Analysis of Tomographic Images of the Soil Pore Space Structure by Integral Geometry Methods

The technique of numerical analysis of three-dimensional tomographic images of the pore space of soil objects has been used in this paper. It applies methods of integral geometry, topology and morphological analysis. To characterize quantitatively the transformation of the pore space structure, tomographic images of four undisturbed soils were analyzed, i.e., heavy loamy agro-gray soil (Retic Phaeozem), agromineral (Sapric Rheic Mineralic Histosols), and hypnum (brown moss Sapric Rheic Histosols) peat soils in dry and wet conditions. For samples of the subplow horizon in agro-gray soil, a decrease in both Betty numbers was observed on wetting, where the zero number (b0) stands for the amount of topologically simple closed pores, and the first number (b1) indicates a decrease in pore connectivity, which varies in a narrower range of pore sizes as compared to b0. When a sample of agromineral peat soil is moistened, the Euler–Poincaré characteristic is negative ​in the pore range of 0.1–0.16 mm, which points to the predominating complicated branched structure of the pore space and high pore connectivity. When hypnum moss is saturated, a lot of tunnel pores get narrower (“collapse”), and the connectivity decreases due to the structural specifics of long-stemmed plant residues. The number of pores and connections between them in peat soils is an order of magnitude higher than those in the subplow horizon A of the agro-gray soil. The provided quantitative changes in the considered parameters of tomographic images of the soil pore space confirm the possibility of applying them for estimating the transformation of the pore space in soils.

The inverse task solution for the calculation of sweeps of sheet parts by integral geometry

This article deals with the search for a rational semi-finished product for parts of the integral form. Here a variant of the solution of the given problem is described based on the inverse solution of the mathematical problem. The reasons and the nature of the problem occurring in standard approaches to the calculation of the semi-finished product are shown using the example of two parts with aerodynamic curvature. The material and sheet thickness of the components are selected to suit the application. The forming process is also the same as the initial situation. The rational shape of the part can be found using the optimization solution based on the discrepancy of the contours with respect to the sweeps of the dynamic calculation and the CAD model. The new shape of the blank makes it possible to realize the forming process of the parts with minimal deviations from the contour of the 3D model.

Integral geometry on discrete matrices

In this note, we study the Radon transform and its dual on the discrete matrices by defining hyperplanes as being infinite sets of solutions of linear Diophantine equations. We then give an inversion formula and a support theorem.