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I know that for a given signal, the sampling frequency $F_s$ must be twice or more than maximum frequency of the signal $F_m$. It is easy to understand the concept for a 1D signal. But I don't know how to calculate sampling frequency or Nyquist rate for a 2D image.

Also what is effect of over-sampling and under-sampling in case of image shown below (dimensions $\text{width} \cdot \text{height}=204\cdot226$)?

enter image description here

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Depends on how the image is created. If by photography, the focus cone and Airy disk for the given lens have a somewhat low-pass filter effect. Figure out the effective transition frequency at the imaging plane, and have a sampler with a bit over twice that resolution to diminish aliasing effects during image capture.

If you undersample (either during capture or with later processing) without low pass filtering, then you may get alias artifacts such as Moire patterns on high contrast fine detail. Classic example is a pin striped shirt at SD TV resolution.

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