surface. The goal of this process is to provide only the
measurement obtained from nadir view, avoiding unneces-
sary measurements that can affect the shape of the lidar
signal. To this end, the distances selected were those taken
with the smallest vertical angle provided by the manufac-
turer (0.00698132 rad) and the horizontal angles equal to
zero. Then, the first lidar distance measured before takeoff
(Figure 2) was applied as a reference to produce a flight
distance from the lidar data.
3. The signals from the
GNSS
data and the lidar data were
compared using the cross-correlation function as a similar-
ity criterion (Equation 1) to compute the highest correla-
tion point between data vectors, enabling identification of
the corresponding vector positions and estimation of the
clock difference. The signals obtained from the beginning
and at the end of the flight trajectory were considered
independently. Therefore, two independent processes
were performed, considering the trajectory signal from the
beginning and the end separately, resulting in two clock
differences whose average was used for synchronization.
4. Finally, the computed clock difference was used to correct the
laser clock to the
GPS
time. The cross-correlation function is
i
A
A
B
ρ
µ
σ
µ
σ
A B
N
i
N
A
i
B
B
,
(
)
=
−
−
−
=
∑
1
1
1
,
(1)
where
μ
A
and
σ
A
are the mean and standard deviation of
A
,
respectively, which correspond to a vector that contains
lidar range values (reference vector); and
μ
B
and
σ
B
are the
mean and standard deviation of
B
, respectively, which
correspond to the vector with
GNSS
height values (search
vector). The maximum correlation coefficient represents a
higher similarity between the flight heights from the two
data sets and is used to identify the position on the
GNSS
data vector that corresponds to the lidar data vector used
as a reference. These correlated heights can be assumed to
have been measured in the same instance, and the posi-
tions identified in the
GNSS
vector can be used to match
the
GPS
time and the local laser clock values in the same
instance. The mean of these values cor
clock difference.
Refinement by the Least-Squares Method
The clock differences estimated as described in the previous
section can be refined with the least-squares method. In the
mathematical model used (Equation 2; adapted from Kraus
1993),
v
are the residuals,
b
is the contrast (scale),
c
is the
brightness (shift),
g
1 and
g
2 are the signal magnitudes,
ξ
is the
pattern position or pixel coordinates, and
a
is the subpixel
shift between the signals:
ν
+
g
2(
ξ
) =
b
.
g
1(
ξ
+
a
) +
c
.
(2)
In our case, the signals can be supposed to have the same
amplitude, and thus the scale and shift can be neglected.
Equation 3 presents the mathematical model implemented in
this research:
ν
+
g
2(
ξ
) =
g
1(
ξ
+
a
).
(3)
Equation 4 is obtained by renaming the parameters de-
scribed in Equation 3 to range, height, and time variables, in
which the parameter
a
is the clock difference or parameter to
be estimated:
ν
+
H
LASER
(
t
) =
H
GPS
(
t
+
Δ
t
).
(4)
This equation was linearized, resulting in Equation 5, in
which
H
LASER
(
t
) is the lidar distance measured at time
t
,
H
GPS
(
t
)
is the
GNSS
flight height measured at the same instance
t
,
H
′
GPS
(
t
) is the first derivative (gradient) of the
GNSS
height, and
∆
t
is the clock difference:
(
H
LASER
(
t
) –
H
GPS
(
t
)) +
ν
=
H
′
GPS
.
Δ
t
.
(5)
The difference between the laser distance and
GNSS
height
at an instance
t
can be treated as an observation, and
Δ
t
the
parameter to be estimated. This value can be estimated with
the least-squares method:
∆
∆
t
H
H H
i
n
i
i
n
i
i
=
′
(
)
(
)
′
=
( )
−
=
( )
( )
∑ ∑
1
2
1
1
GPS
GPS
.
.
(6)
The solution can also be achieved by solving each
Δ
t
in
Equation 4 and averaging the values, which is consistent with
the least-squares solution.
Calibration of the ALS
According to Habib
et al.
(2009), random and systematic er-
rors can affect the point-cloud coordinates because they are
a combination of measurements from different components
of the system and mounting parameters (lever-arm offset and
boresight misalignment angles relating the
IMU
and the laser
scanner). The mounting parameters of the proposed
ALS
sys-
tem were measured in two independent steps.
The lever-arm offsets were directly measured in the labo-
ratory with centimetric accuracy using a precision caliper.
These offsets were not further refined during the calibration
procedure, which involved only boresight-angle estimation.
The lever-arm parameters were used in the
GNSS
data process-
ing performed with the commercial software Inertial Explorer
8.6 (NovAtel, Calgary, Canada), producing the coordinates of
points from the laser unit in the geodetic reference system.
The boresight misalignment angles were estimated in an
indirect
ALS
calibration procedure based on the lidar math-
ematical model (El-Sheimy, Valeo and Habib 2005) adapted
by Torres and Tommaselli (2018):
U
(
t
)
r
LU
IMU
+
R
g
IMU
(
t
)
R
LU
IMU
R
ED
LU
(
t
)
ρ
i
(
t
),
(7)
where
r
i
is the vector of the ground coordinates of point
i
;
r
g
GNSS
(
t
) is the vector of the ground coordinates of the
GNSS
antenna at instant
t
, reduced to the
IMU
coordinate system;
R
g
IMU
(
t
) is the rotation matrix relating the ground and
IMU
coor-
dinate systems at instant
t
, derived after processing the
GNSS
and
IMU
data;
r
LU
IMU
is the offset between the laser-unit and
IMU
origins (lever arm);
R
ED
LU
is the rotation matrix relating the
laser-unit coordinate system and the laser-emitting devices
(mirror scan angles) at instant
t
(as the laser unit used in this
research scans four horizontal layers, the rotation matrix is
expressed as a function of two angles
β
and
θ
); and
ρ
i
is the
vector of the coordinates of point
i
expressed in the emitting-
device reference system.
R
LU
IMU
is the rotation matrix that
relates the laser-unit and
IMU
coordinate systems as a function
of the approximate angles directly measured (
Δ
κ
,
Δ
φ
,
Δ
ω
). The
initial values of the
R
LU
IMU
elements are given as
Δ
κ
= −90°,
Δ
φ
= −90°, and
Δ
ω
= 0°.
Ground control points (
GCPs
) were identified and measured
directly on the terrain and in a preliminary point cloud gener-
ated with the raw lidar data provided by the
UAV
-
LS
to esti-
mate boresight angles. The information obtained from lidar
data concerning these
GCPs
—such as the position, attitude,
scan angles, and distance—and the directly measured 3D
756
October 2019
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
