10

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 ...


1

You can take a 1D Fourier transform of a row of pixels from the image; it will give you the horizontal frequencies present in that row. You could sample one row out of the image, or else average all the rows together to get an overall picture of the horizontal frequencies.


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