(NOTE: I have changed and hopefully simplified this question by removing the section on randomly perturbing lattice points, and instead specifying that the counts at each vertex should be randomly perturbed in the manner described below.)

Imagine an arbitrarily large two-dimensional rectangular lattice with vertices $(v_1, v_2, ...) \in V$ and edges $(e_1, e_2, ...) \in E$, where the edge-length in the lattice is $L$.

Place a point, $Q$, somewhere internal to the lattice (and sufficiently far away from the lattice periphery). Simulate Gaussian noise about $Q$ by sampling $N$ points $(p_1, p_2, ..., p_N) \in P$ according to the PDF of a Gaussian centered on $Q$ with defined values for $\sigma_x$ and $\sigma_y$ that are some fraction of the lattice edge length $L$ (let's simplify things by setting $\sigma_x = \sigma_y$). Bin these points to the nearest lattice vertex, and generate a set of counts $(c_1, c_2, ...) \in C$, where $c_i$ represents the number of points binned to the vertex $v_i$.

Now, define a set of real numbered values over the interval $[0,1]$: $(k_1,...,k_N) \in K$, where $k_i$ values are drawn from a Gaussian distribution with a mean $u_{y}$ and a variance $\sigma_{y}$. Generate a new set of vertex counts, $C^*$, by multiplying each $c_i \in C$ with the corresponding $k_i \in K$: $(c_1*k_1, c_2*k_2, ..., c_N*k_N) \in C^*$.

As a function of $u_{y}$ and $\sigma_{y}$, what will my worst-case error be in terms of determine the true position of $Q$ provided the counts $C^*$?


closed as unclear what you're asking by Anthony Quas, Stefan Kohl, Ricardo Andrade, Daniel Moskovich, David White Nov 16 '13 at 15:19

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    $\begingroup$ I don't think the question is clear enough in its current form. When you say the standard deviation of the distance from vertex to vertex is $\sigma_L$, you (presumably) are talking about the distance between perturbed versions of adjacent vertices in the lattice? I think you need to be more precise about how you're moving the vertices - are you just moving each one by an independent Gaussian for example? $\endgroup$ – Anthony Quas Nov 16 '13 at 7:47
  • $\begingroup$ @AnthonyQuas Thanks for bearing with me here. I updated the question with a few clarifications and comments to help address your comment. Could you let me know if things are still unclear or not sufficiently well defined? $\endgroup$ – SayaSphere Nov 16 '13 at 16:49
  • $\begingroup$ It's just not enough to say if we randomly perturb distances between lattice points. (Does knowing the distances determine the lattice?) You need to be telling us how you're going to move the lattice points themselves... $\endgroup$ – Anthony Quas Nov 16 '13 at 21:37
  • $\begingroup$ @AnthonyQuas In my update I was specifying that the vertices in the lattice are (independently) being moved according to the PDF function for a Gaussian centered on each vertex? I understand your concern that randomly perturbing the edge lengths makes the problem poorly defined, so my hope was that this should address the matter. $\endgroup$ – SayaSphere Nov 16 '13 at 22:21
  • $\begingroup$ @AnthonyQuas My mistake - it was unclear that I made this change because I did not modify the original question posting. $\endgroup$ – SayaSphere Nov 16 '13 at 22:27