Image Processing Reference
In-Depth Information
Now, z
i+1,j
= ⌊a(i + 1) + bj + c⌋≥⌊ai + b(j + 1) + c⌋ = z
i,j+1
.
Therefore, if the second component of the net code is a 1 then the first
component is bound to be a 1, and hence the code 01 can never occur for
a ≥ b. Similarly, if b > a, then the code 10 will not appear. Clearly, only three
codes may appear in the netcode.
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3.4.2 Geometric Characterization
Similar to the chord property to characterize a 2-D DSLS, Kim and Rosen-
feld defined a chordal triangle property to characterize a DPS. The chordal
triangle property is defined in the following.
Definition 3.11. [111] S, A set of digital points in 3-D, is said to have the
chordal triangle property iff for any p
1
, p
2
, p
3
∈S, every point of the triangle
p
1
p
2
p
3
is at L
∞
−distance < 1 from some point of S.
L
∞
− distance between two points p : (x
1
,y
1
,z
1
) and q : (x
2
,y
2
,z
2
) is
defined as max{|x
1
−x
2
|,|y
1
−y
2
|,|z
1
−z
2
|}.
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Kim and Rosenfeld [111] also proved the following theorem.
Theorem 3.14. A simple digital surface is a digital plane iff it has the chordal
triangle property.
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Definition 3.12. Vertical distance V
d
(P,S) between a plane P and a set of
points S is defined as max
p∈S
{ Vertical distance between P and p }.
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Definition 3.13. P is a Nearest Supporting Plane (SP) of a set of points S
if and only if the points in S lie completely on one side of P and V
d
(P,S) < 1.
Further, if S lies below P then P is a nearest upper supporting plane
(NUSP); otherwise, P is a lower nearest supporting plane (NLSP).
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The characterization of a DPS S is obtained following the same approach
as described in [110]. The following lemma is an adaptation of Theorem 15 of
[110].
Lemma 3.11. A set of points S is an n-DPS if and only if there is an nearest
(Upper or Lower) Support Plane P of S.
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Now, let us define a support face of a DPS P.
Definition 3.14. A face F of the convex hull of a DPS P is an Upper Support
Face (USF) of an n-DPS P if and only if the plane Q containing F is an
NUSP of P. The Lower Support Face (LSF) is defined analogously.
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Kim claimed that (Theorem 16, [110]) P is an n-DPS if and only if there
exists a support face of P. It is important to note that there are counter
examples of this claim.
To find the support face, Kim constructed the convex hull CH(P) and
