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Highly oversample FFT2 in one direction for peak detection

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Hello, I have a 2D image and the resulting FFT2 has high peak. Now I can find the maximum to get the resulting peak shift in each dimension. But, in one of the dimensions the resolution of the accuracy bin is 10^4 times worse than what I need for the peak estimation.
So instead of doing an FFT with N samples in that direction, I would require N*10^4 samples. This would require a total amount of samples in that direction of 2^27, which is obviously making me run out of memory.
Is there a smart way to do the FFT with the inital amount of samples in both directions and doing a local oversampling around the peak to get the frequency shift in the required accuracy?

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