I'm a PhD student in image processing, where I've stumbled into a problem that seems to be essentially number theory. I've hunted around online and while I've found many results on similar problems, this particular problem I cannot find a solution for:

Given a line in the plane passing through the origin making angle theta with the x-axis, I am trying to determine the closest nonzero Gaussian integer n+im to the line obeying |n+im| <= r.

I have a conjecture which is backed up by numerous computer tests, but no proof. The conjecture is as follows:

Let \Theta(r) = {\theta_1,\theta_2,...,\theta_N} denote the set of angles representable using Gaussian integers of this form. That is, each \theta_i=Arg(n+im) for some Gaussian integer n+i*m of norm at most r.

Find \theta_i, \theta_{i+1} straddling \theta, i.e. \theta_i <= \theta < \theta_{i+1}. Let n+im be the Gaussian integer that solves our minimization problem. Then either Arg(n+im)=\theta_i or Arg(n+im)=\theta_{i+1}.