obtained from ForeStereo data with each of the three meth-
ods as well as with the caliper measurements, establishing a
bin interval of 0.05 m and a minimum value of
DBH
= 0.075
m. The histograms derived from ForeStereo data and from
caliper measurements were compared through the
quadratic-
form distance
(Equation 6), proposed by Hafner
et al.
(1995)
to assess the similarity between histogram distributions.
d
GroundTruth F
GroundTruth
F
T
GroundTruth
F
H
H H
H A H
H
,
(
)
=
−
(
)
−
(
)
(6)
where (
H
GroundTruth
) is the matrix of histogram bin values as de-
rived from caliper measurements and (
H
F
) the matrix of histo-
gram bin values derived from ForeStereo data.
A
is a similar-
ity matrix with [
a
ij
] denoting the similarity between histogram
bins i and j, calculated as
a
ij
= 1
−
|
i
−
j
|/max(|
i
−
j
|). Lower
values of the quadratic-form distance indicate higher similar-
ity between histogram distributions.
Relaskop-Based Estimation Combined with Correction of Occlusion
Effect Based on Poisson Attenuation Model
The Relaskop-based approach for estimation of plot or stand
level variables is based on the angle-count sampling and
has frequently been used in
TLS
measurement to reduce the
instrument bias (e.g., Strahler
et al.
2008, Lovell
et al.
2011).
Only trees with
DBH
apparently wider than an angular span
κ
(which depends on a predefined
basal area factor
(
BAF
)) are
included in the sample. The
BAF
is typically (and for conve-
nience) set as 0.0002 m
2
/m
2
. With our dataset, the number of
trees per m
2
(
λ
) was calculated from the
n
trees included in
the sample as (Equation 7):
λ
π
=
(
)
=
∑
*
/
i
n
i
1
2
2
BAF
DBH
(7)
where
BAF sin
=
(
)
2
2
κ
/
(8)
The gap probability decreases exponentially w
following a Poisson model of the form P
gap
= e
number of trees
n
corr
expected to actually exist
dius
r
max
(including detected and occluded tree
as a product of the number of trees actually measured
n
and a
factor of attenuation
F
(
t
):
n n F t
corr
=
( )
*
(9)
F
(
t
) depends on
λ
, the effective diameter calculated for all
the trees of the plot (
D
E
) and the distance
r
max
:
F t
t
t
t
( )
= − −
( )
+
(
)
(
)
exp
2
1
1
2
(10)
t
D r
E max
=
λ
The effective diameter takes into account the occlusive
effect of stems, low branches, and understory. Strahler
et al.
(2008) proposed considering a
D
E
which depends on the aver-
age diameter of the trees actually measured and their variabil-
ity (Equation 11, where Cv is a coefficient of variation):
D
E
=
+
(
)
DBH C
v
1
2 2
(11)
(
n
corr
)
i
is then estimated for each detected tree and summa-
rized to estimate
N
:
N n
r
i
n
i
i
=
(
)
(
)
=
∑
1
2
10 000
corr
max
*
π
(12)
Distance-Sampling Based Correction of
Instrument Bias and Occlusion Effect
In this approach, the probability of detecting trees in the plot
is modeled through the
detection function
g
(
r
,
θ
). As in As-
trup
et al.
(2014), we employed the Half-Normal function:
( )
g r
r
,
θ
= −
( )
(
)
exp
2
2
2
σ
(13)
and the Hazard-Rate function:
( )
g r
r
b
,
/
θ
= − −
(
)
(
)
−
1 exp
σ
(14)
In
g
(
r
,
θ
) the parameter
θ
comprises both the scale (
σ
) (in the
Half-Normal and the Hazard-Rate functions) and shape (
b
)
(only in the Hazard-Rate function) parameters. The inclu-
sion of
DBH
as a covariate was explored in both models,
as in Ducey
et al.
(2014), expanding the scale parameter
σ
=
α
0
·exp(
α
1
·
DBH
).
The parameters
θ
ˆ are obtained by maximum likelihood
(Marques and Buckland 2003; Miller and Thomas 2015; Clark
2016). The probability of detection for tree
i
in a plot of radius
R
is given by Equation 15. Our sampling was truncated at 8
m, 9.8 m, and 15 m to analyze the effect of different
R
values
on estimates:
P
i
R
R
r g r
=
( )
∫
2
2
0
* ,
θ
ˆ
(15)
Finally,
N
is estimated as in Equation 16. Note that P
i
is
identical for all
i
in models without covariate:
N
R
i
n
i
=
(
)
=
∑
1
2
10 000
P *
π
(16)
Hemispherical Photogrammetric Correction (
HPC
)
Here we propose a new approach for estimating forest
variables at plot and stand level from data obtained using
ForeStereo. This approach combines the segmentation-based
correction for instrument bias described with detail in Sán-
chez-González
et al.
(2016), and a new method for estimating
the probability of occlusions which adapts the method pro-
posed by Seidel and Ammer (2014) to the case of stereoscopic
hemispherical images.
In the equation for instrument bias correction proposed by
Sánchez-González
et al.
(2016) the range of detection depends
on the stem diameter and the inclination angle of the viewing
direction. Stem sections thinner than the minimum window
size, defined in pixels as
p
sel
, are not detected (as mentioned
in the segmentation process description). Assuming there
are no occlusions, each tree sampling area
A
i
is calculated
integrating a circular area of radius defined by the user (
R
) or
by the maximum horizontal distance
r
max
at which the section
of the tree wider than
p
sel
is projected on the images (Sánchez-
González
et al.
2016) (Equation 17):
A
R r
i
=
(
)
(
)
∫
0
2
2
1
2
π
α
min
d
max
,
(17)
498
July 2019
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
