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Wavelet based multiresolution expectation maximization image reconstruction algorithm for positron emission tomography.

Maximum Likelihood (ML) estimation based Expectation Maximization (EM) [IEEE Trans Med Imag, MI-1 (2) (1982) 113] reconstruction algorithm has shown to provide good quality reconstruction for positron emission tomography (PET). Our previous work [IEEE Trans Med Imag, 7(4) (1988) 273; Proc IEEE EMBS Conf, 20(2/6) (1998) 759] introduced the multigrid (MG) and multiresolution (MR) concept for PET image reconstruction using EM. This work transforms the MGEM and MREM algorithm to a Wavelet based Multiresolution EM (WMREM) algorithm by extending the concept of switching resolutions in both image and data spaces. The MR data space is generated by performing a 2D-wavelet transform on the acquired tube data that is used to reconstruct images at different spatial resolutions. Wavelet transform is used for MR reconstruction as well as adapted in the criterion for switching resolution levels. The advantage of the wavelet transform is that it provides very good frequency and spatial (time) localization and allows the use of these coarse resolution data spaces in the EM estimation process. The MR algorithm recovers low-frequency components of the reconstructed image at coarser resolutions in fewer iterations, reducing the number of iterations required at finer resolution to recover high-frequency components. This paper also presents the design of customized biorthogonal wavelet filters using the lifting method that are used for data decomposition and image reconstruction and compares them to other commonly known wavelets.

Algorithms↗

PubMed2002

Iterative versus filtered backprojection reconstruction for statistical parametric mapping of PET activation measurements: a comparative case study.

The significance of task-induced cerebral blood flow responses, assessed using statistical parametric mapping, depends, among other things, on the signal-to-noise ratio (SNR) of these responses. Generally, positron emission tomography sinograms of H(2)(15)O activation studies are reconstructed using filtered backprojection (FBP). Alternatively, the acquired data can be reconstructed using an iterative reconstruction procedure. It has been demonstrated that the application of iterative reconstruction methods improves image SNR as compared with FBP. The aim of this study was to compare FBP with iterative reconstruction, to assess the statistical power of H(2)(15)O-PET activation studies using statistical parametric mapping. For this case study, PET data originating from a bimanual motor task were reconstructed using both FBP and maximum likelihood expectation maximization (ML-EM), an iterative algorithm. Both resulting data sets were statistically analyzed using statistical parametric mapping. It was found, with this dataset, that the statistical analysis of the iteratively reconstructed data confirm the a priori expected physiological response. In addition, increased Z scores were obtained in the iteratively reconstructed data. In particular, for the expected task-related response, activation of the posterior border of the left angular gyrus, the Z score increased from 3.00 to 3.96. Furthermore, the number of statistically significant clusters doubled while their volume increased by more than 50%. In conclusion, iterative reconstruction has the potential to increase the statistical power in H(2)(15)O-PET activation studies as compared with FBP reconstruction.

Brain Mapping↗

PubMed2006

PET energy-based scatter estimation and image reconstruction with energy-dependent corrections.

In this paper we propose a comprehensive energy-based scatter correction approach for positron emission tomography (PET). We take advantage of the marked difference between the energy spectra of the unscattered and scattered photons, and use the detailed energy information that comes with the list-mode data for the estimation of the scattered events distribution in the data space. Also, inside the maximum-likelihood expectation maximization (ML-EM) image reconstruction algorithm, we introduce energy-dependent factors that individualize the correction terms for each event, given its position and energy information. The central piece of our approach is the two-dimensional detector energy response model represented as a linear combination of four components, each one representing a particular state a PET event can be found in: both photons unscattered, the second scattered while the first not, the first photon scattered while the second not and both photons scattered. For a set of events collected in the vicinity of a point in the projection space, the coefficient of each component is determined by applying a statistical estimator. As a result we obtain the number of scattered events that are in the given set. The model also gives us the variation of scatter fraction with the photon pair energies for that particular position in the data space. A simulation study that demonstrates the proposed methods is presented.

Algorithms↗

PubMed2005

A general model for detecting genetic determinants underlying longitudinal traits with unequally spaced measurements and nonstationary covariance structure.

A mixture model for determining quantitative trait loci (QTL) affecting growth trajectories has been proposed in the literature. In this article, we extend this model to a more general situation in which longitudinal traits for each subject are measured at unequally spaced time intervals, different subjects have different measurement patterns, and the residual correlation within subjects is nonstationary. We derive an EM-simplex hybrid algorithm to estimate the allele frequencies, Hardy-Weinberg disequilibrium, and linkage disequilibrium between QTL in the original population and parameters contained in the growth equation and in the covariance structure. A worked example of head circumference growth in 145 children is used to validate our extended model. A simulation study is performed to examine the statistical properties of the parameter estimation obtained from this example. Finally, we discuss the implications and extensions of our model for detecting QTL that affect growth trajectories.

Algorithms↗

PubMed1999

Estimating linkage disequilibrium between a polymorphic marker locus and a trait locus in natural populations.

Positional cloning of gene(s) underlying a complex trait requires a high-resolution linkage map between the trait locus and genetic marker loci. Recent research has shown that this may be achieved through appropriately modeling and screening linkage disequilibrium between the candidate marker locus and the major trait locus. A quantitative genetics model was developed in the present study to estimate the coefficient of linkage disequilibrium between a polymorphic genetic marker locus and a locus underlying a quantitative trait as well as the relevant genetic parameters using the sample from randomly mating populations. Asymptotic covariances of the maximum-likelihood estimates of the parameters were formulated. Convergence of the EM-based statistical algorithm for calculating the maximum-likelihood estimates was confirmed and its utility to analyze practical data was exploited by use of extensive Monte-Carlo simulations. Appropriateness of calculating the asymptotic covariance matrix in the present model was investigated for three different approaches. Numerical analyses based on simulation data indicated that accurate estimation of the genetic parameters may be achieved if a sample size of 500 is used and if segregation at the trait locus explains not less than a quarter of phenotypic variation of the trait, but the study reveals difficulties in predicting the asymptotic variances of these maximum-likelihood estimates. A comparison was made between the statistical powers of the maximum-likelihood analysis and the previously proposed regression analysis for detecting the disequilibrium.

Genetic Markers↗

PubMed2004

Measuring the fit of sequence data to phylogenetic model: allowing for missing data.

It is fundamentally important to assess the fit of data to model in phylogenetic and evolutionary studies. Phylogenetic methods using molecular sequences typically start with a multiple alignment. It is possible to measure the fit of data to model expectations of data, for example, via the likelihood-ratio (G) test or the X(2) test, if all sites in all sequences have an unambiguous residue. However, nearly all alignments of interest contain sites (columns of the alignment) with missing data, that is, ambiguous nucleotides, gaps, or unsequenced regions, which must presently be removed before using the above tests. Unfortunately, this is often either undesirable or impractical, as it will discard much of the data. Here, we show how iterative ML estimators may directly estimate the site-pattern probabilities for columns with missing data, given only standard i.i.d. assumptions. The optimization may use an EM or Newton algorithm, or any other hill-climbing approach. The resulting optimal likelihood under the unconstrained or multinomial model may be compared directly with the likelihood of the data coming from the model (a G statistic). Alternatively the modified observed and the expected frequencies of site patterns may be compared using a X(2) test. The distribution of such statistics is best assessed using appropriate simulations. The new method is applicable to models using codons or paired sites. The methods are also useful with Hadamard conjugations (spectral analysis) and are illustrated with these and with ML evolutionary models that allow site-rate variability.

Amino Acid Sequence↗

PubMed1986

Nonparametric methods for survival/sacrifice experiments.

In many carcinogenicity studies, the time to disease occurrence is not clinically observable; a survival/sacrifice experiment is considered for nonparametric inference about the rate of disease occurrence. A multistate model for disease development and death is considered and an algorithm of the EM type for maximum likelihood estimation is obtained. Questions of identifiability and estimability are addressed. Under the model, interval hazards for disease occurrence are identifiable for intervals defined by the sacrifice times. A score test is developed appropriate for the comparison of two groups with respect to disease development without need of any assumption concerning lethality of the disease concerned.

Animals↗

PubMed1996

Pattern-mixture models for multivariate incomplete data with covariates.

Pattern-mixture models stratify incomplete data by the pattern of missing values and formulate distinct models within each stratum. Pattern-mixture models are developed for analyzing a random sample on continuous variables y(1), y(2) when values of y(2) are nonrandomly missing. Methods for scalar y(1) and y(2) are here generalized to vector y(1) and y(2) with additional fixed covariates x. Parameters in these models are identified by alternative assumptions about the missing-data mechanism. Models may be underidentified (in which case additional assumptions are needed), just-identified, or overidentified. Maximum likelihood and Bayesian methods are developed for the latter two situations, using the EM and SEM algorithms, direct and interactive simulation methods. The methods are illustrated on a data set involving alternative dosage regimens for the treatment of schizophrenia using haloperidol and on a regression example. Sensitivity to alternative assumptions about the missing-data mechanism is assessed, and the new methods are compared with complete-case analysis and maximum likelihood for a probit selection model.

Algorithms↗

PubMed2000

Performance of ordered-subset reconstruction algorithms under conditions of extreme attenuation and truncation in myocardial SPECT.

UNLABELLED: We studied the bias and variance characteristics of the ordered-subset expectation maximization (OSEM) and rescaled block-iterative EM (RBIEM) iterative reconstruction algorithms in myocardial SPECT under extreme, but realistic, conditions. METHOD: We used the 2-dimensional mathematic cardiac torso phantom to simulate 2 patient anatomies: a large male with a raised diaphragm and a female with large breast size, approximating extreme cases of attenuation conditions found in the clinic. For each anatomy, realistic 201Tl projection data were simulated for a 180 degrees acquisition arc. Three cases of truncation for a 90 degrees-configured dual detector system were simulated: no truncation, moderate truncation, and extreme truncation. For each case, an ensemble of 250 noise simulations was generated, and each noisy dataset was reconstructed with the OSEM and RBIEM algorithms. The reconstructions modeled only the effects of nonuniform attenuation and used a range of subset configurations. Over the ensemble, we computed means and variances of activity in 8 regions of interest (ROIs) in the heart as a function of iteration. RESULTS: Under conditions of no truncation and moderate truncation, the results from OSEM and RBIEM were very close to those from maximum-likelihood EM (MLEM); in all cases, the difference in ROI means was <2.5%. For extreme truncation, the errors increased to as much as 11% with OSEM, but these were no greater than the errors for MLEM under the same conditions. The OSEM algorithm with 2 views per subset was found to result in much higher variance of ROI estimates for the same bias as compared with RBIEM or OSEM with 4 or more views per subset. CONCLUSION: The OSEM and RBIEM algorithms are at least as robust to highly attenuating patients and truncation as MLEM algorithm and can be adequate substitutes for MLEM, even in extreme cases. Clinical users should apply the smallest number of subsets that can be accommodated by allowable processing time to reduce image noise and variance in quantitative estimates.

Algorithms↗

PubMed1998

Iterative and analytical reconstruction algorithms for varying-focal-length cone-beam projections.

In single photon emission computed tomography (SPECT), a varying-focal-length cone-beam collimator can be used to reduce the truncation problem and to maintain sensitivity when imaging the organ of interest. The collimator is constructed so that the collimator holes focus to a circular symmetric, spatially varying, focal point function. The focal length increases radially from the shortest focal length at the centre to the longest focal length at the periphery of the collimator. This paper describes a 3D backprojection-filtering (BF) algorithm for this varying-focal-length cone-beam geometry. The proposed algorithm is compared to an iterative ML-EM (maximum likelihood-expectation maximization) algorithm. The 3D Hoffman brain phantom and Defrise phantom are used in computer simulations. Since the maximum tilt angle of the projection rays is small for most realistic imaging geometries, the proposed algorithm provides a good approximation. When a circular orbit is used, the BF algorithm gives an exact reconstruction of the central slice.

Algorithms↗

PubMed2001

[Optimized logistics in the prehospital management of acute stroke].

OBJECTIVE: Current management of acute stroke is characterised by an aggressive approach including specific therapy i. e. reperfusion therapy. However currently stroke patients often arrive too late in hospitals offering adequate treatment. Therefore optimized logistics play a predominant role in modern stroke management. AIMS OF THE STUDY: 1. Does teaching of EMS staff and the public result in reduced prehospital latencies 2. Will EMS personnel be able to effectively screen patients potentially suitable for thrombolysis? METHODS: During a six week-period all EMS patients presenting with possible signs of an acute stroke were prospectively registered (period 1). Data of interest were age, mode of primary contact, prehospital latencies, mode of transportation, destination and final diagnosis. Next an algorithm was established allowing EMS personnel to transfer patients with an assumed stroke to the best suitable hospital. Teaching comprised clinical signs, indication of CT scanning, pathophysiology, specific therapeutic options (thrombolysis), and criteria to identify patients suitable for thrombolysis. In a second step the public was continuously taught about stroke symptoms and the necessity to instantly seek EMS assistance. After 12 months data were compared to baseline (period 2). RESULTS: (period 2 vs. Period 1): Rate of patients transferred to a stroke center: 60 % vs. 54 %; rate of those transported to hospitals not offering CT scans: 17 % vs. 26 % (p < 0.05). Percentage of patients primarily contacting the EMS system: 33 % vs. 24 %. Median interval between onset of symptoms and emergency call: 54 vs. 263 minutes Median interval between the emergency call and arrival at the emergency department: 44 vs. 58 minutes (p < 0.01). Rate of patients admitted with a diagnosis other than stroke: 18 % vs. 25 % (n. s.). Median interval between onset of symptoms and hospital admission: 140 vs. 368 minutes (p < 0.001). Median age: 69 vs. 75 years (p < 0.01). CONCLUSION: This study demonstrates the efficacy of educational efforts in reducing latencies and in screening patients potentially suitable for thrombolysis. Future efforts will comprise more intense education of a high risk subpopulation.

Age Factors↗

PubMed2005

Clustering genes using gene expression and text literature data.

Clustering of gene expression data is a standard technique used to identify closely related genes. In this paper, we develop a new clustering algorithm, MSC (Multi-Source Clustering), to perform exploratory analysis using two or more diverse sources of data. In particular, we investigate the problem of improving the clustering by integrating information obtained from gene expression data with knowledge extracted from biomedical text literature. In each iteration of algorithm MSC, an EM-type procedure is employed to bootstrap the model obtained from one data source by starting with the cluster assignments obtained in the previous iteration using the other data sources. Upon convergence, the two individual models are used to construct the final cluster assignment. We compare the results of algorithm MSC for two data sources with the results obtained when the clustering is applied on the two sources of data separately. We also compare it with that obtained using the feature level integration method that performs the clustering after simply concatenating the features obtained from the two data sources. We show that the z-scores of the clustering results from MSC are better than that from the other methods. To evaluate our clusters better, function enrichment results are presented using terms from the Gene Ontology database. Finally, by investigating the success of motif detection programs that use the clusters, we show that our approach integrating gene expression data and text data reveals clusters that are biologically more meaningful than those identified using gene expression data alone.

Artificial Intelligence↗

PubMed2007

On classification with incomplete data.

We address the incomplete-data problem in which feature vectors to be classified are missing data (features). A (supervised) logistic regression algorithm for the classification of incomplete data is developed. Single or multiple imputation for the missing data is avoided by performing analytic integration with an estimated conditional density function (conditioned on the observed data). Conditional density functions are estimated using a Gaussian mixture model (GMM), with parameter estimation performed using both Expectation-Maximization (EM) and Variational Bayesian EM (VB-EM). The proposed supervised algorithm is then extended to the semisupervised case by incorporating graph-based regularization. The semisupervised algorithm utilizes all available data-both incomplete and complete, as well as labeled and unlabeled. Experimental results of the proposed classification algorithms are shown.

Algorithms↗

PubMed1999

Iterative X-ray Cone-Beam Tomography for Metal Artifact Reduction and Local Region Reconstruction.

: X-ray cone-beam reconstruction from incomplete projection data has important practical applications, especially in microtomography. We developed expectation maximization (EM)-type and algebraic reconstruction technique (ART)-type iterative cone-beam reconstruction algorithms for metal artifact reduction and local reconstruction from truncated data. These iterative algorithms are adapted from the emission computerized tomography (CT) EM formula and the ART. A key step in our iterative algorithms is introduction of a projection mask and computation of a 3-D spatially varying relaxation factor that allows compensation for beam divergence and data incompleteness. The algorithms are simulated with projection data synthesized from mathematical phantoms. In simulation, the EM-type and ART-type iterative algorithms are demonstrated to be effective for metal artifact reduction and local region reconstruction. They perform similarly in terms of visual quality, image noise, and discrepancy between measured and reprojected data. The EM-type and ART-type iterative cone-beam reconstruction algorithms have potential for metal artifact reduction and local region reconstruction in X-ray CT.

Journal Article↗

PubMed2002

Quantitative analysis of MRI signal abnormalities of brain white matter with high reproducibility and accuracy.

PURPOSE: To assess the reproducibility and accuracy compared to radiologists of three automated segmentation pipelines for quantitative magnetic resonance imaging (MRI) measurement of brain white matter signal abnormalities (WMSA). MATERIALS AND METHODS: WMSA segmentation was performed on pairs of whole brain scans from 20 patients with multiple sclerosis (MS) and 10 older subjects who were positioned and imaged twice within 30 minutes. Radiologist outlines of WMSA on 20 sections from 16 patients were compared with the corresponding results of each segmentation method. RESULTS: The segmentation method combining expectation-maximization (EM) tissue segmentation, template-driven segmentation (TDS), and partial volume effect correction (PVEC) demonstrated the highest accuracy (the absolute value of the Z-score was 0.99 for both groups of subjects), as well as high interscan reproducibility (repeatability coefficient was 0.68 mL in MS patients and 1.49 mL in aging subjects). CONCLUSION: The addition of TDS to the EM segmentation and PVEC algorithms significantly improved the accuracy of WMSA volume measurements, while also improving measurement reproducibility.

Aging↗

PubMed2004

A simple channel estimator for space-time coded OFDM systems in rapid fading channels.

A simple channel estimator for space-time coded orthogonal frequency division multiplexing (OFDM) systems in rapid fading channels is proposed. The channels at the training bauds are estimated using the EM (expectation-maximization) algorithm, while the channels at the data bauds are estimated based on the method for modelling the time-varying channel as the linear combination of several time-invariant "Doppler channels". Computer simulations showed that this estimator outperforms the decision-directed tracking in rapid fading channels and that the performance of this method can be improved by iteration.

Algorithms↗

PubMed1998

Spiral CT image deblurring for cochlear implantation.

Cochlear implantation is the standard treatment for profound hearing loss. Preimplantation and postimplantation spiral computed tomography (CT) is essential in several key clinical and research aspects. The maximum image resolution with commercial spiral CT scanners is insufficient to define clearly anatomical features and implant electrode positions in the inner ear. In this paper, we develop an expectation-maximization (EM)-like iterative deblurring algorithm to achieve spiral CT image super-resolution for cochlear implantation, assuming a spatially invariant linear spiral CT system with a three-dimensional (3-D) separable Gaussian point spread function (PSF). We experimentally validate the 3-D Gaussian blurring model via phantom measurement and profile fitting. The imaging process is further expressed as convolution of an isotropic 3-D Gaussian PSF and a blurred underlying volumetric image. Under practical conditions, an oblique reconstructed section is approximated as convolution of an isotropic two-dimensional (2-D) Gaussian PSF and the corresponding actual cross section. The spiral CT image deblurring algorithm is formulated with sieve and resolution kernels for suppressing noise and edge artifacts. A typical cochlear cross section is used for evaluation, demonstrating a resolution gain up to 30%40% according to the correlation criterion. Physical phantoms, preimplantation and postimplantation patients are reconstructed into volumes of 0.1-mm cubic voxels. The patient images are digitally unwrapped along the central axis of the cochlea and the implanted electrode array respectively, then oblique sections orthogonal to the central axis formed. After deblurring, representation of structural features is substantially improved in all the cases.

Algorithms↗

PubMed2001

Analysis of capture-recapture data with a Rasch-type model allowing for conditional dependence and multidimensionality.

In this article, we show that, if subjects are assumed to be homogeneous within a finite set of latent classes, the basic restrictions of the Rasch model (conditional independence and unidimensionality) can be relaxed in a flexible way by simply adding appropriate columns to a basic design matrix. When discrete covariates are available so that subjects may be classified into strata, we show how a joint modeling approach can achieve greater parsimony. Parameter estimates may be obtained by maximizing the conditional likelihood (given the total number of captures) with a combined use of the EM and Fisher scoring algorithms. We also discuss a technique for obtaining confidence intervals for the size of the population under study based on the profile likelihood.

Algorithms↗