Image Processing Reference
In-Depth Information
I
11
=
l
1=
L
(10.99)
where
L
is the length of the observed line segments along the presumed line. Ac-
cordingly, the inequality
|
I
20
|≤
I
11
(10.100)
holds with equality if all observed directions of the edges are are the same, or else
|
will be less than
I
11
. Accordingly, the
I
20
values computed by Eq. (10.98) along
a given line (2
φ
j
,b
j
) are the total votes in the accumulator cell
A
(2
φ
j
,b
j
). Alter-
natively, a normalized vote
I
20
/I
11
can be cast if there are large contrast variations
in the image. For a given direction 2
φ
j
, the tensor summation in Eq. (10.98) can
be computed for all
b
j
s, yielding one column in the accumulator matrix
A
. Called
orientation radiograms, such columns can encode the shape information, useful for
various applications, e.g., image database retrievals [163].
In the ideal case, the inequality
I
20
|
I
11
will hold with equality, and the count
of votes in the parameter cell
A
(2
φ
j
,b
j
) will be identical to that obtained by use of
the Hough transform. Both methods will thus deliver the same vote count in case all
edge elements have the same direction as the presumed direction.
What happens if equality is not reached, i.e., some of the directions of the edges
are in maximal conflict with each other? To illustrate this, we assume that half of the
edge elements have directions orthogonal to the other half, i.e., we have either the
direction
θ
0
or
θ
0
+
π
2
|
I
20
|≤
, yielding the complex tensor elements:
I
20
=
m
exp(
i
2
θ
0
)+
n
exp(
i
2
θ
0
+
iπ
)=
L
L
2
2
−
=0
(10.101)
The conflicting votes of the orthogonal directions are thus counted as negative votes,
reducing the strength of the total vote for (2
φ
j
,b
j
). By contrast, in the Hough trans-
form an observed edge element with a conflicting direction is only prevented from
voting. Such edge elements are hindered from reducing the accumulator votes.
In fact, the votes of the structure tensor can not only be negative, but can even
be complex-valued. The argument of the complex-valued vote tells which (direction)
candidate the current edge is supporting. Vote counting is a vectorial averaging, and
the result,
, will be as large as
I
11
if the edge directions are collinear, or else the
total vote will be reduced, possibly until
|
I
20
|
reaches 0. In contrast to Hough voting,
tensor voting allows one to fit a line even to a dashed line where the edge elements
share the same direction consistently, but this common direction is different from the
direction of the presumed long line, drawn in magenta in Fig. 10.34. If this is not
desired, the total complex votes not agreeing with the corresponding cell labels, i.e.,
2
φ
j
, can be eliminated by applying a threshold to the total vote arguments (angles).
|
I
20
|
