Image Processing Reference
In-Depth Information
Alternatively, if
α
u
0
, for all
u
such that
u
=
K, K
−
1
,
···
,k,
1
−
α
u
1
.
Therefore,
β
k
1. This means that the amount
α
k
in
b
k
contributes to a
density value of a displayed image with little loss. This corresponds to the
case that material between a point P
u
and P
k
is almost transparent.
f
∗
(c) Visualization of equidensity surface of a density value
Assign suitably an opacity value
α
∗
toadensityvalue
f
∗
,andthengivea
value near to
α
∗
to a point having a density value similar to
f
∗
.Usingthe
gradient, for instance,
=
0
,
and
f
i
=
f
∗
1
,
if
|∇
f
i
|
f
∗
)
/
1
−
(
1
/r
)
·
(
f
i
−
|∇
f
i
|
,
α
i
=
(7.13)
f
∗
≤
if
|∇
f
i
|
>
0
,
and
f
i
−
r
|∇
f
i
|≤
f
i
+
r
|∇
f
i
|
0
,
if otherwise
where
f
i
f
i
is the gradient at a point P
i
,
and
r
is an appropriately given constant [Drebin88].
is a density value at a voxel P
i
,
∇
(d) The visualization of a border surface in a 3D image
If it is known beforehand that typical density values are
f
n
,n
=
1
,
2
,
···
,N
,
where
f
1
<f
N
then opacities
α
k
that are proportional to density
values
f
k
are employed. If a density value
f
i
of a voxel P
i
takes an intermediate
value among them, a value of
α
i
corresponding to it is given as follows.
<f
2
<
···
f
n
)
/
(
f
n
+1
−
f
n
)]
1
,
if
|∇
f
i
|×
α
n
+1
[(
f
i
−
+
α
n
[(
f
n
+1
−
f
i
)
/
(
f
n
+1
−
f
n
)]
,
if
f
n
≤
f
n
+1
α
i
=
f
i
≤
(7.14)
0
,
if otherwise
(
i
=
1
,
2
,...,N
−
1
)
As is known from these examples, the volume rendering also intends to
visualize a surface that may exist in a 3D gray-tone image. A surface to be
visualized (or displayed) is determined by only the spatial distribution of den-
sity values in 3D space. In examples presented above, the existence of a surface
is detected mainly by the gradient. Larger values of the opacity are assigned
to the part of small gradient values in the case of visualizing equidensity sur-
faces and to the part having larger gradient values in the case of visualizing
a border surface, so that those surfaces may become more noticeable.
7.8.3 Front-to-back algorithm
Let us introduce another procedure to perform the volume rendering that is
called
front-to-back algorithm
[ERTL00].
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