Isabelle Bloch, Olivier Colliot, and Roberto M. Cesar, Jr.
IEEE Trans. Syst., Man, Cybern. B, Cybern., Vol. 36, No. 2, April 2006
Presented by Drew Buck
February 22, 2011

We are interested in answering two
questions:
 What is the region of space located between two
objects A1 and A2?
 To what degree is B between A1 and A2?
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Existing Definitions
Convex Hull
Morphological Dilations
Visibility
Satisfaction Measure
Properties
Examples
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Merriam-Webster Dictionary defines
between as “In the time, space, or interval that
separates.”
Some crisp approaches:
 A point is between two objects if it belongs to a
segment with endpoints in each of the objects.
A1
P
A2

Another crisp approach:
 Using bounding spheres, B is between A1 and A2 if
it intersects the line between the centroids of A1
and A2.

Fuzzy approaches:
 For all a1  A1, a2  A2 , b  B calculate the angle θ at b
between the segments [b, a1] and [b, a2]. Define
a function μbetween(θ) to measure the degree to
which b is between a1 and a2.
[12] R. Krishnapuram, J. M. Keller, and Y. Ma, “Quantitative Analysis of Properties and Spatial Relations of Fuzzy Image Regions,” IEEE-Trans. Fuzzy
Syst., vol. 1, no.3, pp. 222-233, Feb. 1993.

Using the histograms of forces:
 The spatial relationship between two objects can
be modeled with force histograms, which give a
degree of support for the statement, “A is in
direction θ from B.”
From P. Matsakis and S. Andréfouët, “The Fuzzy Line Between Among and Surround,”
in Proc. IEEE FUZZ, 2002, pp. 1596-1601.

Limitations of this approach:
 Considers the union of objects, rather than their
individual areas.

For any set X, its complement XC, and its
convex hull CH(X), we define the region of
space between objects A1 and A2 as β(A1,A2).
 CH  A1 , A2   CH ( A1  A2 )  A1C  A2C  CH ( A1  A2 ) \ ( A1  A2 )

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Components of β(A1,A2) which are not
adjacent to both A1 and A2 should be
suppressed.
However, this leads to a continuity problem:

Extension to the fuzzy case:
 CH  A1 , A2   CH (  A   A )  c(  A )  c(  A )
1
2
1
2
 Recommended to chose a t-norm that satisfies the law of excluded
middle, such as Lukasiewicz’ t-norm (a, b)  [0,1], t (a, b)  max( 0, a  b  1).

If A1 can be decomposed into connected
components i A1i we can define the region between
A1 and A2 as   A1 , A2    (i A1i , A2 )  i  ( A1i , A2 )
and similarly if both objects are non-connected.
 This is better than using
CH(A1,A2 ) directly

Definition based on Dilation and Separation:
 Find a “seed” for β(A1,A2) by dilating both objects until they meet and
dilating the resulting intersection.  Dil ( A1 , A2 )  D n D n ( A1 )  D n ( A2 ) A1C  A2C
where Dn denotes a dilation by a disk of radius n.

Using watersheds and SKIZ (skeleton by influence zones):
 Reconstruction of β(A1,A2) by geodesic dilation.

 sep ( A1 , A2 )  DCH
'( A  A ) (WS )
1
2

 DCH
'( A1  A2 ) ( SKIZ )
Where CH ' ( A1  A2 )  CH ( A1  A2 ) \ ( A1  A2 )
from which connected components of the
convex hull not adjacent to both sets are
suppressed.

Removing non-visible cavities
 We could use the convex hulls of A1 and A2
independently, but this approach cannot handle
imbricated objects.
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Fuzzy Directional Dilations:
 Compute the main direction α between A1 and A2 as the average or
maximum value of the histogram of angles.

h( A1 , A2 ) ( )  (a1 , a2 ), a1  A1 , a2  A2 , (a1a2 , u x )  

H ( A1 , A2 ) ( ) 
h( A1 , A2 ) ( )
max  ' h( A1 , A2 ) ( ' )
 Let Dα denote a dilation in direction α using a fuzzy structuring
element, v.
Dv ( )( x)  sup t[ ( y), v( x  y)]
y

Given a fuzzy structuring element in the main
direction from A1 to A2, we can define the
area between A1 and A2 as
 ( A1 , A2 )  D ( A1 )  D  ( A2 )  A1C  A2C

We can consider multiple values for the main
direction by defining β as
 ( A1 , A2 )    ( A1 , A2 )   (   )

Or use the histogram of angles directly as the
fuzzy structuring element.
v1 (r , )  H ( A1 , A2 ) ( )
v2 (r , )  H ( A1 , A2 ) (   )  v1 (r ,   )
 FDil1 ( A1 , A2 )  Dv ( A1 )  Dv ( A2 )  A1C  A2C
2
1

Alternate definition which removes
concavities which are not facing each other
 FDil2 ( A1 , A2 )  Dv ( A1 )  Dv ( A2 )

1
1
 Dv2 ( A1 )  Dv2 ( A2 )

 FDil3 ( A1 , A2 )  Dv ( A1 )  Dv ( A2 )  A1C  A2C
2

 D
1

( A )
 Dv1 ( A1 )  Dv1 ( A2 )
C
v2 ( A1 )  Dv2
C
2

Admissible segments:
 A segment [x1, x2] with x1 in A1 and x2 in A2 is
admissible if it is contained in A1C  A2C .
  Adm ( A1 , A2 ) is defined as the union of all admissible
segments.
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Fuzzy visibility:
 Consider a point P and the segments [a1, P] and [P, a2]
where a1 and a2 come from A1 and A2 respectively.
 For each P, find the angle closest to π between any two
admissible segments included in A1C  A2C
min ( P)  min    ,  [a1 , P], [ P, a2 ]
where [a1, P] and [P, a2] are semiadmissible segments
 FVisib( A1 , A2 )( P)  f  min ( P)
where f :[0, π] → [0,1] is a decreasing
function

Extension to fuzzy objects:
 Compute the fuzzy inclusion of each segment and
intersect with the best angle as before.
incl ([ a1 , a2 ])  inf min 1   A ( y ), 1   A ( y )
y[ a1 , a2 ]
1
2
  f    , a1  Supp( A1 ), a2  Supp( A2 ), 
t 

 FVisib( A1 , A2 )( P)  max   incl ([ a1 , P]), incl ([ P, a2 ])








[
a
,
P
],
[
P
,
a
]
1
2


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Objects with different spatial extensions:
 If A2 has infinite or near-infinite size with respect to A1,
approximate A2 by a segment u and dilate A1 in a direction
orthogonal to u, limited to the closest half plane.

Adding a visibility constraint
 Conjunctively combine projection with admissible
segments. Optionally consider only segments
orthogonal to u.

Once β(A1,A2) has been determined, we want
to know the overlap between β and B.
 Normalized Intersection:
S1 ( B,  ) 
B
B
 Other possible measures:
S2 ( B,  )  1  supB ( x) /  ( x)  0
S3 ( B,  )  inf min 1   B ( x)   ( x), 1
x

Degree of Inclusion:
N ( B,  )  inf T  ( x), 1   B ( x)
x

Degree of Intersection:
( B,  )  sup t  ( x),  B ( x)
x
 Length of this interval gives information on the
ambiguity of the relation.
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If B is extended, we might want to know if B
passes through β.
Let R  Supp(  ) \ Core(  )
R will have two connected components, R1
and R2.
Degree to which B passes through β should
be high if B goes at least from a point in R1 to
a point in R2.


C
C
M 1 ( B,  )  min sup ( B   )( x), sup ( B   )( x)
xR2
 xR1



M 2 ( B,  )  min sup ( B  Core(  )C )( x), sup ( B  Core(  )C )( x)
xR2
 xR1


Anatomical descriptions of brain structures
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Possible to represent the complex spatial
relationship, “between,” using mathematical
morphology and visibility.
Many definitions exist, each with individual
strengths and weaknesses.
Pick the representation most appropriate for
the application.