**Epipolar Geometry**

**Wanho Choi**

**(wanochoi.com)**

**Epipolar Geometry**

**P**

*world*

**camera**

**center**

**camera**

**center**

**epipole**

**epipolar**

**line**

**epipolar**

**plane**

**image**

**plane**

**image**

**plane**

**base line**

**x**

_{L}**y**

_{L}**z**

_{L}**y**

_{R}**x**

_{R}**z**

_{R}**P**

*image*

_{L}_{p}

_{p}

*image*

*R*

**e**

_{L}**e**

*R*

**l**

_{L}_{l}

_{l}

*R*

**F**

_{L}

_{= {x}

_{L}**, y**

_{L}**, z**

_{L}_{}}

**F**

*R*

**= {x**

*R*

**, y**

*R*

**, z**

*R*}

**P***image _{L}*

**= M**

*proj*

_{L}**P**

*world*

*L* **P**

*image*

**Essential Matrix**

• _{A transformation matrix between two different frames}**F**_{L}**= RF**_{R}**+ t**
**t × F**_{L}**= t × RF**_{R}**+ t × t** **0**
**F**_{L}_{⋅ (t × F}_{L}_{) = F}0_{L}_{⋅ (t × RF}_{R}_{)}
**0 = F***T*
*L* **([t**×**] R) F***R*
**:= E** : Essential Matrix

**F**

*T*

_{L}

_{L}

**EF**

_{R}

_{R}

**= 0**

**Fundamental Matrix**

• _{A transformation matrix between two different pixels}**P***pixel _{L}*

**= K**

_{L}

_{[R}

_{L}**|t**

_{L}

_{] P}*world*

**P**

*pixel*

_{R}**= K**

_{R}

_{[R}

_{R}**|t**

_{R}

_{] P}*world*

**P**

*pixel*

_{L}**= K**

_{L}**P**

*camera*

*L*

**P**

*pixel*

_{R}**= K**

_{R}**P**

*camera*

*R*

**F**

*T*

*L*

**EF**

*R*

**= 0**

**(P**

*cameraL*)

*T*

**E (P**

*cameraR*

**) = 0**

*from*

**(K**−1

*L*

**P**

*pixel*)

_{L}*T*

**E (K**−1

*R*

**P**

*pixel*

_{R}**) = 0**

**P**

*camera*

*L*

**= K**−1

*L*

**P**

*pixel*

_{L}**P**

*camera*

*R*

**= K**−1

*R*

**P**

*pixel*

_{R}**(P**

*pixelL*)

*T*

**(K**

*−TL*

**EK**−1

*R*

**) (P**

*pixel*

_{R}**) = 0**

**:= F**: Fundamental Matrix

**(P**

*pixelL*

### )

*T*

**F (P**

*pixelR*

**) = 0**

**Fundamental Matrix**

*[uL*

*vL*1]

*F*

_{11}

*F*

_{12}

*F*

_{13}

*F*

_{21}

*F*

_{22}

*F*

_{23}

*F*

_{31}

*F*

_{32}

*F*

_{33}[

*u*

_{R}*v*1 ] = [ 0 0 0]

_{R}*[uLu*

_{R}*uLv*

_{R}*uL*

*vLu*

_{R}*vLv*

_{R}*vL*

*uR*

*vR*1]

*F*

_{11}

*F*

_{12}

*F*

_{13}

*F*

_{21}

*F*

_{22}

*F*

_{23}

*F*

_{31}

*F*

_{32}

*F*

_{33}= 0

*u*

_{L}u_{R}F_{11}

*+ u*

_{L}v_{R}F_{12}

*+ u*

_{L}F_{13}

*+ v*

_{L}u_{R}F_{21}

*+ v*

_{L}v_{R}F_{22}

*+ v*

_{L}F_{23}

*+ u*

_{R}F_{31}

*+ v*

_{R}F_{32}

*+ F*

_{33}= 0

**We need at least 8 point pairs.**

**Triangulation**

• _{Given: (Essential Matrix) or (Fundamental Matrix)}
• We can determine from .

**E** **F**
**(P***imageL* **, P***imageR* )
**P***world*

**P**

*world*

**epipolar**

**plane**

**image**

**plane**

**image**

**plane**

**P**

*image*

_{L}_{p}

_{p}

*image*

*R*