Image Processing Reference
In-Depth Information
3.3.1.1
Digitization of a 3-D Line Segment
In the following, we define different digitization schemes used in the case
of 3-D CSLS.
Grid Intersection Quantization (for 3-D lines) [4]: Whenever a 3-D line
crosses a coordinate plane the nearest digital point on that plane to the point
of crossing becomes a point of D(l), the digital representation of l. This may
also be termed as Grid Intersection Quantization in (GIQ) 3-D.
Digitization scheme (adapted from [201]): Let the crossing point of the
line (segment) L and the hyperplane x
1
= j be P
j
: (j,p
2
,p
3
). Then we say
that the digital point P
′
j
: (j,r
2
,r
3
), where p
m
− 1/2 < r
m
≤ p
m
+ 1/2, for
m = 2 and 3, is the nearest digital point to P
j
and that P
′
j
is the digital image
of P
j
. The set consisting of all points P
′
j
is said to be the digital image of the
line (segment) L.
Object Boundary Quantization (OBQ): The OBQ digitization for 3-D
CSLS is defined in the following.
Definition 3.6. Let L be a 3-D line and D(L) denote its digitization. For
integral values of i from 0 to n, the point (i,⌊y(i)⌋,⌊z(i)⌋) belongs to D(L) if
and only if (i,y(i),z(i)) is a point on L. In other words, whenever a 3-D line
segment L crosses a coordinate plane x = i at the point (i,y(i),z(i)), the grid
point (i,⌊y(i),⌊z(i)⌋) becomes an element of the digital image D(L) of L for
i = 0 to n.
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The following theorem is restated from [4, 201], which holds for a line
segment in the standard situation.
Theorem 3.12. Let L be a 3 -D DSLS and l be a 3-D CSLS. L is the digital
image of l if and only if the projections of L on the y = 0 and z = 0 planes
are the GIQ images of the projections of l on the same two coordinate planes.
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Next, we consider characterization of DSLS using OBQ digitization as
discussed above. As y(i) = itanφ + y
1
and z(i) = isecφtanθ + z
1
, therefore
the grid points of D(L) can be written as (x
i
,y
i
,z
i
) where
x
i
= i,
y
i
= ⌊itanφ + y
1
⌋, and
z
i
= ⌊isecφtanθ + z
1
⌋ for all i,0 ≤ i≤ n.
We have already seen that the equation of L
Z
is y = xtanφ+y
1
and that
of L
Y
is z = xsecφtanθ + z
1
. Since the slopes of these lines are not greater
than unity, D(L)
Z
is the OBQ image of L
Z
and D(L)
Y
is the OBQ image of
L
Y
. The converse is also true. This leads us to the following theorem.
Theorem 3.13. D(L) is the digital image of L if and only if D(L)
Z
, D(L)
Y
are the OBQ images of L
Z
and L
Y
, respectively.
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