Graphics Reference
In-Depth Information
2.1.1 Primary 3D Surface Representations
Before we provide formal definitions and descriptions of the different types of 3D surface
representations, we will start with an informal description of these representations and indicate
some of their differences.
Scanners usually digitize 3D surfaces in the form of dense 3D point clouds. This is an
intuitive output because each surface measurement results in a 3D point. The point cloud rep-
resentation is capable of
completely
representing open or closed
free-form
(characterized by
an irregular shape) 3D surfaces. In some face-recognition systems, the 3D point cloud repre-
sentation can be used throughout the different modules of the system, namely, preprocessing,
feature extraction, and matching. For example, the point cloud representation is sufficient in
an iterative closest point (ICP) based face recognition system. (
Note
: Some variants of ICP are
based on mesh representation.)
However, converting from the point cloud representation to other
complete
representations
such as 3D mesh, range image, or normal map is often needed or desirable. The 3D mesh
representation provides a flexible and efficient manipulation of surfaces. This because it
stores indexed and precomputed local information of the surface. For instance, once the mesh
representation is computed, it enables a more efficient region growing in comparison to a point
cloud representation. In addition, the deformation of 3D meshes is more flexible than point
clouds. However, 3D meshes require more memory and storage.
For a surface scan of a single view, a range image or a normal map representation can
be adequate. To an extent, they resemble 2D gray-level images and thus many 2D image
processing and computer vision operations can be similarly and directly applied on them,
notably kernel filtering and decimation. These two representations are simpler than 3D meshes,
and yet, they can be handy for some types of surface manipulation such as segmentation,
computation of curvatures, deformation, and translation. Rotating 3D surfaces might, however,
result in the self-occlusion of parts of the surface (in which case some of the 3D points will
be over ridden in the range image) or the exposure of previously self-occluded parts (which
results in the appearance of surface holes). The following sections provide formal definitions
and descriptions of the different 3D surface representations.
Point Cloud Representation
A point cloud representation is merely a set of 3D tuples of the
x
-,
y
- and
z
-coordinates, each
representing a 3D point (or a measurement) on the 3D surface. Let
p
3
denote
=
(
x
,
y
,
z
)
∈
a 3D point in a point cloud
. Alternatively, a 3D point can be represented as a 3D vector
p
,
and a point cloud can be represented as a 3
P
×
matrix
P
, where
N
is the number of points.
[
xyz
]
.
p
=
(2.1)
P
=
[
p
1
...
p
N
]
.
(2.2)
Textured point clouds are usually represented by attaching to each 3D point a pair of
u
and
v
indices pointing to a position on a texture map. In this case, the tuple representation
of a textured 3D point is five dimensional,
p
5
. During the manipulation
of the textured point cloud, such as rigid transformations or deformations, the
x
-,
y
- and
=
(
x
,
y
,
z
,
u
,
v
)
∈
