithms such as the maximum-likelihood gradient type algorithm.

An EM algorithm for estimating SPECT emission and transmission parameters from emissions data only

A maximum-likelihood (ML) expectation-maximization (EM) algorithm (called EM-IntraSPECT) is presented for simultaneously estimating single photon emission computed tomography (SPECT) emission and attenuation parameters from emission data alone. The algorithm uses the activity within the patient as transmission tomography sources, with which attenuation coefficients can be estimated. For this initial study, EM-IntraSPECT was tested on computer-simulated attenuation and emission maps representing a simplified human thorax as well as on SPECT data obtained from a physical phantom. Two evaluations were performed. First, to corroborate the idea of reconstructing attenuation parameters from emission data, attenuation parameters (mu) were estimated with the emission intensities (lambda) fixed at their true values. Accurate reconstructions of attenuation parameters were obtained. Second, emission parameters lambda and attenuation parameters mu were simultaneously estimated from the emission data alone. In this case there was crosstalk between estimates of lambda and mu and final estimates of lambda and mu depended on initial values. Estimates degraded significantly as the support extended out farther from the body, and an explanation for this is proposed. In the EM-IntraSPECT reconstructed attenuation images, the lungs, spine, and soft tissue were readily distinguished and had approximately correct shapes and sizes. As compared with standard EM reconstruction assuming a fix uniform attenuation map, EM-IntraSPECT provided more uniform estimates of cardiac activity in the physical phantom study and in the simulation study with tight support, but less uniform estimates with a broad support. The new EM algorithm derived here has additional applications, including reconstructing emission and transmission projection data under a unified statistical model.

Maximum-likelihood transmission image reconstruction for overlapping transmission beams

In many transmission imaging geometries, the transmitted "beams" of photons overlap on the detector, such that a detector element may record photons that originated in different sources or source locations and thus traversed different paths through the object. Examples include systems based on scanning line sources or on multiple parallel rod sources. The overlap of these beams has been disregarded by both conventional analytical reconstruction methods as well as by previous statistical reconstruction methods. We propose a new algorithm for statistical image reconstruction of attenuation maps that explicitly accounts for overlapping beams in transmission scans. The algorithm is guaranteed to monotonically increase the objective function at each iteration. The availability of this algorithm enables the possibility of deliberately increasing the beam overlap so as to increase count rates. Simulated single photon emission tomography transmission scans based on a multiple line source array demonstrate that the proposed method yields improved resolution/noise tradeoffs relative to "conventional" reconstruction algorithms, both statistical and nonstatistical.

Simultaneous iterative reconstruction for emission and attenuation images in positron emission tomography

The quality of the attenuation correction strongly influences the outcome of the reconstructed emission scan in positron emission tomography. Usually the attenuation correction factors are calculated from the transmission and blank scan and thereafter applied during the reconstruction on the emission data. However, this is not an optimal treatment of the available data, because the emission data themselves contain additional information about attenuation: The optimal treatment must use this information for the determination of the attenuation correction factors. Therefore, our purpose is to investigate a simultaneous emission and attenuation image reconstruction using a maximum likelihood estimator, which takes the attenuation information in the emission data into account. The total maximum likelihood function for emission and transmission is used to derive a one-dimensional Newton-like algorithm for the calculation of the emission and attenuation image. Log-likelihood convergence, mean differences, and the mean of squared differences for the emission image and the attenuation correction factors of a mathematical thorax phantom were determined and compared. As a result we obtain images improved with respect to log likelihood in all cases and with respect to our figures of merit in most cases. We conclude that the simultaneous reconstruction can improve the performance of image reconstruction.

High resolution X-ray computed tomography: an emerging tool for small animal cancer research

Dedicated high-resolution small animal imaging systems have recently emerged as important new tools for cancer research. These new imaging systems permit researchers to noninvasively screen animals for mutations or pathologies and to monitor disease progression and response to therapy. One imaging modality, X-ray microcomputed tomography (microCT) shows promise as a cost-effective means for detecting and characterizing soft-tissue structures, skeletal abnormalities, and tumors in live animals. MicroCT systems provide high-resolution images (typically 50 microns or less), rapid data acquisition (typically 5 to 30 minutes), excellent sensitivity to skeletal tissue and good sensitivity to soft tissue, particularly when contrast-enhancing media are employed. The development of microCT technology for small animal imaging is reviewed, and key considerations for designing small animal microCT imaging protocols are summarized. Recent studies on mouse prostate, lung and bone tumor models are overviewed.

Ordered subsets algorithms for transmission tomography

The ordered subsets EM (OSEM) algorithm has enjoyed considerable interest for emission image reconstruction due to its acceleration of the original EM algorithm and ease of programming. The transmission EM reconstruction algorithm converges very slowly and is not used in practice. In this paper, we introduce a simultaneous update algorithm called separable paraboloidal surrogates (SPS) that converges much faster than the transmission EM algorithm. Furthermore, unlike the 'convex algorithm' for transmission tomography, the proposed algorithm is monotonic even with nonzero background counts. We demonstrate that the ordered subsets principle can also be applied to the new SPS algorithm for transmission tomography to accelerate 'convergence', albeit with similar sacrifice of global convergence properties as for OSEM. We implemented and evaluated this ordered subsets transmission (OSTR) algorithm. The results indicate that the OSTR algorithm speeds up the increase in the objective function by roughly the number of subsets in the early iterates when compared to the ordinary SPS algorithm. We compute mean square errors and segmentation errors for different methods and show that OSTR is superior to OSEM applied to the logarithm of the transmission data. However, penalized-likelihood reconstructions yield the best quality images among all other methods tested.

Monotonic algorithms for transmission tomography

We present a framework for designing fast and monotonic algorithms for transmission tomography penalized-likelihood image reconstruction. The new algorithms are based on paraboloidal surrogate functions for the log likelihood. Due to the form of the log-likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity. Unlike previous methods, the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events, such as scatter and random coincidences. The gradient and the curvature of the likelihood terms are evaluated only once per iteration. Since the problem is simplified at each iteration, the CPU time is less than that of current algorithms which directly minimize the objective, yet the convergence rate is comparable. The simplicity, monotonicity, and speed of the new algorithms are quite attractive. The convergence rates of the algorithms are demonstrated using real and simulated PET transmission scans.

Iterative reconstruction for attenuation correction in positron emission tomography: maximum likelihood for transmission and blank scan

The quality of the attenuation correction strongly influences the outcome of the reconstructed emission scan in positron emission tomography. The calculation of the attenuation correction factors must take into account the Poisson nature of the radioactive decay process, because-for a reasonable scan duration-the transmission measurements contain lines of response with low count numbers in the case of large attenuation factors. Our purpose in this study is to investigate a maximum likelihood estimator for attenuation correction factor calculation in positron emission tomography, which incorporates the Poisson nature of the radioactive decay into transmission and blank measurement. Therefore, the correct maximum likelihood function is used to derive two estimators for the calculation of the attenuation coefficient image and the corresponding attenuation correction factors depending on the measured blank and transmission data. Log likelihood convergence, mean differences, and the mean of squared differences for the attenuation correction factors of a mathematical thorax phantom were determined and compared. The algorithms yield adequate attenuation correction factors, however, the algorithm taking the noise in the blank scan into account can perform better for noisy blank scans. We conclude that maximum likelihood-including blank likelihood-is advantageous to reconstruct attenuation correction factors for low statistic blank and good statistic transmission data. For normal blank and transmission statistics the implementation of the statistical nature of the blank is not mandatory.

Accelerated iterative transmission CT reconstruction using an ordered subsets convex algorithm

Iterative maximum likelihood (ML) transmission computed tomography algorithms have distinct advantages over Fourier-based reconstruction, but unfortunately require increased computation time. The convex algorithm [1] is a relatively fast iterative ML algorithm but it is nevertheless too slow for many applications. Therefore, an acceleration of this algorithm by using ordered subsets of projections is proposed [ordered subsets convex algorithm (OSC)]. OSC applies the convex algorithm sequentially to subsets of projections. OSC was compared with the convex algorithm using simulated and physical thorax phantom data. Reconstructions were performed for OSC using eight and 16 subsets (eight and four projections/subset, respectively). Global errors, image noise, contrast recovery, and likelihood increase were calculated. Results show that OSC is faster than the convex algorithm, the amount of acceleration being approximately proportional to the number of subsets in OSC, and it causes only a slight increase of noise and global errors in the reconstructions. Images and image profiles of the reconstructions were in good agreement. In conclusion, OSC and the convex algorithm result in similar image quality but OSC is more than an order of magnitude faster.

An iterative algorithm for X-ray CT fluoroscopy

X-ray computed tomography fluoroscopy (CTF) enables image guidance of interventions, synchronization of scanning with contrast bolus arrival, and motion analysis. However, filtered backprojection (FB), the current method for CTF image reconstruction, is subject to motion and metal artifacts from implants, needles, or other surgical instruments. Reduced target lesion conspicuity may result from increased image noise associated with reduced tube current. In this report, we adapt the row-action expectation-maximization (EM) algorithm for CTF. Because time-dependent variation in images is localized during CTF, the row-action EM-like algorithm allows rapid convergence. More importantly, this iterative CTF algorithm has fewer metal artifacts and better low-contrast performance than FB.

Choice of initial conditions in the ML reconstruction of fan-beam transmission with truncated projection data

We investigate the effects of initial conditions in the iterative maximum-likelihood (ML) reconstruction of fan-beam transmission projection data with truncation. In an iterative ML reconstruction, the estimate of the transmission reconstructed image in the previous iteration is multiplied by some factors to obtain the current estimate. Normally, a flat initial condition (FIC) or an image with equal positive pixel values is used as initial condition for an ML reconstruction. Usage of FIC has also been perceived as a way of preventing any bias on the reconstruction which may have come from the initial condition. When projection data have truncation, we show that using an FIC in an ML iterative reconstruction can introduce a bias to the reconstruction inside the densely sampled region (DSR), whose projection data have no truncation at any angle. To reduce this bias, we propose to use the largest right singular vector (LRSV) of the system matrix as the initial condition, and demonstrate that the bias can be reduced with the LRSV. When data truncation is reduced, the LRSV approaches the FIC. This result does not contradict to the use of FIC when projection data are not truncated. We also demonstrate that the reconstructed transmission image using LRSV as initial condition provides a more accurate attenuation coefficient distribution than that using FIC. However, the improvement is mostly in the area outside the DSR.

A clinical perspective of accelerated statistical reconstruction

Although the potential benefits of maximum likelihood reconstruction have been recognised for many years, the technique has only recently found widespread popularity in clinical practice. Factors which have contributed to the wider acceptance include improved models for the emission process, better understanding of the properties of the algorithm and, not least, the practicality of application with the development of acceleration schemes and the improved speed of computers. The objective in this article is to present a framework for applying maximum likelihood reconstruction for a wide range of clinically based problems. The article draws particularly on the experience of the three authors in applying an acceleration scheme involving use of ordered subsets to a range of applications. The potential advantages of statistical reconstruction techniques include: (a) the ability to better model the emission and detection process, in order to make the reconstruction converge to a quantitative image, (b) the inclusion of a statistical noise model which results in better noise characteristics, and (c) the possibility to incorporate prior knowledge about the distribution being imaged. The great flexibility in adapting the reconstruction for a specific model results in these techniques having wide applicability to problems in clinical nuclear medicine.

Grouped-coordinate ascent algorithms for penalized-likelihood transmission image reconstruction

This paper presents a new class of algorithms for penalized-likelihood reconstruction of attenuation maps from low-count transmission scans. We derive the algorithms by applying to the transmission log-likelihood a version of the convexity technique developed by De Pierro for emission tomography. The new class includes the single-coordinate ascent (SCA) algorithm and Lange's convex algorithm for transmission tomography as special cases. The new grouped-coordinate ascent (GCA) algorithms in the class overcome several limitations associated with previous algorithms. 1) Fewer exponentiations are required than in the transmission maximum likelihood-expectation maximization (ML-EM) algorithm or in the SCA algorithm. 2) The algorithms intrinsically accommodate nonnegativity constraints, unlike many gradient-based methods. 3) The algorithms are easily parallelizable, unlike the SCA algorithm and perhaps line-search algorithms. We show that the GCA algorithms converge faster than the SCA algorithm, even on conventional workstations. An example from a low-count positron emission tomography (PET) transmission scan illustrates the method.

EM algorithms without missing data

Most problems in computational statistics involve optimization of an objective function such as a loglikelihood, a sum of squares, or a log posterior function. The EM algorithm is one of the most effective algorithms for maximization because it iteratively transfers maximization from a complex function to a simple, surrogate function. This theoretical perspective clarifies the operation of the EM algorithm and suggests novel generalizations. Besides simplifying maximization, optimization transfer usually leads to highly stable algorithms with well-understood local and global convergence properties. Although convergence can be excruciatingly slow, various devices exist for accelerating it. Beginning with the EM algorithm, we review in this paper several optimization transfer algorithms of substantial utility in medical statistics.

PET regularization by envelope guided conjugate gradients

The authors propose a new way to iteratively solve large scale ill-posed problems and in particular the image reconstruction problem in positron emission tomography by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain iteratively approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows the authors to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired solution can be reconstructed with a small number of iterations.

A unified approach to statistical tomography using coordinate descent optimization

Over the past years there has been considerable interest in statistically optimal reconstruction of cross-sectional images from tomographic data. In particular, a variety of such algorithms have been proposed for maximum a posteriori (MAP) reconstruction from emission tomographic data. While MAP estimation requires the solution of an optimization problem, most existing reconstruction algorithms take an indirect approach based on the expectation maximization (EM) algorithm. We propose a new approach to statistically optimal image reconstruction based on direct optimization of the MAP criterion. The key to this direct optimization approach is greedy pixel-wise computations known as iterative coordinate decent (ICD). We propose a novel method for computing the ICD updates, which we call ICD/Newton-Raphson. We show that ICD/Newton-Raphson requires approximately the same amount of computation per iteration as EM-based approaches, but the new method converges much more rapidly (in our experiments, typically five to ten iterations). Other advantages of the ICD/Newton-Raphson method are that it is easily applied to MAP estimation of transmission tomograms, and typical convex constraints, such as positivity, are easily incorporated.

Hybrid Poisson/polynomial objective functions for tomographic image reconstruction from transmission scans

This paper describes rapidly converging algorithms for computing attenuation maps from Poisson transmission measurements using penalized-likelihood objective functions. We demonstrate that an under-relaxed cyclic coordinate-ascent algorithm converges faster than the convex algorithm of Lange (see ibid., vol.4, no.10, p.1430-1438, 1995), which in turn converges faster than the expectation-maximization (EM) algorithm for transmission tomography. To further reduce computation, one could replace the log-likelihood objective with a quadratic approximation. However, we show with simulations and analysis that the quadratic objective function leads to biased estimates for low-count measurements. Therefore we introduce hybrid Poisson/polynomial objective functions that use the exact Poisson log-likelihood for detector measurements with low counts, but use computationally efficient quadratic or cubic approximations for the high-count detector measurements. We demonstrate that the hybrid objective functions reduce computation time without increasing estimation bias.

On the convergence of an EM-type algorithm for penalized likelihood estimation in emission tomography

Recently, we proposed an extension of the expectation maximization (EM) algorithm that was able to handle regularization terms in a natural way. Although very general, convergence proofs were not valid for many possibly useful regularizations. We present here a simple convergence result that is valid assuming only continuous differentiability of the penalty term and can be also extended to other methods for penalized likelihood estimation in tomography.