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 ,
For a 3-D Gaussian random field, the average MFs per unit volume can be expressed analytically as follows:
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. , and further details may be found in [57, 32]. Analytic expressions of MFs in weakly non-Gaussian fields are derived in .
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