Image Processing Reference
In-Depth Information
v
j
) of v is an n-DSLS. The i-th row DSLS v
i
consists of the following digital
points:
v
i
= (⌊bj + (ai + c)⌋ where 0 ≤ j ≤ n).
Similarly, the j-th column DSLS v
j
is defined as follows:
v
j
= (⌊ai + (bj + c)⌋ where 0 ≤ i ≤n).
Definition 3.9. v is an n-DPS if there exists a plane p such that D(p) = v
in 0 ≤ x,y ≤ n.
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Let P and P
′
be two planes given by the equations z = ax + by + c and
z = a
′
x + b
′
y + c
′
, respectively. Let v be D(P) and v
′
be D(P
′
). Then the
′
two n-DPSs v and v
are identical if and only if all the row and the column
n-DSLSs are the same for them. That is, we have this lemma:
′
Lemma 3.9. ∀i,j v(i,j) = v′(i,j) if and only if ∀i(∀j(v
i
(j) = v
i
(j))) and
′
∀j(∀i(v
j
(i) = v
j
(i))).
Proof: The proof follows from simple rearrangement.
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Similar to the chain code description of a digital line segment we describe a
net code representation of a DPS in 3-D. If p is a plane and v is its digitization
D(P), then v consists of only those grid points that are nearest and below p on
the vertical grid lines. Now consider a point p : (i,j,k) of v on the vertical line
x = i, y = j. Looking forward from P we encounter two points p
1
and p
2
on
vertical lines x = i+1, y = j and x = i, y = j+1. Depending on the difference
between the Z-coordinates of p and p
1
we associate a '0' or a '1' to p. Similarly
another '0' or '1' is attached to p by considering the Z-coordinates of p and
p
2
. Associating a netcode element to every point v
i,j
of v, 0 ≤i,j ≤ n−1, in
this fashion we construct a forward net of the n-DPS v. A backward net of v
is analogously defined.
Definition 3.10. Let v
i,j
= (i,j,z
i,j
),0 ≤i,j ≤n−1 and P : z = ax+by+c.
If the netcode at v
i,j
is N
i,j
, then N
i,j
=< z
i+1,j
−z
i,j
,z
i,j+1
−z
i,j
>. We also
write N
i,j
(1) = the first component = z
i+1,j
−z
i,j
and N
i,j
(2) = the second
component = z
i,j+1
−z
i,j
.
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It may seem at a first glance that four types of net code elements are
required to describe the forward net but we prove in the next lemma that one
out of the four possible codes is not needed.
Lemma 3.10. In the forward net of an n-DPS v, only three types of netcode
elements viz. 00, 10, 11 can occur.
Proof: Let P be given by z = ax + by + c and a ≥ b. In this case we claim
that the code 01 cannot appear.
Let us assume that there exists some i,j such that N
i,j
= 01. So z
i,j+1
=
⌊ai + b(j + 1) + c⌋ = z
i,j+1
.
