transformation mathematical model. Then, the transformation
parameters are retrieved by minimizing the volume formed
between the points that represent the conjugate control and
low-density lidar planes.
Volume Minimization Algorithm
The basic idea of the volume minimization algorithm is
to find transformation parameters that generate minimum
volume between corresponding 3D planes in two coordinate
systems. Figure 4a shows an example where a control surface
is marked in black and low-density lidar surface is marked in
gray. The volume that needs to be minimized is the light gray
box shown in Figure 4b. It should be noted that this example
figure assumed only a Z-translation exists between the sur-
faces. In reality, it can have six degrees of freedom, including
three translations and three rotations which refers to boresight
translations and rotations, respectively. The volume compu-
tation between surfaces is not a trivial problem. Unlike area
minimization used in Nagarajan and Schenk (2016), there are
no closed form solutions available for volume computation.
Hence, 3D Delaunay triangulation is used to form a surface
that represents the volume that needs to be minimized be-
tween conjugate planes. 2D Delaunay triangulation generates
unique triangular edges from given set of points that does not
have more than three points lie on a circumcircle. The union
of all edges from Delaunay triangulation generates convex
hull (Cheng
et al
., 2012) of given points. The area contained
by the convex hull is the sum of areas of individual triangles
bounded by the given set of points. Similar to 2D Delaunay
triangulation, 3D Delaunay triangulation generates tetrahe-
drons from the given set of points. The total volume of all
tetrahedrons created through 3D Delaunay triangulation is a
representation of volume contained by the given set of points.
The total volume is given by Equation 4 for
n
-tetrahedrons
created from 3D Delaunay triangulation. In the boresight
calibration problem, the lidar point cloud point will carry
boresight translation and rotation as unknowns, which will be
determined by minimizing the volume between control and
lidar surfaces. The control surface will be in
IMU
body frame
and whereas lidar surface will be in a local lidar
SBF
. Though
any free-form surface can be used in Volume Minimization,
only planar patches are used in this research for simplicity.
V
X i Y i Z i
X i Y i Z i
X i Y i Z i
X
i
n
=
( )
( )
( )
( )
( )
( )
( )
( )
( )
=
1
6
1
1
1
1
1
1
1
2
2
2
3
3
3
Σ
4
4
4
1
i Y i Z i
( )
( )
( )
(4)
The proposed approach for the determination of boresight
translation/rotation as 3D rigid body transformation is imple-
mented using the following steps.
• Identify at least three mutually perpendicular conjugate
planes in both lidar and control point clouds. In the ab-
sence of such planes, sloping planes can be chosen.
• Select 3 or more coplanar points on each corresponding
pair of each lidar and control planes.
• Establish the 3D Delaunay triangulation of both lidar
plane and control plane points. This process is performed
independently for each corresponding pair of planar
surfaces. The tetrahedrons formed between the surfaces is
the representation of volume formed between the planes.
The generated tetrahedrons are classified into three types.
The three types of tetrahedrons are illustrated in Figure 5.
• Type I -one point from control plane and three points
from lidar plane
• Type II -two points from control plane and two points
from lidar plane
• Type III -three points from control plane and one point
from lidar plane
Figure 5. Three types (Type I, II, and III ) of tetrahedrons
that are possible between corresponding control (dark gray)
and lidar (light gray)planes.
All three types of tetrahedron have at least one point in
the lidar plane. The rigid body transformation parameters
that transform the lidar points to control plane coordinates is
determined by minimizing the volume of each category tetra-
hedrons. It should be noted that type I, II and III tetrahedrons
have three, two and one point, respectively, in the lidar co-
ordinate system that need to be transformed to control plane
coordinate system by minimizing the volume between them.
Derivation of Volume Minimization Mathematical Model
The total volume between two planar surfaces from lidar and
control data is the sum of volume of tetrahedrons that are
formed between them. In order to determine the 3D rigid-
body transformation parameters that transforms the lidar
points into control plane coordinate system or
IMU
body
frame, the volume equation needs to be written in terms of
(a)
(b)
Figure 4. Illustration of Volume Minimization: (a) An
example of control (black) and lidar surfaces (dark gray), and
(b) Light gray box is the volume that needs to be minimized
to recover Z translation.
622
October 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
