Digital Signal Processing Reference
In-Depth Information
7.4.5 2D Rigid Transformations
Contrary to the general case, the bi-invariant mean of 2D rigid-body transformations
have a
closed form
. The underlying reason is that
SO
, the group of 2D rotations,
is
commutative
. As a consequence, one can compute explicitly the bi-invariant mean
of the rotation parts of the data and deduce from it the translation part using the
barycentric equation, like in the proof of Theorem 7.15 More precisely, we have:
(
2
)
Proposition 7.10
be a set of 2D rigid-body transformations, such that
the angles of rotation of the rotations R
i
R
j
Let
{
(
R
i
,
t
i
)
}
are all strictly less than
π
. Then the
(
R
,
t
bi-invariant mean
)
associated to the weights
{
w
i
}
is given explicitly by:
R
1
exp
+
i
w
i
log
R
1
R
i
,
R
=
=
i
w
i
Z
(
−
1
)
M
log
¯
RR
i
(
−
1
)
R
i
(7.19)
t
t
i
,
with the following formulas for M and Z :
M
0
(
−
1
)
def
=
θ
sin
(θ)
2
−
θ
2
(
1
−
cos
(θ))
θ
sin
(θ)
θ
0
−
2
2
(
1
−
cos
(θ))
log
¯
RR
i
(
−
1
)
def
=
Z
w
i
.
i
Example of Bi-Invariant Mean
√
2
Le
t
us take a look at the exa
m
ple chosen in [
56
, p. 31]. Let
T
1
=
(R(π/
4
),
[−
/
2
,
√
2
√
2
√
2
√
2
T
T
T
/
)
]
)
=
(
Id
2
,
[
,
]
)
=
(R(
−
π/
),
[−
/
,
−
/
]
)
2
,
T
2
0
and
T
3
4
2
2
be
three rigid-body transformations in 2D.
We can compute exactly the bi-invariant mean of these rigid-body transformations
with Eq. (
7.19
). A left-invariant Fréchet mean can also be computed explicitly in this
case thanks to the simple form taken by the corresponding geodesics. And finally,
thanks to Proposition 7.1, the analogous right-invariant Fréchet mean can be com-
puted by inverting the data, computing their left-invariant mean and then inverting
this Fréchet mean. The log-Euclidean mean can also easily be computed in closed
form. This yields (after a number of simple but tedious algebraic manipulations):
T
•
Left-invariant Fréchet mean:
(
Id
2
,
[
0
,
0
]
)
,
Log-Euclidean mean:
Id
2
,
0
T
√
2
−
4
3
Id
2
,
[
T
,
•
,
0
.
2096
,
0
]
⎛
0
T
⎞
√
2
⎠
Id
2
,
[
T
,
−
4
⎝
Id
2
,
•
Bi-invariant mean:
1
,
0
.
2171
,
0
]
+
4
√
2
1
+
