reference object and the corresponding object by computing
the distance between their centroids.
qLoc
ij
=
dist
(
Centroid of r
i
,
Centroid of s
j
),
s
j
∈
S
*
i
1
(21)
where
dist
(
a,b
) is the
Euclidean distance
between
a
and
b
.
Zero is the minimum and optimal value of
qLoc
ij
. The maxi-
mum value of
qLoc
ij
depends on the segmentation method
and object matching method.
Möller
et al
. (2007) improved on
qLoc
, and proposed the
Relative Position
(
RP
)
by normalizing the
qLoc
result.
RP
dist Centroidof r Centroidof s
qLoc
s S
ij
i
j
j
ij
j
i
=
(
)
(
)
∈
,
max
,
*
1
(22)
RP
ij
’s value is continuous in the range of [0,1], and positively
correlated with the location similarity.
Based on the object-fate matching method, Cheng
et al
.
(2014) put forward the Position Discrepancy Index
(PDI)
which evaluates the location similarity by calculating the
average distance between the Expanding, Good, and the refer-
ence objects.
PDI
dist Centroidof r Centroidof s
dist Cen
i
j
N
i
j
k
M
=
(
)
+
=
=
∑
∑
1
1
,
troidof r Centroidof s
N M
i
k
,
(
)
+
(23)
s
j
∈
Good
i
s
k
∈
Expanding
i
where
N
and
M
stand for the number of good objects and
expanding objects, respectively; a zero value of
PDI
i
indicates
an ideal location similarity between the reference object and
the corresponding object.
Based on Boundary
The boundary based evaluation indexes evaluate the seg-
mentation result by measuring the degree of coincidence of
the boundaries and the shape similarity directly between the
reference object and the corresponding object.
Lucieer and Stein (2002) described the Distance-Based
Measure
(D)
to reflect the degree of boundary coincidence
between the reference object and the corresponding object.
This index computes the shortest
Euclidean distance
from
each pixel on a vector boundary of the reference object to the
boundary pixel of the corresponding object.
D
dist PR PS
N
s S
ij
n
N
in j
j
i
=
(
)
(
)
∈
=
∑
0
1
min
,
,
*
(24)
where
N
is the number of pixels on the boundary of the refer-
ence object.
PR
stands for the pixel on the boundary of the
reference object and
PS
represents the pixel on the boundary
of the corresponding object which has the shortest Euclid-
ean distance to
PR
. The smaller the
D
ij
value, the higher the
boundary coincidence degree between the reference and the
corresponding object.
Yu
et al
. (2010) defined the Vector Distance
(
VD
)
index
which is the sum of the
Euclidian Distance
between the vec-
tor boundary of the reference object and the corresponding
object in both vertical and horizontal directions.
VD
H
n
V
n
s S
ij
a
n
ija
b
n
ijb
j
i
=
+
∈
=
=
∑ ∑
1
1
1
2
1
1
2
,
*
(25)
where
H
ij
a
is the length of the
a
th
transverse ray,
n
1
is the num-
ber of transverse rays;
V
ij
b
is the length of the
b
th
longitudinal
ray, and
n
2
is the sum of the longitudinal rays. In addition, the
intervals of transverse and longitudinal rays are equidistant.
VD
ij
is negatively correlated with the quality of the segmenta-
tion result. When
VD
ij
is 0, the boundaries of the reference
object and the corresponding object coincide completely, so
the segmentation result is optimal.
Liu
et al
. (2013) proposed the Shape Similarity
(
SS
)
index
in order to measure the similarity and contact ratio of the
boundary between the reference object and the correspond-
ing object. Using the
SS
index requires shooting rays from the
centroid to the boundary of both, the reference object and
the corresponding object. After, the length discrepancy of the
reference object rays and corresponding object rays are com-
puted and discrepancy result is normalized.
( )
SS
f
f
max
f
f
ij
k
N
r k s k
k
N
r k
k
N
s
i
j
i
j
=
( )
−
( )
(
)
( )
=
−
=
−
=
−
∑
∑ ∑
0
1
2
0
1
0
1
θ
θ
θ
,
θ
k
j
i
s S
∈
,
*
1
(26)
where
θ
is the angle interval of ray shotting from the object’s
centroid to the object’s boundary. The number of rays is 2
π
/
θ
,
denoted as
N
.
f
(
θ
k
) is the length of ray with
θ
k
rotation angle.
The range of
SS
ij
is [0,1]. The lower the
SS
, the smaller the
difference between the reference object and the corresponding
object.
Pros and Cons of Geometric Discrepancy Indexes
The metrics based on overlap region are some of the most
commonly used measure indexes given their low compu-
tational complexity and intuitive definition. However, this
method indirectly evaluates the existence of over-segmenta-
tion and under-segmentation, which is not absolutely reliable
under some special cases, such as when the overlapping area
is small.
The evaluation indexes based on over-segmentation and
under-segmentation regions were able to directly quantita-
tively reflect these phenomena of segmentation results. These
indexes not only enable the evaluation of the performance of
the segmentation algorithms and the comparison between in-
dexes’ advantages and disadvantages through their evaluation
results, but also select the optimal segmentation scale through
the adjustment of scale parameters according to the degree of
under-segmentation and over-segmentation. This is why the
indexes based on over-segmentation and under-segmentation
regions have gradually replaced the indexes based on the
overlap region as the mainstream index.
Indexes based on location, with low computation complex-
ity, are theoretically simple and easily carried out. However,
due to only measuring the location similarity, the evaluation
results based on these indexes are insufficient. Serious over-
segmentation and under-segmentation phenomenon can exist
with these indexes, especially when the location similarity
between the reference object and the corresponding object
is exactly the same, and vice versa. Therefore, the location
based indexes have to be combined with another kind of
index, instead of using them on its own.
The evaluation indexes based on boundary can be solely
used to evaluate the segmentation result by measuring the de-
gree of coincidence of the boundaries and the shape similarity
directly between the reference object and the corresponding
object, due to that the ideal segmented object should exactly
coincide with the reference object on the vector boundary.
However, the algorithms for boundary-based indexes are
complex, with high computation complexly. Moreover, these
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
October 2018
635
