Absorption-contrast synchrotron-radiation-based micro computed tomography in medicine, biology, and materials science

Abstract

Keywords: 3D-image analysis, porous materials, characterizationIn materials' science, as well as in other research fields, the Euler number (or its density) is used as a characteristic describing the connectivity of the components (constituents) of a composite material or the pore space of a porous medium, see [1]. From a theoretical point of view the Euler number of a 3-dimensional set X in the Euclidean space is a basic quantity of integral geometry. By means of Crofton's intersection formulae, the intrinsic volumes (e.g. the surface area, the total length of curved fibres, the mean breadth of a particle, the integral of Germain's curvature) can be expressed in terms of the Euler numbers of intersections of X with lines and planes, respectively. This is the basis of the measurement of the quantities in 3d-image analysis. The observation of a set X on a homogeneous point lattice implies a discretization of the Crofton's formulas. This leads to an efficient expression of the intrinsic volumes as a scalar product of two vectors. The fist vector depends on the image. Its components (the frequencies of pixel configurations) can easily be measured by a marching cube method. The second vector is independent of the image. Its components depends on the lattice spacings (lateral resolution), the adjacency system assumed for the pixels and the quadrature rule applied in the numerical computation of the integrals occuring in Crofton's formulas [2,3]. The approach basing on Crofton's intersection formulas has clear advantages over other methods where the weights are determined more or less empirically. It leads to higher accuracy of measurement, it can be applied to a wide range of features and it supplies multigrid convergence under weak conditions for the discretization of the set X. The applicability of the method is demonstrated for a wide range of materials, e.g. open foams, fleeces and autoclaved aerated concrete.