Keywords: Gaussian, segmentation, concrete, micro-structure, porosity, backscatter, electron, image.

Properties of hardened concrete strongly depend on its microstructure and the analysis of images from optical and electron microscope is a vital tool in the search of knowledge on the microstructure of concrete. Work undertaken at Queen's [1] has demonstrated the role the analysis of scanning electron microscope (SEM) images can play.

For a given backscatter electron (BSE) image materials with differing molecular weight appear as different shades of grey; the greater the molecular weight the lighter the shade. Typically within an image the porosity of the sample will be depicted by black along with the darkest shades of grey while the unhydrated cement shows up as white along with the lightest shades of grey. The hydrated phases of the cement reside in the middle shades of grey between the extreme shades of the voids and the unhydrated cement. To date very limited work has gone into the phase segmentation process. Currently a simple thresholding approach has been adopted. However the boundaries between phases can be rather arbitrary. The segmentation outlined below provides a rigorous and repeatable alternative. It is assumed that each phase in the cement paste will be represented on the backscatter image obtained from an electron microscope by a Gaussian distribution of greyscale, with each pixel from the image being associated with a particular Gaussian distribution. Through parameterization of the Gaussian distribution, this provides for very general modelling. Other distributions are eminently feasible also, including Poisson to model background shot noise. The overall population of pixel values has the mixture density:

where is the Gaussian distribution associated with the th phase, is the mixing or prior probabilities, which will sum to , is the number of phases considered and is a vector of parameters which define the Gaussian distributions i.e. and for to .

The likelihood function in the Gaussian mixture case is defined as [2]:

The solution to this function is obtained by computing the values of and that maximize . This allows the Gaussian distributions to be defined and hence the phases. An iterative solution is provided by the expectation-maximization (EM) algorithm of Dempster et al. [3].

A Bayes factor, the ratio of one model's odds to another, is an appropriate means of assessing the suitability of such a model [4]. A model in this context is usually the number of phases, . In order to assess the effectiveness of the fit of the segmented models (mixture models for given ) against another, a Bayes factor of a mixture model is compared against an alternative mixture model . The Bayes factor can be approximated by the Bayes information criterion (BIC):

BIC = ( maximized likelihood function) - (no. of parameters)
BIC =

The maximized likelihood value is obtained from the EM algorithm used to fit the model parameters.

The paper describes this innovative approach to segmentation and illustrates its use on a sample image. The major emphasis is on the identification of the porosity phase. However the segmentation provides information regarding the other phases, this will also be discussed.

1

P.A.M. Basheer, L. Basheer D.A. Lange and A.E. Long, "Role of thresholding to determine the size of interfacial transition Zone", American Concrete Institute, Special publication SP 189, 1999.

2

J.D. Banfield and A.E. Raftery, "Model-based Gaussian and non-Gaussian clustering", Biometrics 49, 803-821,1993. doi:10.2307/2532201

3

A.P. Dempster, N.M. Laird and D.B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm". Journal of the Royal Statistical Society Series B 39,1-22, 1977.

4

R.E. Kass and A.E. Raftert, "Bayes factors", Journal of the American Statistical Association 90, 773-795, 1995. doi:10.2307/2291091

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