多相机系统通用姿态估计与自校准重建方法
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"这篇文章是关于多相机束调整（Multi-ColBundle Adjustment）的，它提出了一种通用、模块化的算法，用于估计多相机系统的姿态、同时进行自我校准和重建。作者Steffen Urban等人在2017年发表在《国际计算机视觉杂志》上，提出了扩展的共线性方程，能直接表达所有图像观测值与未知参数的关系，包括相机间的相对定向，从而实现对相机系统、单个相机和场景重建的联合优化。"
在多相机系统中，束调整（Bundle Adjustment, BA）是一种关键的技术，用于优化相机参数和三维结构。传统的束调整方法通常基于单个相机的几何特性，如针孔模型，而本文提出的“MultiColBundleAdjustment”则引入了一种更通用的方法，不仅考虑了相机的内在参数，如焦距、主点坐标，还考虑了相机间的相对姿态，以及多相机系统的全局框架。
该方法的核心是扩展的共线性方程。在传统的共线性方程中，图像点、三维点和相机光心之间的关系被表示出来。而这里，作者将相机模型扩展为更一般的形式，并且包含了相机相对于固定多相机系统框架的相对定向。这样，每个相机的图像观测可以直接表示为所有未知参数（包括相机参数、三维点坐标和系统姿态）的函数，从而实现了对整个系统的联合优化。
自我校准是指在没有先验知识的情况下，通过图像数据自动确定相机的内参和外参。在多相机系统中，每个相机可能有不同的畸变和安装位置，自我校准能够消除这些不确定性，提高重建和定位的精度。同时，文章还讨论了如何利用这种方法进行场景的三维重建，即根据多视图图像恢复场景中物体的三维坐标。
论文强调了这种方法的模块化特性，这意味着它可以适应各种多相机系统，不论其结构如何复杂。此外，它还可以单独用于估计系统的姿态，这对于机器人导航和SLAM（Simultaneous Localization And Mapping，同步定位与建图）等应用至关重要。
"MultiColBundleAdjustment" 提供了一个强大的工具，用于处理多相机系统的优化问题，它能够同时解决相机参数校准、系统姿态估计和三维重建，对多相机系统的理论研究和实际应用都有着重要的贡献。
Int J Comput Vis (2017) 121:234–252 237
scene points in the camera reference frame to the image plane.
In the classical case π
c
is expressed in terms of the calibration
matrix K
c
:
π
c
= K
c
=
⎡
⎣
ccso
u
0 c(1 + m) o
v
00 1
⎤
⎦
=
⎡
⎣
c
u
c
u
so
u
0 c
v
o
v
001
⎤
⎦
(3)
where c is the focal length, (1 + m) is a s cale factor for the
v axis, o
u
, o
v
are the coordinates of the principal point and
s models the skewness of the u and v axis. The rigid homo-
geneous transformation from camera t o reference frame is:
M
t
=
R
t
Z
0t
0 1
, R
t
=
⎡
⎣
r
11
r
12
r
13
r
21
r
22
r
23
r
31
r
32
r
33
⎤
⎦
, Z
0t
=
⎡
⎣
X
0t
Y
0t
Z
0t
⎤
⎦
(4)
with Z
0t
being the projection center in world frame coordi-
nates, and R
t
the rotation matrix of world to camera system.
Substituting the above transformation and projection matrix
into Eq. 2 the classical collinearity equations become:
λ
it
m
it
= K
c
R
T
t
[I|−Z
0t
]X
i
(5)
To express the observations in the image plane as a function
of all unknowns, we divide Eq. 5 by the third row yielding:
u = c
u
·
r
11
(X − X
0
) + r
21
(Y − Y
0
) + r
31
(Z − Z
0
)
r
13
(X − X
0
) + r
23
(Y − Y
0
) + r
33
(Z − Z
0
)
+ o
u
+ d
r
(ρ) + d
t
u
(ρ)
v = c
v
·
r
12
(X − X
0
) + r
22
(Y − Y
0
) + r
32
(Z − Z
0
)
r
13
(X − X
0
) + r
23
(Y − Y
0
) + r
33
(Z − Z
0
)
+ o
v
+ d
r
(ρ) + d
t
v
(ρ) (6)
with ρ =
√
˜u
2
+˜v
2
being the radial distance from the ori-
gin of the sensor coordinate system, and
˜
m =[˜u, ˜v]
T
=
[u − o
u
,v − o
v
]
T
. Note that we now added radial and tan-
gential distortion functions d
r
and d
t
respectively and omitted
the skew factor s as well as the indices i and t to facilitate the
reading. The most common characterization for both distor-
tion effects can be found in Brown (1966):
d
r
(ρ) = R
1
ρ
2
+ R
2
ρ
4
+ R
3
ρ
6
d
t
u
(ρ) = 2T
1
uv + T
2
(ρ
2
+ 2u
2
)
d
t
v
(ρ) = T
1
(ρ
2
+ 2v
2
) + 2T
2
uv
(7)
where R
1
, R
2
, R
3
are the coefficients of the radial-symmetric
distortion and T
1
, T
2
model the tangential-asymmetric distor-
tion. Analyzing Eq. 6 we can identify the following possible
challenges for the use in multi-camera systems:
1. Point with Incident Angles Greater 90
◦
We can not dis-
tinguish, and hence project, points that lie behind the
cameras, given that a perspective camera model is used.
2. Camera Model To overcome the first issue, we could use
an omnidirectional camera model and include it in Eq. 6.
If multiple cameras, e.g. fisheye and perspective cam-
eras would be combined to a camera system, we have to
provide different versions of Eq. 6. This challenge could
either be eluded by transforming image coordinates to
bearing vectors (see Sect. 3.4) using the corresponding
intrinsics of each camera or by using a general cam-
era model that models all prevalent cameras. This work
utilizes the latter approach, but see Sect. 3.4 for a com-
parison to Schneider et al. (2012) and Kneip et al. (2013)
who use bearing vectors as observations and a reasoning
why new challenges arise when optimizations are carried
out over camera rays instead of image coordinates.
3. Multiple Cameras Finally, Eq. 6 expresses s olely the pro-
jection of scene points i to a camera at time t.InaMCS
the projection has to be expanded by a transformation of
scene point X
i
to a MCS pose at time t and finally to a
camera c within some MCS coordinate system.
In the following, the general camera model is introduced. We
subsequently show how the classical collinearity equation
is expanded with it and how the transformation from the
MCS coordinate system to each camera coordinate system is
modeled.
3.2 Camera Model
In order to be able to utilize arbitrary cameras in the MCS,
a suitable camera model is necessary. We chose to include
the camera model proposed in Scaramuzza et al. (2006a,b),
since it is not limited to specific cases and allows us to employ
all prevalent cameras, that are currently used in applied com-
puter vision and robotics, e.g., perspective, dioptric (fisheye),
as well as catadioptric cameras. This section provides a brief
compilation of the model as well as a comparison to perspec-
tive cameras. This is supposed to emphasize the differences
and show how the classical perspective model is generalized.
Again, given a point m =[u,v]
T
on the image plane,
the corresponding point on the sensor plane is
˜
m. Those two
points, are related by an affine transformation:
m = A
˜
m + O
c
(8)
where matrix A =[a
11
, a
12
;a
21
, 1]
T
accounts for small mis-
alignments between sensor and lens axis and the digitization
process (see Scaramuzza et al. 2006a). The principal point
O
c
=[o
u
, o
v
]
T
relates all coordinates to the center of distor-
tion. Now let X
c
=[X
c
, Y
c
, Z
c
]
T
be a scene point already
transformed into the camera frame. Then the following for-
123
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