Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms

Block-Iterative Reconstruction from Dynamically Selected Sparse Projection Views Using Extended Power-Divergence Measure

Iterative reconstruction of density pixel images from measured projections in computed tomography has attracted considerable attention. The ordered-subsets algorithm is an acceleration scheme that uses subsets of projections in a previously decided order. Several methods have been proposed to improve the convergence rate by permuting the order of the projections. However, they do not incorporate object information, such as shape, into the selection process. We propose a block-iterative reconstruction from sparse projection views with the dynamic selection of subsets based on an estimating function constructed by an extended power-divergence measure for decreasing the objective function as much as possible. We give a unified proposition for the inequality related to the difference between objective functions caused by one iteration as the theoretical basis of the proposed optimization strategy. Through the theory and numerical experiments, we show that nonuniform and sparse use of projection views leads to a reconstruction of higher-quality images and that an ordered subset is not the most effective for block-iterative reconstruction. The two-parameter class of extended power-divergence measures is the key to estimating an effective decrease in the objective function and plays a significant role in constructing a robust algorithm against noise.

Noise-Robust Image Reconstruction Based on Minimizing Extended Class of Power-Divergence Measures

The problem of tomographic image reconstruction can be reduced to an optimization problem of finding unknown pixel values subject to minimizing the difference between the measured and forward projections. Iterative image reconstruction algorithms provide significant improvements over transform methods in computed tomography. In this paper, we present an extended class of power-divergence measures (PDMs), which includes a large set of distance and relative entropy measures, and propose an iterative reconstruction algorithm based on the extended PDM (EPDM) as an objective function for the optimization strategy. For this purpose, we introduce a system of nonlinear differential equations whose Lyapunov function is equivalent to the EPDM. Then, we derive an iterative formula by multiplicative discretization of the continuous-time system. Since the parameterized EPDM family includes the Kullback-Leibler divergence, the resulting iterative algorithm is a natural extension of the maximum-likelihood expectation-maximization (MLEM) method. We conducted image reconstruction experiments using noisy projection data and found that the proposed algorithm outperformed MLEM and could reconstruct high-quality images that were robust to measured noise by properly selecting parameters.

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.

A Novel Low-Dose Dual-Energy Imaging Method for a Fast-Rotating Gantry-Type CT Scanner

CT scan by use of a beam-filter placed between the x-ray source and the patient allows a single-scan low-dose dual-energy imaging with a minimal hardware modification to the existing CT systems. We have earlier demonstrated the feasibility of such imaging method with a multi-slit beam-filter reciprocating along the direction perpendicular to the CT rotation axis in a cone-beam CT system. However, such method would face mechanical challenges when the beam-filter is supposed to cooperate with a fast-rotating gantry in a diagnostic CT system. In this work, we propose a new scanning method and associated image reconstruction algorithm that can overcome these challenges. We propose to slide a beam-filter that has multi-slit structure with its slits being at a slanted angle with the CT gantry rotation axis during a scan. A streaky pattern would show up in the sinogram domain as a result. Using a notch filter in the Fourier domain of the sinogram, we removed the streaks and reconstructed an image by use of the filtered-backprojection algorithm. The remaining image artifacts were suppressed by applying l₀ norm based smoothing. Using this image as a prior, we have reconstructed low- and high-energy CT images in the iterative reconstruction framework. An image-based material decomposition then followed. We conducted a simulation study to test its feasibility using the XCAT phantom and also an experimental study using the Catphan phantom, a head phantom, an iodine-solution phantom, and a monkey in anesthesia, and showed its successful performance in image reconstruction and in material decomposition.

DVDeconv: An Open-Source MATLAB Toolbox for Depth-Variant Asymmetric Deconvolution of Fluorescence Micrographs

To investigate the cellular structure, biomedical researchers often obtain three-dimensional images by combining two-dimensional images taken along the z axis. However, these images are blurry in all directions due to diffraction limitations. This blur becomes more severe when focusing further inside the specimen as photons in deeper focus must traverse a longer distance within the specimen. This type of blur is called depth-variance. Moreover, due to lens imperfection, the blur has asymmetric shape. Most deconvolution solutions for removing blur assume depth-invariant or x-y symmetric blur, and presently, there is no open-source for depth-variant asymmetric deconvolution. In addition, existing datasets for deconvolution microscopy also assume invariant or x-y symmetric blur, which are insufficient to reflect actual imaging conditions. DVDeconv, that is a set of MATLAB functions with a user-friendly graphical interface, has been developed to address depth-variant asymmetric blur. DVDeconv includes dataset, depth-variant asymmetric point spread function generator, and deconvolution algorithms. Experimental results using DVDeconv reveal that depth-variant asymmetric deconvolution using DVDeconv removes blurs accurately. Furthermore, the dataset in DVDeconv constructed can be used to evaluate the performance of microscopy deconvolution to be developed in the future.

Modification of Green's one-step-late algorithm for attenuated emission data

Green's one-step-late (OSL) algorithm is a popular image reconstruction algorithm in emission-data tomography, in spite of its convergence issues. One drawback of Green's algorithm is that the algorithm exhibits non-stationary regularization when the algorithm's projector and backprojector model the attenuation effects in single photon emission computed tomography (SPECT). This paper suggests a remedy to improve Green's OSL algorithm so that stationary regularization can be obtained. This paper also observes some similarities between the modified Green's OSL algorithm and the gradient descent algorithm.

Extension of emission expectation maximization lookalike algorithms to Bayesian algorithms

We recently developed a family of image reconstruction algorithms that look like the emission maximum-likelihood expectation-maximization (ML-EM) algorithm. In this study, we extend these algorithms to Bayesian algorithms. The family of emission-EM-lookalike algorithms utilizes a multiplicative update scheme. The extension of these algorithms to Bayesian algorithms is achieved by introducing a new simple factor, which contains the Bayesian information. One of the extended algorithms can be applied to emission tomography and another to transmission tomography. Computer simulations are performed and compared with the corresponding un-extended algorithms. The total-variation norm is employed as the Bayesian constraint in the computer simulations. The newly developed algorithms demonstrate a stable performance. A simple Bayesian algorithm can be derived for any noise variance function. The proposed algorithms have properties such as multiplicative updating, non-negativity, faster convergence rates for bright objects, and ease of implementation. Our algorithms are inspired by Green's one-step-late algorithm. If written in additive-update form, Green's algorithm has a step size determined by the future image value, which is an undesirable feature that our algorithms do not have.

Resolution modeling in projection space using a factorized multi-block detector response function for PET image reconstruction

Positron emission tomography (PET) images usually suffer from limited resolution and statistical uncertainties. However, a technique known as resolution modeling (RM) can be used to improve image quality by accurately modeling the system's detection process within the iterative reconstruction. In this study, we present an accurate RM method in projection space based on a simulated multi-block detector response function (DRF) and evaluate it on the Siemens hybrid MR-BrainPET system. The DRF is obtained using GATE simulations that consider nearly all the possible annihilation photons from the field-of-view (FOV). Intrinsically, the multi-block DRF allows the block crosstalk to be modeled. The RM blurring kernel is further generated by factorizing the blurring matrix of one line-of-response (LOR) into two independent detector responses, which can then be addressed with the DRF. Such a kernel is shift-variant in 4D projection space without any distance or angle compression, and is integrated into the image reconstruction for the BrainPET insert with single instruction multiple data (SIMD) and multi-thread support. Evaluation of simulations and measured data demonstrate that the reconstruction with RM yields significantly improved resolutions and reduced mean squared error (MSE) values at different locations of the FOV, compared with reconstruction without RM. Furthermore, the shift-variant RM kernel models the varying blurring intensity for different LORs due to the depth-of-interaction (DOI) dependencies, thus avoiding severe edge artifacts in the images. Additionally, compared to RM in single-block mode, the multi-block mode shows significantly improved resolution and edge recovery at locations beyond 10 cm from the center of BrainPET insert in the transverse plane. However, the differences have been observed to be low for patient data between single-block and multi-block mode RM, due to the brain size and location as well as the geometry of the BrainPET insert. In conclusion, the RM method proposed in this study can yield better reconstructed images in terms of resolution and MSE value, compared to conventional reconstruction without RM.

A Convergence Proof of MLEM and MLEM-3 With Fixed Background

Maximum likelihood expectation-maximization (MLEM) is a popular algorithm to reconstruct the activity image in positron emission tomography. This paper introduces a "fundamental equality" for the MLEM complete data from which two key properties easily follow that allows us to: 1) prove in an elegant and compact way the convergence of MLEM for a forward model with fixed background (i.e., counts such as random and scatter coincidences) and 2) generalize this proof for the MLEM-3 algorithm. Moreover, we give necessary and sufficient conditions for the solution to be unique.

Technical Advances in Image Guidance of Radionuclide Therapy

Internal radiation therapy with radionuclides (i.e., radionuclide therapy) owes its success to the many advantages over other, more conventional, treatment options. One distinct advantage of radionuclide therapies is the potential to use (part of) the emitted radiation for imaging of the radionuclide distribution. The combination of diagnostic and therapeutic properties in a set of matched radiopharmaceuticals (sometimes combined in a single radiopharmaceutical) is often referred to as theranostics and allows accurate diagnostic imaging before therapy. The use of imaging benefits treatment planning, dosimetry, and assessment of treatment response. This paper focuses on a selection of advances in imaging technology relevant for image guidance of radionuclide therapy. This involves developments in nuclear imaging modalities, as well as other anatomic and functional imaging modalities. The quality and quantitative accuracy of images used for guidance of radionuclide therapy is continuously being improved, which in turn may improve the therapeutic outcome and efficiency of radionuclide therapies.

Deep reconstruction model for dynamic PET images

Accurate and robust tomographic reconstruction from dynamic positron emission tomography (PET) acquired data is a difficult problem. Conventional methods, such as the maximum likelihood expectation maximization (MLEM) algorithm for reconstructing the activity distribution-based on individual frames, may lead to inaccurate results due to the checkerboard effect and limitation of photon counts. In this paper, we propose a stacked sparse auto-encoder based reconstruction framework for dynamic PET imaging. The dynamic reconstruction problem is formulated in a deep learning representation, where the encoding layers extract the prototype features, such as edges, so that, in the decoding layers, the reconstructed results are obtained through a combination of those features. The qualitative and quantitative results of the procedure, including the data based on a Monte Carlo simulation and real patient data demonstrates the effectiveness of our method.

Edge-preserving denoising for intra-operative cone beam CT in endovascular aneurysm repair

C-arm cone-beam computed tomography (CBCT) acquisition during endovascular aneurysm repair (EVAR) is an emergent technology with more and more applications. It offers real time imaging with a stationary patient and provides 3-D information to achieve guidance of intervention. However, there is growing concern on the overall radiation doses delivered to patients all along the endovascular management due to pre-, intra-, and post-operative X-ray imaging. Manufactures may have their low dose protocols to realize reduction of radiation dose, but CBCT with a low dose protocol has too many artifacts, particularly streak artifacts, and decreased contrast-to-noise ratio (CNR). To reduce noise and artifacts, a penalized weighted least-squares (PWLS) algorithm with an edge-preserving penalty is proposed. The proposed method is evaluated by quantitative parameters including a defined signal-to-noise ratio (SNR), CNR, and modulation transfer function (MTF) on clinical CBCT. Comparisons with PWLS algorithms with isotropic, TV, Huber, anisotropic penalties demonstrate that the proposed edge-preserving penalty performs well not only on edge preservation, but also on streak artifacts suppression, which may be crucial for observing guidewire and stentgraft in EVAR.

Theoretical Analysis of Penalized Maximum-Likelihood Patlak Parametric Image Reconstruction in Dynamic PET for Lesion Detection

Detecting cancerous lesions is a major clinical application of emission tomography. In a previous work, we studied penalized maximum-likelihood (PML) image reconstruction for lesion detection in static PET. Here we extend our theoretical analysis of static PET reconstruction to dynamic PET. We study both the conventional indirect reconstruction and direct reconstruction for Patlak parametric image estimation. In indirect reconstruction, Patlak parametric images are generated by first reconstructing a sequence of dynamic PET images, and then performing Patlak analysis on the time activity curves (TACs) pixel-by-pixel. In direct reconstruction, Patlak parametric images are estimated directly from raw sinogram data by incorporating the Patlak model into the image reconstruction procedure. PML reconstruction is used in both the indirect and direct reconstruction methods. We use a channelized Hotelling observer (CHO) to assess lesion detectability in Patlak parametric images. Simplified expressions for evaluating the lesion detectability have been derived and applied to the selection of the regularization parameter value to maximize detection performance. The proposed method is validated using computer-based Monte Carlo simulations. Good agreements between the theoretical predictions and the Monte Carlo results are observed. Both theoretical predictions and Monte Carlo simulation results show the benefit of the indirect and direct methods under optimized regularization parameters in dynamic PET reconstruction for lesion detection, when compared with the conventional static PET reconstruction.

The Use of Anatomical Information for Molecular Image Reconstruction Algorithms: Attenuation/Scatter Correction, Motion Compensation, and Noise Reduction

PET and SPECT are important tools for providing valuable molecular information about patients to clinicians. Advances in nuclear medicine hardware technologies and statistical image reconstruction algorithms enabled significantly improved image quality. Sequentially or simultaneously acquired anatomical images such as CT and MRI from hybrid scanners are also important ingredients for improving the image quality of PET or SPECT further. High-quality anatomical information has been used and investigated for attenuation and scatter corrections, motion compensation, and noise reduction via post-reconstruction filtering and regularization in inverse problems. In this article, we will review works using anatomical information for molecular image reconstruction algorithms for better image quality by describing mathematical models, discussing sources of anatomical information for different cases, and showing some examples.

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.

PET image segmentation using a Gaussian mixture model and Markov random fields

Background

Classification algorithms for positron emission tomography (PET) images support computational treatment planning in radiotherapy. Common clinical practice is based on manual delineation and fixed or iterative threshold methods, the latter of which requires regression curves dependent on many parameters.

Methods

An improved statistical approach using a Gaussian mixture model (GMM) is proposed to obtain initial estimates of a target volume, followed by a correction step based on a Markov random field (MRF) and a Gibbs distribution to account for dependencies among neighboring voxels. In order to evaluate the proposed algorithm, phantom measurements of spherical and non-spherical objects with the smallest diameter being 8 mm were performed at signal-to-background ratios (SBRs) between 2.06 and 9.39. Additionally ⁶⁸Ga-PET data from patients with lesions in the liver and lymph nodes were evaluated.

Results

The proposed algorithm produces stable results for different reconstruction algorithms and different lesion shapes. Furthermore, it outperforms all threshold methods regarding detection rate, determines the spheres’ volumes more accurately than fixed threshold methods, and produces similar values as iterative thresholding. In a comparison with other statistical approaches, the algorithm performs equally well for larger volumes and even shows improvements for small volumes and SBRs. The comparison with experts’ manual delineations on the clinical data shows the same qualitative behavior as for the phantom measurements.

Conclusions

In conclusion, a generic probabilistic approach that does not require data measured beforehand is presented whose performance, robustness, and swiftness make it a feasible choice for PET segmentation.

Evaluation of penalty design in penalized maximum-likelihood image reconstruction for lesion detection

Detecting cancerous lesions is a major clinical application in emission tomography. Previously, we developed a method to design a shift-variant quadratic penalty function in penalized maximum-likelihood (PML) image reconstruction to improve the lesion detectability. We used a multiview channelized Hotelling observer (mvCHO) to assess the lesion detectability in three-dimensional images and validated the penalty design using computer simulations. In this study, we evaluate the benefit of the proposed penalty function for lesion detection using real patient data and artificial lesions. A high-count real patient dataset with no identifiable tumor inside the field of view is used as the background data. A Na-22 point source is scanned in air at variable locations and the point source data are superimposed onto the patient data as artificial lesions after being attenuated by the patient body. Independent Poisson noise is introduced to the high-count sinograms to generate 200 pairs of lesion-present and lesion-absent datasets, each mimicking a 5-min scan. Lesion detectability is assessed using a mvCHO and a human observer two-alternative forced choice (2AFC) experiment. The results show improvements in lesion detection by the proposed method compared with the conventional first-order quadratic penalty function and a total variation (TV) edge-preserving penalty function.

Statistical model based iterative reconstruction (MBIR) in clinical CT systems: experimental assessment of noise performance

PURPOSE

To reduce radiation dose in CT imaging, the statistical model based iterative reconstruction (MBIR) method has been introduced for clinical use. Based on the principle of MBIR and its nonlinear nature, the noise performance of MBIR is expected to be different from that of the well-understood filtered backprojection (FBP) reconstruction method. The purpose of this work is to experimentally assess the unique noise characteristics of MBIR using a state-of-the-art clinical CT system.

METHODS

Three physical phantoms, including a water cylinder and two pediatric head phantoms, were scanned in axial scanning mode using a 64-slice CT scanner (Discovery CT750 HD, GE Healthcare, Waukesha, WI) at seven different mAs levels (5, 12.5, 25, 50, 100, 200, 300). At each mAs level, each phantom was repeatedly scanned 50 times to generate an image ensemble for noise analysis. Both the FBP method with a standard kernel and the MBIR method (Veo(®), GE Healthcare, Waukesha, WI) were used for CT image reconstruction. Three-dimensional (3D) noise power spectrum (NPS), two-dimensional (2D) NPS, and zero-dimensional NPS (noise variance) were assessed both globally and locally. Noise magnitude, noise spatial correlation, noise spatial uniformity and their dose dependence were examined for the two reconstruction methods.

RESULTS

(1) At each dose level and at each frequency, the magnitude of the NPS of MBIR was smaller than that of FBP. (2) While the shape of the NPS of FBP was dose-independent, the shape of the NPS of MBIR was strongly dose-dependent; lower dose lead to a "redder" NPS with a lower mean frequency value. (3) The noise standard deviation (σ) of MBIR and dose were found to be related through a power law of σ ∝ (dose)(-β) with the component β ≈ 0.25, which violated the classical σ ∝ (dose)(-0.5) power law in FBP. (4) With MBIR, noise reduction was most prominent for thin image slices. (5) MBIR lead to better noise spatial uniformity when compared with FBP. (6) A composite image generated from two MBIR images acquired at two different dose levels (D1 and D2) demonstrated lower noise than that of an image acquired at a dose level of D1+D2.

CONCLUSIONS

The noise characteristics of the MBIR method are significantly different from those of the FBP method. The well known tradeoff relationship between CT image noise and radiation dose has been modified by MBIR to establish a more gradual dependence of noise on dose. Additionally, some other CT noise properties that had been well understood based on the linear system theory have also been altered by MBIR. Clinical CT scan protocols that had been optimized based on the classical CT noise properties need to be carefully re-evaluated for systems equipped with MBIR in order to maximize the method's potential clinical benefits in dose reduction and/or in CT image quality improvement.

Compensation of missing wedge effects with sequential statistical reconstruction in electron tomography

Electron tomography (ET) of biological samples is used to study the organization and the structure of the whole cell and subcellular complexes in great detail. However, projections cannot be acquired over full tilt angle range with biological samples in electron microscopy. ET image reconstruction can be considered an ill-posed problem because of this missing information. This results in artifacts, seen as the loss of three-dimensional (3D) resolution in the reconstructed images. The goal of this study was to achieve isotropic resolution with a statistical reconstruction method, sequential maximum a posteriori expectation maximization (sMAP-EM), using no prior morphological knowledge about the specimen. The missing wedge effects on sMAP-EM were examined with a synthetic cell phantom to assess the effects of noise. An experimental dataset of a multivesicular body was evaluated with a number of gold particles. An ellipsoid fitting based method was developed to realize the quantitative measures elongation and contrast in an automated, objective, and reliable way. The method statistically evaluates the sub-volumes containing gold particles randomly located in various parts of the whole volume, thus giving information about the robustness of the volume reconstruction. The quantitative results were also compared with reconstructions made with widely-used weighted backprojection and simultaneous iterative reconstruction technique methods. The results showed that the proposed sMAP-EM method significantly suppresses the effects of the missing information producing isotropic resolution. Furthermore, this method improves the contrast ratio, enhancing the applicability of further automatic and semi-automatic analysis. These improvements in ET reconstruction by sMAP-EM enable analysis of subcellular structures with higher three-dimensional resolution and contrast than conventional methods.

Alternating direction method of multiplier for tomography with nonlocal regularizers

The ordered subset expectation maximization (OSEM) algorithm approximates the gradient of a likelihood function using a subset of projections instead of using all projections so that fast image reconstruction is possible for emission and transmission tomography such as SPECT, PET, and CT. However, OSEM does not significantly accelerate reconstruction with computationally expensive regularizers such as patch-based nonlocal (NL) regularizers, because the regularizer gradient is evaluated for every subset. We propose to use variable splitting to separate the likelihood term and the regularizer term for penalized emission tomographic image reconstruction problem and to optimize it using the alternating direction method of multiplier (ADMM). We also propose a fast algorithm to optimize the ADMM parameter based on convergence rate analysis. This new scheme enables more sub-iterations related to the likelihood term. We evaluated our ADMM for 3-D SPECT image reconstruction with a patch-based NL regularizer that uses the Fair potential function. Our proposed ADMM improved the speed of convergence substantially compared to other existing methods such as gradient descent, EM, and OSEM using De Pierro's approach, and the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm.

Statistical model based iterative reconstruction (MBIR) in clinical CT systems. Part II. Experimental assessment of spatial resolution performance

PURPOSE

Statistical model based iterative reconstruction (MBIR) methods have been introduced to clinical CT systems and are being used in some clinical diagnostic applications. The purpose of this paper is to experimentally assess the unique spatial resolution characteristics of this nonlinear reconstruction method and identify its potential impact on the detectabilities and the associated radiation dose levels for specific imaging tasks.

METHODS

The thoracic section of a pediatric phantom was repeatedly scanned 50 or 100 times using a 64-slice clinical CT scanner at four different dose levels [CTDIvol =4, 8, 12, 16 (mGy)]. Both filtered backprojection (FBP) and MBIR (Veo(®), GE Healthcare, Waukesha, WI) were used for image reconstruction and results were compared with one another. Eight test objects in the phantom with contrast levels ranging from 13 to 1710 HU were used to assess spatial resolution. The axial spatial resolution was quantified with the point spread function (PSF), while the z resolution was quantified with the slice sensitivity profile. Both were measured locally on the test objects and in the image domain. The dependence of spatial resolution on contrast and dose levels was studied. The study also features a systematic investigation of the potential trade-off between spatial resolution and locally defined noise and their joint impact on the overall image quality, which was quantified by the image domain-based channelized Hotelling observer (CHO) detectability index d'.

RESULTS

(1) The axial spatial resolution of MBIR depends on both radiation dose level and image contrast level, whereas it is supposedly independent of these two factors in FBP. The axial spatial resolution of MBIR always improved with an increasing radiation dose level and/or contrast level. (2) The axial spatial resolution of MBIR became equivalent to that of FBP at some transitional contrast level, above which MBIR demonstrated superior spatial resolution than FBP (and vice versa); the value of this transitional contrast highly depended on the dose level. (3) The PSFs of MBIR could be approximated as Gaussian functions with reasonably good accuracy. (4) Thez resolution of MBIR showed similar contrast and dose dependence. (5) Noise standard deviation assessed on the edges of objects demonstrated a trade-off with spatial resolution in MBIR. (5) When both spatial resolution and image noise were considered using the CHO analysis, MBIR led to significant improvement in the overall CT image quality for both high and low contrast detection tasks at both standard and low dose levels.

CONCLUSIONS

Due to the intrinsic nonlinearity of the MBIR method, many well-known CT spatial resolution and noise properties have been modified. In particular, dose dependence and contrast dependence have been introduced to the spatial resolution of CT images by MBIR. The method has also introduced some novel noise-resolution trade-off not seen in traditional CT images. While the benefits of MBIR regarding the overall image quality, as demonstrated in this work, are significant, the optimal use of this method in clinical practice demands a thorough understanding of its unique physical characteristics.

Regularization design in penalized maximum-likelihood image reconstruction for lesion detection in 3D PET

Detecting cancerous lesions is a major clinical application in emission tomography. In previous work, we have studied penalized maximum-likelihood (PML) image reconstruction for the detection task and proposed a method to design a shift-invariant quadratic penalty function to maximize detectability of a lesion at a known location in a two dimensional image. Here we extend the regularization design to maximize detectability of lesions at unknown locations in fully 3D PET. We used a multiview channelized Hotelling observer (mvCHO) to assess the lesion detectability in 3D images to mimic the condition where a human observer examines three orthogonal views of a 3D image for lesion detection. We derived simplified theoretical expressions that allow fast prediction of the detectability of a 3D lesion. The theoretical results were used to design the regularization in PML reconstruction to improve lesion detectability. We conducted computer-based Monte Carlo simulations to compare the optimized penalty with the conventional penalty for detecting lesions of various sizes. Only true coincidence events were simulated. Lesion detectability was also assessed by two human observers, whose performances agree well with that of the mvCHO. Both the numerical observer and human observer results showed a statistically significant improvement in lesion detection by using the proposed penalty function compared to using the conventional penalty function.

Model-Based Iterative Reconstruction for Dual-Energy X-Ray CT Using a Joint Quadratic Likelihood Model

Dual-energy X-ray CT (DECT) has the potential to improve contrast and reduce artifacts as compared to traditional CT. Moreover, by applying model-based iterative reconstruction (MBIR) to dual-energy data, one might also expect to reduce noise and improve resolution. However, the direct implementation of dual-energy MBIR requires the use of a nonlinear forward model, which increases both complexity and computation. Alternatively, simplified forward models have been used which treat the material-decomposed channels separately, but these approaches do not fully account for the statistical dependencies in the channels. In this paper, we present a method for joint dual-energy MBIR (JDE-MBIR), which simplifies the forward model while still accounting for the complete statistical dependency in the material-decomposed sinogram components. The JDE-MBIR approach works by using a quadratic approximation to the polychromatic log-likelihood and a simple but exact nonnegativity constraint in the image domain. We demonstrate that our method is particularly effective when the DECT system uses fast kVp switching, since in this case the model accounts for the inaccuracy of interpolated sinogram entries. Both phantom and clinical results show that the proposed model produces images that compare favorably in quality to previous decomposition-based methods, including FBP and other statistical iterative approaches.

Depth-variant deconvolution of 3D widefield fluorescence microscopy using the penalized maximum likelihood estimation method

We investigated the deconvolution of 3D widefield fluorescence microscopy using the penalized maximum likelihood estimation method and the depth-variant point spread function (DV-PSF). We build the DV-PSF by fitting a parameterized theoretical PSF model to an experimental microbead image. On the basis of the constructed DV-PSF, we restore the 3D widefield microscopy by minimizing an objective function consisting of a negative Poisson likelihood function and a total variation regularization function. In simulations and experiments, the proposed method showed better performance than existing methods.

Quantitative statistical methods for image quality assessment

Quantitative measures of image quality and reliability are critical for both qualitative interpretation and quantitative analysis of medical images. While, in theory, it is possible to analyze reconstructed images by means of Monte Carlo simulations using a large number of noise realizations, the associated computational burden makes this approach impractical. Additionally, this approach is less meaningful in clinical scenarios, where multiple noise realizations are generally unavailable. The practical alternative is to compute closed-form analytical expressions for image quality measures. The objective of this paper is to review statistical analysis techniques that enable us to compute two key metrics: resolution (determined from the local impulse response) and covariance. The underlying methods include fixed-point approaches, which compute these metrics at a fixed point (the unique and stable solution) independent of the iterative algorithm employed, and iteration-based approaches, which yield results that are dependent on the algorithm, initialization, and number of iterations. We also explore extensions of some of these methods to a range of special contexts, including dynamic and motion-compensated image reconstruction. While most of the discussed techniques were developed for emission tomography, the general methods are extensible to other imaging modalities as well. In addition to enabling image characterization, these analysis techniques allow us to control and enhance imaging system performance. We review practical applications where performance improvement is achieved by applying these ideas to the contexts of both hardware (optimizing scanner design) and image reconstruction (designing regularization functions that produce uniform resolution or maximize task-specific figures of merit).

Penalized maximum likelihood estimation of lifetime and amplitude images from multi-exponentially decaying fluorescence signals

We investigated the penalized maximum likelihood estimation of lifetime and amplitude images for fluorescence lifetime imaging microscopy. The proposed method penalizes large variations in the lifetimes and amplitudes in the spatial domain to reduces noise in the images, which is a serious problem in the conventional maximum likelihood estimation method. For an effective optimization of the objective function, we applied an optimization transfer method that is based on a separable surrogate function. Simulations show that the proposed method outperforms the conventional MLE method in terms of the estimation accuracy, and the proposed method yielded less noisy images in real experiments.

Statistical image reconstruction from limited projection data with intensity priors

The radiation dose generated from x-ray computed tomography (CT) scans and its responsibility for increasing the risk of malignancy became a major concern in the medical imaging community. Accordingly, investigating possible approaches for image reconstruction from low-dose imaging protocols, which minimize the patient radiation exposure without affecting image quality, has become an issue of interest. Statistical reconstruction (SR) methods are known to achieve a superior image quality compared with conventional analytical methods. Effective physical noise modeling and possibilities of incorporating priors in the image reconstruction problem are the main advantages of the SR methods. Nevertheless, the high computation cost limits its wide use in clinical scanners. This paper presents a framework for SR in x-ray CT when the angular sampling rate of the projection data is low. The proposed framework is based on the fact that, in many CT imaging applications, some physical and anatomical structures and the corresponding attenuation information of the scanned object can be a priori known. Therefore, the x-ray attenuation distribution in some regions of the object can be expected prior to the reconstruction. Under this assumption, the proposed method is developed by incorporating this prior information into the image reconstruction objective function to suppress streak artifacts. We limit the prior information to only a set of intensity values that represent the average intensity of the normal and expected homogeneous regions within the scanned object. This prior information can be easily computed in several x-ray CT applications. Considering the theory of compressed sensing, the objective function is formulated using the ℓ(1) norm distance between the reconstructed image and the available intensity priors. Experimental comparative studies applied to simulated data and real data are used to evaluate the proposed method. The comparison indicates a significant improvement in image quality when the proposed method is used.

Joint L1 and total variation regularization for fluorescence molecular tomography

Fluorescence molecular tomography (FMT) is an imaging modality that exploits the specificity of fluorescent biomarkers to enable 3D visualization of molecular targets and pathways in vivo in small animals. Owing to the high degree of absorption and scattering of light through tissue, the FMT inverse problem is inherently ill-conditioned making image reconstruction highly susceptible to the effects of noise and numerical errors. Appropriate priors or penalties are needed to facilitate reconstruction and to restrict the search space to a specific solution set. Typically, fluorescent probes are locally concentrated within specific areas of interest (e.g., inside tumors). The commonly used L(2) norm penalty generates the minimum energy solution, which tends to be spread out in space. Instead, we present here an approach involving a combination of the L(1) and total variation norm penalties, the former to suppress spurious background signals and enforce sparsity and the latter to preserve local smoothness and piecewise constancy in the reconstructed images. We have developed a surrogate-based optimization method for minimizing the joint penalties. The method was validated using both simulated and experimental data obtained from a mouse-shaped phantom mimicking tissue optical properties and containing two embedded fluorescent sources. Fluorescence data were collected using a 3D FMT setup that uses an EMCCD camera for image acquisition and a conical mirror for full-surface viewing. A range of performance metrics was utilized to evaluate our simulation results and to compare our method with the L(1), L(2) and total variation norm penalty-based approaches. The experimental results were assessed using the Dice similarity coefficients computed after co-registration with a CT image of the phantom.

L0 constrained sparse reconstruction for multi-slice helical CT reconstruction

In this paper, we present a Bayesian maximum a posteriori method for multi-slice helical CT reconstruction based on an L0-norm prior. It makes use of a very low number of projections. A set of surrogate potential functions is used to successively approximate the L0-norm function while generating the prior and to accelerate the convergence speed. Simulation results show that the proposed method provides high quality reconstructions with highly sparse sampled noise-free projections. In the presence of noise, the reconstruction quality is still significantly better than the reconstructions obtained with L1-norm or L2-norm priors.

Fast model-based X-ray CT reconstruction using spatially nonhomogeneous ICD optimization

Recent applications of model-based iterative reconstruction (MBIR) algorithms to multislice helical CT reconstructions have shown that MBIR can greatly improve image quality by increasing resolution as well as reducing noise and some artifacts. However, high computational cost and long reconstruction times remain as a barrier to the use of MBIR in practical applications. Among the various iterative methods that have been studied for MBIR, iterative coordinate descent (ICD) has been found to have relatively low overall computational requirements due to its fast convergence. This paper presents a fast model-based iterative reconstruction algorithm using spatially nonhomogeneous ICD (NH-ICD) optimization. The NH-ICD algorithm speeds up convergence by focusing computation where it is most needed. The NH-ICD algorithm has a mechanism that adaptively selects voxels for update. First, a voxel selection criterion VSC determines the voxels in greatest need of update. Then a voxel selection algorithm VSA selects the order of successive voxel updates based upon the need for repeated updates of some locations, while retaining characteristics for global convergence. In order to speed up each voxel update, we also propose a fast 1-D optimization algorithm that uses a quadratic substitute function to upper bound the local 1-D objective function, so that a closed form solution can be obtained rather than using a computationally expensive line search algorithm. We examine the performance of the proposed algorithm using several clinical data sets of various anatomy. The experimental results show that the proposed method accelerates the reconstructions by roughly a factor of three on average for typical 3-D multislice geometries.

Restoration of Poissonian images using alternating direction optimization

Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using state-of-the-art regularizers (such as those based upon multiscale representations or total variation) is still an active research area, since the associated optimization problems are quite challenging. In this paper, we propose an approach to deconvolving Poissonian images, which is based upon an alternating direction optimization method. The standard regularization [or maximum a posteriori (MAP)] restoration criterion, which combines the Poisson log-likelihood with a (nonsmooth) convex regularizer (log-prior), leads to hard optimization problems: the log-likelihood is nonquadratic and nonseparable, the regularizer is nonsmooth, and there is a nonnegativity constraint. Using standard convex analysis tools, we present sufficient conditions for existence and uniqueness of solutions of these optimization problems, for several types of regularizers: total-variation, frame-based analysis, and frame-based synthesis. We attack these problems with an instance of the alternating direction method of multipliers (ADMM), which belongs to the family of augmented Lagrangian algorithms. We study sufficient conditions for convergence and show that these are satisfied, either under total-variation or frame-based (analysis and synthesis) regularization. The resulting algorithms are shown to outperform alternative state-of-the-art methods, both in terms of speed and restoration accuracy.

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.

Application and evaluation of a measured spatially variant system model for PET image reconstruction

Accurate system modeling in tomographic image reconstruction has been shown to reduce the spatial variance of resolution and improve quantitative accuracy. System modeling can be improved through analytic calculations, Monte Carlo simulations, and physical measurements. The purpose of this work is to improve clinical fully-3-D reconstruction without substantially increasing computation time. We present a practical method for measuring the detector blurring component of a whole-body positron emission tomography (PET) system to form an approximate system model for use with fully-3-D reconstruction. We employ Monte Carlo simulations to show that a non-collimated point source is acceptable for modeling the radial blurring present in a PET tomograph and we justify the use of a Na22 point source for collecting these measurements. We measure the system response on a whole-body scanner, simplify it to a 2-D function, and incorporate a parameterized version of this response into a modified fully-3-D OSEM algorithm. Empirical testing of the signal versus noise benefits reveal roughly a 15% improvement in spatial resolution and 10% improvement in contrast at matched image noise levels. Convergence analysis demonstrates improved resolution and contrast versus noise properties can be achieved with the proposed method with similar computation time as the conventional approach. Comparison of the measured spatially variant and invariant reconstruction revealed similar performance with conventional image metrics. Edge artifacts, which are a common artifact of resolution-modeled reconstruction methods, were less apparent in the spatially variant method than in the invariant method. With the proposed and other resolution-modeled reconstruction methods, edge artifacts need to be studied in more detail to determine the optimal tradeoff of resolution/contrast enhancement and edge fidelity.

Performance comparison between total variation (TV)-based compressed sensing and statistical iterative reconstruction algorithms

Of all available reconstruction methods, statistical iterative reconstruction algorithms appear particularly promising since they enable accurate physical noise modeling. The newly developed compressive sampling/compressed sensing (CS) algorithm has shown the potential to accurately reconstruct images from highly undersampled data. The CS algorithm can be implemented in the statistical reconstruction framework as well. In this study, we compared the performance of two standard statistical reconstruction algorithms (penalized weighted least squares and q-GGMRF) to the CS algorithm. In assessing the image quality using these iterative reconstructions, it is critical to utilize realistic background anatomy as the reconstruction results are object dependent. A cadaver head was scanned on a Varian Trilogy system at different dose levels. Several figures of merit including the relative root mean square error and a quality factor which accounts for the noise performance and the spatial resolution were introduced to objectively evaluate reconstruction performance. A comparison is presented between the three algorithms for a constant undersampling factor comparing different algorithms at several dose levels. To facilitate this comparison, the original CS method was formulated in the framework of the statistical image reconstruction algorithms. Important conclusions of the measurements from our studies are that (1) for realistic neuro-anatomy, over 100 projections are required to avoid streak artifacts in the reconstructed images even with CS reconstruction, (2) regardless of the algorithm employed, it is beneficial to distribute the total dose to more views as long as each view remains quantum noise limited and (3) the total variation-based CS method is not appropriate for very low dose levels because while it can mitigate streaking artifacts, the images exhibit patchy behavior, which is potentially harmful for medical diagnosis.

PET transmission tomography using a novel nonlocal MRF prior

In positron emission tomography, transmission scans can be performed to estimate attenuation correction factors (ACFs) which are in turn used to correct the emission scans. And such an attenuation correction is crucial for quantitatively accurate PET reconstructions. The prior model used in this work was based on our assumption that the attenuation values vary smoothly, with occasional discontinuities at anatomical borders. And on the other hand, long acquisition or scan times, although alleviating the noise effect of the count-limited scans, are blamed for patient uncomfortableness and movements. So, transmission tomography often suffers from the noise effect because of the short scan time. Thus reconstruction which is capable of overcoming the noise effect is highly needed. In this article, we apply the nonlocal prior Bayesian reconstruction method in PET transmission tomography. Resulting experimentations validate that the reconstructions using the nonlocal prior can reconstruct better transmission images and overcome noise effect even when the scan time is relatively short.

Adaptive wavelet-based deconvolution method for remote sensing imaging

Fourier-based deconvolution (FoD) techniques, such as modulation transfer function compensation, are commonly employed in remote sensing. However, the noise is strongly amplified by FoD and is colored, thus producing poor visual quality. We propose an adaptive wavelet-based deconvolution algorithm for remote sensing called wavelet denoise after Laplacian-regularized deconvolution (WDALRD) to overcome the colored noise and to preserve the textures of the restored image. This algorithm adaptively denoises the FoD result on a wavelet basis. The term "adaptive" means that the wavelet-based denoising procedure requires no parameter to be estimated or empirically set, and thus the inhomogeneous Laplacian prior and the Jeffreys hyperprior are proposed. Maximum a posteriori estimation based on such a prior and hyperprior leads us to an adaptive and efficient nonlinear thresholding estimator, and therefore WDALRD is computationally inexpensive and fast. Experimentally, textures and edges of the restored image are well preserved and sharp, while the homogeneous regions remain noise free, so WDALRD gives satisfactory visual quality.

Iterative image reconstruction for CBCT using edge-preserving prior

On-board cone-beam computed tomography (CBCT) is a new imaging technique for radiation therapy guidance, which provides volumetric information of a patient at treatment position. CBCT improves the setup accuracy and may be used for dose reconstruction. However, there is great concern that the repeated use of CBCT during a treatment course delivers too much of an extra dose to the patient. To reduce the CBCT dose, one needs to lower the total mAs of the x-ray tube current, which usually leads to reduced image quality. Our goal of this work is to develop an effective method that enables one to achieve a clinically acceptable CBCT image with as low as possible mAs without compromising quality. An iterative image reconstruction algorithm based on a penalized weighted least-squares (PWLS) principle was developed for this purpose. To preserve edges in the reconstructed images, we designed an anisotropic penalty term of a quadratic form. The algorithm was evaluated with a CT quality assurance phantom and an anthropomorphic head phantom. Compared with conventional isotropic penalty, the PWLS image reconstruction algorithm with anisotropic penalty shows better resolution preservation.

Denoising for 3-d photon-limited imaging data using nonseparable filterbanks

In this paper, we present a novel frame-based denoising algorithm for photon-limited 3-D images. We first construct a new 3-D nonseparable filterbank by adding elements to an existing frame in a structurally stable way. In contrast with the traditional 3-D separable wavelet system, the new filterbank is capable of using edge information in multiple directions. We then propose a data-adaptive hysteresis thresholding algorithm based on this new 3-D nonseparable filterbank. In addition, we develop a new validation strategy for denoising of photon-limited images containing sparse structures, such as neurons (the structure of interest is less than 5% of total volume). The validation method, based on tubular neighborhoods around the structure, is used to determine the optimal threshold of the proposed denoising algorithm. We compare our method with other state-of-the-art methods and report very encouraging results on applications utilizing both synthetic and real data.

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.

Signal-processing approaches for image-resolution restoration for TOMBO imagery

Thin observation module by bounded optics (TOMBO) is an optical system that achieves compactness and thinness by replacing a conventional large full aperture by a lenslet array with several smaller apertures. This array allows us to collect diverse low-resolution measurements. Finding an efficient way of combining these diverse measurements to make a high-resolution image is an important research problem. We focus on finding a computational method for performing the resolution restoration and evaluating the method via simulations. Our approach is based on advanced signal-processing concepts: we construct a computational data model based on Fourier optics and propose restoration algorithms based on minimization of an information-theoretic measure, called Csiszár's I divergence between two nonnegative quantities: the measured data and the hypothetical images that are induced by our algorithms through the use of our computational data model. We also incorporate Poisson and Gaussian noise processes to model the physical measurements. To solve the optimization problem, we adapt the popular expectation-maximization method. These iterative algorithms, in a multiplicative form, preserve powerful nonnegativity constraints. We further incorporate a regularization based on minimization of total variation to suppress incurring artifacts such as roughness on the surfaces of the estimates. Two sets of simulation examples show that the algorithms can produce very high-quality estimates from noiseless measurements and reasonably good estimates from noisy measurements, even when the measurements are incomplete. Several interesting and useful avenues for future work such as the effects of measurement selection are suggested in our conclusional remarks.

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.

Statistical image reconstruction from correlated data with applications to PET

Most statistical reconstruction methods for emission tomography are designed for data modeled as conditionally independent Poisson variates. In reality, due to scanner detectors, electronics and data processing, correlations are introduced into the data resulting in dependent variates. In general, these correlations are ignored because they are difficult to measure and lead to computationally challenging statistical reconstruction algorithms. This work addresses the second concern, seeking to simplify the reconstruction of correlated data and provide a more precise image estimate than the conventional independent methods. In general, correlated variates have a large non-diagonal covariance matrix that is computationally challenging to use as a weighting term in a reconstruction algorithm. This work proposes two methods to simplify the use of a non-diagonal covariance matrix as the weighting term by (a) limiting the number of dimensions in which the correlations are modeled and (b) adopting flexible, yet computationally tractable, models for correlation structure. We apply and test these methods with simple simulated PET data and data processed with the Fourier rebinning algorithm which include the one-dimensional correlations in the axial direction and the two-dimensional correlations in the transaxial directions. The methods are incorporated into a penalized weighted least-squares 2D reconstruction and compared with a conventional maximum a posteriori approach.

An expanded theoretical treatment of iteration-dependent majorize-minimize algorithms

The majorize-minimize (MM) optimization technique has received considerable attention in signal and image processing applications, as well as in statistics literature. At each iteration of an MM algorithm, one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate. The next iterate is obtained by minimizing this tangent majorant function, resulting in a sequence of iterates that reduces the cost function monotonically. A well-known special case of MM methods are expectation-maximization algorithms. In this paper, we expand on previous analyses of MM, due to Fessler and Hero, that allowed the tangent majorants to be constructed in iteration-dependent ways. Also, this paper overcomes an error in one of those earlier analyses. There are three main aspects in which our analysis builds upon previous work. First, our treatment relaxes many assumptions related to the structure of the cost function, feasible set, and tangent majorants. For example, the cost function can be nonconvex and the feasible set for the problem can be any convex set. Second, we propose convergence conditions, based on upper curvature bounds, that can be easier to verify than more standard continuity conditions. Furthermore, these conditions allow for considerable design freedom in the iteration-dependent behavior of the algorithm. Finally, we give an original characterization of the local region of convergence of MM algorithms based on connected (e.g., convex) tangent majorants. For such algorithms, cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods. This expanded treatment widens the scope of the MM algorithm designs that can be considered for signal and image processing applications, allows us to verify the convergent behavior of previously published algorithms, and gives a fuller understanding overall of how these algorithms behave.