Image Processing Reference
In-Depth Information
defining an analytic expression for the autocorrelation. This expression can be obtained by
considering the local approximation of intensity changes.
We can consider that the points
P
x
+
i
,
y
+
j
and
P
x
+
i
+
u
,
y
+
j
+
v
define a vector (
u
,
v
) in the image.
Thus, in a similar fashion to the development given in Equation 4.53, the increment in the
image function between the points can be approximated by the directional derivative
u
∂
x
+
v
∂
P
x
+
i
,
y
+
j
/∂
P
x
+
i
,
y
+
j
/∂
y
. Thus, the intensity at
P
x
+
i
+
u
,
y
+
j
+
v
can be approximated as
P
P
xiy j
+, +
xiy j
+, +
P
=
P
+
+
u
v
(4.54)
v
xiuy j
++,++
xiy j
+,+
x
y
where this expression corresponds to the three first terms of the Taylor expansion around
P
x
+
i
,
y
+
j
(an expansion to first-order). If we consider this approximation in Equation 4.53 we
have that
2
w
w
P
P
xiy j
+, +
xiy j
+, +
E
( , ) =
xy
+
u
v
(4.55)
v
u
,
x
y
i
=-
w
j
=-
w
By expansion of the squared term (and since
u
and
v
are independent of the summations),
we obtain,
E
u
,
v
(
x
,
y
) =
A
(
x
,
y
)
u
2
+ 2
C
(
x
,
y
)
u
v
+
B
(
x
,
y
)
v
2
(4.56)
where
2
2
P
P
w
w
w
w
xiy j
+, +
xiy j
+, +
Axy
(
, ) =
Bxy
(
, ) =
x
y
i
=-
w
j
=-
w
i
=-
w
j
=-
w
(4.57)
w
w
P
P
xiy j
y
xiy j
+, +
+, +
Cxy
(
, ) =
x
i
=-
w
j
=-
w
That is, the summation of the squared components of the gradient direction for all the
pixels in the window. In practice, this average can be weighted by a Gaussian function to
make the measure less sensitive to noise (i.e. by filtering the image data). In order to
measure the curvature at a point (
x
,
y
), it is necessary to find the vector (
u
,
v
) that minimises
E
u
,
v
(
x
,
y
) given in Equation 4.56. In a basic approach, we can recall that the minimum is
obtained when the window is displaced in the direction of the edge. Thus, we can consider
that
u
= cos (ϕ (
x
,
y
)) and
v
= sin(ϕ (
x
,
y
)). These values were defined in Equation 4.48.
Accordingly, the minima values that define curvature are given by
2
2
AxyM
( , )
+ 2
CxyMM
( , )
+ ( , )
BxyM
y
x
y
x
(
xy
, ) = min
E
(
xy
, ) =
v
v
u
,
u
,
MM
2
+
2
x
y
(4.58)
In a more sophisticated approach, we can consider the form of the function
E
u
,
v
(
x
,
y
). We
can observe that this is a quadratic function, so it has two principal axes. We can rotate the
function such that its axes have the same direction as the axes of the co-ordinate system.
That is, we rotate the function
E
u
,
v
(
x
,
y
) to obtain
F
u
,
v
(
x
,
y
) = α
(
x
,
y
)
2
u
2
+ β
(
x
,
y
)
2
v
2
(4.59)
