Graphics Reference
In-Depth Information
and elevation
⊂
angles within the view range of the 3D digitizer,
r
:
×
→
.
Alternatively, it can be the sets of
x
- and
y
-coordinates,
X
⊂
and
Y
⊂
, that is
r
:
X
×
Y
→
. The range image from a data structure point view is a 2D matrix,
R
, entries of
which correspond to the range data, where its horizontal and vertical indices implicitly define
the azimuth and elevation angles (for an angle-based range image) or the
x
- and
y
-coordinates
(for the
XY
-based range image), see Figure 2.2.
The angle-based range image representation—which can be the default representation of
some 3D digitizers because of the direct relationship with the sensor orientation—suffers from
the limitation that it undergoes a perspective-like transformation. The range readings are as if
they were projected on a spherical surface in a way similar to a 2D image plane in the case of
2D imaging. In contrast, the
XY
-based range image does not suffer from this transformation,
which makes it a better choice for 3D face recognition. Nevertheless, an angle-based range
image is capable of representing multi view surfaces (e.g., closed surfaces). A cylindrical
form of a range image can represent the 3D surface around one direction, often the azimuth,
r
:
×
Y
→
.
Conversion from an angle-based to an
XY
-based range representation:
The range is first
converted to a point cloud representation. Then the point cloud is converted to an
XY
-based
range image. Let an
N
1 matrix,
R
θφ
, represent an angle-based range image with
implicit azimuth angles ranging from
+
1
×
M
+
(
θ
max
−
θ
min
)
i
N
θ
min
to
θ
max
,
={
θ
i
=
θ
min
+
|
i
=
0
...
N
−
(
φ
max
−
φ
min
)
j
M
1
}
, and elevation angles ranging from
φ
min
to
φ
max
,
={
φ
j
=
φ
min
+
|
j
=
0
...
M
−
1
}
. The point cloud representation is shown in Equation 2.10.
x
=
R
ij
cos
φ
j
cos
θ
i
,
(2.7)
y
=
R
ij
sin
φ
i
,
(2.8)
z
=
R
ij
cos
φ
j
sin
θ
i
,
(2.9)
P
={
p
i
,
j
=
(
x
,
y
,
z
)
|
(
θ
i
,φ
j
)
∈
×
}
.
(2.10)
Conversion from a point cloud to an
XY
-based range representation:
First, the resolution
of the
XY
-based range image
R
xy
is decided, let it be
N
1. The implicit information
about the
x
- and
y
-coordinates are then decided. One choice is to let the horizontal indices
represent the set
−
1
×
M
−
X
of the
x
-coordinates varying from minimum
x
min
=
min
x
P
to maxi-
(
x
max
−
x
min
)
i
N
mum
x
max
=
max
x
P
, i.e.,
X
={
x
i
=
x
min
+
|
i
=
0
...
N
−
1
}
. Similarly, the ver-
(
x
max
−
x
min
)
j
M
Y
Y
={
y
j
=
x
min
+
|
=
...
−
}
tical indices represent the
.
The range image pixels correspond to the implicit
x
- and
y
-coordinates according to
R
ij
=
-coordinates,
j
0
M
1
. The pixel values are the interpolation of the range (
z
-coordinate)
at the implicit
x
- and
y
-coordinates. Those pixels not in the 2D convex hull formed by the
neighboring
x
- and
y
-coordinates (of 3D points in
(
x
i
,
y
j
)
∈
X
×
Y
) are masked out because not all the
implicit
x
- and
y
-coordinates correspond to the 3D surface (or 3D points in
P
P
).
Normal Map Representation
A normal map representation can be defined as a partial binary function that maps the horizontal
and vertical coordinates to unit normal vectors (or tuples),
n
:
×
3
. Similar to range
→
