Speeding up the EM algorithm for mixture model-based segmentation of magnetic resonance images

Mixture models implemented via the expectation-maximization (EM) algorithm are being increasingly used in a wide range of problems in pattern recognition such as image segmentation. However, the EM algorithm requires considerable computational time in its application to huge data sets such as a three-dimensional magnetic resonance (MR) image of over 10 million voxels. Recently, it was shown that a sparse, incremental version of the EM algorithm could improve its rate of convergence. In this paper, we show how this modified EM algorithm can be speeded up further by adopting a multiresolution kd-tree structure in performing the E-step. The proposed algorithm outperforms some other variants of the EM algorithm for segmenting MR images of the human brain. (C) 2004 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved.

Mixture modelling for cluster analysis

Cluster analysis via a finite mixture model approach is considered. With this approach to clustering, the data can be partitioned into a specified number of clusters g by first fitting a mixture model with g components. An outright clustering of the data is then obtained by assigning an observation to the component to which it has the highest estimated posterior probability of belonging; that is, the ith cluster consists of those observations assigned to the ith component (i = 1,..., g). The focus is on the use of mixtures of normal components for the cluster analysis of data that can be regarded as being continuous. But attention is also given to the case of mixed data, where the observations consist of both continuous and discrete variables.

Eutectic modification of Al-Si alloys with rare earth metals

The effects of different concentrations of individual additions of rare earth metals (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu) on eutectic modification in Al-10mass%Si has been studied by thermal analysis and optical microscopy. According to the twin-plane re-entrant edge (TPRE) and impurity induced twinning mechanism, rare earth metals with atomic radii of about 1.65 times larger than that of silicon, are possible candidates for eutectic modification. All of the rare earth elements caused a depression of the eutectic growth temperature, but only Eu modified the eutectic silicon to a fibrous morphology. At best, the remaining elements resulted in only a small degree of refinement of the plate-like silicon. The samples were also quenched during the eutectic arrest to examine the eutectic solidification modes. Many of the rare-earth additions significantly altered the eutectic solidification mode from that of the unmodified alloy. It is concluded that the impurity induced twinning model of modification, based on atomic radius alone, is inadequate and other mechanisms are essential for the modification process. Furthermore, modification and the eutectic nucleation and growth modes are controlled independently of each other.

Eutectic solidification and its role in casting porosity formation

Understanding and controlling the eutectic solidification process in Al-Si alloys permits prediction of the formation of casting porosity, eventually leading to methods for its control and elimination. In addition, it enables control of eutectic structure, silicon morphology, and eutectic grain size to further improve the alloy properties. This paper presents the current understanding of eutectic solidification in hypoeutectic Al-Si foundry alloys and the relationship between eutectic solidification and porosity formation. New concepts in engineering eutectic solidification are also explored.

Eutectic nucleation in Al-Si alloys

In addition to a change in silicon morphology, modification of aluminium-silicon alloys with strontium or sodium increases the size of the eutectic grains. To determine the mechanism responsible, eutectic solidification in commercial purity and ultra-high purity aluminium-si I icon alloys, with and without strontium additions, was examined by a quenching technique. In the commercial unmodified alloy, nucleation was prolific while in the high-purity unmodified alloy few eutectic grains nucleated. The addition of strontium to the commercial alloy reduced the number of eutectic grains that nucleated. Addition of strontium to the high-purity alloy did not significantly alter nucleation. It is concluded that commercial purity alloys contain a large number of potent nuclei that are susceptible to poisoning by impurity modification. The flake-to-fibre transition that occurs with impurity modification is shown to be independent of any change in eutectic nucleation mode and frequency. (C) 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Aluminium phosphide as a eutectic grain nucleus in hypoeutectic Al-Si alloys

Aluminium phosphide (AlP) particles arc often suggested to be the nucleation site for eutectic silicon in Al-Si alloys, since both the crystal structure and lattice parameter of AlP (crystal structure: cubic K(4) over bar m; lattice parameter: 5.421 Angstrom) are close to that of silicon (cubic Fd3m, 5.431 Angstrom), and the melting point is higher than the Al-Si eutectic temperature. However, the crystallographic relationships between AlP particles and the surrounding eutectic silicon are seldom reported due to the difficulty in analysing the AlP particles, which react with water during sample preparation for polishing. in this study, the orientation relationships between AlP and Si are analysed by transmission electron microscopy using focused ion-beam milling for sample preparation to investigate the nucleation mechanism of eutectic silicon on AlP. The results show a clear and direct lattice relationship between centrally located AlP particles and the surrounding silicon in the hypoeutectic Al-Si alloy.

Eutectic grains in unmodified and strontium-modified hypoeutectic aluminum-silicon alloys

Additions of strontium to hypoeutectic aluminum-silicon alloys modify the morphology of the eutectic silicon phase from a coarse platelike structure to a fine fibrous structure. Thermal analysis, interrupted solidification, and microstructural examination of sand castings in this work revealed that, in addition to a change in silicon morphology, modification with strontium also causes an increase in the size of eutectic grains. The eutectic grain size increases because fewer grains nucleate, possibly due to poisoning of the phosphorus-based nucleants, that are active in the unmodified alloy. A simple growth model is developed to estimate the interface velocity during solidification of a eutectic grain. The model confirms, independent of microstructural observations, that the addition of 100 ppm strontium increases the eutectic grain size by at least an order of magnitude compared with the equivalent unmodified alloy. The model predicts that the growth velocity varies significantly during eutectic growth. At low strontium levels, these variations may be sufficient to cause transitions between flake and fibrous silicon morphologies depending on the casting conditions. The model can be used to rationally interpret the eutectic grain structure and silicon morphology of fully solidified aluminum-silicon castings and, when coupled with reliable thermal data, can be used to estimate the eutectic grain size.

Adsorption of N-2, Ar, acetone, chloroform and acetone-chloroform mixture on carbon black in the framework of molecular layer structure theory (MLST)

Adsorption of pure nitrogen, argon, acetone, chloroform and acetone-chloroform mixture on graphitized thermal carbon black is considered at sub-critical conditions by means of molecular layer structure theory (MLST). In the present version of the MLST an adsorbed fluid is considered as a sequence of 2D molecular layers, whose Helmholtz free energies are obtained directly from the analysis of experimental adsorption isotherm of pure components. The interaction of the nearest layers is accounted for in the framework of mean field approximation. This approach allows quantitative correlating of experimental nitrogen and argon adsorption isotherm both in the monolayer region and in the range of multi-layer coverage up to 10 molecular layers. In the case of acetone and chloroform the approach also leads to excellent quantitative correlation of adsorption isotherms, while molecular approaches such as the non-local density functional theory (NLDFT) fail to describe those isotherms. We extend our new method to calculate the Helmholtz free energy of an adsorbed mixture using a simple mixing rule, and this allows us to predict mixture adsorption isotherms from pure component adsorption isotherms. The approach, which accounts for the difference in composition in different molecular layers, is tested against the experimental data of acetone-chloroform mixture (non-ideal mixture) adsorption on graphitized thermal carbon black at 50 degrees C. (C) 2005 Elsevier Ltd. All rights reserved.

Application of mixture models to detect differentially expressed genes

An important and common problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. As this problem concerns the selection of significant genes from a large pool of candidate genes, it needs to be carried out within the framework of multiple hypothesis testing. In this paper, we focus on the use of mixture models to handle the multiplicity issue. With this approach, a measure of the local FDR (false discovery rate) is provided for each gene. An attractive feature of the mixture model approach is that it provides a framework for the estimation of the prior probability that a gene is not differentially expressed, and this probability can subsequently be used in forming a decision rule. The rule can also be formed to take the false negative rate into account. We apply this approach to a well-known publicly available data set on breast cancer, and discuss our findings with reference to other approaches.

Using mixture models to detect differentially expressed genes

An important and common problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. As this problem concerns the selection of significant genes from a large pool of candidate genes, it needs to be carried out within the framework of multiple hypothesis testing. In this paper, we focus on the use of mixture models to handle the multiplicity issue. With this approach, a measure of the local false discovery rate is provided for each gene, and it can be implemented so that the implied global false discovery rate is bounded as with the Benjamini-Hochberg methodology based on tail areas. The latter procedure is too conservative, unless it is modified according to the prior probability that a gene is not differentially expressed. An attractive feature of the mixture model approach is that it provides a framework for the estimation of this probability and its subsequent use in forming a decision rule. The rule can also be formed to take the false negative rate into account.

Three-dimensional analysis of eutectic grains in hypoeutectic Al-Si alloys

An approach to the qualitative analysis of quenched microstructures in three dimensions is presented and demonstrated on unmodified and Sr-modified Al-10% Si samples. The samples were repeatedly polished to obtain a series of digital images through the depth of the microstructure. A three-dimensional reconstruction of the microstructure was obtained by assembling the images of the serial sections. Reconstructions were made of unmodified and Sr-modified Al-Si eutectic grains that were quenched during eutectic solidification. The three-dimensional reconstructions show that strontium modification changes the size and morphology of the Al-Si eutectic grains. Sr-modified eutectic grains are large approximately spherical grains and grow with a high interface velocity. In the unmodified alloy, many small eutectic grains grow from the dendrite arm tips. The unmodified eutectic grains appear to grow from the dendrite tips into the undercooled liquid, rather than back-filling the dendrite envelope, possibly continuing to grow in the same manner as the equiaxed dendrites. (C) 2004 Elsevier B.V. All rights reserved.