  The genus statistics of isodensity contours has become a well-established
tool in cosmology. In this Letter we place the genus in the wider framework of
a complete family of morphological descriptors. These are known as the
Minkowski functionals, and we here apply them for the first time to isodensity
contours of a continuous random field. By taking two equivalent approaches, one
through differential geometry, the other through integral geometry, we derive
two complementary formulae suitable for numerically calculating the Minkowski
functionals. As an example we apply them to simulated Gaussian random fields
and compare the outcome to the analytically known results, demonstrating that
both are indeed well suited for numerical evaluation. The code used for
calculating all Minkowski functionals is available from the authors.
