Image Processing Reference
In-Depth Information
x
(
t
) =
a
x
+
b
x
t
+
c
x
t
2
(4.41)
y
(
t
) =
a
y
+
b
y
t
+
c
y
t
2
A simplification can be obtained by considering that
v
(0) is the point where we want to
evaluate curvature. Thus, the lower order values of the polynomial are known and are given
by the pixel's co-ordinates. That is,
a
x
=
x
(0) and
a
y
=
y
(0). If the parameter
t
enumerates
the points in the sequence, then this means that the pixels must be indexed by negative and
positive values of
t
in the trailing and leading curve segments, respectively. That is, we
need to index a sequence of pixels relative to the pixel where we are computing curvature.
We can obtain a definition of curvature at the point
v
(0) by considering the derivatives
of Equations 4.41 within Equation 4.34. Accordingly, the value of curvature for the pixel
v
(0) is given by
-
[ +
cb cb
bb
yx
xy
κ
(0) = 2
(4.42)
2
2
]
3/2
x
y
In order to evaluate this expression, we need to estimate a pair of parameters for each
component in Equation 4.41. These parameters can be obtained by least squares fitting
(Appendix 2, Section 9.2). This fitting strategy will minimise the average error when the
error in the position of the points in the digital curve has a Gaussian distribution with
constant standard deviation. The main advantage of this fitting process is its simplicity and
in practice even when the error assumption is not completely true, the result can provide
a useful value. To estimate the four parameters of the curve it is necessary to minimise the
squared error given by
2
2
ε
=
wt x
( )(
(0) +
bt
+
ct
- ( ))
xt
x
x
x
t
(4.43)
2
2
ε
=
wt y
( )(
(0) +
bt
+
ct
- ( ))
yt
y
y
y
t
where the weighting function
w
(
t
) takes values between 0 and 1. Generally, these values
are used to limit the fitting to a small rectangular area in the image (i.e. a window). That
is, for a region of size 2
w
+ 1, the weight takes a value of one when
x
(
t
) -
x
(0) ≤
w
or
y
(
t
)
-
y
(0) ≤
w
and zero otherwise. Alternatively, weights with a Gaussian distribution can also
be used to increase the importance of points close to the fitting curve. By considering that
the derivatives with respect to the four unknown parameters vanish at the minimum of
Equation 4.43, we have that
S
S
S
S
S
S
S
S
b
c
b
c
yt
2
3
xt
2
3
y
x
t
t
t
t
=
and
=
(4.44)
S
S
S
S
2
3
4
2
3
4
y
x
yt
t
t
xt
t
t
where
2
3
4
S
=
wtt
( )
S
=
wtt
( )
S
=
wtt
( )
2
3
4
t
t
t
t
t
t
2
(4.45)
S
=
wt
( )(
x t
( ) - (0))
x
t
S
=
wt
( )(
x t
( ) - (0))
x
t
xt
2
xt
t
t
2
S
=
wt
( )(
y t
( ) - (0))
y
t
S
=
wt
( )(
yyt
( ) - (0))
y
t
yt
2
yt
t
t
Therefore, the solution for the best-fit model parameters is given by
