Direction in 2D - Vision with Direction - page 155

Fig. 10.8. Left : a real image that is linearly symmetric. It shows a close-up view of the blinds.
Right : The Fourier transform magnitude of a neighborhood in the central part of the original
(brightness) image. Notice that the power is concentrated to a line orthogonal to isocurves in
the original

Lemma 10.1 is proven in the Appendix of this chapter, Sect 10.17. It states that the
function g ( k T r ), which is in general a “spread” function such as a sinusoid or an
edge, is compressed to a line having the direction k in the Fourier domain. Even
more important, it says that as far as k is concerned the choice of G , and thereby
g , has no significance. This is because, k , the angle at which all nonzero F reside,
remains the same no matter what G is. This is achieved by the Dirac pulse δ , which
becomes a line pulse along the infinite line u T

).
Because u is the normal direction of this line and u is orthogonal to k , the direction
of the spectral line u T

ω =0by the expression δ ( u T ω

ω =0coincides with the vector k . We have already observed
this line in red-colored images of Examples 10.1-10.5, as a concentration of the
magnitudes to the central line, in the same direction as the same (green) k vector
shown in the gray images, regardless g .
Along this central spectral line, not only the magnitude but also the complex
values conform to that of the 1D Fourier transform. This is because G is the 1D
Fourier transform of g , and Eq. (10.8) is a formula for how to produce the 2D Fourier
transform of the linearly symmetric functions only from the 1D Fourier transform G
and the isocurve normal k . According to the lemma, the vector u can always be
deduced from k up to a sign factor, because it is orthogonal to k . Due to limitations
of the illustration methods, Examples 10.1-10.5 could only be indicative about this
more powerful result, and this only as far as the Fourier transform magnitudes are
concerned.
Consequently, to determine whether or not an image is linearly symmetric is the
same thing as to quantitate to what extent the Fourier transform vanishes outside of
a line, a property which will be exploited to construct computer algorithms below.
Such algorithms can be constructed conveniently to describe textures possessing a