How can I estimate the centre of mass with obscured/missing data? How can I estimate the centre of mass for a sample of simple point objects with uniform mass distributed in 2 dimensions when parts of the sample are obscured?
For example consider a collection of points where we have absolute knowledge over most of the space but inside a rectangular region we have no knowledge. We cannot just assume there are no points in the space.

Does the answer to the problem change if we know the overall number of points? For instance if we know there are 1000 points in total but we can only observe 800.
This question may be overly broad. To clarify: I am asking what methods can be used to solve questions like this, what are their limitations and what are the recent advances in mathematics related to solving these types of questions.
(I am not a mathematician)
 A: Premised that the answer depends on many factors, and on the "rigourness" you want for your estimation, let me just lay some guidelines.
The barycenter of two figures is the weighted average of the single barycenters.   
Thus clearly, knowing how many points are missing has much influence.
You need to have some knowledge/hypothesis about the probability with which the missing points might be distributed in the black rectangle, and of course that will depend on where the rectangle is placed wrt the global cloud.
For instance, if the knowledge of the underlying physical phenomena authorizes you to assume a distribution symmetrical wrt the barycenter, and the black rectangle is enough "small" and "peripherical", then you can compute the barycenter of the visible points, reproduce in the black rectangle the points laying in a visible symmetrical rectangle, recompute the barycenter and reiterate the process vs. the new barycenter. 
I do not think you can expect more than that, and do not expect to assess the accuracy of such estimation, unless you know the expected distribution of the points (2D Gaussian or whatever).
