Graphics Reference
In-Depth Information
The rotation resulting from the unit quaternion,
q
, is equivalent to that given by the rotation
matrix,
R
, as shown in Equation 2.21.
⎡
⎣
⎤
⎦
.
w
2
x
2
y
2
z
2
+
−
−
2
xy
−
2
wz
2
xz
+
2
wy
w
2
x
2
y
2
z
2
R
=
2
xy
+
2
wz
−
+
−
2
yz
−
2
wx
(2.21)
w
2
x
2
y
2
z
2
2
xz
−
2
wy
2
yz
+
2
wx
−
−
+
2.1.3 Decimation of 3D Surfaces
Decimation refers to the process of reducing the resolution of digitized signals or, in our case,
3D surface scans/representations. The resolution has impacts on the accuracy, memory storage,
and the computational complexity of 3D face recognition. A high resolution representation
enables the capture of facial surface details and generally yields to a better recognition accuracy
unless it is beyond the finest details. However, in the case of overly high resolution, in addition
to the high storage requirement and computational burden, a diminished or even adverse effect
on the recognition accuracy may result (possibly as result of the curse of high dimensionality).
Acquired facial surfaces are usually of a higher and possibly varying resolution (the resolution
depends on the distance of the face from the 3D digitizer). By decimating the 3D surface,
a compromise between these requirements can be achieved. Generally, a resolution of 1 or
0.5 mm suffices for 3D face recognition.
Mesh Decimation
Mesh decimation, or mesh simplification as it is sometimes referred to, aims to reduce the
number of mesh elements such that the reduced mesh remains as approximate to the original
mesh as possible. An overview of some well-known of the various approaches to mesh
decimation are provided here. For more in-depth discussions and a wider exposure to mesh
decimation algorithms, the reader is referred to the surveys by Botsch et al. (2007), Heckbert
and Garland (1997), Luebke (2001), and Renze and Oliver (1996).
Mesh decimation algorithms can be contrasted from each other depending on the following
factors: iterativeness (iterative versus non iterative), ability to preserve the mesh topology,
faithfulness to preserve the mesh details, error measures used to prioritize elements removal
or quantification of the closeness of the decimated mesh to the original mesh, regularity (as
opposed to adaptiveness) of the decimated mesh, and/or efficiency.
Vertex clustering:
Vertex clustering was proposed by Rossignac and Borrel (1993) for the
rendering of scenes in computer graphics. Vertex clustering is fast and produces regular
decimated meshes that can be of large decimation ratios. Despite the relatively low mesh
quality and the possibility of introducing topological changes (e.g., producing handles), the
algorithm has potential applications in 3D face recognition. It can be used, for example, along
with suitable feature extraction techniques (that are tolerant to its drawbacks) to perform fast
rejection classification in which matching gallery facial surfaces are short-listed not only for
a more accurate but also a more computationally expensive classifier.
The 3D space around the surface is first divided into regular cells (a 3D grid). The vertices
of the original surface that are within each cell is replaced by a representative vertex associated
