We consider the problem of detecting anomalies in the directional distribution of fibre materials observed in 3D images. We divide the image into a set of scanning windows and classify them into two clusters: homogeneous material and anomaly. Based on a sample of estimated local fibre directions, for each scanning window we compute several classification attributes, namely the coordinate wise means of local fibre directions, the entropy of the directional distribution, and a combination of them. We also propose a new spatial modification of the Stochastic Approximation Expectation-Maximization (SAEM) algorithm. Besides the clustering we also consider testing the significance of anomalies. To this end, we apply a change point technique for random fields and derive the exact inequalities for tail probabilities of a test statistics. The proposed methodology is first validated on simulated images. Finally, it is applied to a 3D image of a fibre reinforced polymer.

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We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet. This Gaussian field is an extension of fractional Brownian motion. We prove some properties of covariance function for self-similar fields with rectangular increments. Using Lamperti transformation we obtain properties of covariance function for the corresponding stationary fields. We present an example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet.

The paper considers the problem of anomaly detection in 3D images of fibre materials. The spatial Stochastic Expectation Maximisation algorithm and Adaptive Weights Clustering are applied to solve this problem. The initial 3D grey scale image was divided into small cubes subject to clustering. For each cube clustering attributes values were calculated: mean local direction and directional entropy. Clustering is conducted according to the given attributes. The proposed methods are tested on the simulated images and on real fibre materials. Introduction.Nowadays, there exists a large amount of novel materials with interesting physical properties. For instance, reinforcement of polymers with fibres significantly increases the mechanical properties of the materials. The materials' performance is determined mostly by their composition as well as by allocation and directions of the reinforcing fibres.Due to the production process, an anomaly region may be formed in the material. We define it as an area where the distribution of fibre directions differs from the remaining material. To keep the stated material's properties, it is necessary to identify regions with untypical fibre distribution. For this purpose, high-resolution microcomputer tomography reaching a level of microns [1,2] is used to observe the fibre system in a composite sample, cf. Figure 1 (right).Our main task is then to find the areas with anomalous directional properties of fibres in the 3D image. This is done by means of cluster analysis dividing the whole image volume into two clusters: the smaller "anomaly" region and the bigger "normal" material.

We have obtained some upper bounds for the probability distribution of extremes of a selfsimilar Gaussian random field with stationary rectangular increments that are defined on the compact spaces. The probability distributions of extremes for the normalized self-similar Gaussian random fields with stationary rectangular increments defined in R 2 + have been presented. In our work we have used the techniques developed for the self-similar fields and based on the classical series analysis of the maximal probability bounding from below for the Gaussian fields.

In this paper, we use the concept of excursion sets for the extrapolation of stationary random fields. Doing so, we define excursion sets for the field and its linear predictor, and then minimize the expected volume of the symmetric difference of these sets under the condition that the univariate distributions of the predictor and of the field itself coincide. We illustrate the new approach on Gaussian random fields.

In this paper, we provide the proof of L 2 consistency for the kth nearest neighbour distance estimator of the Shannon entropy for an arbitrary fixed k ≥ 1. We construct the non-parametric test of goodness-of-fit for a class of introduced generalized multivariate Gaussian distributions based on a maximum entropy principle. The theoretical results are followed by numerical studies on simulated samples.