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x
2
Affinity
region
i
c
i
c
x
1
Fig. 9.
A non-coherent affinity region
aff
can also be defined by a bell-shaped function like the following:
()
−
γ
⋅
d
i
,
x
e
(21)
(
)
( )
aff
i
,
x
,
t
=
a
t
⋅
(
)
2
()
−
γ
⋅
d
i
,
x
1
+
e
This function never adopts the value 0 but its value is very low outside some limited
circular region. Thus, by diminishing the function by some small constant we can get
the desired version.
To illustrate the definitions in this section, let us consider the complementarity
based version of affinity for the two-dimensional shape-space and with linear affinity
function. If
d
is the Euclidean metric, the affinity function is
(
)
()
(
)
(
)
()
2
2
(22)
aff
c
i
,
i
,
x
,
x
,
t
=
−
a
t
⋅
2
c
−
i
−
x
+
2
c
−
i
−
x
+
b
t
1
2
1
2
1
1
1
2
2
2
The affinity region is defined by the condition
aff
c
(
i
1
,
i
2
,
x
1
,
x
2
,
t
) = 0 or
() ()
(
)
(
)
which means that it is a circle with
radius
b
(
t
)/
a
(
t
). The points inside the circle have positive values and the maximum
value is at the point (2
c
1
-
i
1
, 2
c
2
-
i
2
), i.e. at the center of the circle. The affinity
function has the form of a cone. If
d
is the Manhattan metric, the affinity function is
2
2
b
t
a
t
=
2
c
−
i
−
x
+
2
c
−
i
−
x
1
1
1
2
2
2
(
)
()
(
()
)
(23)
aff
c
i
,
i
,
x
,
x
,
t
=
−
a
t
⋅
2
c
−
i
−
x
+
2
c
−
i
−
x
+
b
t
1
2
1
2
1
1
1
2
2
2
The rim of the affinity region is given by
b
(
t
)/
a
(
t
) = (|2
c
1
-
i
1
-
x
1
| + |2
c
2
-
i
2
-
x
2
|).
Thus it is a rhombus with center (2
c
1
-
i
1
, 2
c
2
-
i
2
) and a pyramid as the form of the
affinity function, as was shown in [6]. Let us consider Bersini's version of the
function in [1] written in the notation used throughout this paper:
(
)
(
)
(
(
)
)
(24)
aff
c
i
,
i
,
x
,
x
,
t
=
C
i
,
i
,
t
⋅
L
−
2
c
−
i
−
x
+
2
c
−
i
−
x
1
2
1
2
1
2
1
1
1
2
2
2
This equation can be slightly transformed such that it is more similar to (23):
(
)
(
)
(
(
)
(25)
)
aff
c
i
,
i
,
x
,
x
,
t
=
−
C
i
,
i
,
t
⋅
2
c
−
i
−
x
+
2
c
−
i
−
x
+
L
⋅
C
i
,
i
,
t
1
2
1
2
1
2
1
1
1
2
2
2
1
2
C
(
i
1
,
i
2
,
t
) the two equations become identical.
The difference between them is that in Bersini's version
b
is just the
L
-fold of
a
for
With
a
(
t
) =
C
(
i
1
,
i
2
,
t
) and
b
(
t
) =
L
⋅
