Image Processing Reference
In-Depth Information
7.2 Voxel data and the polygon model
To begin with, let us clarify the form (it may be called
model
and
data struc-
ture
) used to represent an image to be visualized.
We will discuss a 3D continuous gray-tone image and a 3D digitized gray-
tone image. As explained in Chapter 2, these are represented by the notation
f
(
x, y, z
)and
F
=
{
f
ijk
}
, respectively. A digitized image is stored in a 3D
array.
These data are called
voxel data
,orarenameda
voxel model
.Another
way to represent data, frequently used in visualization, is approximating the
surface of an object to be visualized by polygons.This is called a
polygon model
.
To simplify, a triangle is most frequently adopted to represent a polygon
model. As a result the object surface is expressed by a set of many small
triangular elements (called
triangular patches
).
To apply a polygon model to voxel data, we first segment an object from a
3D image, after which we approximate the object surface by a set of polygons,
that is, we transform the object surface to a polygon representation. There are
various methods to perform this procedure corresponding to different proper-
ties of objects and the specific purpose. For digitized data, features specific to
voxel data will be effectively used. It is important that each density value is
put on the regular cubic lattice, and that an object itself is regarded as a set
of cuboids. Let us discuss two examples.
7.2.1 Voxel surface representation
Here we regard each voxel as a small 3D continuous figure and render visible
faces of cuboid voxels directly. In this case, all faces forming the object surface
are squares of the same size. If we assume that the viewpoint is infinitely far,
that is, if we adopt the orthogonal projection, only three kinds of faces appear
around every vertex, looking to the same direction except for specific locations
(Fig. 7.1). The spatial resolution of the imaging system used to obtain an
original image is preserved, and shape deformation is not caused during the
rendering procedure. These properties greatly contribute to the simplification
of the visualization procedure.
On the other hand, artifacts are likely occur in rendering due to the fact
that the surface is represented by only three kinds of parallelogram patches.
Image quality of a visualization result may be not so good in such a resolution
where individual voxels can be seen. The surface of a rendered figure lacks
smoothness. The computation time is not always short compared with the
computation time for triangular patches.
7.2.2 Marching cubes algorithm
Fitting triangular patches to the surface of an object is done by placing a
triangular patch so that a set of patches makes a border surface between a set
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