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The very strong 45º lines in the rotated image are probably mostly artifacts caused by the image edges seen in the rotated version. A rotation in real-space is

The very strong 45º lines in the rotated image are probably mostly artifacts caused by the image edges seen in the rotated version. A rotation in real-space is equivalent to one in reciprocal space, so the two transforms should look the same (barring the interpolation differences, etc.) but rotated.

wirrbel 9y ago
I wonder whether one could show this. maybe by pretending the image is actually a cell in a periodic grid, then transform this grid, and rotate this grid. The periodicity would make roation possible without getting empty spots.
- mturmon 9y ago
  That's a good idea. The condition implicitly assumed to get the result mentioned ("A rotation in real-space is equivalent to one in reciprocal space") is that the original image is periodically tiled in all directions. Your suggestion mimics this condition. Since your other comment mentioned limitations of optical apertures, maybe it's worth noting that the way the 45 degree rotation was done here (plunking the rotated image into a black background) mimics the action of an optical aperture -- that is, clipping the observed image along that rectangle. This causes 45 degree spikes, similar to: https://en.wikipedia.org/wiki/Diffraction_spike#Diffraction_spikes_due_to_non-circular_aperture https://en.wikipedia.org/wiki/Diffraction_spike#Diffraction_....
  - craigyk 9y ago
    Yes. I was going to add that the lines in the original image are more likely due to the discontinuity between the top, bottom, left, right edges. The other comment wasn’t mine, but is right that an aperture in the back focal plane of a lens would cut out image information at higher resolution.
- kmill 9y ago
  Something I find interesting is that, although on the surface this seems like a way to deal with rotations in general, it only works for certain angles. For instance, with a 45 degree rotation, you have to scale the image, too, and the resulting tile contains two of the original image. Topologically, an underlying problem is that there is not a continuous family of rotations for a torus (the topological space corresponding to two independent periodic dimensions).
brianberns 9y ago
> The very strong 45º lines in the rotated image are probably mostly artifacts caused by the image edges seen in the rotated version. I think this is correct, and there are also some (more subtle) artifacts in the unrotated version for the same reason. The thing I don't understand is why this occurs for 2D images, but there are no artifacts created by the "edges" of a 1D sound (i.e. the start and end of the clip).
- robryk 9y ago
  When you do a Fourier transform of sound, you'd usually apply a windowing function: multiply the sound with a function that descends continuously to zero at the boundary of the "window", so that there is no discontinuity there. That function is chosen in such a way that it should affect frequencies of interest in the transform's output only mildly.
  - brianberns 9y ago
    That makes sense. Seems like the same technique could be used with images, yes?
    - robryk 9y ago
      Yes, although not with very low-resolution images (8x8 or 16x16), because then you run out of frequency gap between interesting frequencies and ones that are contributed to by the windowing function. EDIT: Another facet of the same issue viewed from pixel space is that you lose a lot of contribution of pixels close to the edge, and in such small images majority of pixels are close to the edge. There's also an interesting alternative for images, if you don't care about inverting the transform to get the original image: you can express an image as a sum of a periodic component and a very slowly changing one. The periodic component looks like the original to the human eye, so taking DFT of it is usually sufficient for analysis. The paper that discovered that technique: https://hal.archives-ouvertes.fr/hal-00388020v1/document https://hal.archives-ouvertes.fr/hal-00388020v1/document
  - gregfjohnson 9y ago

wirrbel 9y ago

I wonder whether one could show this. maybe by pretending the image is actually a cell in a periodic grid, then transform this grid, and rotate this grid. The periodicity would make roation possible without getting empty spots.

mturmon 9y ago
That's a good idea. The condition implicitly assumed to get the result mentioned ("A rotation in real-space is equivalent to one in reciprocal space") is that the original image is periodically tiled in all directions. Your suggestion mimics this condition. Since your other comment mentioned limitations of optical apertures, maybe it's worth noting that the way the 45 degree rotation was done here (plunking the rotated image into a black background) mimics the action of an optical aperture -- that is, clipping the observed image along that rectangle. This causes 45 degree spikes, similar to: https://en.wikipedia.org/wiki/Diffraction_spike#Diffraction_spikes_due_to_non-circular_aperture https://en.wikipedia.org/wiki/Diffraction_spike#Diffraction_....
- craigyk 9y ago
  Yes. I was going to add that the lines in the original image are more likely due to the discontinuity between the top, bottom, left, right edges. The other comment wasn’t mine, but is right that an aperture in the back focal plane of a lens would cut out image information at higher resolution.
kmill 9y ago
Something I find interesting is that, although on the surface this seems like a way to deal with rotations in general, it only works for certain angles. For instance, with a 45 degree rotation, you have to scale the image, too, and the resulting tile contains two of the original image. Topologically, an underlying problem is that there is not a continuous family of rotations for a torus (the topological space corresponding to two independent periodic dimensions).

brianberns 9y ago

> The very strong 45º lines in the rotated image are probably mostly artifacts caused by the image edges seen in the rotated version. I think this is correct, and there are also some (more subtle) artifacts in the unrotated version for the same reason. The thing I don't understand is why this occurs for 2D images, but there are no artifacts created by the "edges" of a 1D sound (i.e. the start and end of the clip).

robryk 9y ago
When you do a Fourier transform of sound, you'd usually apply a windowing function: multiply the sound with a function that descends continuously to zero at the boundary of the "window", so that there is no discontinuity there. That function is chosen in such a way that it should affect frequencies of interest in the transform's output only mildly.
- brianberns 9y ago
  That makes sense. Seems like the same technique could be used with images, yes?
  - robryk 9y ago
    Yes, although not with very low-resolution images (8x8 or 16x16), because then you run out of frequency gap between interesting frequencies and ones that are contributed to by the windowing function. EDIT: Another facet of the same issue viewed from pixel space is that you lose a lot of contribution of pixels close to the edge, and in such small images majority of pixels are close to the edge. There's also an interesting alternative for images, if you don't care about inverting the transform to get the original image: you can express an image as a sum of a periodic component and a very slowly changing one. The periodic component looks like the original to the human eye, so taking DFT of it is usually sufficient for analysis. The paper that discovered that technique: https://hal.archives-ouvertes.fr/hal-00388020v1/document https://hal.archives-ouvertes.fr/hal-00388020v1/document
- gregfjohnson 9y ago