Image Processing Reference
In-Depth Information
Arithmetic geometry, as briefly introduced in [85] and developed in [172],
provides a uniform approach to the study of digitized hyperplanes in n dimen-
sions. Basic definitions follow the general idea of specifying lower and upper
supporting planes. We discuss here the three-dimensional case. Let a,b,c,µ
and ω > 0 be integers.
Definition 3.16. D
(a,b,c,µ,ω)
= {(i,j,k) in 3 − D digital space : µ <
ai+bj +ck < µ+ω} is called an arithmetic plane with normal n = (a,b,c)
T
,
intercept µ, and arithmetic thickness ω.
€
An arithmetic plane is a generalization of an arithmetic line D
(a,b,µ,ω)
=
{(i,j) in 2−D digital space : µ < ai+bj < µ+ω}. From Reveilles' theorem
on arithmetic lines [172] we know that naive lines with ω = max{|a|,|b|} are
the same as digital lines. If ω = max{|a|,|b|,|c|}, then the arithmetic plane
D
(a,b,c,µ,ω)
is called a naive plane. They have shown that a finite DPS γ in
the grid- point model is characterized by the property that it is between two
supporting planes
ai + bj + ck = µ and ai + bj + ck = µ + c.
The upper supporting plane is a translation of the lower supporting plane (by
translation vector (0,0,1)). The main diagonal direction of both (under the
assumption 0 < a < b < c) is (−1,−1,+1), and the main diagonal distance
between both planes is less than or equal to
√
3. In [46], Coeurjolly et al. also
presented a theorem to compute all the pre-images of a set of digital points
representing a 3-D digital plane segment.
3.4.4 Area Estimators
This subsection introduces various estimators for the actual area of a pre-
digitized planar segment corresponding to a given netcode of an n-DPS. As
in the case of 3-D line segments we shall characterize the netcode by a tuple
t that consists of a fixed number of parameters extracted from the digital
image. The parameters of these tuples are then combined in several ways to
formulate various estimators A(t).
In order to measure the performance quantitatively, we define an error
function associated with an estimator A(t), which we call the Relative De-
viation RDEV (A(t),N) where N is a netcode. It is the square root of the
normalized (with respect to n
2
) mean square error (MSE) of the estimator
A(t)) in the area measurement averaged over all plane segments producing
the tuple t. Formally, if A = n
2
(1 +a
2
+b
2
) is the area of the original plane
segment (we use m = n here), then
(A(t) −A)
2
)p(a,b,c)dadbdc)/n
2
RDEV (A(t),N) =
(
Domain(t)
where Domain(t) is the set of continuous plane segments (i.e., a,b,c triplet)
having the same characterization t of their digitizations.
