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Spherical harmonics really are the "spherical Fourier transform" you're looking for. The kind of hack you mention in comments, of doing a 2D Fourier transform on a lat-long projection, suffers from all the problems you usually have when you try to project a sphere onto a plane: not all spatial relations in the sphere are well-represented in the plane. If you ...


8

Fourier transforms wouldn't help you with a rotation. You'd just end up having to rotate the matrix of Fourier coefficients, instead of rotating the original image. Consider for example an image made of a perfect sine wave along the x-axis with wave-vector $(k, 0)$. (The wave-vector is the spacial frequencies along the $x$ and $y$ axes). The Fourier ...


7

A 2D Fourier transform is performed by first doing a 1D Fourier transform on each row of the image, then taking the result and doing a 1D Fourier transform on each column. Or vice versa; it doesn't matter. Just as a 1D Fourier transform allows you to decompose a function into a sum of (1D) sine waves at various frequencies, a 2D Fourier transform ...


4

Yes, it is possible. Remember that a shift in space is equivalent to a linear-phase multiplication in frequency. A rotation can be accomplished by a shearing operation in one direction followed by a shearing operation in the perpendicular direction followed by a final shear in the original direction (Alan Paeth, ``A Fast Algorithm for General Raster ...


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