# Zero-order-hold for two dimensional signal $x(t_1,t_2)$

From Wikipedia:

A zero-order hold reconstructs the following continuous-time waveform from a sample sequence $x[n]$, assuming one sample per time interval $T$: $$x_{\mathrm{ZOH}}(t)\,= \sum_{n=-\infty}^{\infty} x[n]\cdot \mathrm{rect} \left(\frac{t-T/2 -nT}{T} \right) \$$ where $\mathrm{rect}()$ is the rectangular function.

We can refer to $\mathrm{rect}()$ also as one-dimensional rectangular function. For example, the product of two one-dimensional rect functions can be viewed as a rect function in two dimension, i.e. a function that has the value 1 on the square of side length 1 centered at the origin, and has the value 0 outside this square. I am wondering if there exists, for example, a two dimensional version of ZOH for a signal in two variable, $x(t_1,t_2)$.

Yes there exists a 2D analog of the 1D zero order hold for image reconstruction, whose mathematical formulation follows closely that of 1D case as: $$I_{\mathrm{ZOH}}(x,y)\,= \sum_{n=-\infty}^{\infty} \sum_{m=-\infty}^{\infty} x[n,m]\cdot \operatorname{rect} \left( {x-nT_x,y-mT_y} \right)$$
Where $x[n,m]$ is the spatial samples of 2D image to be reconstructed, $T_x$ and $T_y$ are sampling periods in $x$ and $y$ dimensions (which are generally equal in practice) and the $\operatorname{rect}()$ function is a 2D rectengular pulse of height one and durations $T_x$ and $T_y$ along the dimensions.