基礎矩陣有許多種推導方式，下面介紹其中一種。
圖1 在雙相機的拍攝場景中建立一個空間直角坐標系，稱為世界坐標系（如圖1中藍色坐標系）。物點就是場景中物體表面上的點，比如說點
P
{\displaystyle P}
(
X
,
Y
,
Z
)
⊤
{\displaystyle (X,Y,Z)^{\top }}
相機的光心從物理上講就是相機鏡頭組的光學中心。以光心為原點，主光軸為Z軸建立空間直角坐標系，稱為相機坐標系（如圖1中綠色和紅色坐標系）。像平面在相機坐標系中的方程即為z=1，像點就是在物點在像平面上的投影 ，這個投影關係是透視投影 。
用一句話來概括相機的拍攝模型，就是物點、像點、光心三點一線，此模型稱為針孔相機模型。在此模型中，世界坐標繫到左右相機坐標系的變換是剛性變換，即只包含旋轉和平移，因此我們分別用增廣矩陣
[
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
(
X
C
,
Y
C
,
Z
C
)
⊤
=
[
R
|
t
]
P
~
{\displaystyle P_{cam}(X_{C},Y_{C},Z_{C})^{\top }=[R|t]{\widetilde {P}}}
P
c
a
m
′
(
X
C
′
,
Y
C
′
,
Z
C
′
)
⊤
=
[
R
′
|
t
′
]
P
~
{\displaystyle P'_{cam}(X_{C'},Y_{C'},Z_{C'})^{\top }=[R'|t']{\widetilde {P}}}
圖2 以一台相機為例，如圖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
=
(
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
)
{\displaystyle (x_{0},y_{0},f)}
設兩相機內參數矩陣同為：
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],}
那麼物點與像點之間的關係為：
p
=
1
Z
C
K
P
c
a
m
=
1
Z
C
K
[
R
|
t
]
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
′
|
t
′
]
P
,
{\displaystyle p'={\frac {1}{Z_{C}}}KP'_{cam}={\frac {1}{Z_{C}}}K[R'|t']P,}
將
P
=
[
R
|
t
]
+
P
c
a
m
=
Z
C
′
K
[
R
|
t
]
+
p
{\displaystyle P=[R|t]^{+}P_{cam}=Z'_{C}K[R|t]^{+}p}
H
π
=
K
[
R
′
|
t
′
]
K
[
R
|
t
]
+
{\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}
由於物點、像點、光心三點一線，那麼物點、一對同名點和2個光心這5個點一定處於同一個平面上，我們將這個平面稱為𝜋平面。𝜋平面和像平面的交線稱為極線
l
′
{\displaystyle l'}
p
{\displaystyle p}
l
′
{\displaystyle l'}
p
′
{\displaystyle p'}
l
′
{\displaystyle l'}
e
′
{\displaystyle e'}
在右片像平面內，極線
l
′
{\displaystyle l'}
A
x
+
B
y
+
C
=
0
{\displaystyle Ax+By+C=0}
(
A
,
B
,
C
)
⊤
⋅
(
x
,
y
,
1
)
⊤
=
0
{\displaystyle (A,B,C)^{\top }\cdot (x,y,1)^{\top }=0}
因此，我們可以用一個三維向量
(
A
,
B
,
C
)
{\displaystyle (A,B,C)}
l
′
{\displaystyle l'}
l
′
{\displaystyle l'}
e
′
{\displaystyle e'}
p
′
{\displaystyle p'}
l
′
:
e
′
×
p
′
=
[
e
′
]
×
p
′
{\displaystyle l':e'\times p'=[e']_{\times }p'}
[
e
′
]
×
=
[
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}
因為
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}
^
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.
![{\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)