基础矩阵有许多种推导方式，下面介绍其中一种。
图1 在双相机的拍摄场景中建立一个空间直角坐标系，称为世界坐标系（如图1中蓝色坐标系）。物点就是场景中物体表面上的点，比如说点
P
{\displaystyle P}
(
X
,
Y
,
Z
)
⊤
{\displaystyle (X,Y,Z)^{\top }}
相机的光心从物理上讲就是相机镜头组的光学中心。以光心为原点，主光轴为Z轴建立空间直角坐标系，称为相机坐标系（如图1中绿色和红色坐标系）。像平面在相机坐标系中的方程即为z=1，像点就是在物点在像平面上的投影 ，这个投影关系是透视投影 。
用一句话来概括相机的拍摄模型，就是物点、像点、光心三点一线，此模型称为针孔相机模型。在此模型中，世界坐标系到左右相机坐标系的变换是刚性变换，即只包含旋转和平移，因此我们分别用增广矩阵
[
R
|
t
]
{\displaystyle [R|t]}
[
R
′
|
t
′
]
{\displaystyle [R'|t']}
R
{\displaystyle R}
R
′
{\displaystyle R'}
3
×
3
{\displaystyle 3\times 3}
t
{\displaystyle t}
t
′
{\displaystyle t'}
P
~
{\displaystyle {\widetilde {P}}}
P
c
a
m
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X
C
,
Y
C
,
Z
C
)
⊤
=
[
R
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t
]
P
~
{\displaystyle P_{cam}(X_{C},Y_{C},Z_{C})^{\top }=[R|t]{\widetilde {P}}}
P
c
a
m
′
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X
C
′
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Y
C
′
,
Z
C
′
)
⊤
=
[
R
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′
]
P
~
{\displaystyle P'_{cam}(X_{C'},Y_{C'},Z_{C'})^{\top }=[R'|t']{\widetilde {P}}}
图2 以一台相机为例，如图2所示，
C
{\displaystyle C}
Z
{\displaystyle Z}
P
~
{\displaystyle {\widetilde {P}}}
p
{\displaystyle p}
x
=
(
f
x
X
C
Z
C
+
x
0
,
)
⊤
{\displaystyle x=\left({\frac {f_{x}X_{C}}{Z_{C}}}+x_{0},\right)^{\top }}
y
=
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f
y
Y
C
Z
C
+
y
0
,
)
⊤
{\displaystyle y=\left({\frac {f_{y}Y_{C}}{Z_{C}}}+y_{0},\right)^{\top }}
式中
(
x
0
,
y
0
,
f
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{\displaystyle (x_{0},y_{0},f)}
设两相机内参数矩阵同为：
K
=
[
f
x
0
p
c
x
0
f
y
p
c
y
0
0
1
]
,
{\displaystyle K=\left[{\begin{array}{ccc}f_{x}&0&{p_{c}}_{x}\\0&f_{y}&{p_{c}}_{y}\\0&0&1\end{array}}\right],}
那么物点与像点之间的关系为：
p
=
1
Z
C
K
P
c
a
m
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1
Z
C
K
[
R
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P
,
{\displaystyle p={\frac {1}{Z_{C}}}KP_{cam}={\frac {1}{Z_{C}}}K[R|t]P,}
p
′
=
1
Z
C
K
P
c
a
m
′
=
1
Z
C
K
[
R
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P
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{\displaystyle p'={\frac {1}{Z_{C}}}KP'_{cam}={\frac {1}{Z_{C}}}K[R'|t']P,}
将
P
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[
R
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t
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+
P
c
a
m
=
Z
C
′
K
[
R
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t
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+
p
{\displaystyle P=[R|t]^{+}P_{cam}=Z'_{C}K[R|t]^{+}p}
H
π
=
K
[
R
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t
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]
K
[
R
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+
{\displaystyle H_{\pi }=K[R'|t']K[R|t]^{+}}
p
′
=
Z
C
′
Z
C
H
π
p
{\displaystyle p'={\frac {Z'_{C}}{Z_{C}}}H_{\pi }p}
由于物点、像点、光心三点一线，那么物点、一对同名点和2个光心这5个点一定处于同一个平面上，我们将这个平面称为𝜋平面。𝜋平面和像平面的交线称为极线
l
′
{\displaystyle l'}
p
{\displaystyle p}
l
′
{\displaystyle l'}
p
′
{\displaystyle p'}
l
′
{\displaystyle l'}
e
′
{\displaystyle e'}
在右片像平面内，极线
l
′
{\displaystyle l'}
A
x
+
B
y
+
C
=
0
{\displaystyle Ax+By+C=0}
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C
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⋅
(
x
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y
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1
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⊤
=
0
{\displaystyle (A,B,C)^{\top }\cdot (x,y,1)^{\top }=0}
因此，我们可以用一个三维向量
(
A
,
B
,
C
)
{\displaystyle (A,B,C)}
l
′
{\displaystyle l'}
l
′
{\displaystyle l'}
e
′
{\displaystyle e'}
p
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{\displaystyle p'}
l
′
:
e
′
×
p
′
=
[
e
′
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×
p
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{\displaystyle l':e'\times p'=[e']_{\times }p'}
[
e
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×
=
[
0
−
1
y
0
1
0
−
x
0
−
y
0
x
0
0
]
,
{\displaystyle [e']_{\times }=\left[{\begin{array}{ccc}0&-1&y_{0}\\1&0&-x_{0}\\-y_{0}&x_{0}&0\end{array}}\right],}
令
[
e
′
]
x
{\displaystyle [e']x}
l
′
:
Z
C
′
Z
C
[
e
′
]
×
H
π
p
{\displaystyle l':{\frac {Z'_{C}}{Z_{C}}}[e']_{\times }H_{\pi }p}
因为
p
′
{\displaystyle p'}
l
′
{\displaystyle l'}
p
′
l
′
=
p
′
[
e
′
]
×
H
π
p
=
0
{\displaystyle p'l'=p'[e']_{\times }H_{\pi }p=0}
令
F
=
[
e
′
]
×
H
π
{\displaystyle \mathrm {F} =[e']_{\times }H_{\pi }}
p
′
⊤
F
p
=
0
{\displaystyle p'^{\top }\mathrm {F} p=0}
根据立体像对估算基础矩阵
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在三维重建技术中的应用
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参考文献
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^
Q. T. Luong. Matrice fondamentale et auto-calibration en vision par ordinateur. PhD Thesis, University of Paris, Orsay. 1992.
^
Olivier D. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig?. Proceedings of European Conference on Computer Vision. 1992.
外部链接
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![{\displaystyle [R|t]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1487e9584a457b3ab6e7461a6947fd9af5782d2d)
![{\displaystyle [R'|t']}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3544db39c2af9f5985731580676afcd66353ec38)
![{\displaystyle P_{cam}(X_{C},Y_{C},Z_{C})^{\top }=[R|t]{\widetilde {P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edd5acc6c3b11d13d337bb520be2fd0a520151bc)
![{\displaystyle P'_{cam}(X_{C'},Y_{C'},Z_{C'})^{\top }=[R'|t']{\widetilde {P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f89994807de475cd67b3b6957584cee4a4952b5e)
![{\displaystyle K=\left[{\begin{array}{ccc}f_{x}&0&{p_{c}}_{x}\\0&f_{y}&{p_{c}}_{y}\\0&0&1\end{array}}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10d752e2cb039e167ee15f1190e625cc55029108)
![{\displaystyle p={\frac {1}{Z_{C}}}KP_{cam}={\frac {1}{Z_{C}}}K[R|t]P,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/402261a53d8fb5d9331e8961bd9de8a64936ac86)
![{\displaystyle p'={\frac {1}{Z_{C}}}KP'_{cam}={\frac {1}{Z_{C}}}K[R'|t']P,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d29bbf7b4a1524cc996cc7f334d174ca87f7d606)
![{\displaystyle P=[R|t]^{+}P_{cam}=Z'_{C}K[R|t]^{+}p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3c913eb1dada9221472dccc76578c6e470e1bb)
![{\displaystyle H_{\pi }=K[R'|t']K[R|t]^{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd3e78367d071958d8812109c24991e328430a39)
![{\displaystyle l':e'\times p'=[e']_{\times }p'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d165a016e62d80f03b2ea78c70f503b6c314038)
![{\displaystyle [e']_{\times }=\left[{\begin{array}{ccc}0&-1&y_{0}\\1&0&-x_{0}\\-y_{0}&x_{0}&0\end{array}}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/757d215842aef06f59c02c848530e4f3d6656629)
![{\displaystyle [e']x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aadd443ba23b258f43fd306623d1b1cce5645dc2)
![{\displaystyle l':{\frac {Z'_{C}}{Z_{C}}}[e']_{\times }H_{\pi }p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79549286397d9a14814dd8c33af44b09bcdfaa31)
![{\displaystyle p'l'=p'[e']_{\times }H_{\pi }p=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6663c80c342d0e565e6926c40bd44e7290a7758f)
![{\displaystyle \mathrm {F} =[e']_{\times }H_{\pi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b21746cb0317082fa26c71adb389f1e018415b6)