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 seem to find a solution to:

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 $|n+im| \le r$.

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 $\theta_k=Arg(n+im)$ for some non-zero Gaussian integer $n+im$ of norm at most $r$.

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