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Hexagonal signal processing

Usually in image processing or other 2-dimensional (2-d) signal processing a rectangular grid is used, most typically a square grid. Let a function $f(x,y)$ with $x,y\in {\mathbb{R}}^2$ be sampled at integer locations $x,y\in {\mathbb{Z}}^2$. A way to describe the sampling process is to multiply $f(x,y)$ by a grid of multi-variate Dirac delta functions shifted to integer locations $a,b\in {\mathbb{Z}}^2$:

$$\begin{gathered} f(x,y) \\ ,\text{sampled}=f(x,y)\sum _{a=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{b=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x-a,y-b) \end{gathered}$$

If there was no other information about $f(x,y)$ the information about its value at non-integer $x,y$ would be lost.

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