Image Processing Reference
In-Depth Information
which also immediately defines the energy-invariant measure for the
lack of linear
symmetry
C
f
3
:
−
|
I
20
|
I
11
C
f
3
=1
(balanced directions)
(10.52)
At the heart of such measures is how well the Schwartz inequality,
I
11
,
is fulfilled. The case of equality happens if and only if one has linear symmetry
(
|
I
20
|≤
=
I
11
). The left-hand side of it vanishes and the right-hand side becomes as
large as the energy permits it, if and only we have balanced directions (0=
|
I
20
|
|
I
20
|
).
Having this in mind, then other functionals that measure the distance
I
11
−|
can
be used to quantitate the balancedness of the directions in the image. The popular
detector of Harris and Stephen [97] (a similar measure is that of Forstner and Gulch
[74]) used to measure cornerness, quantitates this distance as well
I
20
|
0
.
04(
λ
max
+
λ
min
)
2
C
hs
=
λ
max
λ
min
−
(10.53)
0
.
04
I
11
=(
I
11
+
|
I
20
|
)(
I
11
−|
I
20
|
)
/
4
−
=(0
.
84
I
11
−|
2
)
/
4
I
20
|
(10.54)
albeit in the quadratic scale, which is most obvious if the empirical constant 0.84 is
replaced by 1. Because of the constant, the measure
C
hs
must be combined with a
threshold to reject the negative values. This will happen at (local) images that have
only the linear symmetry component (e.g., on lines and edges) where
I
11
,
yielding
C
hs
=
−
0
.
04
|I
20
| <
0. The measure
C
hs
responds strongly to many corner
types, including a corner that consists of the junction of two orthogonal directions, or
a corner that consists of the intersection of several lines. A word of caution is in place
because
C
hs
will also respond strongly to other patterns, including at every point in a
texture image that lacks direction. This may be a desirable property for an application
at hand. However, it is also possible that the application is actually unintentionally
accepting (false acceptence) many patterns as corners by using
C
f
3
or
C
hs
. The
texture images shown in Fig. 10.10 are perfectly balanced everywhere, meaning that
every point is a “
balanced directions corner
”or“
Stephen-Harris corner
”. Likewise,
all boundary points, except the boundary corners, between region A and region B
in Fig. 10.11 are the strongest corners in either of the two corner senses above. All
points of these four lines are, in fact, stronger “corners” than the four boundary corner
points, as discussed in Sect. 10.7!
=
|I
20
|
10.10 The Total Least Squares Direction and Tensors
It is in place to ask what makes the matrices
J
,
S
,oreven
Z
(second-order) tensors.
We recall that the basic difference between a second-order tensor and a matrix is
subtle and lies in that a tensor represents a physical quantity on which the coordinate
system has no real influence except for a numerical representation. The numerical
representation of a tensor is then a matrix that corresponds to physical measurements
