View source for Fourier transforms rotation effects

<html>In
general, rotation of the image results in equivalent rotation of its
FT. To see that this is true, we will take the FT of a simple cosine
and also the FT of a rotated version of the same function. The results
can be seen by:<br /><br /><img
src="http://www.cs.unm.edu/%7Ebrayer/vision/slant.gif" /><br
/> 
<p style="text-align: left;"> At first, the results seem
rather surprising. The horizontal cosine has its normal, very simple
FT. But the rotated cosine seems to have an FT that is much 
more
complicated, with strong diagonal components, and also strong
"plus sign" shaped horizontal and vertical
components. The question is, where did these 
horizontal and vertical
components come from? The answer is that the FT always treats an image
as if it were part of a periodically replicated array of 
identical
images extending horizontally and vertically to infinity. And there are
strong edge effects between the neighbors of such a periodic array as
can be seen by:<br /><br /></p><p
style="text-align: left;"><img
src="http://www.cs.unm.edu/%7Ebrayer/vision/slant_group.gif"
/><br /></p><p style="text-align: left;">Thus,
what we see as the FT in the "slant" image (lower
right of the image before last) is actually the combination of the
actual FT of the cosine function and that caused by the edge effects of
looking at a finite part of the image. These edge effects can be
significantly reduced by "windowing" the image with a

function that slowly tapers off to a medium gray at the edge. The
result can be seen by:</p><center><br /><p
style="text-align: left;"><img
src="http://www.cs.unm.edu/%7Ebrayer/vision/window.gif" /><br
/></p><p style="text-align: left;">The windowed image is
shown in the upper left. Its FT is shown in the lower left. The
non-windowed FT is shown in the upper right and the actual, true FT of
a cosine is shown in the lower right. These images are all scaled
differently and the comparison is only qualitative, but it can be seen
that the windowed image FT is much closer to the true FT and liminates
many of the edge effects. 
</p></center></html>
[[Category:microscopy]]