### 3.5 Minkowski functionals

In fact, genus is one of the complete sets of quantities, known as the Minkowski functionals
(MFs), which determine the morphological properties of a pattern in -dimensional space. In the analysis
of galaxy redshift survey data, one considers isodensity contours from the three-dimensional density
contrast field by taking its excursion set , i.e., the set of all points where the density contrast
exceeds the threshold level as was the case in the case of genus described in the above
subsection.
All MFs can be expressed as integrals over the excursion set. While the first MF is simply given by the
volume integration of a Heaviside step function normalized to the total volume ,

the other MFs () are calculated by the surface integration of the local MFs . The
general expression is
with the local Minkowski functionals for given by
where and are the principal radii of curvature of the isodensity surface.
For a 3-D Gaussian random field, the average MFs per unit volume can be expressed analytically as
follows:

where , , , and is the density contrast.
The above MFs can be indeed interpreted as well-known geometric quantities: the volume fraction
, the total surface area , the integral mean curvature , and the integral Gaussian
curvature, i.e., the Euler characteristic . In our current definitions (see Equations (101, 108),
or Equations (102, 115)), one can easily show that reduces simply to . The
MFs were first introduced to cosmological studies by Mecke et al. [57], and further details may
be found in [57, 32]. Analytic expressions of MFs in weakly non-Gaussian fields are derived
in [52].