Image Processing Reference
In-Depth Information
3.3.3.2 (n,n
o1
,n
o2
)-characterization [39]
Here, we use (n,n
o1
) and (n,n
o2
) separately to estimate the length of
the projection on the XY and XZ planes. Let these estimates be denoted
by |L
Z
| and |L
Y
|, respectively, and the estimator for the 3-D line be |L| =
(|L
Y
|
2
+|L
Z
|
2
−n
2
). |L
Y
| and |L
Z
| may be evaluated using several methods.
To design a very good estimator for the length of L, |L
Y
| and |L
Z
| are
to be chosen properly so that the RDEV for L is minimized. This global
minimization seems to be very di
cult. Instead, we discuss a simple method
of estimation. We select good estimators for L
Y
locally, and for simplicity,
we use a similar functional form for L
Z
. It is true that the estimators so
obtained are not optimal but the performance of these estimators is seen to
be satisfactory both theoretically and practically.
Depending on different choices of |L
Y
| and |L
Z
|, we may use different
√
length estimators. In one method, |L
Y
| may be (n−n
o1
) +
2n
o1
and|L
Z
|
√
may be (n−n
o2
) +
2n
o2
. The 3-D estimator so formed is called L
1
. In the
other method, the 2-D estimators are computed using the formula L
k
(n,n
o
) =
0.948(n−n
o
) + 1.343n
o
and the resultant 3-D estimator is named L
2
.
Another interesting estimator is the Euclidean length between the first
and the last digital point. This turns out to be the length between (0,0,0)
and (n,n
o1
,n
o2
). Thus, the estimator is L
0
=
(n
2
+ n
o1
+ n
o2
). As this
characterization is not a faithful one, the image of the line joining the above
mentioned points is not the given digital set. However, this line is close to
one probable line producing the same set of discrete points. This closeness
arises as a consequence of the famous chord property, which holds for the
projections on 2-D. This additional property of this estimator has made it
worth investigating.
3.3.3.3 (n,n
o1
,n
c1
,n
o2
,n
c2
)-characterization [39]
In this case also, we combine two 2-D estimators to obtain the desired 3-D
estimator. The 2-D estimators are calculated using the formula L
c
(n,n
o
,n
c
) =
0.980(n−n
o
) + 1.406n
o
− 0.091n
c
.
3.3.3.4 (n,q
1
,p
1
,s
1
,q
2
,p
2
,s
2
)-characterization
Since this is the faithful characterization, in most cases, the line defined
by the slopes tanφ = (p
1
/q
1
) and secφtanθ = (p
2
/q
2
) reproduces the same
digital set on quantization. Therefore, in most cases, this line turns out to be
one probable pre-image of the image data. Thus, another three-dimensional
estimator of interest is given by
(1 + (p
1
/q
1
)
2
+ (p
2
/q
2
)
2
).
L
0
= n
For different values of n, the average error of each length estimator has
been reported in [39].
