Graphics Reference
In-Depth Information
algebra respectively. The adjoint operator in a Lie group allows elements of the Lie
algebra to be moved from the right tangent space of a transformation to the left. Thus,
the reversed edge
e
−
1
, pointing in the opposite direction in the graph but encoding
the same transformation constraint, is given by
μ
−
1
T
Adj
μ
−
1
Adj
e
−
1
μ
−
1
=
,
·
Σ
·
(2.3)
2.8.2.2 Edge Composition
Given an edge
e
0
from node
a
to node
b
and an edge
e
1
=
(μ
1
,Σ
1
)
from node
b
to node
c
, the two edges may be composed into one edge from
a
to
c
by composing the uncertain transformations, as in a Kalman filter motion update:
=
(μ
0
,Σ
0
)
μ
1
]
T
e
1
·
e
0
=
μ
1
·
μ
0
,Σ
1
+
Adj
[
μ
1
]
·
Σ
0
·
Adj
[
(2.4)
2.8.2.3 Edge Combination
Given two edges
e
0
=
(μ
0
,Σ
0
)
and
e
1
=
(μ
1
,Σ
1
)
connecting the same two nodes
in the same direction, their constraints may be combined by multiplying the asso-
ciated Gaussian distributions together to yield the resulting Gaussian. Because the
exponential map from the tangent space to the transformation manifold is nonlinear,
the combination procedure for the mean is iterative. The combined covariance
Σ
C
is computed by summing the information of the two edges:
−
1
Σ
−
1
0
+
Σ
−
1
1
Σ
C
=
(2.5)
Let the initial estimate of the combined mean be the first edge's mean:
0
μ
C
=
μ
0
(2.6)
Then the combined transformation is updated by taking the information-weighted
average between the two transformations and exponentiating the correction into the
Lie group:
ln
v
i
j
i
C
·
μ
−
1
=
μ
,
j
∈{
0
,
1
}
(2.7)
j
v
i
1
Σ
−
1
0
v
i
0
+
Σ
−
1
δ
i
=
Σ
C
·
·
·
(2.8)
1
i
+
1
i
C
μ
=
exp
(δ
i
)
·
μ
(2.9)
C
