Minkowski functionals provide morphological measures for characterization of size, shape, and connectivity of spatial patterns in d dimensions [101]. These functionals turned out to be an appropriate tool to quantify clustering substructures in astronomy, e.g. from galaxies [102]. They also give insight into the morphology of random inter-faces in microemulsions [103].
The scalar Minkowski valuations V applied to patterns P and Q in Euclidian space are dened by three types of covariances [102]:
1. Invariance to motion;
2. Additivity: V(P SQ) = V(P) +V(Q)−V(P TQ);
3. Continuity: continuous change for slight distortions in pattern P.
It is guaranteed by the theorem of Hadwiger that inddimensions there are exactlyd+1 morphological measures V that are linearly independent [101]. For d = 2, the three functionals have intuitive correspondences3: The surface area, the circumference of
3For d=3 the functionals correspond to: volume, area, integral mean curvature, and Euler charac-teristic.
H1
Figure B.2: A: Example of a simple 2D pattern. The 'Euler characteristic' is calculated by χ = S−H, where S is the number of separated surfaces Si and H the number of holes Hi. In the shown case the 'Euler characteristic' is χ = 3−2 = 1. B: Sketch of 'Euler characteristic' χ/N for a conguration with cover discs of dierent radii R normalized with the number of points N. The pictograph of partially overlapping discs illustrates this morphology for a xed value R.
the surface area, and the Euler characteristic χ. Obviously the rst and the second fulll the necessary covariances. Also the third measure does so, what will be clear after the following introduction of the Euler characteristic.
In two dimensions the Euler characteristic χfor a pattern P is dened as
χ=S−H (B.13)
where S is the number of surfaces and H the number of holes. This is illustrated for a simple example in Figure B.2A. There, three surfaces S1, S2 and S3 are drawn and surface S1 has two holes H1 and H2. Thus, the Euler characteristic of this example adds up to χ= 3−2 = 1.
Morphological information can also be obtained from particle congurations. As con-gurations only consist of a set of coordinates, a cover disc is placed on each coordinate to construct a pattern that can be evaluated in the mentioned way. A sketch of such a decorated disc scenario is given in the inset of Figure B.2B. The Minkowski measures are then determined for dierent cover radii R, leading to a characteristic curve for a given conguration: The rst Minkowski measure (disc area normalized with total area) increases from 0% to 100%for increasing radius R with a decreasing slope when discs start overlapping. The second Minkowski measure (circumference) increases with cover radius R, reaches a maximum, and then decays to zero when all holes are over-lapped. The third Minkowski measure (Euler characteristic) is very subtle and requires
more detailed explanation as given in the following. A typical devolution of an Eu-ler characteristic χ/N with N particles is sketched in the right graph of Figure B.2.
The curve is divided in three characteristic parts for continuously increasing cover disc radius R:
1. For small R the curve is constant at χ/N = 1. Discs are not touching and thus S is equal to the number of particles. No holes are present. As χ is normalized with the number of points, the Euler characteristic amounts to χ/N = 1.
2. With increasing R the curve drops and χ/N can become negative when cover discs are large enough to overlap and holes are formed. Therefore, the number of surfaces S decreases and holes are forming which further decreasesχ/N. The minimum is reached when discs are connected to a percolating network and the maximum number of holes has formed.
3. For large R the curve starts to raise again because the holes are collapsing until the whole plane is covered with overlapping discs and χ/N →0 for R→ ∞. This qualitative behavior is typical for congurations in 2D. However, the individual morphological information is obtained from specic features in the three regions as the onset of the fall and raise, characteristic kinks or plateaus, and the slope of the fall and raise.
There are sophisticated ways to calculate the Minkowski measures for a given conguration [51, 103, 104]. However, in this work a simple method based on image processing is used. It is briey sketched as follows.
Particle congurations are obtained from image processing as described in chapter 3.
Cover discs are placed with their center at the coordinates on a matrix of 1392×1040 cells4. The cells are assigned with value 1 if the center of a cell is within the cover disc area generating the desired pattern on the grid. For the area (rst Minkowski measure), the matrix elements are simply summed up and normalized with the total number of matrix elements. The circumference (second Minkowski measure) is determined by eroding5 the pattern and subsequently subtracting it from the original pattern. The residual structure corresponds to a prescription of the circumference. Summation of all elements of the matrix leads to the total circumference6. Surface eects at the matrix edge have been accounted for. For the Euler characteristic (third Minkowski measure) a simple blob analysis7 counts the
4This is the resolution of the CCD camera, but any other grid can be chosen if only ne enough to reveal the desired eect.
5For detailed description of the erosion procedure see chapter 3.2.2, especially the footnote speci-fying the erode/dilate functions.
6As pixels with unit length 1 are square shaped, an eective diameter d with1 < d <√
2 has to be used to calibrate the contour length. As an approximation the mean valued≈1.207was used.
7For detailed description of the blob analysis see chapter 3.2.2.
number of connected surfaces S. The number of holes H is calculated by applying the same procedure to the inverted pattern Sinv as H = Sinv − 1. For the data tested in this work this leads to the same results as with the method used by N. Homann [51].
The Minkowski measures provide a morphological ngerprint of particle congurations.
It is used complementary to pair correlation functions and static structure factors, because it reveals additional structural information.