The surface properties of newsprint and other paper qualities are to a great extent determined by the properties of the cellulose fibres. An appropriate description of these fibres as they appear in the paper is therefore important and can be used for quality classification and process monitoring. We suggest a model that considers the fibre geometry and appearance. It is based on a two-dimensional shot-noise process. The model is fit by minimizing a weighted least squares distance between the model-based and estimated covariance functions and this provides estimates of the fibre size, intensity and the non-uniform distribution of the fibre orientation. The model is applied to simulated and real data.

Detta är en rapport över en kvantitativ undersökning om förhållanden i matematikundervisningen i grundskolan med hjälp av datorer. Syftet har varit att få en bild av omfattning, tillämpning samt lärarnas utbildning för undervisning med hjälp av dator. Resultaten skall kunna användas som underlag för flera studier i datoranvändning i matematikundervisningen.

Kurtosis is generally associated with measurements of peakedness of a distribution. In this paper, we suggest a method where kurtosis can be used as a measure of homogeneity of any quantifiable property on a planar surface. A 2-dimensional, continuous and uniform distribution has kurtosis equal to 5.6. This value is also the limiting value for a discrete uniform distribution defined on a regular, rectangular grid when the number of grid points tend to infinity. Measurements of a planar surface, taken at regular grid points, are considered as realizations of random fields. These are associated with 2-dimensional random variables from which the value of kurtosis can be computed and used as a measure of the homogeneity of the field. A deviation from 5.6 indicates that the stochastic variable is not uniformly distributed and that the corresponding random field is not homogeneous. The model is applied on the spatial variation of the roughness on the surface of newsprint, an application where homogeneity is very important.

The Gibbs distribution is often used to model micro-textures. This includes a definition of a neighbourhood system. If a micro-texture contains a large-scale variation, the neighbourhood system will be large, which implies many parameters in the corresponding Gibbs distribution. The estimation of the parameters for such models will be difficult and time consuming. I suggest, in this paper, a separation of the micro-texture into a large-scale variation and a small-scale variation and model each source of variation with a Gibbs distribution. This method is applied on full-tone print of newsprint to model the variation caused by print mottle. In this application, the large-scale variation is mainly caused by fibre flocculation and clustering and the small-scale variation contains the variation of the fibres and fines on and between the clusters. The separate description of these two variations makes it possible to relate different kinds of paper qualities to the appropriate source of variation.

In texture analysis, the Gibbs sampler constitutes an important tool in the generation of synthetic textures. The textures are modeled as distributions with specified parameters. In this paper, we study the estimation process of the parameters in such distributions and compare Besags coding method with a pseudo-likelihood method. We also compare simulated annealing with the Newton-Raphson method to find the global maximum of a likelihood or pseudo-likelihood function. For some textures, the two methods differ but in most case there are no important differences between them. The two maximization methods find the same maximum, but the Newton-Raphson method is much faster. However, the Newton-Raphson method cannot be applied in some cases when the location of the maximum differs too much from the starting points. Here, it is often possible to find the global maximum using simulated annealing. The methods have been used in an application with newsprint.