Image Processing Reference
In-Depth Information
which asymptotically approximates to
√
√
A
2
= n
1
+
2n
2
+
3n
3
for large n.
Though A
2
is expectedly better than A
0
or A
1
, this is again a biased
estimator usually overestimating the original. Actually all these estimators
A
0
, A
1
, A
2
are expressed as linear combinations of n
1
, n
2
, and n
3
. Depending
on the choice of the coe
cients we can think of a class of simple estimators,
which are of the form µ
1
n
1
+ µ
2
n
2
+ µ
3
n
3
, where µ
1
,µ
2
,µ
3
are constants.
Important among them is the one for which the RDEV is the least. There is
an effort [42] to obtain better estimates by minimizing RDEV with respect to
the coe
cients.
3.4.4.3
Non-linear Estimators
A major common drawback of all simple estimators discussed so far is
that the RDEV of them does not decrease beyond a certain value (i.e., the
asymptotic bound obtained theoretically), even if we go on increasing the
sampling density. To offer a solution to this problem, two non-linear estimators
are used [42].
An example of these estimators is given below:
= n
2
(1 + (n
3
/n
2
)
2
+ ((n
2
+ n
3
)/n
2
)
2
)
A
4
((n
1
+ n
2
+ n
3
)
2
+ (n
2
+ n
3
)
2
+ n
3
).
=
The RDEV of A
4
is seen to be zero as n tends to infinity.
The RDEV and bias of the different area estimators and their bias are
pictorially presented in Fig. 3.10 [42]. They establish that the performance of
all these estimators except A
4
cannot be improved beyond a certain limit by
increasing the sampling density alone.
3.5 Summary
In this chapter, we have dealt with digitized straight line segments in 2
and 3 dimensions and digitized plane segments. In each case, we discussed
digitization schemes and various characterizations of the digitized figures. We
also discussed about several property estimators for these discrete shapes.
For 2-D and 3-D digital straight line segments, we outlined algorithms to
compute one probable continuous figure that could be the pre-image of a given
digital set of points. In fact, we described algorithms to compute the domain
of a digitized 2-D straight line segment, which is the set of all continuous
