Penalized weighted least-squares image reconstruction for positron emission tomography

Presents an image reconstruction method for positron-emission tomography (PET) based on a penalized, weighted least-squares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, the author argues statistically that a least-squares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. The author proposes a simple data-based method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive over-relaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood expectation-maximization (ML-EM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminated by the PWLS+SOR method, and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over FBP.     

Stationary positron emission tomography and its image reconstruction

Feasibility of stationary positron emission tomography (PET) using discrete detectors has been investigated by simulation studies. To enable full utilization of detector resolution, a "bank array" of detectors is proposed and an EM algorithm is adopted for image reconstruction. The bank array consists of an odd number of detector banks arranged on a circular ring with a gap equal to one half the detector width. The EM algorithm [11] is used with some modifications for reducing the quantity of computation, improving the convergence speed, and suppressing statistical noise, so as to meet the present purpose. Simulation studies involving several phantoms show that the stationary PET with the new detector array provides image quality which is good enough for clinical applications. For fast dynamic studies with low spatial resolution, the convolution-backprojection method is efficient, but for high-resolution static imaging, resolution enhancement by an iterative method is required. Problems arising in the corrections for attenuation of photons and detector sensitivity, etc., are also discussed. A totally stationary PET avoids the mechanical problems associated with accurate movement of heavy assemblies and is particularly advantageous in gated cardiac imaging or in fast dynamic studies. Elimination of a scan along the detector plane allows a quick scan in the axial direction to achieve three-dimensional imaging with a small number of detector rings.     

Performance evaluation of an iterative image reconstruction algorithm for positron emission tomography

An image reconstruction method motivated by positron emission tomography (PET) is discussed. The measurements tend to be noisy and so the reconstruction method should incorporate the statistical nature of the noise. The authors set up a discrete model to represent the physical situation and arrive at a nonlinear maximum a posteriori probability (MAP) formulation of the problem. An iterative approach which requires the solution of simple quadratic equations is proposed. The authors also present a methodology which allows them to experimentally optimize an image reconstruction method for a specific medical task and to evaluate the relative efficacy of two reconstruction methods for a particular task in a manner which meets the high standards set by the methodology of statistical hypothesis testing. The new MAP algorithm is compared to a method which maximizes likelihood and with two variants of the filtered backprojection method.     

Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms

We present two types of globally convergent relaxed ordered subsets (OS) algorithms for penalized-likelihood image reconstruction in emission tomography: modified block sequential regularized expectation-maximization (BSREM) and relaxed OS separable paraboloidal surrogates (OS-SPS). The global convergence proof of the existing BSREM (De Pierro and Yamagishi, 2001) required a few a posteriori assumptions. By modifying the scaling functions of BSREM, we are able to prove the convergence of the modified BSREM under realistic assumptions. Our modification also makes stepsize selection more convenient. In addition, we introduce relaxation into the OS-SPS algorithm (Erdo?an and Fessler, 1999) that otherwise would converge to a limit cycle. We prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed OS-SPS is a special case; main results of the proofs are from (Nedi? and Bertsekas, 2001) and (Correa and Lemaréchal, 1993). Simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms.     

Detector response restoration in image reconstruction of high resolution positron emission tomography

A mathematical method was studied to model the detector response of high spatial-resolution positron emission tomography systems consisting of close-packed small crystals, and to restore the resolution deteriorated due to crystal penetration and/or nonuniform sampling across the field-of-view (FOV). The simulated detector system had 600 bismuth germanate crystals of 3.14 mm width and 30 mm length packed on a single ring of 60 cm diameter. The space between crystals was filled up with lead (i.e., septa). Each crystal was in coincidence with 200 opposite crystals so that the FOV had a radius of 30 cm. The detector response was modeled based on the attenuating properties of the crystals and the septa, as well as the geometry of the detector system. The modeled detector-response function was used to restore the projections from the sinogram of the ring-detector system. The restored projections had a uniform sampling of 1.57 mm across the FOV. The crystal penetration and/or the nonuniform sampling were compensated in the projections. A penalized maximum-likelihood algorithm was employed to accomplish the restoration. The restored projections were then filtered and backprojected to reconstruct the image. A chest phantom with a few small circular "cold" objects ( approximately 4 mm diameter) located at the center and near the periphery of FOV was computer generated and used to test the restoration. The reconstructed images from the restored projections demonstrated resolution improvement off the FOV center, while preserving the resolution near the center.     

Block-iterative Fisher scoring algorithms for maximum penalized likelihood image reconstruction in emission tomography

This paper introduces and evaluates a block-iterative Fisher scoring (BFS) algorithm. The algorithm provides regularized estimation in tomographic models of projection data with Poisson variability. Regularization is achieved by penalized likelihood with a general quadratic penalty. Local convergence of the block-iterative algorithm is proven under conditions that do not require iteration dependent relaxation. We show that, when the algorithm converges, it converges to the unconstrained maximum penalized likelihood (MPL) solution. Simulation studies demonstrate that, with suitable choice of relaxation parameter and restriction of the algorithm to respect nonnegative constraints, the BFS algorithm provides convergence to the constrained MPL solution. Constrained BFS often attains a maximum penalized likelihood faster than other block-iterative algorithms which are designed for nonnegatively constrained penalized reconstruction.      

A multigrid expectation maximization reconstruction algorithm for positron emission tomography

The problem of reconstruction in positron emission tomography (PET) is basically estimating the number of photon pairs emitted from the source. Using the concept of the maximum-likelihood (ML) algorithm, the problem of reconstruction is reduced to determining an estimate of the emitter density that maximizes the probability of observing the actual detector count data over all possible emitter density distributions. A solution using this type of expectation maximization (EM) algorithm with a fixed grid size is severely handicapped by the slow convergence rate, the large computation time, and the nonuniform correction efficiency of each iteration, which makes the algorithm very sensitive to the image pattern. An efficient knowledge-based multigrid reconstruction algorithm based on the ML approach is presented to overcome these problems.     

Regularized image reconstruction algorithms for dual-isotope myocardial perfusion SPECT (MPS) imaging using a cross-tracer prior

In simultaneous dual-isotope myocardial perfusion SPECT (MPS) imaging, data are simultaneously acquired to determine the distributions of two radioactive isotopes. The goal of this work was to develop penalized maximum likelihood (PML) algorithms for a novel cross-tracer prior that exploits the fact that the two images reconstructed from simultaneous dual-isotope MPS projection data are perfectly registered in space. We first formulated the simultaneous dual-isotope MPS reconstruction problem as a joint estimation problem. A cross-tracer prior that couples voxel values on both images was then proposed. We developed an iterative algorithm to reconstruct the MPS images that converges to the maximum a posteriori solution for this prior based on separable surrogate functions. To accelerate the convergence, we developed a fast algorithm for the cross-tracer prior based on the complete data OS-EM (COSEM) framework. The proposed algorithm was compared qualitatively and quantitatively to a single-tracer version of the prior that did not include the cross-tracer term. Quantitative evaluations included comparisons of mean and standard deviation images as well as assessment of image fidelity using the mean square error. We also evaluated the cross tracer prior using a three-class observer study with respect to the three-class MPS diagnostic task, i.e., classifying patients as having either no defect, reversible defect, or fixed defects. For this study, a comparison with conventional ordered subsets-expectation maximization (OS-EM) reconstruction with postfiltering was performed. The comparisons to the single-tracer prior demonstrated similar resolution for areas of the image with large intensity changes and reduced noise in uniform regions. The cross-tracer prior was also superior to the single-tracer version in terms of restoring image fidelity. Results of the three-class observer study showed that the proposed cross-tracer prior and the convergent algorithms improved the image quality of dual-isotope MPS images compared to OS-EM.     

Effects of detector wobble motion on image noise in positron emission tomography

The effects of detector wobble motion on image variance is described analytically and verified by simulation studies. Simulation results in the form of variance images are presented for various phantoms, sampled under both uniform and wobble-motion conditions. It is shown that under most measurement conditions in a typical PET study, the image precision is degraded due to the nonuniform projection variance introduced by the wobble motion. Using the wobble parameters of the UBC-TRIUMF development of PETT VI, the rms noise is found to be increased by 9 percent due to wobbling.     

Accelerated iterative reconstruction for positron emission tomography based on the em algorithm for maximum likelihood estimation

The EM method that was originally developed for maximum likelihood estimation in the context of mathematical statistics may be applied to a stochastic model of positron emission tomography (PET). The result is an iterative algorithm for image reconstruction that is finding increasing use in PET, due to its attractive theoretical and practical properties. Its major disadvantage is the large amount of computation that is often required, due to the algorithm's slow rate of convergence. This paper presents an accelerated form of the EM algorithm for PET in which the changes to the image, as calculated by the standard algorithm, are multiplied at each iteration by an overrelaxation parameter. The accelerated algorithm retains two of the important practical properties of the standard algorithm, namely the selfnormalization and nonnegativity of the reconstructed images. Experimental results are presented using measured data obtained from a hexagonal detector system for PET. The likelihood function and the norm of the data residual were monitored during the iterative process. According to both of these measures, the images reconstructed at iterations 7 and 11 of the accelerated algorithm are similar to those at iterations 15 and 30 of the standard algorithm, for two different sets of data. Important theoretical properties remain to be investigated, namely the convergence of the accelerated algorithm and its performance as a maximum likelihood estimator.     

A fast reconstruction algorthm for stationary positron emission tomography based on a modified em algorithm

An efficient iterative reconstruction method for positron emission tomography (PET) is presented. The algorithm is basically an enhanced EM (expectation maximization) algorithm with improved frequency response. High-frequency components of the ratio of measured to calculated projections are extracted and are taken into account for the iterative correction of image density in such a way that the correction is performed with a uniform efficiency over the image plane and with a flat frequency response. As a result, the convergence speed is not so sensitive to the image pattern or matrix size as the standard EM algorithm, and nonuniformity of the spatial resolution is significantly improved. Nonnegativity of the reconstructed image is preserved. Simulation studies have been made assuming two PET systems: a scanning PET with ideal sampling and a stationary PET with sparse sampling. In the latter, a "bank array" of detectors is employed to improve the sampling in the object plane. The new algorithm provides satisfactory images by two or three iterations starting from a flat image in either case. The behavior of convergence is monitored by evaluating the root mean square of C(b)-1 where C(b) is the correction factor for pixel b in the EM algorithm. The value decreases rapidly and monotonically with iteration number. Although the theory is not accurate enough to assure the stability of convergence, the algorithm is promising to achieve significant saving in computation compared to the standard EM algorithm.     

Algebraic reconstruction techniques can be made computationally efficient [positron emission tomography application

Algebraic reconstruction techniques (ART) are iterative procedures for recovering objects from their projections. It is claimed that by a careful adjustment of the order in which the collected data are accessed during the reconstruction procedure and of the so-called relaxation parameters that are to be chosen in an algebraic reconstruction technique, ART can produce high-quality reconstructions with excellent computational efficiency. This is demonstrated by an example based on a particular (but realistic) medical imaging task, showing that ART can match the performance of the standard expectation-maximization approach for maximizing likelihood (from the point of view of that particular medical task), but at an order of magnitude less computational cost.     

Maximum likelihood reconstruction for emission tomography

Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density lambda = lambda(x, y, z) generates, and is to be reconstructed from, the number of counts n(*)(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate lambdainsertion mark of lambda which maximizes the probability p(n(*)|lambda) of observing the actual detector count data n(*) over all possible densities lambda. Let independent Poisson variables n(b) with unknown means lambda(b), b = 1, ..., B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ..., D with p(b,d) a one-step transition matrix, assumed known. We observe the total number n(*) = n(*)(d) of emissions in each detector unit d and want to estimate the unknown lambda = lambda(b), b = 1, ..., B. For each lambda, the observed data n(*) has probability or likelihood p(n(*)|lambda). The EM algorithm of mathematical statistics starts with an initial estimate lambda(0) and gives the following simple iterative procedure for obtaining a new estimate lambdainsertion mark(new), from an old estimate lambdainsertion mark(old), to obtain lambdainsertion mark(k), k = 1, 2, ..., lambdainsertion mark(new)(b)= lambdainsertion mark(old)(b)Sum of (n(*)p(b,d) from d=1 to D/Sum of lambdainsertion mark()old(b('))p(b('),d) from b(')=1 to B), b=1,...B.     

Grouped-coordinate ascent algorithms for penalized-likelihoodtransmission image reconstruction

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     

基于MRF二次Membrane-Plate混合自适应先验的PET图像的收敛Bayesian重建算法

对于如何抑制正电子发射成像(positron emission tomography,PET)中的噪声效果的问题,Bayesian重建或者最大化后验估计(maximum a posteriori,MAP)的方法在重建图像质量和收敛性方面具有相对于其他方法的优越性.基于Bayesian理论,本文提出了一种新的能够保持其先验能量函数凸性的马尔可夫随机场(Markov Random Fields,MRF)混合多阶二次先验(quadratic hybrid multi-order,QHM),该QHM先验综合了二次-阶(quadratic membrane,QM)先验和二次二阶(quadratic plate,QP)先验,且能够根据不同阶数的二次先验和待重建表面的性质自适应的发挥QM先验和QP先验的作用.文中还给出了使用该新的混合先验的收敛重建算法.模拟实验结果的视觉和量化比较证明了对于PET重建,该先验在抑制背景噪声和保持边缘方面具有很好的表现.     

A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography

The expectation maximization method for maximum likelihood image reconstruction in emission tomography, based on the Poisson distribution of the statistically independent components of the image and measurement vectors, is extended to a maximum aposteriori image reconstruction using a multivariate Gaussian a priori probability distribution of the image vector. The approach is equivalent to a penalized maximum likelihood estimation with a special choice of the penalty function. The expectation maximization method is applied to find the a posteriori probability maximizer. A simple iterative formula is derived for a penalty function that is a weighted sum of the squared deviations of image vector components from their a priori mean values. The method is demonstrated to be superior to pure likelihood maximization, in that the penalty function prevents the occurrence of irregular high amplitude patterns in the image with a large number of iterations (the so-called "checkerboard effect" or "noise artifact").     

Multimodality Bayesian algorithm for image reconstruction inpositron emission tomography: a tissue composition model

The use of anatomical information to improve the quality of reconstructed images in positron emission tomography (PET) has been extensively studied. A common strategy has been to include spatial smoothing within boundaries defined from the anatomical data. The authors present an alternative method for the incorporation of anatomical information into PET image reconstruction, in which they use segmented magnetic resonance (MR) images to assign tissue composition to PET image pixels. The authors model the image as a sum of activities for each tissue type, weighted by the assigned tissue composition. The reconstruction is performed as a maximum a posteriori (MAP) estimation of the activities of each tissue type. Two prior functions, defined for tissue-type activities, are considered. The algorithm is tested in realistic simulations employing a full physical model of the PET scanner     

Super-resolution in respiratory synchronized positron emission tomography

Respiratory motion is a major source of reduced quality in positron emission tomography (PET). In order to minimize its effects, the use of respiratory synchronized acquisitions, leading to gated frames, has been suggested. Such frames, however, are of low signal-to-noise ratio (SNR) as they contain reduced statistics. Super-resolution (SR) techniques make use of the motion in a sequence of images in order to improve their quality. They aim at enhancing a low-resolution image belonging to a sequence of images representing different views of the same scene. In this work, a maximum a posteriori (MAP) super-resolution algorithm has been implemented and applied to respiratory gated PET images for motion compensation. An edge preserving Huber regularization term was used to ensure convergence. Motion fields were recovered using a B-spline based elastic registration algorithm. The performance of the SR algorithm was evaluated through the use of both simulated and clinical datasets by assessing image SNR, as well as the contrast, position and extent of the different lesions. Results were compared to summing the registered synchronized frames on both simulated and clinical datasets. The super-resolution image had higher SNR (by a factor of over 4 on average) and lesion contrast (by a factor of 2) than the single respiratory synchronized frame using the same reconstruction matrix size. In comparison to the motion corrected or the motion free images a similar SNR was obtained, while improvements of up to 20% in the recovered lesion size and contrast were measured. Finally, the recovered lesion locations on the SR images were systematically closer to the true simulated lesion positions. These observations concerning the SNR, lesion contrast and size were confirmed on two clinical datasets included in the study. In conclusion, the use of SR techniques applied to respiratory motion synchronized images lead to motion compensation combined with improved image SNR and contrast, without any increase in the overall acquisition times.     

Weighted expectation maximization reconstruction algorithms for thermoacoustic tomography

Thermoacoustic tomography (TAT) is an emerging imaging technique with potential for a wide range of biomedical imaging applications. In this correspondence, we propose an infinite family of weighted expectation maximization (EM) algorithms for reconstruction of images from temporally truncated TAT measurement data. The weighted EM algorithms are equivalent mathematically to the conventional EM algorithm, but are shown to propagate data inconsistencies in different ways. Using simulated and experimental TAT measurement data, we demonstrate that suitable choices of weighted EM algorithms can effectively mitigate image artifacts that are attributable to temporal truncation of the TAT data function.