Direction in 2D - Vision with Direction - page 147

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Fig. 10.1. The graph represents the 1D function g ( t )=sin( ωt ) that will be used to construct
a linearly symmetric 2D function

f ( x, y )= g ( k T r )= g ( k x x + k y y ) (10.1)

The direction of the linear symmetry is ± k .

The term is justified in that F , the 2D Fourier transform of f , is concentrated to a line,
as will be shown below. In addition, all isocurves of linearly symmetric functions are
lines that have a common direction k , i.e., they are parallel to each other. Note that the
term isocurves refers to the fact that the values of g and thereby f are invariant when
one moves along certain curves in the argument domain. For linearly symmetric
images, these curves are lines.
It should be noted that while g ( t ) is a function of one free variable, g ( k t r ) is a
function of two free variables, ( x, y ) T since k is constant. In the rest of this section
we will assume that the argument domain of f is two-dimensional, whereas that of g
is one-dimensional and g is a “constructor” of f via Eq. (10.1) and k whenever f is
linearly symmetric. Therefore g ( k T r ) generates an image despite that g by itself is a
one-dimensional function. By definition, images with the linear symmetry property
have the same gray value at all points r satisfying k T r = C for a given value C .
Because k T r = C describes a line in the ( x, y )-plane, it follows that along this line
the gray values of the image do not change and this gray value equals to g ( C ). In such
images, the only occasion when g can change is when the argument of g changes,
i.e., when the constant C assumes another value. However, the curves k T r = C 1 ,
k T r = C 2 , ... k T r = C n , with C i being different constants, represent lines that
are shifted versions of each other, all having the same direction, k . Consequently,
the one-dimensional function g is a profile of the two-dimensional function g ( k T r )
along any line perpendicular to the line k T r = C .
Local images, which can be extracted by multiplying the original image with an
appropriate window function, are, from mathematical viewpoint, no different than
the larger original image from which these local images are “cut”. For notational