CSE445Computer Vision
ModulesMulti-View & MotionEpipolar Geometry
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Epipolar Geometry

The fundamental matrix, epipoles and epipolar lines, the 8-point algorithm, and rank-2 enforcement.

L14CO2M04 · Multi-View & MotionFinal Exam

Key Terms

Fundamental MatrixEpipoleEpipolar Line8-Point AlgorithmRank-2 ConstraintSampson Distance

Python Tools

OpenCVNumPyMatplotlib

Textbook Reference

Szeliski Ch.11Hartley & Zisserman Ch.9-11

Topics Covered

  1. 1Two-view geometry, the baseline, and the epipolar constraint x'^T F x = 0
  2. 2Epipoles and epipolar lines as null spaces and images of points
  3. 3The normalized 8-point algorithm
  4. 4Rank-2 enforcement of F by SVD truncation
  5. 5Geometric error, the Sampson distance, and RANSAC for F

Learning Outcomes

  • State and interpret the epipolar constraint and the meaning of F
  • Estimate F with the normalized 8-point algorithm and enforce rank 2
  • Use epipolar lines to constrain the correspondence search

Two-view geometry lab

Epipolar Geometry

Two-camera geometry through epipoles, epipolar lines, and the fundamental matrix.

e off-screenCAMERA 1x = (464.4, 182.7, 1)
e off-screenCAMERA 2x = (366.1, 193.6, 1)
Without F: ≈256,000 pixelsWith F: ≈640 candidates on one line

Step 18/18 · Correspondence search is reduced from 2D to one line.

cyan ray · camera 1violet ray · camera 2orange plane / epipole✓ valid correspondence

Live geometry

x₁ = [464.444, 182.667, 1.000]ᵀ

x₂ = [366.081, 193.629, 1.000]ᵀ

l₁ = [-1.16e-20, -1.47e-2, 2.69e+0]ᵀ

l₂ = [1.54e-4, 1.56e-2, -3.08e+0]ᵀ

Correspondence satisfies the epipolar constraint

Sampson distance: 0.00e+0

Fundamental matrix F

rank 2 · valid
5.91e-24-8.86e-61.77e-30.00e+0-5.73e-71.57e-2-1.37e-20-1.14e-2-1.00e+0
det ≈ 3.0e-27scale ‖M‖F = 1residual 0.00e+0distance 0.00 px‖t‖ 2.20rotation -17.2° yaw
x2⊤Fx1=0\mathbf{x}_2^{\top}\mathbf{F}\mathbf{x}_1 = 0
l2=Fx1\mathbf{l}_2=\mathbf{F}\mathbf{x}_1
l1=F⊤x2\mathbf{l}_1=\mathbf{F}^{\top}\mathbf{x}_2
E=[t]×R\mathbf{E}=[\mathbf{t}]_{\times}\mathbf{R}
F=K2−⊤EK1−1\mathbf{F}=\mathbf{K}_2^{-\top}\mathbf{E}\mathbf{K}_1^{-1}
d(x2,l2)=∣x2⊤Fx1∣l2,12+l2,22d(\mathbf{x}_2,\mathbf{l}_2)=\frac{|\mathbf{x}_2^{\top}\mathbf{F}\mathbf{x}_1|}{\sqrt{l_{2,1}^2+l_{2,2}^2}}

Lecture PDFs

Epipolar Geometry

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