Image Processing Reference
In-Depth Information
The values of α and β are proportional to the
autocorrelation
function along the principal
axes. Accordingly, if the point (
x
,
y
) is in a region of constant intensity, then we will have
that both values are small. If the point defines a straight border in the image, then one value
is large and the other is small. If the point defines an edge with high curvature, then both
values are large. Based on these observations a measure of curvature is defined as
κ
)
2
(4.60)
The first term in this equation makes the measure large when the values of α and β
increase. The second term is included to decrease the values in flat borders. The parameter
k
must be selected to control the sensitivity of the detector. The higher the value, the more
sensitive to changes in the image (and therefore to noise) computed curvature will be.
In practice, in order to compute κ
k
(
x
,
y
) it is not necessary to compute explicitly the
values of α and β , but the curvature can be measured from the coefficient of the quadratic
expression in Equation 4.56. This can be derived by considering the matrix forms of
Equations 4.56 and 4.59. That is,
k
(
x
,
y
) = αβ
-
k
(α
+ β
T
,
T
E
(
xy
, ) =
P
MP
and
F
(
xy
, ) =
P
QP
(4.61)
u
,
v
x y
x y
,
u
,
v
x y
x y
,
where
T
denotes the transpose and where
Axy
(,
)
Cxy
(,
)
0
M
=
)
and
Q
=
(4.62)
Cxy
(,
)
Bxy
(,
0
In order to relate the matrices
M
and
Q
we consider the rotation transformation
′
P
x
,
=
RP
x
,
y
(4.63)
Thus, the rotated system is obtained by substitution of the rotated point in
E
u
,
v
(
x
,
y
). That
is,
T
F
(,
xy
) = [
RP
]
MRP
(4.64)
v
u
,
x y
,
x y
,
,
TT
By arranging terms, we obtain
F
(,
xy
) =
P
R
MRP
. By comparison with Equation
v
u
,
x y
x y
,
4.61, we have that
Q
=
R
T
MR
(4.65)
which means that
Q
is an orthogonal decomposition of
M
. If we compute the determinant
of the matrices in each side of this equation, we have that det(
Q
) = det(
R
T
) det(
M
) det(
R
).
Since det(
R
T
) det(
R
) = 1, thus,
αβ
=
A
(
x
,
y
)
B
(
x
,
y
) -
C
(
x
,
y
)
2
(4.66)
which defines the first term in Equation 4.60. The second term can be obtained by taking
the trace of the matrices in each side of this equation. Thus, we have that
α + β =
A
(
x
,
y
) +
B
(
x
,
y
) (4.67)
If we substitute these values in Equation 4.60, we have that curvature is measured by
κ
k
(
x
,
y
) =
A
(
x
,
y
)
B
(
x
,
y
) -
C
(
x
,
y
)
2
-
k
(
A
(
x
,
y
) +
B
(
x
,
y
))
2
(4.68)
Code
4.19
shows an implementation for Equations 4.57 and 4.67. The equation to be used
is selected by the
op
parameter. Curvature is only computed at edge points. That is, at
pixels whose edge magnitude is different of zero after applying maximal suppression. The
