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PubMed1997

An Efficient EM-based Training Algorithm for Feedforward Neural Networks.

A fast training algorithm is developed for two-layer feedforward neural networks based on a probabilistic model for hidden representations and the EM algorithm. The algorithm decomposes training the original two-layer networks into training a set of single neurons. The individual neurons are then trained via a linear weighted regression algorithm. Significant improvement on training speed has been made using this algorithm for several bench-mark problems. Copyright 1997 Elsevier Science Ltd. All Rights Reserved.

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PubMed1989

Applications of the expectation-maximization algorithm to quantal analysis of postsynaptic potentials.

The expectation-maximization (EM) algorithm is a robust method for maximum likelihood estimation of the parameters of an incompletely sampled distribution. It has been used to resolve the trial-to-trial amplitude fluctuations of postsynaptic potentials, when these are recorded in the presence of noise. Its use has however been limited by the need for different recursion equations for each set of conditions defined by the signal and noise processes. These equations are derived for the following conditions which arise in studies of synaptic transmission: non-gaussian noise process; quantal fluctuation; quantal variability. In addition, a constraint can be incorporated to accommodate simple and compound binomial models of transmitter release. Some advantages of these methods are illustrated by Monte Carlo simulations.

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PubMed2002

Pinhole single-photon emission tomography reconstruction based on median root prior.

The maximum likelihood expectation maximisation (ML-EM) algorithm can be used to reduce reconstruction artefacts produced by filtered backprojection (FBP) methods in pinhole single-photon emission tomography (SPET). However, ML-EM suffers from noise propagation along iterations, which leads to quantitatively unpleasant reconstruction results. To avoid this increase in noise, the median root prior (MRP) algorithm for pinhole SPET was implemented. Projection data of a line source and Picker's thyroid phantom were collected using a single-head gamma camera with a pinhole collimator. MRP was added to existing pinhole ML-EM reconstruction algorithm and the phantom studies were reconstructed using MRP, ML-EM and FBP for comparison. Coefficients of variation, contrasts and full-widths at half-maximum were calculated and showed a clear reduction in noise without significant loss of resolution or decrease in contrast when MRP was applied. MRP also produced visually pleasing images even with high iteration numbers, free of the checkerboard-type noise patterns which are typical of ML-EM images.

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PubMed2002

High-breakdown estimation of multivariate mean and covariance with missing observations.

We consider the problem of outliers in incomplete multivariate data when the aim is to estimate a measure of mean and covariance, as is the case, for example, in factor analysis. The ER algorithm of Little and Smith which combines the EM algorithm for missing data and a robust estimation step based on an M-estimator could be used in such a situation. However, the ER algorithm as originally proposed can fail to be robust in some cases, especially in high dimensions. We propose here two alternatives to avoid the problem. One is to combine a small modification of the ER algorithm with a so-called high-breakdown estimator as the starting point for the iterative procedure, and the other is to base the estimation step of the ER algorithm on a high-breakdown estimator. Among the high-breakdown estimators which are actually built to keep their robustness properties even if the number of variables is relatively large, we consider here the minimum covariance determinant estimator and the t-biweight S-estimator. Simulated and real data are used to compare and illustrate the different procedures.

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PubMed1986

Unbalanced repeated-measures models with structured covariance matrices.

The question of how to analyze unbalanced or incomplete repeated-measures data is a common problem facing analysts. We address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within-subject covariances. Models that can be fit include standard univariate and multivariate models with incomplete data, random-effects models, and models with time-series and factor-analytic error structures. We describe Newton-Raphson and Fisher scoring algorithms for computing maximum likelihood estimates, and generalized EM algorithms for computing restricted and unrestricted maximum likelihood estimates. An example fitting several models to a set of growth data is included.

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PubMed1998

Hybrid interior point training of modular neural networks.

Modular neural networks use a single gating neuron to select the outputs of a collection of agent neurons. Expectation-maximization (EM) algorithms provide one way of training modular neural networks to approximate non-linear functionals. This paper introduces a hybrid interior-point (HIP) algorithm for training modular networks. The HIP algorithm combines an interior-point linear programming (LP) algorithm with a Newton-Raphson iteration in such a way that the computational efficiency of the interior point LP methods is preserved. The algorithm is formally proven to converge asymptotically to locally optimal networks with a total computational cost that scales in a polynomial manner with problem size. Simulation experiments show that the HIP algorithm produces networks whose average approximation error is better than that of EM-trained networks. These results also demonstrate that the computational cost of the HIP algorithm scales at a slower rate than the EM-procedure and that, for small-size networks, the total computational costs of both methods are comparable.

Journal Article↗

PubMed2005

A new simple iterative reconstruction algorithm for SPECT transmission measurement.

This paper proposes a new iterative reconstruction algorithm for transmission tomography and compares this algorithm with several other methods. The new algorithm is simple and resembles the emission ML-EM algorithm in form. Due to its simplicity, it is easy to implement and fast to compute a new update at each iteration. The algorithm also always guarantees non-negative solutions. Evaluations are performed using simulation studies and real phantom data. Comparisons with other algorithms such as convex, gradient, and logMLEM show that the proposed algorithm is as good as others and performs better in some cases.

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PubMed1999

Ordered subsets algorithms for transmission tomography.

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

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PubMed1994

Fisher's method of scoring in statistical image reconstruction: comparison of Jacobi and Gauss-Seidel iterative schemes.

Many algorithms for medical image reconstruction adopt versions of the expectation-maximization (EM) algorithm. In this approach, parameter estimates are obtained which maximize a complete data likelihood or penalized likelihood, in each iteration. Implicitly (and sometimes explicitly) penalized algorithms require smoothing of the current reconstruction in the image domain as part of their iteration scheme. In this paper, we discuss alternatives to EM which adapt Fisher's method of scoring (FS) and other methods for direct maximization of the incomplete data likelihood. Jacobi and Gauss-Seidel methods for non-linear optimization provide efficient algorithms applying FS in tomography. One approach uses smoothed projection data in its iterations. We investigate the convergence of Jacobi and Gauss-Seidel algorithms with clinical tomographic projection data.

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PubMed1995

An E-M algorithm and testing strategy for multiple-locus haplotypes.

This paper gives an expectation maximization (EM) algorithm to obtain allele frequencies, haplotype frequencies, and gametic disequilibrium coefficients for multiple-locus systems. It permits high polymorphism and null alleles at all loci. This approach effectively deals with the primary estimation problems associated with such systems; that is, there is not a one-to-one correspondence between phenotypic and genotypic categories, and sample sizes tend to be much smaller than the number of phenotypic categories. The EM method provides maximum-likelihood estimates and therefore allows hypothesis tests using likelihood ratio statistics that have chi 2 distributions with large sample sizes. We also suggest a data resampling approach to estimate test statistic sampling distributions. The resampling approach is more computer intensive, but it is applicable to all sample sizes. A strategy to test hypotheses about aggregate groups of gametic disequilibrium coefficients is recommended. This strategy minimizes the number of necessary hypothesis tests while at the same time describing the structure of disequilibrium. These methods are applied to three unlinked dinucleotide repeat loci in Navajo Indians and to three linked HLA loci in Gila River (Pima) Indians. The likelihood functions of both data sets are shown to be maximized by the EM estimates, and the testing strategy provides a useful description of the structure of gametic disequilibrium. Following these applications, a number of simulation experiments are performed to test how well the likelihood-ratio statistic distributions are approximated by chi 2 distributions. In most circumstances the chi 2 grossly underestimated the probability of type I errors. However, at times they also overestimated the type 1 error probability. Accordingly, we recommended hypothesis tests that use the resampling method.

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PubMed1992

Maximum-penalized-likelihood estimation for independent and Markov-dependent mixture models.

This paper concerns the use and implementation of maximum-penalized-likelihood procedures for choosing the number of mixing components and estimating the parameters in independent and Markov-dependent mixture models. Computation of the estimates is achieved via algorithms for the automatic generation of starting values for the EM algorithm. Computation of the information matrix is also discussed. Poisson mixture models are applied to a sequence of counts of movements by a fetal lamb in utero obtained by ultrasound. The resulting estimates are seen to provide plausible mechanisms for the physiological process.

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PubMed1994

Pinhole collimation for ultra-high-resolution, small-field-of-view SPECT.

The objective of this investigation was to evaluate small-field-of-view, ultra-high-resolution pinhole collimation for a rotating-camera SPECT system that could be used to image small laboratory animals. Pinhole collimation offers distinct advantages over conventional parallel-hole collimation when used to image small objects. Since geometric sensitivity increases markedly for points close to the pinhole, small-diameter and high-magnification pinhole geometries may be useful for selected imaging tasks when used with large-field-of-view scintillation cameras. The use of large magnifications can minimize the loss of system resolution caused by the intrinsic resolution of the scintillation camera. A pinhole collimator has been designed and built that can be mounted on one of the scintillation cameras of a triple-head SPECT system. Three pinhole inserts with approximate aperture diameters of 0.6, 1.2 and 2.0 mm have been built and can be mounted individually on the collimator housing. When a ramp filter is used with a three-dimensional (3D) filtered backprojection (FBP) algorithm, the three apertures have in-plane SPECT spatial resolutions (FWHM) at 4 cm of 1.5, 1.9 and 2.8 mm, respectively. In-air point source sensitivities at 4 cm from the apertures are 0.9, 2.6 and 5.7 counts s(-1) microCi(-1) (24, 70 and 154 counts s(-1) MBq(-1)) for the 0.6, 1.2 and 2.0 mm apertures, respectively. In vitro image quality was evaluated with a micro-cold-rod phantom and a micro-Defrise phantom using both the 3D FBP algorithm and a 3D maximum likelihood-expectation maximization (ML-EM) algorithm. In vivo image quality was evaluated using two (315 and 325 g) rats. Ultra-high-resolution pinhole SPECT is an inexpensive and simple approach for imaging small animals that can be used with existing rotating-camera SPECT system.

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PubMed1994

On mixtures of three normal populations caused by monogenic inheritance: application to desipramine metabolism.

For a mixture of three normal distributions, which represent genotypes AA, Aa and aa, a method of estimation of the seven unknown parameters is proposed which works well whenever the phenotype (aa) is sufficiently well separated from the phenotype (AA, Aa). It is based on p-values of Kolmogorov's test of goodness of fit to normality. Initial parameter values for this iterative algorithm can be found by visual check and/or by using the EM algorithm. In an example of a data set of size 59 from a study of the metabolic rate of desipramine, the usefulness of this method is demonstrated. Extensions to more complex situations are feasible and are indicated at the end.

Adolescent↗

PubMed2003

Accelerated median root prior reconstruction for pinhole single-photon emission tomography (SPET).

Pinhole collimation can be used to improve spatial resolution in SPET. However, the resolution improvement is achieved at the cost of reduced sensitivity, which leads to projection images with poor statistics. Images reconstructed from these projections using the maximum likelihood expectation maximization (ML-EM) algorithms, which have been used to reduce the artefacts generated by the filtered backprojection (FBP) based reconstruction, suffer from noise/bias trade-off: noise contaminates the images at high iteration numbers, whereas early abortion of the algorithm produces images that are excessively smooth and biased towards the initial estimate of the algorithm. To limit the noise accumulation we propose the use of the pinhole median root prior (PH-MRP) reconstruction algorithm. MRP is a Bayesian reconstruction method that has already been used in PET imaging and shown to possess good noise reduction and edge preservation properties. In this study the PH-MRP algorithm was accelerated with the ordered subsets (OS) procedure and compared to the FBP, OS-EM and conventional Bayesian reconstruction methods in terms of noise reduction, quantitative accuracy, edge preservation and visual quality. The results showed that the accelerated PH-MRP algorithm was very robust. It provided visually pleasing images with lower noise level than the FBP or OS-EM and with smaller bias and sharper edges than the conventional Bayesian methods.

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PubMed2004

Incorporating genotyping uncertainty in haplotype inference for single-nucleotide polymorphisms.

The accuracy of the vast amount of genotypic information generated by high-throughput genotyping technologies is crucial in haplotype analyses and linkage-disequilibrium mapping for complex diseases. To date, most automated programs lack quality measures for the allele calls; therefore, human interventions, which are both labor intensive and error prone, have to be performed. Here, we propose a novel genotype clustering algorithm, GeneScore, based on a bivariate t-mixture model, which assigns a set of probabilities for each data point belonging to the candidate genotype clusters. Furthermore, we describe an expectation-maximization (EM) algorithm for haplotype phasing, GenoSpectrum (GS)-EM, which can use probabilistic multilocus genotype matrices (called "GenoSpectrum") as inputs. Combining these two model-based algorithms, we can perform haplotype inference directly on raw readouts from a genotyping machine, such as the TaqMan assay. By using both simulated and real data sets, we demonstrate the advantages of our probabilistic approach over the current genotype scoring methods, in terms of both the accuracy of haplotype inference and the statistical power of haplotype-based association analyses.

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PubMed1995

A fast and stable maximum a posteriori conjugate gradient reconstruction algorithm.

We have derived a maximum a posteriori (MAP) approach for iterative reconstruction based on a weighted least-squares conjugate gradient (WLS-CG) algorithm. The WLS-CG algorithm has been shown to have initial convergence rates up to 10x faster than the maximum-likelihood expectation maximization (ML-EM) algorithm, but WLS-CG suffers from rapidly increasing image noise at higher iteration numbers. In our MAP-CG algorithm, the increasing noise is controlled by a Gibbs smoothing prior, resulting in stable, convergent solutions. Our formulation assumes a Gaussian noise model for the likelihood function. When a linear transformation of the pixel space is performed (the "relaxation" acceleration method), the MAP-CG algorithm obtains a low-noise, stable solution (one that does not change with further iterations) in 10-30 iterations, compared to 100-200 iterations for MAP-EM. Each iteration of MAP-CG requires approximately the same amount of processing time as one iteration of ML-EM or MAP-EM. We show that the use of an initial image estimate obtained from a single iteration of the Chang method helps the algorithm to converge faster when acceleration is not used, but does not help when acceleration is applied. While both the WLS-CG and MAP-CG methods suffer from the potential for obtaining negative pixel values in the iterated image estimates, the use of the Gibbs prior substantially reduces the number of pixels with negative values and restricts them to regions of little or no activity. We use SPECT data from simulated hot-sphere phantoms and from patient studies to demonstrate the advantages of the MAP-CG algorithm. We conclude that the MAP-CG algorithm requires 10%-25% of the processing time of EM techniques, and provides images of comparable or superior quality.

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PubMed2006

Model-based image reconstruction for four-dimensional PET.

Positron emission tonography (PET) is useful in diagnosis and radiation treatment planning for a variety of cancers. For patients with cancers in thoracic or upper abdominal region, the respiratory motion produces large distortions in the tumor shape and size, affecting the accuracy in both diagnosis and treatment. Four-dimensional (4D) (gated) PET aims to reduce the motion artifacts and to provide accurate measurement of the tumor volume and the tracer concentration. A major issue in 4D PET is the lack of statistics. Since the collected photons are divided into several frames in the 4D PET scan, the quality of each reconstructed frame degrades as the number of frames increases. The increased noise in each frame heavily degrades the quantitative accuracy of the PET imaging. In this work, we propose a method to enhance the performance of 4D PET by developing a new technique of 4D PET reconstruction with incorporation of an organ motion model derived from 4D-CT images. The method is based on the well-known maximum-likelihood expectation-maximization (ML-EM) algorithm. During the processes of forward- and backward-projection in the ML-EM iterations, all projection data acquired at different phases are combined together to update the emission map with the aid of deformable model, the statistics is therefore greatly improved. The proposed algorithm was first evaluated with computer simulations using a mathematical dynamic phantom. Experiment with a moving physical phantom was then carried out to demonstrate the accuracy of the proposed method and the increase of signal-to-noise ratio over three-dimensional PET. Finally, the 4D PET reconstruction was applied to a patient case.

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PubMed2003

Estimation of the dominance merit in noninbred populations without recourse to its inverted relationship matrix.

This study presents a new approach to obtain dominance estimates without using the full Henderson's mixed model equations (MMEs) related to an additive plus dominance animal model. This reduction could decrease substantially the computing time and hence its cost. In contrast to a procedure that we proposed before, the method developed in this paper does not require D(-1) and provides best linear unbiased prediction (BLUP) of genetic values that is close to that given by processing the full MMEs. In the previous study, we also elaborated an algorithm (denoted xi-REML) in order to approximate restricted maximum likelihood estimation of variance components via the expectation maximization (EM) algorithm. The xi-REML algorithm has been modified to be adapted to our new resolution approach. Through a numerical example, we show that there is a good agreement between REML-(EM), xi-REML and modified xi-REML estimates and that the latter algorithm is more efficient than our first proposition in terms of computing time and memory conservation.

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