Fractal analysis and stereological evaluation of microstructures

Asymptotic fractals in the context of grey-scale images

The estimation of the fractal dimension in the case of concave log-log Richardson-Mandelbrot plots can be obtained by using asymptotic fractal equations. We demonstrate here, under asymptotic fractal conditions, that additional derivations making use of the Minkowski dilation in grey-scales lead to two asymptotes, one having a slope of 1 and the other a slope of DT-D + 1 (where DT is the topological dimension and D the fractal dimension). The resulting equation offers important advantages. It allows: (i) evaluation of scaling properties of a grey-scale image; (ii) estimation of D without any iteration and (iii) generation of texture and heterogeneity models. We concentrate here on the first two possibilities. Images from cultured cells in studies of cytoskeleton intermediate filaments and kinetic deformability of endothelial cells were used as examples.

The application of fractal geometric analysis to microscopic images

Fractal geometry is a relatively new tool for the quantitative microscopist that is a more valid way of measuring dimensions of complex irregular objects than the integer-dimensional geometries (such as Euclidean geometry). This review discusses the theory of fractal geometry using the classic examples of the Von Koch curve, the Cantor set and the Sierpinski gasket. The problems of describing the dimensions of these objects are discussed and the concept of fractal dimensionality is introduced. Methods for measuring fractal dimensions are discussed, including their implementation on microcomputer-based image analysis systems . The advantages and problems of fractal geometric analysis are discussed and current applications in the field of microscopy are reviewed.

Determination of the lengths of nonisotropic linear features in micrographs

The method described was devised to facilitate rapid and reasonably accurate estimations of the length of a nonisotropic linear feature in a micrograph. It arose from studies in which we were determining the length of the Purkinje cell layer in each of a set of serial sections through the rat cerebellum. The Purkinje cell layer appears as a linear feature in such sections and the simplest and most rapid method of estimating the length of this type of feature is to count the number of intersections that it makes with a series of equally spaced parallel test lines (see, e.g., Weibel, E.R., 1979: Stereological Methods Vol. 1, Practical Methods for Biological Morphometry). In many sections, the Purkinje cell layer was markedly nonisotropic, and the length obtained by this method varied very considerably depending on the orientation of the section relative to the test lines. The present method employs two orthogonal sets of parallel test lines, and the length of the feature is estimated from the square root of the sum of the squares of the counts of the number of intersections with each of the two sets of lines. The result obtained varies very little with the relative orientations of the feature and the test grid and a good estimate of the length can be obtained from the counts from a single random placement of the grid on the section. It has been found that the technique can be carried out efficiently and reliably by relatively inexperienced personnel, and the results are obtained more rapidly than when alternative methods for estimating dimensions of nonisotropic features are used.