Finding a local maximum of an interpolation splines is an easy enough thing;note that you should generally use a true splines here not a pchip interpolant

I think you re using few values for the interpolation by changing to i get the following and changint the splines degree to i get the following i think a good starting point for the interpolation could be n 2 and as the interpolation presents less data deformation

With a lower order splines that works better but then you lose the advantage of cubic interpolation

I did the math for interpolation with trig functions rather than polynomials as mdurant suggested;it turns out to be very similar to the cubic splines but requires more computation and produces worse results so i won t be doing that

Using a zero-padded ifft to upsample produces a similar high quality interpolation except for circular boundary effects;if you have to use 1 of your 3 available methods splines is less worse than pchip as the latter may clip a bit and both of those are less worse than linear

That is you could use to give but for the more interesting case of images like you mention at the end of your question then the axes probably won t be integer multiples and you ll be better served by a splines or other type interpolation

I am quite free in terms of the type of the interpolation but it shouldn t be too fancy nothing crazier than a splines or a low order polynomial