How to estimate a linear function in $\mathbb R$, given rounded linearly spaced measurements?

Question

Given measurements $$\left \lfloor t+i\cdot x\right \rfloor \\\\ i\in\left\{ 0,1,\cdots ,N\right\} \\\\ x,t \in \mathbb R$$ Where $\left \lfloor \cdot \right \rfloor $ denotes integer part, how to estimate $t$ and $x$?

Methods of any level of difficulty and branch of mathematics can feel heartily welcomed. I will add my own thoughts a bit later, but am sadly in a hurry right now.

Answer 1

I am going to assume $N$ is large and $x$ is small $(<1)$. Let $f:\{0,\dots,N\}\to\mathbb{Z}$ be the function $f(n)=\lfloor t+ n x\rfloor$. Let $n_1\geq 1$ be the first $n$ such that $f(n)>f(n-1)$ and $n_2$ be the last such $n$. Then $x\approx \frac{f(n_2)-f(n_1)}{n_2-n_1}$ and $t\approx f(0)+1-n_1x$.