Interpolating and Integrating Three-point Correlation Functions on a Lattice

Various methods of estimating eeective properties of composite materials require geometrical or topological information contained in the statistical three-point correlation functions. If these three-point correlation functions are measured using digital image processing techniques, the values are computed for a discrete set of admissible triangular arguments corresponding to triangles whose vertices are commensurate with a simple cubic lattice. To extract the desired geometrical information from the correlation functions, methods of interpolating and integrating between these lattice-based correlation function values must be developed. In previous work, a minimal subset of the lattice-commensurate triangles was proposed as the primary data set. However, to obtain suucient accuracy in the interpolations, it has been found necessary to expand the primary set of lattice-commensurate triangles to include other triangles. The size of the new set of triangles is less than a factor of two larger than that of the previous set; yet this set has the distinct advantage that a sorting and storing algorithm mapping the triangular arguments onto a compact one-dimensional array depends on formulas known in closed form. The spline interpolation algorithm maintains the shape of the argument triangle while (1) scaling its longest side to an integer number of pixel widths, (2) following with either bilinear or biquadratic interpolation through lattice points surrounding the third vertex, and nally (3) completing the estimate with a Newton forward-diierence interpolation in the triangle scale size. Statistical comparison with exact values for the penetrable sphere model shows that this interpolation scheme provides accurate estimates of S 3 oo the grid points; better than 0:02% accuracy is typical for a quadratic interpolation scheme. These interpolated values are subsequently used in a Monte Carlo integration scheme developed previously for various 3-D integrals of S 3 and the results are comparable to those obtained with the exact S 3 for penetrable spheres.