Epipolar Geometry and the Fundamental Matrix F

The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates this

geometry

x F x’ = 0

Epipolar geometry

• The Epipolar geometry depends only on the internal parameters of the cameras and the relative pose.

• A point X in 3 space is imaged in 2 views: x and x’

• X, x, x’ and the camera centre C are coplanar in the plane

• The rays back-projected from x and x’ meet at X

Point correspondence geometry

Fig. 8.1

Point correspondence geometry

Epipolar GeometryFig. 8.2

Epipolar geometry

The geometric entities involved in epipolar geometry

Fig 8.3

Converging cameras

Fig 8.4

Motion parallel to the image plane

Geometric derivationFig. 8.5

Point transfer via a plane

The fundamental matrix F

• x l’

• Geometric Derivation

• Step 1: Point transfer via a plane

There is a 2D homography H mapping

each xi to xi’

Step 2: Constructing the epipolar line

Constructing the epipolar line

Cross products

• If a = ( a1, a2 , a3)T is a 3-vector, then one define a corresponding skew-sysmmetric matrix as follows:

0aa-

a-0a

aa-0

a

12

13

23

X

Cross products 2

• Matrix [a]x is singular and a is its null vector

• a x b = ( a2b3 - a3b2, a3b1 - a1b3 , a1b2 – a2b1)T

• a x b = [a]x b =( aT [b]x )T

Algebraic derivation

Algebraic derivation 2

Example 8.2

Example 8.2 b

Properties of the fundamental matrix (a)

Properties of the fundamental matrix (b)

Summary of the Properties of the fundamental matrix 1

Summary of the properties of the fundamental matrix 2

Epipolar line homography 1

Fig. 8.6a

Epipolar line homography 2

Fig. 8.6 b

Epipolar line homography

The epipolar line homography

A pure camera motion

Pure translation

Fig. 8.8

Pure translation motion

Example of pure translation

General camera motion

Fig. 8.9

General camera motion

Example of general motion

Pure planar motion

Retrieving the camera matricesUsing F to determine the camera matrices of 2 views

• Projective invariance and canonical cameras• Since the relationships l’ = Fx and• x’ F x = 0 are projective relationships• which

Projective invariance and canonical cameras

• The camera matrix relates 3-space measurements to image measurements and so depends on both the image coordinate frame and the choice of world coordinate frame.

• F is unchanged by a projective transformation of 3-space.

Projective invariance and canonical cameras 2

Canonical form camera matrices

Projective ambiguity of cameras given F

Projective ambiguity of cameras given F2

Projective ambiguity of cameras given F3

Canonical cameras given F

Canonical cameras given F 2

Canonical cameras given F 3

Canonical cameras given F 4

The Essential Matrix

Normalized Coordinates

Normalized coordinates 2

Normalized coordinates 3

Properties of the Essential Matrix

Result 8.17 on Essential matrix

Result 8.17 on Essential matrix 2

Extraction of cameras from the Essential Matrix

Determine the t part of the camera matrix P’

Result 8.19

Geometrical interpretation of the four solutions

Geometrical interpretation of the four solutions 2

The 4 possible solutions for calibrated reconstruction from E