Dear Rene Westerholt,

I am not sure of what you call the Euler-Poincaré number. If it is the usual Euler Characteristic, probably more structure is needed to compute it, can you define something like a simplicial complex with your spatiotemporal data?

The most usual way to compute the homology with points cloud is investigated in persistent Homology. See for example: https://team.inria.fr/geometrica/

They have free soft there

Best

piero

PS: if you cannot define a more refined complex structure the Euler chararcteristic is the number of points (but that s probably not what you are looking for).

to add to Pierre Baudot's answer, there  are nice Bulletin articles on persistent homology and data

http://www.ams.org/journals/bull/2008-45-01/S0273-0979-07-01191-3/S0273-0979-07-01191-3.pdf

and the more elaborate 

http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf

You can compute probably compute the Euler (Euler-Poincare) characteristics yourself however:

Let x_1, x_2 … x_n be your cloud of points and suppose we are given a distance d(x_i, x_j) for each pair (x_i, x_j) (it must satisfy all the usual axioms of a metric. The persistent homology is based on the Cech complex or a variation thereof the Rips complex. The complexes depend on a scale epsilon. The Cech complex computes the homology of the space you get by fattening your points to balls of radius epsilon and the Rips complex computes something between the epsilon and 2 epsilon scale i.e. essentially the same. The homology is the simplicial homology of the Cech or the Rips simplicial complex. The Rips simplicial complex is probably easiest to define. The k-simplices are the subsets of (k+1) points {x_{i_1},… x_{i_k}} such that d(x_{i_a}, x_{i_b}) < epsilon. The

SInce the Euler characteristic of the complex is the Euler characteristic of the homology, it is particularly easy to compute

chi_Rips( eps) = #0-simplices - #1-simplices + #2-simplices

                                  - #3-simplices + …...

                       = #points  -

                          #pairs of points  with mutual distance < eps +

                          #triples of points with all mutual distances < eps  -

                          #quadruples of points with all mutual distances < eps  +

                          ….  

Dear Pierre and Rogier,

Thank you very much for your detailed answers! I think your answers are quite close to what I was asking. I should probably provide some more context:

Let's assume we have some points in a 3D Euclidean space. One might expand the points to balls of radius r and take the union of all these balls. We could do that over a range of radii, which would lead to a function producing growing sets of unions of balls.

My understanding of the Euler Poincaré characteristic (I am not a trained Mathematician) is that it reflects the number of components formed by these sets plus the number of enclosed holes minus tunnels minus outer boundary components. The specific Euler Poincaré number is the expectation of the function explained above divided by the volume of the observation window of the point process. These explanations should be consistent with Mecke & Stoyan (2005) and Illian et al. (2008) (see references below).

Thus, I think that your answers are quite close to what I am looking for. As far as I got it correct, both, Cech and Rips complexes are constructed by building a graph based on the topology of the closed balls centered on the points. Afterwards, k-simplices are constructed from these graph edges. However, I am looking for a very similar thing. The difference is that I am not looking for faces made up of the edges of a graph. Instead, the set I am talking about is made up of the union of the balls themselves.

I hope I stated my problem properly, so that you can follow my points.

Mecke, K.R., and Stoyan, D., 2005. Morphological Characterization of Point Patterns. Biometrical Journal, 47, 473-488.

Illian, J., Penttinen, A., Stoyan, H., and Stoyan, D., 2008. Statistical Analysis and Modelling of Point Patterns. Chichester: Wiley and Sons.

if you want to compute the Euler characteristic of a 3D image, there is an OpenSource implementation in BoneJ ( http://bonej.org/ ) which is available as plugin in ImageJ and FIJI

Not sure, if this is still relevant, but this might be useful for you or anyone with a similar problem: https://morphometry.org/software/karambola/

You might have to convert your point-pattern into a polygonal 3D geometry. Either, as you wrote, by blowing the points up into small spheres and unify them (using something like CGAL or if it's not to computationally intensive you could do it in openscad), or maybe using their pointpattern2voronoi demo.