Minkowski functionals for composite smooth random fields

Minkowski functionals quantify the morphology of smooth random fields. They are widely used to probe statistical properties of cosmological fields. Analytic formulas for ensemble expectations of Minkowski functionals are well known for Gaussian and mildly non-Gaussian fields. In this paper, we extend the formulas to composite fields which are sums of two fields and explicitly derive the expressions for the sum of uncorrelated mildly non-Gaussian and Gaussian fields. These formulas are applicable to observed data which is usually a sum of the true signal and one or more secondary fields that can be either noise, or some residual contaminating signal. Our formulas provide explicit quantification of the effect of the secondary field on the morphology and statistical nature of the true signal. As examples, we apply the formulas to determine how the presence of Gaussian noise can bias the morphological properties and statistical nature of Gaussian and non-Gaussian cosmic microwave background temperature maps.

Figure 1

Plot of the factor $B(\epsilon ,p)=(1+{\epsilon }^2{p}^2)/(1+{\epsilon }^2)$ that relates ${(r_c^f)}^{-2}$ and ${(r_c^u)}^{-2}$ in Eq. (25). The green regions are where $B(\epsilon ,p)$ has values $\sim 1$. Blue regions are where it has values less than 1, while the other color bands toward the top left indicate regions where it has values larger than 1.

Figure 2

Top: simulated Gaussian CMB (left) and noise (right) maps. Middle: the composite of CMB and noise maps. Bottom: the MFs $V_1$ (left) and $V_2$ (right) for CMB SMICA (red), noise (purple), and their composite (cyan) maps. The plot for the analytic formula is shown by the black dashed lines which overlap the cyan lines.

Figure 3

The relative change of $f_{\mathrm{NL}}$, given by $({\sigma }^u/{\sigma }^f{)}^4=1/(1+{\epsilon }^2{)}^2$, that is induced by the presence of Gaussian noise is shown as a function of the SNR ($={\epsilon }^{-1}$).

Figure 4

$\mathrm{\Delta }{V}_k$ are shown for the non-Gaussian CMB temperature field $u$ (red hollow dots), and the composite field $f$ (blue solid dots). The corresponding $\mathrm{\Delta }{V}_k^{\mathrm{ana}}$ are also shown for $u$ (black solid line), and for $f$ (black dashed line).

Figure 5

$\mathrm{\Delta }{V}_k/A_k$ are shown for the non-Gaussian CMB temperature field $u$ (red hollow dots), and the composite field $f$ (blue solid dots). The corresponding $\mathrm{\Delta }{V}_k^{\mathrm{ana}}/A_k$ are also shown for $u$ and $f$ by the black solid and dashed lines, respectively.