Image Processing Reference
In-Depth Information
L exp(i2
φ
)
L exp(i2
φ j )
exp(i2
θ
(x l ,y l ))
j ,b j 2
(
exp(i2
θ
(x l ,y l ))
j ,b j 2
(
Fig. 10.34. The graphs illustrate the structure tensor voting along an infinitely long line drawn
in magenta .( Left ) The directions of the green edges are approximately consistent with the long
magenta line , ( φ j ,b j ). A few lines corresponding to the same φ j but different b j are shown
in magenta .( Right ) The coherent directions of the green lines are in maximal conflict with
the long line. The linear symmetry tensors of the edges are shown in green as the complex
numbers exp( i 2 θ ( x l ,y l ))
parametrization that yields a unique representation of lines. Squaring both sides of
Eq. (10.91) yields:
( k T r ) 2 = r T kk T r = b 2
(10.96)
where
= cos 2 φ
= 1
2
I + cos 2 φ sin 2 φ
sin 2 φ cos 2 φ
sin φ cos φ
kk T
(10.97)
cos 2 φ
sin φ cos φ
The parameters (2 φ , b 2 ) will uniquely represent lines since ( φ + π,
b ) and ( φ, b )
map to the same parameter, the one and the same line.
The problem is thus to find how much support or votes there are along a presumed
line represented by (2 φ j ,b j ), given the observed edge points having the directions
{
} l along that line. In other words, we wish to know if a “subimage”, which
only consists of the long narrow line (2 φ j ,b j ), is composed of edge elements having
the same direction. The subimage is illustrated by the magenta line in Fig. 10.34.
This is equivalent to investigating if the image is linearly symmetric in the subimage.
A few other subimages, i.e., lines, corresponding to the same 2 φ j but different b j
are shown in magenta. According to the structure tensor theory, the solution to this
problem is to measure I 20
exp( i 2 θ l )
and I 11
in the image, the long line. Accordingly, we can
compute I 20 :
I 20 =
l
exp( i 2 θ ( x l ,y l ))
(10.98)
where ( x l ,y l ) T
are edge points along the presumed line. Likewise, summing the
magnitudes
|
exp( i 2 θ l )
|
=1yields:
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