I was not able to find good answers in Google, or perhaps I am just missing the correct key words. Any help or suggetions are welcome!
My problem is the following: I want to calculate the area a certain point cloud covers (in 2D). I know that mathematically speaking the area is 0, but I am only able to take sample points out of the correct distribution. Additionally I do not have any information about the boundary of the point cloud, every shape is possible, including holes etc. So algorithms using the boundary of a manifold will not work?!.
Since the functions I am working with are smooth I can assume that the space in between points also belongs to the area I want to calculate.
At the moment I divide the space into a lot of small boxes and count how many boxes are populated with one or more points. The count multiplied with the box size gives me an area.
Is there a more elegant solution to this? Any ideas?
Thanks Thomas
EDIT:
What I do is projecting high dimensional points to a low dimensional embedding. I can determine the number of points in the high dimensional space and therefore also the number of points in the low dimensional space which form the area I want to calculate. If I increase the number of points it turns out that they are positionned between the "old" points, that is what I mean by smooth. Given a certain point I can assume that in some proximity around that point I will be able to find new points belonging to the area if I sample more dense.
Additionally I have a threshold value at which I can consider two points to be "equal", or in other words I know which resulution I want to achieve.
EDIT 2:
I use GPLVM's to do the mapping from high dimensional space to low dimensional space. So I think analysing that directly is two difficult/not possible. They are not very intuitive and I think in that case it is easier to work directly with the two dimensional points...
One option would be to find the convex hull of your set of points, that is, the convex polygon that contains all your points. Once you have the polygon, you can then find the area covered.
Of course, this won't handle the case where your underlying distribution has holes though, in which case I can't think of a better solution than your box-laying variant.