Image Processing Reference
In-Depth Information
3.4.4.1 n
2
-characterization
The simplest way to characterize a netcode N is to count the number of
netcode elements. Obviously, it is n
2
. This is also the crudest estimator of the
actual area.
A
0
= n
2
Clearly, A
0
is a biased estimator and therefore the unbiased estimator is
of the form
A
1
= µn
2
,where µ is a constant.
3.4.4.2 (n
1
,n
2
,n
3
) -characterization
From Lemma 3.10, we can say that there are three types of codes in a
netcode N. Using the number of each of the three codes, we provide a new
characterization for N as follows:
Definition 3.17. A netcode N is characterized by (n
1
,n
2
,n
3
) if n
1
= |((i,j) :
N
i,j
= 00)|,n
2
= |((i,j) : N
i,j
= 10)| and n
3
= |((i,j) : N
i,j
= 11)|.
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This is a direct extension of the (n
e
,n
o
)- characterization for straight lines.
Also, (n
1
+ n
2
+ n
3
) = n
2
.
The exact relationship between (n
1
,n
2
,n
3
) and a,b,c is not yet reported.
However, we provide here a relation between them for large n.
Lemma 3.12. For an n-DPS , n
3
= bn
2
, n
2
= (a−b)n
2
and n
1
= (1−a)n
2
when n →∞ [42].
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We can also characterize the backward net by three similar parameters,
namely, n
1
, n
2
and n
3
. As a corollary to the previous lemma, we conclude that
for large n, n
i
≈ n
i
, where i = 1,2,3. From the description of v in terms of
forward and backward nets, we may visualize the digitization as a triangular
tessellation in 3-D. The areas of these triangles are better estimators for the
original area. Three types of triangles δ
1
, δ
2
, and δ
3
, may arise in our context
depending on the three types of netcode present. The vertices of these triangles
are:
δ
1
: {(i,j,k),(i + 1,j,k),(i,j + 1,k)}
δ
2
: {(i,j,k),(i + 1,j,k + 1),(i,j + 1,k)}
δ
3
: {(i,j,k),(i + 1,j,k + 1),(i,j + 1,k + 1)}
√
√
The areas of these triangles are |δ
1
| = 1/2, |δ
2
| = 1/
2, and |δ
3
| =
3/2.
Thus, the corresponding estimator is given by
√
(n
1
+ n
1
)|δ
1
|+ (n
2
+ n
2
)|δ
2
|+
3(n
3
+ n
3
)|δ
3
|,
