Minkowski functionals

3.5 Minkowski functionals

In fact, genus is one of the complete sets of $N + 1$ quantities, known as the Minkowski functionals (MFs), which determine the morphological properties of a pattern in $N$ -dimensional space. In the analysis of galaxy redshift survey data, one considers isodensity contours from the three-dimensional density contrast field $δ$ by taking its excursion set $F ν$ , i.e., the set of all points where the density contrast $δ$ exceeds the threshold level $ν$ 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 $Θ$ normalized to the total volume $V tot$ ,

the other MFs $V k$ ( $k = 1,2,3$ ) are calculated by the surface integration of the local MFs $vloc k$ . The general expression is

with the local Minkowski functionals for $k = 1,2,3$ given by

loc 1 v1 (ν,x) = --, (109 ) 6 ( ) vloc(ν,x) = -1- -1- + -1- , (110 ) 2 6π R1 R2 1 1 vl3oc(ν,x) = ---------, (111 ) 4π R1R2

where $R1$ and $R2$ 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:

1 1 ∫ ν ( x2) V0(ν ) = --− √---- exp − --- dx, (112 ) 2 2π 0( ) 2 2--λ-- 1-2 V1(ν ) = 3√ ---exp − 2ν , (113 ) 2π2 ( ) 2√λ--- 1- 2 V2(ν ) = 3 2π ν exp − 2ν , (114 ) 3 ( ) V3(ν ) = √λ--(ν2 − 1)exp − 1ν2 , (115 ) 2π 2

where $λ = ∘ σ2-∕6πσ2- 1$ , $σ ≡ ⟨δ2⟩1∕2$ , $σ ≡ ⟨|∇ δ|2⟩1∕2 1$ , and $δ$ is the density contrast.

The above MFs can be indeed interpreted as well-known geometric quantities: the volume fraction $V0(ν )$ , the total surface area $V1(ν)$ , the integral mean curvature $V2(ν )$ , and the integral Gaussian curvature, i.e., the Euler characteristic $V3(ν)$ . In our current definitions (see Equations (101 , 108 ), or Equations (102 , 115 )), one can easily show that $V3(ν )$ reduces simply to $− G (ν)$ . 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 ].