object. The definition of the invading object is similar to the
expanding object, however, its border is out of the range of
the reference object and the overlapping area between the
expanding object and the reference object is less than 50% of
the invading object. The merged region of Good and Expand-
ing objects is the desired corresponding object for a successful
evaluation task.
Good
i
= {
s
j
:
area
(
r
i
∩
s
j
)=
area
(
s
j
),
s
j
∈
S
i
}
(5)
Expanding s
area r s
area s
Centroidof s isin
i
k
i
k
k
k
=
(
)
( )
>
:
%
∩
∩
50
,
the insideof r s S
i k i
∈
(6)
Invading s
area r s
area s
Centroidof s is in
i
l
i
l
l
l
=
(
)
( )
<
:
%
∩
∩
50
theoutsideof r s S
i l
i
,
∈
(7)
Discrepancy Calculation
After the object matching process, the corresponding objects
are obtained and then the discrepancy calculation process,
which calculates the differences between the reference object
and its corresponding objects, is completed. Various kinds of
discrepancy indexes have been proposed by scholars through
extensive research and experiments. Discrepancy indexes
can be calculated based on the objects’ shape, size, loca-
tion, boundary, gray-scale, and the number of segmentation
objects and reference objects. The higher the measured value
of the discrepancy indexes, the greater the deviation between
the experimental segmentation and the ideal segmentation
results, which indicates that the performance of the proposed
segmentation algorithm is poor for experimental images.
Discrepancy Indexes
Discrepancy indexes can be divided into three groups, geo-
metric discrepancy indexes, arithmetic discrepancy indexes,
and mixed indexes. Also, geometric discrepancy indexes can
be further divided into two sub-groups, area-based indexes
and location and boundary based indexes. Arithmetic dis-
crepancy indexes capture the notion of fragmentation, num-
ber of segments ratio as well as ratio index of good objects
and invading objects number. Mixed indexes mean compre-
hensive use of the aboved indexes.
Geometric Discrepancy Indexes
Area-Based Indexes
Area-based indexes measure the geometric discrepancy
between reference objects and corresponding objects. There
are three basic type of geometric relationships: overlapping
(
r
i
∩
s
j
), over-segmentation (
r
i
–
s
j
) and under-segmentation
(
s
j
–
r
i
) (Figure 4). Studies (Blaschke
et al
., 2008; Bowyer,
2000; Liu
et al
. 2012; Marpu
et al
. 2010) have defined and
elaborated on these three relationships.
Figure 4. Geometric relationship between reference object
and corresponding object.
Overlapping
Based on the Maximum Overlapping Area Object Matching
Method, Fram and Deutsch (1975) and Chen
et al
. (2006)
proposed the Fraction of Correctly Segmented Pixels
(
FCSP
)
index. The overlap region of the reference object and the cor-
responding object is the area where the reference objects are
correctly segmented. However, this method only describes the
over-segmentation phenomenon of the reference objects, and
does not evaluate the degree of under-segmentation.
FCSP
area r s
area r
s S
i
i
j
i
j
i Max
=
(
)
∈
∩
( )
,
*
(8)
The value of
FCSP
i
is positively correlated with the overlap-
ping area between the reference object and the corresponding
object. The value of
FCSP
i
ranges from 0 to 1. A
FCSP
i
value of 1
defines the ideal segmentation.
Similar to
FCSP
, Lucieer and Stein (2002) defined the Area
Fit Index
(
AFI
)
based on the Maximum Overlapping Area
Object Matching Method. The
AFI
evaluates the segmentation
result by measuring the area differences between the refer-
ence and corresponding objects:
AFI
area r area s
area r
s S
i
i
j
i
j
i Max
=
( )
−
( )
( )
∈
,
*
(9)
AFI
i
values range from [-1, 1], and the value of the ideal index
is 0. If the value of
AFI
i
is larger than 0, the reference object
is over segmented, while a negative value represents the
reference object being under segmented. However, when the
overlap region represents a small area of the reference object
or corresponding object,
AFI
i
is unreliable.
Zhan
et al
. (2005) described the SimSize index for evaluating
the area similarity of the reference object and the correspond-
ing object. The
SimSize
index only takes into consideration the
similarity of area. In a one-to-many matching relationship, espe-
cially when multiple corresponding objects have the same area
similarity but different overlapping areas with the reference
object, SimSize does not perform well. Although it is obvious
that the corresponding object with a large overlap is better, this
cannot be inferred from the
SimSize
index.
SimSize
area r area s
area r area s
s
ij
i
j
i
j
j
=
( )
( )
(
)
( )
( )
(
)
min
,
max
,
,
∈
S
i
*
1
(10)
SimSize
ij
is continuous in [0,1] and value 1 represents the best
segmentation result.
According to the area ratio of the overlap region to the ref-
erence object and the overlapping object, Möller
et al
. (2007)
proposed the Relative Area in Super-Object
(
RA
super
)
index and
the Relative Area in Sub-Object
(
RA
sub
)
index, which is extend-
ing
FCSP
for evaluating the over-segmentation and under-seg-
mentation phenomena.
RA
area r s
area r
s S
sub
i
j
i
j
i
ij
=
(
)
∈
∩
( )
,
(11)
RA
area r s
area s
s S
super
i
j
j
j
i
ij
=
(
)
∈
∩
( )
,
(12)
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
October 2018
633
