Region-Line Association Modeling
We classify the direction relationships of a region to a line as
“above,” “bilateral,” and “below.” These classifications indi-
cate that the region is located above the line, on the two sides
of the line, and below the line, respectively. In particular, if the
line is vertical, then “above” denotes the right side of the line.
The topology of a region to a line includes separation, intersec-
tion, tangent, and inclusion. The last is regarded as a special
case of intersection. The combination of different direction
and topology relationships creates several region-line relation-
ships. Relationships, in which regions and lines are in contact
(including the topologies of intersection, tangent, and inclu-
sion), are more important than cases where they are separated.
Thus, the following model is mainly built for cases in contact.
Let regions
Q
and
R
be two subpixel sets within image
I
.
Q
= {
q
i
= (
x
i
,
y
i
)|
i
∈
[1,
k
],
k
=|
Q
|} and,
R
= {
r
i
= (
x
i
,
y
i
)|
i
∈
[1,
k
],
k
=|
R
|}where
x
and
y
are the pixel coordinates, and |·| is the
potential of a set. Let straight line segment
L
be a pixel set in
I
:
L
= {
l
i
= (
x
i
,
y
i
)|
i
∈
[1,
k
],
k
=|
L
|}.
L
fits the following straight
line equation where
a
and
b
are the coefficients:
Z
L
=
y
–
ax
–
b
= 0,
(1)
Regions
Q
and
R
need to intersect line
L
as follows:
Q
Ç
L
≠
Ø,
R
Ç
L
≠
Ø
(2)
As shown in Figure 2a, we define the direction and topol-
ogy operator sets, which work on regions and straight lines
and produce their subsets. The direction operator set is
defined as follows:
Dir
(
Q,L
) = {
Neg
(
Q,L
),
Zero
(
Q,L
),
Pos
(
Q,L
)}
(3)
which denotes a subset extracted from
Q
that is located above,
on, or below line
L
. For example, operator
Neg
(
Q,L
) is defined
as follows:
Neg
(
Q,L
)= {
q
i
= (
x
i
,
y
i
)|
y
i
–
ax
i
–
b
< 0}
(4)
The other two operators exhibit similar forms. Let
B
Q
be
the boundary pixels of
Q
defined by four connected neighbor-
hoods. We define the topology operator set as follows:
Top
(
L,Q
) = {
In
(
L,Q
),
Touch
(
L,Q
),
Out
(
L,Q
),
Proj
(
L,Q
)}. (5)
The first three operators denote a subset extracted from
L
,
which is contained by, touched by, or outside
Q
. For example,
operator
Touch
(
L,Q
) is defined as follows:
Touch
(
L,Q
) = {
l
i
|
l
i
∈
B
Q
,
l
i
∈
L
}.
(6)
Operator
Proj
(
L,Q
) denotes the straight line segment ob-
tained by vertically projecting
Q
onto
L
, as shown in Figure
2a and 2b.
Concepts based on Region-Line Association Model
A set of line-based concepts is proposed and used in sub-
sequent
OBIA
s. In general, the line should be long and the
projected length of the intersecting region should not consid-
erably exceed those of the parts of the line that fall within and
intersect the region to investigate only meaningful relation-
ships between a pair of region and line. That is,
|
L
|
≥
T
a
(7-1)
and
Proj L Q
In L Q Touch L Q
,
,
| ( , )|
(
)
(
)
+
≤
T
b
,
(7-2)
where
T
a
and
T
b
are two user defined thresholds. Based on
Equation 7, the following concepts or indices are defined by
considering the topology and directional relationships among
regions and lines.
1. Unilateral and Tangent Relationship
Region
Q
is unilateral to line
L
if it satisfies Equation 8, as
follows:
|
Pos
(
Q,L
)|+|
Zero
(
Q,L
)|=|
Q
|or|
Neg
(
Q,L
)|+|
Zero
(
Q,L
)|=|
Q
|.(8)
The combination of Equations 2 and 8 indicate that
|
Zero
(
Q,L
)| is not zero. In this case, region
Q
is also tangent
to line
L
in topology. Thus, region
Q
is called unilateral and
tangent to
L.
Considering the possible errors in image seg-
mentation and straight line extraction, the aforementioned
conditions are relaxed.
Q
is unilateral and tangent to
L
when
threshold
T
c
is close to 1.0, that is,
Figure 2. Region–line relationship modeling. (a) A pair of touch-
ing region and line. (b) A unilateral case that does not satisfy
Equation 7. (c) A tangent relationship that is not unilateral. (d) An
IPSL-neighborhood relationship. (e) Regions that are not IPSL-
neighbors because they are located bilaterally to line
L
, and (f) An
IPSL chain.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
February 2016
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