A commonly used tool for estimating the parameters of a mixture model is the Expectation–Maximization (EM) algorithm, which is an iterative procedure that can serve as a maximum-likelihood estimator. The EM algorithm has well-documented drawbacks, such as the need for good initial values and the possibility of being trapped in local optima. Nevertheless, because of its appealing properties, EM plays an important role in estimating the parameters of mixture models. To overcome these initialization problems with EM, in this paper, we propose the Rough-Enhanced-Bayes mixture estimation (REBMIX) algorithm as a more effective initialization algorithm. Three different strategies are derived for dealing with the unknown number of components in the mixture model. These strategies are thoroughly tested on artificial datasets, density–estimation datasets and image–segmentation problems and compared with state-of-the-art initialization methods for the EM. Our proposal shows promising results in terms of clustering and density-estimation performance as well as in terms of computational efficiency. All the improvements are implemented in the rebmix R package.

Cowper–Symonds parameters were estimated for the complex-phase high-strength steel with a commercial name of SZBS800. The parameter estimation was based on a series of conventional tensile tests and unconventional high-strain rate experiments. The parameters were estimated using a reverse engineering approach. LS-Dyna was used for numerical simulations, and the material’s response was modelled using a piece-wise linear plasticity model with a visco-plastic formulation of the Cowper–Symonds material model. A multi-criteria cost function was defined and applied to obtain a response function for the parameters p and C. The cost function was modelled with a response surface, and the optimal parameters were estimated using a real-valued genetic algorithm. The main novelty and innovation of this article is the definition of a cost function that measures a deviation between the deformed geometry of the flat plate-like specimens and the results of the numerical simulations. The results are compared to the relevant literature. A critical evaluation of our results and references is another novelty of this article.

A maximum-likelihood estimation of a multivariate mixture model’s parameters is a difficult problem. One approach is to combine the REBMIX and EM algorithms. However, the REBMIX algorithm requires the use of histogram estimation, which is the most rudimentary approach to an empirical density estimation and has many drawbacks. Nevertheless, because of its simplicity, it is still one of the most commonly used techniques. The main problem is to estimate the optimum histogram-bin width, which is usually set by the number of non-overlapping, regularly spaced bins. For univariate problems it is usually denoted by an integer value; i.e., the number of bins. However, for multivariate problems, in order to obtain a histogram estimation, a regular grid must be formed. Thus, to obtain the optimum histogram estimation, an integer-optimization problem must be solved. The aim is therefore the estimation of optimum histogram binning, alone and in application to the mixture model parameter estimation with the REBMIX&EM strategy. As an estimator, the Knuth rule was used. For the optimization algorithm, a derivative based on the coordinate-descent optimization was composed. These proposals yielded promising results. The optimization algorithm was efficient and the results were accurate. When applied to the multivariate, Gaussian-mixture-model parameter estimation, the results were competitive. All the improvements were implemented in the rebmix R package.

Unsupervised image segmentation is one of the most important and fundamental tasks in many computer vision systems. Mixture model is a compelling framework for unsupervised image segmentation. A segmented image is obtained by clustering the pixel color values of the image with an estimated mixture model. Problems arise when the selected optimal mixture model contains a large number of mixture components. Then, multiple components of the estimated mixture model are better suited to describe individual segments of the image. We investigate methods for merging the components of the mixture model and their usefulness for unsupervised image segmentation. We define a simple heuristic for optimal segmentation with merging of the components of the mixture model. The experiments were performed with gray-scale and color images. The reported results and the performed comparisons with popular clustering approaches show clear benefits of merging components of the mixture model for unsupervised image segmentation.

The demand for accurate and reliable unsupervised image segmentation methods is high. Regardless of whether we are faced with a problem for which we do not have a usable training dataset, or whether it is not possible to obtain one, we still need to be able to extract the desired information from images. In such cases, we are usually gently pushed towards the best possible clustering method, as it is often more robust than simple traditional image processing methods. We investigate the usefulness of combining two clustering methods for unsupervised image segmentation. We use the mixture models to extract the color and spatial image features based on the obtained output segments. Then we construct a similarity matrix (adjacency matrix) based on these features to perform spectral clustering. In between, we propose a label noise correction using Markov random fields. We investigate the usefulness of our method on many hand-crafted images of different objects with different shapes, colorization, and noise. Compared to other clustering methods, our proposal performs better, with 10% higher accuracy. Compared to state-of-the-art supervised image segmentation methods based on deep convolutional neural networks, our proposal proves to be competitive.