Essential Matrix, Fundamental Matrix

 

 

• X: A 3D point projected to two cameras.
• u_i​, u_j: Corresponding image points in the two views.
• The line joining the two camera centers and X forms the epipolar plane.
• The intersections of this plane with each image plane form epipolar lines.
• The key idea: the corresponding point must lie on the epipolar line.

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• Essential Matrix 𝐸

 

\begin{equation}\mathbf{u}_j^\top\mathbf{E}\mathbf{u}_i=0\end{equation}

 

This is the epipolar constraint:

• It means: if you know the point ui in image 1, then its corresponding point in image 2 must lie on the epipolar line, which is defined by E.
• Valid only when intrinsic camera parameters are known (i.e., the cameras are calibrated).

 

Feature	Description
Epipolar constraint	Describes the geometric relation of two images
Degrees of Freedom (DoF)	5 (3 for rotation, 2 for translation direction)
Applies to	Calibrated cameras

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• Fundamental Matrix 𝐹

 

\begin{equation}\mathbf{x}_j^\top\mathbf{F}\mathbf{x}_i=0\end{equation}

 

• This is the general form of the epipolar constraint.
• It applies when the cameras are uncalibrated.
• The coordinates xi,xj​ are in pixel coordinates (not normalized).

 

Feature	Description
Calibration Needed	❌ No (works with raw pixel coordinates)
Degrees of Freedom (DoF)	7 (due to scale ambiguity and rank 2 constraint)
Relationship to E	E=K⊤FK

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• Homography vs Essential Matrix vs Fundamental Matrix

 

Property / Matrix	Homography (H)	Essential Matrix (E)	Fundamental Matrix (F)
Purpose	Maps points between images assuming a planar scene or pure camera rotation	Encodes epipolar geometry (calibrated cameras)	Encodes epipolar geometry (uncalibrated cameras)
Requires camera calibration?	❌ No	✅ Yes	❌ No
Works for	Planar scenes or pure rotation	General 3D scenes, stereo with calibrated cameras	General 3D scenes, stereo with uncalibrated cameras
Degrees of Freedom (DoF)	8	5 (rotation + translation direction only)	7 (rank 2, scale-invariant 3×3 matrix)
Geometric meaning	Maps image points via a planar transformation	Defines epipolar lines in second image (normalized coords)	Defines epipolar lines in second image
Related to calibration matrix KKK?	No	Yes: E=K⊤FK	No (but used to convert to E)
Common use case	Augmented reality, image stitching	Calibrated stereo, structure from motion	Uncalibrated stereo matching
Minimum point correspondences	4 point pairs	5 point pairs (with calibrated cameras)	8 point pairs (or 7 with 7-point algorithm)