Abstract of the paper ``Integrating and interpolating three-point
correlation functions on a lattice''
Various methods of estimating effective 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 sufficient 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 finally (3) completing the estimate with a Newton
forward-difference 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 off the
grid points; better than plus/minus 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.
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