Graphics Reference
In-Depth Information
z
-coordinates of the 3D points change, but the
u
and
v
indices should remain fixed. For this
reason, it is preferred to separate the shape
S
from texture
T
in the matrix representation,
namely,
[
xyz
]
,
s
=
(2.3)
S
=
[
s
1
...
s
N
]
,
(2.4)
[
uv
]
,
t
=
and
(2.5)
=
[
t
1
...
.
T
t
N
]
(2.6)
In this matrix representation, the geometric manipulation should be performed on the shape
matrix, whereas, the texture matrix stores and maintains the one-to-one correspondence of
the 3D points to the texture information. An example of a point cloud and its corresponding
texture map is shown in Figure 2.1.
3D Polygonal Mesh Representation
A3Dmesh
M
represents a 3D surface using sets of mesh elements; vertices
V
, edges
E
, and
polygons (facets)
F
along with incidence and/or adjacency relations,
M
=
(
V, E, F
). The
3
. Each edge,
e
i
∈
E
mesh vertices are 3D points,
V
⊂
, is defined by two distinct vertices,
E
⊂{
, is defined by three or more
edges such that each pair of edges share a vertex. (The vertex is incident to both edges.) In
the case of a triangular mesh (which is the most widely used type of mesh due its relative
simplicity), the facet
f
i
is exactly defined by three edges
e
i
={
v
j
,
v
k
}|
v
j
,
v
k
∈
V,
j
=
k
}
. While each facet,
f
i
∈
F
F
⊂{
f
i
={
e
j
,
e
k
,
e
l
}|
e
j
,
e
k
,
e
l
∈
E,
j
=
k
,
j
=
l
,
k
=
l
,
∀
e
1
,
e
2
∈
f
i
∃
v
(
v
∈
e
1
∧
v
∈
e
2
)
}
. Alternatively, each facet is defined
by three distinct vertices,
. The use
of subsets in the definitions of the mesh elements signifies that (1) the mesh representation of a
3D surface is not unique. In fact, there are several valid mesh representations for the same 3D
surface. (2) For a valid (and to a less extent optimal) mesh representation, further constraints
should be imposed on the selection of the mesh element sets and their adjacency relations. For
3D meshes representing compact 2-manifold surfaces (which is the case with facial surfaces),
each point of the 3D mesh and its neighborhood should be homeomorphic to an open disc
and no holes should be introduced (open discs are removed from the representation). Such
topological changes occur for example when (1) adding an excessive number of polygons
to
F
⊂{
f
i
={
v
j
,
v
k
,
v
l
}|
v
j
,
v
k
,
v
l
∈
V,
j
=
k
,
j
=
l
,
k
=
l
}
that are incident to a common vertex or edge (2) dropping internal polygons (away
from the border) from a valid mesh representation (creates a hole). Additionally, constraints
on the angles of the polygons and the lengths of the edges can result in more optimal mesh
representations. Polygons with acute angles are not desirable in mesh representation. There are
different types of mesh representations, depending on how the data of the mesh are stored and
organized in data structures. Pros and cons of some well-known meshes are briefly discussed
in the following paragraphs:
F
Polygon mesh:
It is the simplest 3D mesh in which the polygon data are stored in a table.
Each row of the table stores the
x
-,
y
- and
z
-coordinates of all vertices of a polygon. As
the vertices that are incident to a polygon can also be incident to many other polygons, this
