Stereological analysis and modelling of gradient structures

Gradient structures are inhomogeneous along a particular gradient direction but homogeneous perpendicular to that direction. Consequently, structural parameters such as volume fraction or surface area density are local characteristics which depend on the 'vertical' coordinate with respect to the 'vertical' gradient axis. Analogously, models for gradient structures have model parameters depending on the vertical coordinate z. For example, a Voronoi tessellation with a gradient is generated by a gradient point process with a local intensity which is a function of z. Similarly, a gradient germ grain model is obtained from a gradient point process where the grain size distribution may also depend on z. For a gradient Boolean model, local volume fraction VV(z) and local surface area density SV(z) can be calculated from the model parameters. Stereological methods for gradient structures are based on vertical sections parallel to the gradient direction. Estimation of VV(z), SV(z) and local length density LV(z) is done by lineal analysis using horizontal test lines with vertical coordinate z. Similarly, lineal analysis is used to estimate local mean cell volume of gradient tessellations. For the estimation of local particle number density and size in the spirit of the Wicksell problem the use of kernel methods and distributional assumptions is required.