Maximum likelihood, least squares, and penalized least squares for PET

Compressed-sensing-inspired reconstruction algorithms in low-dose computed tomography: A review

BACKGROUND

Optimization of the dose the patient receives during scanning is an important problem in modern medical X-ray computed tomography (CT). One of the basic ways to its solution is to reduce the number of views. Compressed sensing theory helped promote the development of a new class of effective reconstruction algorithms for limited data CT. These compressed-sensing-inspired (CSI) algorithms optimize the Lp (0 ≤ p ≤ 1) norm of images and can accurately reconstruct CT tomograms from a very few views. The paper presents a review of the CSI algorithms and discusses prospects for their further use in commercial low-dose CT.

METHODS

Many literature references with the CSI algorithms have been were searched. To structure the material collected the author gives a classification framework within which he describes Lp regularization methods, the basic CSI algorithms that are used most often in few-view CT, and some of their derivatives. Lots of examples are provided to illustrate the use of the CSI algorithms in few-view and low-dose CT.

RESULTS

A list of the CSI algorithms is compiled from the literature search. For better demonstrativeness they are summarized in a table. The inference is done that already today some of the algorithms are capable of reconstruction from 20 to 30 views with acceptable quality and dose reduction by a factor of 10.

DISCUSSION

In conclusion the author discusses how soon the CSI reconstruction algorithms can be introduced in the practice of medical diagnosis and used in commercial CT scanners.

Anatomy-guided PET reconstruction usingl1bowsher prior

Advances in simultaneous positron emission tomography/magnetic resonance imaging (PET/MRI) technology have led to an active investigation of the anatomy-guided regularized PET image reconstruction algorithm based on MR images. Among the various priors proposed for anatomy-guided regularized PET image reconstruction, Bowsher's method based on second-order smoothing priors sometimes suffers from over-smoothing of detailed structures. Therefore, in this study, we propose a Bowsher prior based on thel1-norm and an iteratively reweighting scheme to overcome the limitation of the original Bowsher method. In addition, we have derived a closed solution for iterative image reconstruction based on this non-smooth prior. A comparison study between the originall2and proposedl1Bowsher priors was conducted using computer simulation and real human data. In the simulation and real data application, small lesions with abnormal PET uptake were better detected by the proposedl1Bowsher prior methods than the original Bowsher prior. The originall2Bowsher leads to a decreased PET intensity in small lesions when there is no clear separation between the lesions and surrounding tissue in the anatomical prior. However, the proposedl1Bowsher prior methods showed better contrast between the tumors and surrounding tissues owing to the intrinsic edge-preserving property of the prior which is attributed to the sparseness induced byl1-norm, especially in the iterative reweighting scheme. Besides, the proposed methods demonstrated lower bias and less hyper-parameter dependency on PET intensity estimation in the regions with matched anatomical boundaries in PET and MRI. Therefore, these methods will be useful for improving the PET image quality based on the anatomical side information.

Characterization of tissue-specific pre-log Bayesian CT reconstruction by texture-dose relationship

PURPOSE

Tissue textures have been recognized as biomarkers for various clinical tasks. In computed tomography (CT) image reconstruction, it is important but challenging to preserve the texture when lowering x-ray exposure from full- toward low-/ultra-low dose level. Therefore, this paper aims to explore the texture-dose relationship within one tissue-specific pre-log Bayesian CT reconstruction algorithm.

METHODS

To enhance the texture in ultra-low dose CT (ULdCT) reconstruction, this paper presents a Bayesian type algorithm. A shifted Poisson model is adapted to describe the statistical properties of pre-log data, and a tissue-specific Markov random field prior (MRFt) is used to incorporate tissue texture from previous full-dose CT, thus called SP-MRFt algorithm. Utilizing the SP-MRFt algorithm, we investigated tissue texture degradation as a function of x-ray dose levels from full dose (100 mAs/120 kVp) to ultralow dose (1 mAs/120 kVp) by using quantitative texture-based evaluation metrics.

RESULTS

Experimental results show the SP-MRFt algorithm outperforms conventional filtered back projection (FBP) and post-log domain penalized weighted least square MRFt (PWLS-MRFt) in terms of noise suppression and texture preservation. Comparable results are also obtained with shifted Poisson model with 7 × 7 Huber MRF weights (SP-Huber7). The investigation on texture-dose relationship shows that the quantified texture measures drop monotonically as dose level decreases, and interestingly a turning point is observed on the texture-dose response curve.

CONCLUSIONS

This important observation implies that there exists a minimum dose level, at which a given CT scanner (hardware configuration and image reconstruction software) can achieve without compromising clinical tasks. Moreover, the experiment results show that the variance of electronic noise has higher impact than the mean to the texture-dose relationship.

Constructing a tissue-specific texture prior by machine learning from previous full-dose scan for Bayesian reconstruction of current ultralow-dose CT images

Purpose: Bayesian theory provides a sound framework for ultralow-dose computed tomography (ULdCT) image reconstruction with two terms for modeling the data statistical property and incorporating a priori knowledge for the image that is to be reconstructed. We investigate the feasibility of using a machine learning (ML) strategy, particularly the convolutional neural network (CNN), to construct a tissue-specific texture prior from previous full-dose computed tomography. Approach: Our study constructs four tissue-specific texture priors, corresponding with lung, bone, fat, and muscle, and integrates the prior with the prelog shift Poisson (SP) data property for Bayesian reconstruction of ULdCT images. The Bayesian reconstruction was implemented by an algorithm called SP-CNN-T and compared with our previous Markov random field (MRF)-based tissue-specific texture prior algorithm called SP-MRF-T. Results: In addition to conventional quantitative measures, mean squared error and peak signal-to-noise ratio, structure similarity index, feature similarity, and texture Haralick features were used to measure the performance difference between SP-CNN-T and SP-MRF-T algorithms in terms of the structure and tissue texture preservation, demonstrating the feasibility and the potential of the investigated ML approach. Conclusions: Both training performance and image reconstruction results showed the feasibility of constructing CNN texture prior model and the potential of improving the structure preservation of the nodule comparing to our previous regional tissue-specific MRF texture prior model.

3D Tensor Based Nonlocal Low Rank Approximation in Dynamic PET Reconstruction

Reconstructing images from multi-view projections is a crucial task both in the computer vision community and in the medical imaging community, and dynamic positron emission tomography (PET) is no exception. Unfortunately, image quality is inevitably degraded by the limitations of photon emissions and the trade-off between temporal and spatial resolution. In this paper, we develop a novel tensor based nonlocal low-rank framework for dynamic PET reconstruction. Spatial structures are effectively enhanced not only by nonlocal and sparse features, but momentarily by tensor-formed low-rank approximations in the temporal realm. Moreover, the total variation is well regularized as a complementation for denoising. These regularizations are efficiently combined into a Poisson PET model and jointly solved by distributed optimization. The experiments demonstrated in this paper validate the excellent performance of the proposed method in dynamic PET.

Algorithms for joint activity-attenuation estimation from positron emission tomography scatter

Background

Attenuation correction in positron emission tomography remains challenging in the absence of measured transmission data. Scattered emission data may contribute missing information, but quantitative scatter-to-attenuation (S2A) reconstruction needs to input the reconstructed activity image. Here, we study S2A reconstruction as a building block for joint estimation of activity and attenuation.

Methods

We study two S2A reconstruction algorithms, maximum-likelihood expectation maximization (MLEM) with one-step-late attenuation (MLEM-OSL) and a maximum-likelihood gradient ascent (MLGA). We study theoretical properties of these algorithms with a focus on convergence and convergence speed and compare convergence speeds and the impact of object size in simulations using different spatial scale factors. Then, we propose joint estimation of activity and attenuation from scattered and nonscattered (true) emission data, combining MLEM-OSL or MLGA with scatter-MLEM as well as trues-MLEM and the maximum-likelihood transmission (MLTR) algorithm.

Results

Shortcomings of MLEM-OSL inhibit convergence to the true solution with high attenuation; these shortcomings are related to the linearization of a nonlinear measurement equation and can be linked to a new numerical criterion allowing geometrical interpretations in terms of low and high attenuation. Comparisons using simulated data confirm that while MLGA converges largely independent of the attenuation scale, MLEM-OSL converges if low-attenuation data dominate, but not with high attenuation. Convergence of MLEM-OSL can be improved by isolating data satisfying the aforementioned low-attenuation criterion. In joint estimation of activity and attenuation, scattered data helps avoid local minima that nonscattered data alone cannot. Combining MLEM-OSL with trues-MLEM may be sufficient for low-attenuation objects, while MLGA, scatter-MLEM, and MLTR may additionally be needed with higher attenuation.

Conclusions

The performance of S2A algorithms depends on spatial scales. MLGA provides lower computational complexity and convergence in more diverse setups than MLEM-OSL. Finally, scattered data may provide additional information to joint estimation of activity and attenuation through S2A reconstruction.

Optimization-Based Image Reconstruction From Low-Count, List-Mode TOF-PET Data

OBJECTIVE

We investigate an optimization-based approach to image reconstruction from list-mode data in digital time-of-flight (TOF) positron emission tomography (PET) imaging.

METHOD

In the study, the image to be reconstructed is designed as a solution to a convex, non-smooth optimization program, and a primal-dual algorithm is developed for image reconstruction by solving the optimization program. The algorithm is first applied to list-mode TOF-PET data of a typical count level from physical phantoms and a human subject. Subsequently, we explore the algorithm's potential for image reconstruction in low-dose and/or fast TOF-PET imaging of practical interest by applying the algorithm to list-mode TOF-PET data of different, low-count levels from the same physical phantoms and human subject.

RESULTS

Visual inspection and quantitative-metric analysis reveal that the optimization reconstruction approach investigated can yield images with enhanced spatial and contrast resolution, suppressed image noise, and increased axial volume coverage over the reference images obtained with a standard clinical reconstruction algorithm especially for low-dose TOF-PET data.

SIGNIFICANCE

The optimization-based reconstruction approach can be exploited for yielding insights into potential quality upper bound of reconstructed images in, and design of scanning protocols of, TOF-PET imaging of practical significance.

Evaluation of tomographic image quality of extended and conventional parallel hole collimators using maximum likelihood expectation maximization algorithm by Monte Carlo simulations

OBJECTIVE

One of the major problems associated with parallel hole collimators (PCs) is the trade-off between their resolution and sensitivity. To solve this problem, a novel PC - namely, extended parallel hole collimator (EPC) - was proposed, in which particular trapezoidal denticles were increased upon septa on the side of the detector.

MATERIALS AND METHODS

In this study, an EPC was designed and its performance was compared with that of two PCs, PC35 and PC41, with a hole size of 1.5 mm and hole lengths of 35 and 41 mm, respectively. The Monte Carlo method was used to calculate the important parameters such as resolution, sensitivity, scattering, and penetration ratio. A Jaszczak phantom was also simulated to evaluate the resolution and contrast of tomographic images, which were produced by the EPC6, PC35, and PC41 using the Monte Carlo N-particle version 5 code, and tomographic images were reconstructed by using maximum likelihood expectation maximization algorithm.

RESULTS

Sensitivity of the EPC6 was increased by 20.3% in comparison with that of the PC41 at the identical spatial resolution and full-width at tenth of maximum here. Moreover, the penetration and scattering ratio of the EPC6 was 1.2% less than that of the PC41. The simulated phantom images show that the EPC6 increases contrast-resolution and contrast-to-noise ratio compared with those of PC41 and PC35.

CONCLUSION

When compared with PC41 and PC35, EPC6 improved trade-off between resolution and sensitivity, reduced penetrating and scattering ratios, and produced images with higher quality. EPC6 can be used to increase detectability of more details in nuclear medicine images.

Three-Dimensional Reconstruction of Three-Way FRET Microscopy Improves Imaging of Multiple Protein-Protein Interactions

Fluorescence resonance energy transfer (FRET) microscopy is a powerful tool for imaging the interactions between fluorescently tagged proteins in two-dimensions. For FRET microscopy to reach its full potential, it must be able to image more than one pair of interacting molecules and image degradation from out-of-focus light must be reduced. Here we extend our previous work on the application of maximum likelihood methods to the 3-dimensional reconstruction of 3-way FRET interactions within cells. We validated the new method (3D-3Way FRET) by simulation and fluorescent protein test constructs expressed in cells. In addition, we improved the computational methods to create a 2-log reduction in computation time over our previous method (3DFSR). We applied 3D-3Way FRET to image the 3D subcellular distributions of HIV Gag assembly. Gag fused to three different FPs (CFP, YFP, and RFP), assembled into viral-like particles and created punctate FRET signals that become visible on the cell surface when 3D-3Way FRET was applied to the data. Control experiments in which YFP-Gag, RFP-Gag and free CFP were expressed, demonstrated localized FRET between YFP and RFP at sites of viral assembly that were not associated with CFP. 3D-3Way FRET provides the first approach for quantifying multiple FRET interactions while improving the 3D resolution of FRET microscopy data without introducing bias into the reconstructed estimates. This method should allow improvement of widefield, confocal and superresolution FRET microscopy data.

Sparse/Low Rank Constrained Reconstruction for Dynamic PET Imaging

In dynamic Positron Emission Tomography (PET), an estimate of the radio activity concentration is obtained from a series of frames of sinogram data taken at ranging in duration from 10 seconds to minutes under some criteria. So far, all the well-known reconstruction algorithms require known data statistical properties. It limits the speed of data acquisition, besides, it is unable to afford the separated information about the structure and the variation of shape and rate of metabolism which play a major role in improving the visualization of contrast for some requirement of the diagnosing in application. This paper presents a novel low rank-based activity map reconstruction scheme from emission sinograms of dynamic PET, termed as SLCR representing Sparse/Low Rank Constrained Reconstruction for Dynamic PET Imaging. In this method, the stationary background is formulated as a low rank component while variations between successive frames are abstracted to the sparse. The resulting nuclear norm and l₁ norm related minimization problem can also be efficiently solved by many recently developed numerical methods. In this paper, the linearized alternating direction method is applied. The effectiveness of the proposed scheme is illustrated on three data sets.

Quantitative comparison of OSEM and penalized likelihood image reconstruction using relative difference penalties for clinical PET

Ordered subset expectation maximization (OSEM) is the most widely used algorithm for clinical PET image reconstruction. OSEM is usually stopped early and post-filtered to control image noise and does not necessarily achieve optimal quantitation accuracy. As an alternative to OSEM, we have recently implemented a penalized likelihood (PL) image reconstruction algorithm for clinical PET using the relative difference penalty with the aim of improving quantitation accuracy without compromising visual image quality. Preliminary clinical studies have demonstrated visual image quality including lesion conspicuity in images reconstructed by the PL algorithm is better than or at least as good as that in OSEM images. In this paper we evaluate lesion quantitation accuracy of the PL algorithm with the relative difference penalty compared to OSEM by using various data sets including phantom data acquired with an anthropomorphic torso phantom, an extended oval phantom and the NEMA image quality phantom; clinical data; and hybrid clinical data generated by adding simulated lesion data to clinical data. We focus on mean standardized uptake values and compare them for PL and OSEM using both time-of-flight (TOF) and non-TOF data. The results demonstrate improvements of PL in lesion quantitation accuracy compared to OSEM with a particular improvement in cold background regions such as lungs.

Generalized EM-type reconstruction algorithms for emission tomography

We provide a general form for many reconstruction estimators of emission tomography. These estimators include Shepp and Vardi's maximum likelihood (ML) estimator, the quadratic weighted least squares (WLS) estimator, Anderson's WLS estimator, and Liu and Wang's multi-objective estimator, and others. We derive a generic update rule by constructing a surrogate function. This work is inspired by the ML-EM (EM, expectation maximization), where the latter naturally arises as a special case. A regularization with a specific form can also be incorporated by De Pierro's trick. We provide a general and quite different convergence proof compared with the proofs of the ML-EM and De Pierro. Theoretical analysis shows that the proposed algorithm monotonically decreases the cost function and automatically meets nonnegativity constraints. We have introduced a mechanism to provide monotonic, self-constraining, and convergent algorithms, from which some interesting existing and new algorithms can be derived. Simulation results illustrate the behavior of these algorithms in term of image quality and resolution-noise tradeoff.

MULTILAYER NONNEGATIVE MATRIX FACTORIZATION USING PROJECTED GRADIENT APPROACHES

The most popular algorithms for Nonnegative Matrix Factorization (NMF) belong to a class of multiplicative Lee-Seung algorithms which have usually relative low complexity but are characterized by slow-convergence and the risk of getting stuck to in local minima. In this paper, we present and compare the performance of additive algorithms based on three different variations of a projected gradient approach. Additionally, we discuss a novel multilayer approach to NMF algorithms combined with multi-start initializations procedure, which in general, considerably improves the performance of all the NMF algorithms. We demonstrate that this approach (the multilayer system with projected gradient algorithms) can usually give much better performance than standard multiplicative algorithms, especially, if data are ill-conditioned, badly-scaled, and/or a number of observations is only slightly greater than a number of nonnegative hidden components. Our new implementations of NMF are demonstrated with the simulations performed for Blind Source Separation (BSS) data.

Tomographic reconstruction of gated data acquisition using DFT basis functions

In image reconstruction gated acquisition is often used in order to deal with blur caused by organ motion in the resulting images. However, this is achieved almost inevitably at the expense of reduced signal-to-noise ratio in the acquired data. In this work, we propose a reconstruction procedure for gated images based upon use of discrete Fourier transform (DFT) basis functions, wherein the temporal activity at each spatial location is regulated by a Fourier representation. The gated images are then reconstructed through determination of the coefficients of the Fourier representation. We demonstrate this approach in the context of single photon emission computed tomography (SPECT) for cardiac imaging, which is often hampered by the increased noise due to gating and other degrading factors. We explore two different reconstruction algorithms, one is a penalized least-square approach and the other is a maximum a posteriori approach. In our experiments, we conducted a quantitative evaluation of the proposed approach using Monte Carlo simulated SPECT imaging. The results demonstrate that use of DFT-basis functions in gated imaging can improve the accuracy of the reconstruction. As a preliminary demonstration, we also tested this approach on a set of clinical acquisition.

Image reconstruction for PET/CT scanners: past achievements and future challenges

PET is a medical imaging modality with proven clinical value for disease diagnosis and treatment monitoring. The integration of PET and CT on modern scanners provides a synergy of the two imaging modalities. Through different mathematical algorithms, PET data can be reconstructed into the spatial distribution of the injected radiotracer. With dynamic imaging, kinetic parameters of specific biological processes can also be determined. Numerous efforts have been devoted to the development of PET image reconstruction methods over the last four decades, encompassing analytic and iterative reconstruction methods. This article provides an overview of the commonly used methods. Current challenges in PET image reconstruction include more accurate quantitation, TOF imaging, system modeling, motion correction and dynamic reconstruction. Advances in these aspects could enhance the use of PET/CT imaging in patient care and in clinical research studies of pathophysiology and therapeutic interventions.

Weighted iterative reconstruction for magnetic particle imaging

Magnetic particle imaging (MPI) is a new imaging technique capable of imaging the distribution of superparamagnetic particles at high spatial and temporal resolution. For the reconstruction of the particle distribution, a system of linear equations has to be solved. The mathematical solution to this linear system can be obtained using a least-squares approach. In this paper, it is shown that the quality of the least-squares solution can be improved by incorporating a weighting matrix using the reciprocal of the matrix-row energy as weights. A further benefit of this weighting is that iterative algorithms, such as the conjugate gradient method, converge rapidly yielding the same image quality as obtained by singular value decomposition in only a few iterations. Thus, the weighting strategy in combination with the conjugate gradient method improves the image quality and substantially shortens the reconstruction time. The performance of weighting strategy and reconstruction algorithms is assessed with experimental data of a 2D MPI scanner.

Exact feature extraction using finite rate of innovation principles with an application to image super-resolution

The accurate registration of multiview images is of central importance in many advanced image processing applications. Image super-resolution, for example, is a typical application where the quality of the super-resolved image is degrading as registration errors increase. Popular registration methods are often based on features extracted from the acquired images. The accuracy of the registration is in this case directly related to the number of extracted features and to the precision at which the features are located: images are best registered when many features are found with a good precision. However, in low-resolution images, only a few features can be extracted and often with a poor precision. By taking a sampling perspective, we propose in this paper new methods for extracting features in low-resolution images in order to develop efficient registration techniques. We consider, in particular, the sampling theory of signals with finite rate of innovation and show that some features of interest for registration can be retrieved perfectly in this framework, thus allowing an exact registration. We also demonstrate through simulations that the sampling model which enables the use of finite rate of innovation principles is well suited for modeling the acquisition of images by a camera. Simulations of image registration and image super-resolution of artificially sampled images are first presented, analyzed and compared to traditional techniques. We finally present favorable experimental results of super-resolution of real images acquired by a digital camera available on the market.

Noniterative sequential weighted least squares algorithm for positron emission tomography reconstruction

This paper proposes a new sequential weighted least squares (SWLS) method for positron emission tomography (PET) reconstruction. The SWLS algorithm is noniterative and can be considered as equivalent to the penalized WLS (PWLS) method under certain initial conditions. However, a full implementation of SWLS is computationally intensive. To overcome this problem, we propose a simplified SWLS as a reasonable alternative to the SWLS. The performance of this SWLS method is evaluated in experiments using both simulated and clinical data. The results show that the method can be advantageously compared with the original SWLS both in computation time and reconstruction quality.

Three-dimensional FRET reconstruction microscopy for analysis of dynamic molecular interactions in live cells

Analysis of cellular pathways requires concentration measurements of dynamically interacting molecules within the three-dimensional (3D) space of single living cells. Förster resonance energy transfer (FRET) microscopy from widefield, from confocal, and potentially from superresolution microscopes can access this information; however, these measurements are distorted by the inherent 3D blurring of optical imaging, spectral overlap of fluorophores, and detection noise. We propose a mathematical model of these processes and demonstrate, through simulation, how these distortions limit the dynamic range and sensitivity of conventional FRET microscopy. Using this model, we devise and validate a new approach (called 3D-FRET stoichiometry reconstruction, 3DFSR) for reconstructing 3D distributions of bound and free fluorescent molecules. Previous attempts to reconstruct 3D-FRET data relied on sequential spectral unmixing and deconvolution, a process that corrupts the detection statistics. We demonstrate that 3DFSR is superior to these approaches since it simultaneously models spectral mixing, optical blurring, and detection noise. To achieve the full potential of this technique, we developed an instrument capable of acquiring 3D-FRET data rapidly and sensitively from single living cells. Compared with conventional FRET microscopy, our 3D-FRET reconstruction technique and new instrumentation provides orders of magnitude gains in both sensitivity and accuracy wherein sustained high-resolution four-dimensional (x,y,z,t) imaging of molecular interactions inside living cells was achieved. These results verify previous observations that Cdc42 signaling is localized to the advancing margins of forming phagosomes in macrophages.

Advances in PET Image Reconstruction

Until recently, the most widely used methods for image reconstruction were direct analytic techniques. Iterative techniques, although computationally much more intensive, produce improved images (principally arising from more accurate modeling of the acquired projection data), enabling these techniques to replace analytic techniques not only in research settings but also in the clinic. This article offers an overview of image reconstruction theory and algorithms for PET, with a particular emphasis on statistical iterative reconstruction techniques. Future directions for image reconstruction in PET are considered, which concern mainly improving the modeling of the data acquisition process and task-specific specification of the parameters to be estimated in image reconstruction.

Wavelet domain Bayesian method for high noise level PET image reconstruction

In this paper, a new maximum a posterior(MAP) method for PET image reconstruction defined in wavelet domain is proposed. Compared to the conventional MAP methods with Markov Random Field (MRF) prior models, the proposed method, named WD-MAP method, has better performance in characterize both local and global feature of reconstructed image due to the wavelet transform. Wavelet packet decomposition strategy is applied to further improve the reconstruction quality. The convergence speed of WD-MAP method is accelerated by adopting conjugate gradient(CG) technique. Simulated experiment suggests that the proposed method offers competitive performance in PET image reconstruction.

An efficient analytical calculation of probability matrix in 2D SPECT

Expectation Maximization iterative reconstruction algorithms are being widely used in PET and SPECT imaging. The system probability matrix is usually calculated by using Monte Carlo simulations, since the analytical calculation is a rather complicated problem, especially in 3D reconstruction; however, realistic Monte Carlo simulations in 3D are time consuming and simplifications are necessary. In this paper, the probability matrix in the case of 2D SPECT is analytically calculated, using only two basic parameters: (i) the total number of image pixels and (ii) the number of projection angles. In the more general phase three more parameters will be taken into account: (i) the distance between the detectors and the object to be imaged; (ii) the collimator parameters; (iii) scintillator cells parameters (in the case of pixilated scintillators) and relative position between scintillator cell and collimator hole. It is shown that the accuracy of the probability matrix affects the quality of reconstructed images, especially in the case of pixilated scintillators, which are used in many dedicated SPECT systems. The methods presented here can be extended to 3D SPECT and also 2D and 3D PET. In addition, this analytically calculated matrix can be a reference matrix in order to be compared with Monte Carlo generated matrices.

Design of a very high-resolution small animal PET scanner using a silicon scatter detector insert

A small animal positron emission tomography (PET) instrument using a high-resolution solid-state detector insert in a conventional PET system was investigated for its potential to achieve sub-millimeter spatial resolution for mouse imaging. Monte Carlo simulations were used to estimate the effect of detector configurations (thickness, length and radius) on sensitivity. From this initial study, a PET system having an inner cylindrical silicon detector (4 cm ID, 4 cm length and 1.6 cm thickness composed of 16 layers of 300 microm x 300 microm x 1 mm pads), for scattering, surrounded by an outer cylindrical BGO scintillation detector (17.6 cm ID, 16 cm length and 2 cm thickness segmented into 3 mm x 3 mm x 20 mm crystals), for capture was evaluated in detail. In order to evaluate spatial resolution, sensitivity and image quality of the PET system, 2D images of multiple point and cylinder sources were reconstructed with the simulation data including blurring from positron range and annihilation photon acollinearity using filtered backprojection (FBP). Simulation results for (18)F demonstrate 340 microm FWHM at the center of the field of view with 1.0% sensitivity from the coincidence of single scattering events in both silicon detectors and 1.0 mm FWHM with 9.0% sensitivity from the coincidence of single scattering in the silicon and full energy absorption of the second photon in the BGO detector.

Evaluation of optimal density weighting for regridding

Density weighting is a necessary component of the regridding algorithm for interpolating nonuniformly sampled data points onto a regular grid. Differing concepts of optimality for the density weighting have been proposed previously. The present study reviews some of these concepts and evaluates the accuracy of different techniques by comparison with the image obtained by a computationally intensive least squares minimization. A variant on one of the techniques is proposed that yields the highest accuracy of those studied.

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.

Kronecker product approximation for preconditioning in three-dimensional imaging applications

We derive Kronecker product approximations, with the help of tensor decompositions, to construct approximations of severely ill-conditioned matrices that arise in three-dimensional (3-D) image processing applications. We use the Kronecker product approximations to derive preconditioners for iterative regularization techniques; the resulting preconditioned algorithms allow us to restore 3-D images in a computationally efficient manner. Through examples in microscopy and medical imaging, we show that the Kronecker approximation preconditioners provide a powerful tool that can be used to improve efficiency of iterative image restoration algorithms.

A modified OSEM algorithm for PET reconstruction using wavelet processing

Ordered subset expectation-maximization (OSEM) method in positron emission tomography (PET) has been very popular recently. It is an iterative algorithm and provides images with superior noise characteristics compared to conventional filtered backprojection (FBP) algorithms. Due to the lack of smoothness in images in OSEM iterations, however, some type of inter-smoothing is required. For this purpose, the smoothing based on the convolution with the Gaussian kernel has been used in clinical PET practices. In this paper, we incorporated a robust wavelet de-noising method into OSEM iterations as an inter-smoothing tool. The proposed wavelet method is based on a hybrid use of the standard wavelet shrinkage and the robust wavelet shrinkage to have edge preserving and robust de-noising simultaneously. The performances of the proposed method were compared with those of the smoothing methods based on the convolution with Gaussian kernel using software phantoms, physical phantoms, and human PET studies. The results demonstrated that the proposed wavelet method provided better spatial resolution characteristic than the smoothing methods based on the Gaussian convolution, while having comparable performance in noise removal.

A fast image reconstruction algorithm based on penalized-likelihood estimate

Statistical iterative methods for image reconstruction like maximum likelihood expectation maximization (ML-EM) are more robust and flexible than analytical inversion methods and allow for accurately modeling the counting statistics and the photon transport during acquisition. They are rapidly becoming the standard for image reconstruction in emission computed tomography. The maximum likelihood approach provides images with superior noise characteristics compared to the conventional filtered back projection algorithm. But a major drawback of the statistical iterative image reconstruction is its high computational cost. In this paper, a fast algorithm is proposed as a modified OS-EM (MOS-EM) using a penalized function, which is applied to the least squares merit function to accelerate image reconstruction and to achieve better convergence. The experimental results show that the algorithm can provide high quality reconstructed images with a small number of iterations.

Quantitative comparison of FBP, EM, and Bayesian reconstruction algorithms for the IndyPET scanner

We quantitatively compare filtered backprojection (FBP), expectation-maximization (EM), and Bayesian reconstruction algorithms as applied to the IndyPET scanner--a dedicated research scanner which has been developed for small and intermediate field of view imaging applications. In contrast to previous approaches that rely on Monte Carlo simulations, a key feature of our investigation is the use of an empirical system kernel determined from scans of line source phantoms. This kernel is incorporated into the forward model of the EM and Bayesian algorithms to achieve resolution recovery. Three data sets are used, data collected on the IndyPET scanner using a bar phantom and a Hoffman three-dimensional brain phantom, and simulated data containing a hot lesion added to a uniform background. Reconstruction quality is analyzed quantitatively in terms of bias-variance measures (bar phantom) and mean square error (lesion phantom). We observe that without use of the empirical system kernel, the FBP, EM, and Bayesian algorithms give similar performance. However, with the inclusion of the empirical kernel, the iterative algorithms provide superior reconstructions compared with FBP, both in terms of visual quality and quantitative measures. Furthermore, Bayesian methods outperform EM. We conclude that significant improvements in reconstruction quality can be realized by combining accurate models of the system response with Bayesian reconstruction algorithms.

Generalization of median root prior reconstruction

Penalized iterative algorithms for image reconstruction in emission tomography contain conditions on which kind of images are accepted as solutions. The penalty term has commonly been a function of pairwise pixel differences in the activity in a local neighborhood, such that smooth images are favored. Attempts to ensure better edge and detail preservation involve difficult tailoring of parameter values or the penalty function itself. The previously introduced median root prior (MRP) favors locally monotonic images. MRP preserves sharp edges while reducing locally nonmonotonic noise at the same time. Quantitative properties of MRP are good, because differences in the neighboring pixel values are not penalized as such. The median is used as an estimate for a penalty reference, against which the pixel value is compared when setting the penalty. In order to generalize the class of MRP-type of priors, the standard median was replaced by other order statistic operations, the L and finite-impluse-response median hybrid (FMH) filters. They allow for smoother appearance as they apply linear weighting together with robust nonlinear operations. The images reconstructed using the new MRP-L and MRP-FMH priors are visually more conventional. Good quantitative properties of MRP are not significantly altered by the new priors.

Spatiotemporal reconstruction of list-mode PET data

We describe a method for computing a continuous time estimate of tracer density using list-mode positron emission tomography data. The rate function in each voxel is modeled as an inhomogeneous Poisson process whose rate function can be represented using a cubic B-spline basis. The rate functions are estimated by maximizing the likelihood of the arrival times of detected photon pairs over the control vertices of the spline, modified by quadratic spatial and temporal smoothness penalties and a penalty term to enforce nonnegativity. Randoms rate functions are estimated by assuming independence between the spatial and temporal randoms distributions. Similarly, scatter rate functions are estimated by assuming spatiotemporal independence and that the temporal distribution of the scatter is proportional to the temporal distribution of the trues. A quantitative evaluation was performed using simulated data and the method is also demonstrated in a human study using 11C-raclopride.

Interior-point methodology for 3-D PET reconstruction

Interior-point methods have been successfully applied to a wide variety of linear and nonlinear programming applications. This paper presents a class of algorithms, based on path-following interior-point methodology, for performing regularized maximum-likelihood (ML) reconstructions on three-dimensional (3-D) emission tomography data. The algorithms solve a sequence of subproblems that converge to the regularized maximum likelihood solution from the interior of the feasible region (the nonnegative orthant). We propose two methods, a primal method which updates only the primal image variables and a primal-dual method which simultaneously updates the primal variables and the Lagrange multipliers. A parallel implementation permits the interior-point methods to scale to very large reconstruction problems. Termination is based on well-defined convergence measures, namely, the Karush-Kuhn-Tucker first-order necessary conditions for optimality. We demonstrate the rapid convergence of the path-following interior-point methods using both data from a small animal scanner and Monte Carlo simulated data. The proposed methods can readily be applied to solve the regularized, weighted least squares reconstruction problem.

Fast maximum entropy approximation in SPECT using the RBI-MAP algorithm

In this work, we present a method for approximating constrained maximum entropy (ME) reconstructions of SPECT data with modifications to a block-iterative maximum a posteriori (MAP) algorithm. Maximum likelihood (ML)-based reconstruction algorithms require some form of noise smoothing. Constrained ME provides a more formal method of noise smoothing without requiring the user to select parameters. In the context of SPECT, constrained ME seeks the minimum-information image estimate among those whose projections are a given distance from the noisy measured data, with that distance determined by the magnitude of the Poisson noise. Images that meet the distance criterion are referred to as feasible images. We find that modeling of all principal degrading factors (attenuation, detector response, and scatter) in the reconstruction is critical because feasibility is not meaningful unless the projection model is as accurate as possible. Because the constrained ME solution is the same as a MAP solution for a particular value of the MAP weighting parameter, beta, the constrained ME solution can be found with a MAP algorithm if the correct value of beta is found. We show that the RBI-MAP algorithm, if used with a dynamic scheme for estimating beta, can approximate constrained ME solutions in 20 or fewer iterations. We compare results for various methods of achieving feasible images on a simulation of Tl-201 cardiac SPECT data. Results show that the RBI-MAP ME approximation provides images and quantitative estimates close to those from a slower algorithm that gives the true ME solution. Also, we find that the ME results have higher spatial resolution and greater high-frequency noise content than a feasibility-based stopping rule, feasibility-based low-pass filtering, and a quadratic Gibbs prior with beta selected according to the feasibility criterion. We conclude that fast ME approximation is possible using either RBI-MAP with the dynamic procedure or a feasibility-based stopping rule, and that such reconstructions may be particularly useful in applications where resolution is critical.

Algorithms for density and composition-discrimination imaging for fourth-generation CT systems

This paper shows that if the off-beam idle detectors in the detection ring of a fourth-generation x-ray computed tomography (CT) system are used to measure the scattered radiation, it is numerically feasible to reconstruct electron-density images to supplement the conventional attenuation-coefficient images of transmitted radiation. It is also shown that by combining these two images, composition changes can be detected with the aid of an effective-atomic-number indicator. The required image-reconstruction algorithms are developed and tested against Monte Carlo simulated measurements, for a variety of phantom configurations. In spite of the relatively poor statistical quality of scattering measurements, it is demonstrated that electron-density images of reasonable quality can be obtained. In addition, it is shown that composition discrimination is possible for materials of effective atomic number greater than five, in the photon energy range of a typical medical x-ray CT system operating at 102 kVp. The obtained supplementary electron-density and composition images can be useful in radiotherapy planning and for studying tumour histology, as well as in industrial and security applications where identification of materials based on density and composition is important.

Simultaneous maximum a posteriori reconstruction of attenuation and activity distributions from emission sinograms

In order to perform attenuation correction in emission tomography an attenuation map is required. We propose a new method to compute this map directly from the emission sinogram, eliminating the transmission scan from the acquisition protocol. The problem is formulated as an optimization task where the objective function is a combination of the likelihood and an a priori probability. The latter uses a Gibbs prior distribution to encourage local smoothness and a multimodal distribution for the attenuation coefficients. Since the attenuation process is different in positron emission tomography (PET) and single photon emission tomography (SPECT), a separate algorithm for each case is derived. The method has been tested on mathematical phantoms and on a few clinical studies. For PET, good agreement was found between the images obtained with transmission measurements and those produced by the new algorithm in an abdominal study. For SPECT, promising simulation results have been obtained for nonhomogeneous attenuation due to the presence of the lungs.

Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.