3 Considerations about the Structure of the Linearized System
According to Eq. 13, the matrix H and the vector b are obtained by summing up a set of matr ice s and
vectors, one for every constraint. If we set b
k
= J
T
k
Ω
k
e
k
and H
k
= J
T
k
Ω
k
J
k
we can rewrite H and b as
b =
X
k∈C
b
ij
(17)
H =
X
k∈C
H
ij
. (18)
Every cons tr aint will contribute to the system with an addend term. The structure of this adden d
depends on the Jacobian of the error function. Since the e r r or function of a constraint depends only on
the values of the nodes x
i
∈ x
k
, the Jacobian in Eq. 5 has the following form:
J
k
=
0 · · · 0 J
k
1
· · · J
k
i
· · · 0 · · · J
k
q
0 · · · 0
. (19)
Here J
k
i
=
∂e(x
k
)
∂x
k
i
are the derivatives of the error function with respect to the nodes connected by the
k
th
hyper-edge, with respect to the par amete r block x
k
i
∈ x
k
.
From Eq. 9 we obtain the following structure for the block matrix H
ij
:
H
k
=
.
.
.
J
T
k
1
Ω
k
J
k
1
· · · J
T
k
1
Ω
k
J
k
i
· · · J
T
k
1
Ω
k
J
k
q
.
.
.
.
.
.
.
.
.
J
T
k
i
Ω
k
J
k
1
· · · J
T
k
i
Ω
k
B
k
i
· · · J
T
k
i
Ω
k
J
k
q
.
.
.
.
.
.
.
.
.
J
T
k
q
Ω
k
J
k
1
· · · J
T
k
q
Ω
k
B
k
i
· · · J
T
k
q
Ω
k
J
k
q
.
.
.
(20)
b
k
=
.
.
.
J
k
1
Ω
k
e
k
.
.
.
J
T
k
i
Ω
k
e
k
.
.
.
J
T
k
q
Ω
k
e
k
.
.
.
(21)
For simplicity of notation we omitte d the zero blocks. The reader might notice that the b lock structur e
of the matrix H is the adjacency matrix of the hyper graph. Additionally the Hessian H is a symmetric
matrix, since all th e H
k
are symmetric. A single hyper-edge connecting q vertices will introduce q
2
non
zero blocks in the Hessian, in correspondence of each pair
x
k
i
, x
k
j
, of nodes connected.
4 Least Squar es on Manifold
To deal with parameter blocks that s pan over a non-Euclidean spaces, it is common to ap ply the error
minimization on a manifold. A manifold is a mathematical space that is n ot necessarily Euclidean on a
global scale, but can be seen as Euclidean on a local scale [6].
For example, in the context of SLAM problem, each parameter block x
i
consists of a translation vector
t
i
and a rotational component α
i
. The translation t
i
clearly forms a Euclid ean sp ace. In contrast to that,
the rotational components α
i
span over the non-Euclidean 2D or 3D rotation group SO(2) or SO(3).
To avoid singularities, these spaces are usually described in an over-parameterized way, e.g., by rotation
matrices or quaternions. Directly applying Eq. 15 to these over-parameterized representations breaks
the constraints induced by the over-parameterization. The over-parameterization results in additional
4
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