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(grant 086901/Z/08/Z) and the Marie Curie IEF pro-
gramme (grant 299605, SP-MORPH).
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APPENDIX
Starting from the definition of A and S :
A (m,a) = 1− S (m,a) = 1 −
∑
j
min(m
j
,m
j+a
)
∑
j
m
j
=
∑
j
m
j
−
∑
j
min(m
j
,m
j+a
)
∑
j
m
j
(22)
Now we use the following equality:
2
∑
j
m
j
=
∑
j
max(m
j
,m
j+a
) + min(m
j
,m
j+a
)
(23)
which holds because for every pair (m
j
,m
j+a
) one el-
ement is the maximum and the other one the mini-
mum, hence adding both guarantees to include each
element of m exactly twice in the summation (recall
that j + a is an addition module the cardinality of m).
Then, in (23):
∑
j
m
j
−
∑
j
min(m
j
,m
j+a
) =
=
1
2
∑
j
max(m
j
,m
j+a
) − min(m
j
,m
j+a
)
=
1
2
∑
j
|m
j
− m
j+a
| (24)
which directly leads to our final result:
A (m,a) =
1
2
∑
j
|m
j
− m
j+a
|
∑
j
m
j
(25)
RotationallyInvariant3DShapeContextsusingAsymmetryPatterns
17