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Hexagonal signal processing

2022-09-02, Olli Niemitalo

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 \(a, b \in\mathbb{Z}^2\):

\[f(x, y)\\, \text{sampled} = f(x, y)\sum_{a=-\infty}^\infty\sum_{b=-\infty}^\infty\delta(x-a, y-b)\]

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