
Higher-Dimensional Spaces and Marks 657
When comparing the estimate
b
F(r) with the benchmark value for a Poisson process, w e need to
be aware of a digital artefact. The ‘sphere’ of radius r in the distance tra nsform — tha t is, the set of
voxels that lie at most r units from a given voxel according to the distance transform algorithm — is
a p olyhedron of fixed shape, even for large values of r. For the spatstat algorithm the volume v(r )
of the d igital sphere o f radius r is roughly 0.78 times the volume of the euc lidean sphere, (4
π
/3)r
3
.
This is a considerable deficit, so the benchmark value sho uld really be calculated b y replacing the
ideal