Best 2D interpolation for a set of points (x/y)

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Given a set of X/Y points in a bounded area ( 0<=X<=Xmax and 0<=Y<=Ymax ), what are the best 'interpolation' methods/algorithms to compute any other point of that plane?

I do have an arbitrary set of pixels I(x,y) in an image (simple grey-value pixels) and would like to interpolate the whole image, i.e. all other points P(x,y) in that image which are not yet of defined value.

What are reliable mathmatical algorithms to do so?