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 xL yL zL yR xR zRP
imageLp
image Re
Le
Rl
Ll
R FL = {xL, yL, zL} FR = {xR, yR, zR}PimageL = MprojL Pworld
L P
image
Essential Matrix
• A transformation matrix between two different frames FL = RFR + t t × FL = t × RFR + t × t 0 FL ⋅ (t × FL) = F0 L ⋅ (t × RFR) 0 = FT L ([t×] R) FR := E : Essential Matrix
F
T
L
EF
R
= 0
Fundamental Matrix
• A transformation matrix between two different pixels PpixelL = KL[RL|tL] Pworld PpixelR = KR[RR|tR] Pworld PpixelL = KLPcamera L PpixelR = KRPcamera R FT LEFR = 0 (PcameraL )T E (PcameraR ) = 0 from (K−1L PpixelL ) T E (K−1 R PpixelR ) = 0 Pcamera L = K−1L PpixelL Pcamera R = K−1R PpixelR (PpixelL ) T (K−TL EK−1R ) (PpixelR ) = 0 := F : Fundamental Matrix
(P
pixelL)
TF (P
pixelR) = 0
Fundamental Matrix
[uL vL 1] F11 F12 F13 F21 F22 F23 F31 F32 F33 [ uR vR 1 ] = [ 0 0 0] [uLuR uLvR uL vLuR vLvR vL uR vR 1] F11 F12 F13 F21 F22 F23 F31 F32 F33 = 0 uLuRF11 + uLvRF12 + uLF13 + vLuRF21 + vLvRF22 + vLF23 + uRF31 + vRF32 + F33 = 0We need at least 8 point pairs.
Triangulation
• Given: (Essential Matrix) or (Fundamental Matrix) • We can determine from .
E F (PimageL , PimageR ) Pworld
