Identifying specific pattern in a noisy dataset

Question

I am working on a project where I have the following forward model. There are multiple point sources sparsely distributed in 3D space with coordinate $[x_0,y_0,z_0]$ for each of them. The detected data consists of 8 points in the form of $[x,y,k]=[x_0,y_0]+r_0[\sin(k\pi/4),\cos(k\pi/4)]+\mathrm{Gauss}(0,\sigma^2), k\in\{1,...,8\}$. One challenge is that sometimes one measurement could be missing, e.g., we sometimes only detect 3 points for a certain $[x_0,y_0,z_0]$. Is there a way to correctly identify and estimate all the $[x_0,y_0,z_0]$?

Answer 1

The answer depends on what you mean by the word detect. The inability to detect points may be due to imperfection of the device or imperfection of the processing algorithm. If the device does not allow you to receive additional data on the missing points, then in your formulation of the problem the answer will be no. However, if you know any additional theory about the relative position of these points, then perhaps the solution to the problem can be obtained in the presence of incomplete information.