Before I forget I thought I would do a brief post on the subject of Minkowski Functionals, as used in the paper we recently published in the Open Journal of Astrophysics. As as has been pointed out, the Wikipedia page on Minkowski Functionals is somewhat abstract and impenetrable so here is a much simplified summary of their application in a cosmological setting.
One of things we want to do with a cosmological data set to characterize its statistical properties to compare theoretical predictions with observations. One interesting way of doing this is to study the morphology of the patterns involved using quantitative measures based on topology.
The approach normally used deals with Excursion Sets, i.e. regions where a field exceeds a certain level usually given in terms of the rms fluctuation or defined by the fraction of space above the threshold. The field could, for example, be the temperature field on the CMB Sky or the density field traced by galaxies. In general the excursion set will consist of a number of disjoint pieces which may be simply or multiply connected. As the threshold is raised, the connectivity of the excursion set will shrink but also its connectivity will change, so we need to study everything as a function of threshold to get a full description.
One can think of lots of ways of defining measures related to an excursion set. The Minkowski Functionals are the topological invariants that satisfy four properties:
- Additivity
- Continuity
- Rotation Invariance
- Translation Invariance
In D dimensions there are (D+1) invariants so defined. In cosmology we usually deal with D=2 or D=3. In 2D, two of the characteristics are obvious: the total area of the excursion set and the total length of its boundary (perimeter). These are clearly additive.
In order to understand the third invariant we need to invoke the Gauss-Bonnet theorem, shown in this graphic:
The Euler-Poincare characteristic (χ) is our third invariant. The definition here allows one to take into account whether or not the data are defined on a plane or curved surface such as the celestial sphere. In the simplest case of a plane we get:
As an illustrative example consider this familiar structure:
Instead of using a height threshold let’s just consider the structure defined by land versus water. There is one obvious island but in fact there are around 80 smaller islands surrounding it. That illustrates the need to define a resolution scale: structures smaller than the resolution scale do not count. The same goes with lakes. If we take a coarse resolution scale of 100 km2 then there are five large lakes (Lough Neagh, Lough Corrib, Lough Derg, Lough Ree and Lower Lough Erne) and no islands. At this resolution, the set consists of one region with 5 holes in it: its Euler-Poincaré characteristic is therefore χ=-4. The change of χ with scale in cosmological data sets is of great interest. Note also that the area and length of perimeter will change with resolution too.
One can use the Gauss-Bonnet theorem to extend these considerations to 3D by applying to the surfaces bounding the pieces of the excursion set and consequently defining their corresponding Euler-Poincaré. characteristics, though for historical reasons many in cosmology refer not to χ but the genus g.
A sphere has zero genus (χ=1) and torus has g=1 (χ=0).
In 3D the four Minkowski Functionals are: the volume of the excursion set; the surface area of the boundary of the excursion set; the mean curvature of the boundary; and χ (or g).
Great advantage of these measures is that they are quite straightforward to extract from data (after suitable smoothing) and their mean values are calculable analytically for the cosmologically-relevant case of a Gaussian random field.
Here endeth the lesson.
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