Epipolar Geometry and the Fundamental Matrix F The

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

Fig. 8. 1 Point correspondence geometry

Point correspondence geometry

Fig. 8. 2 Epipolar Geometry

Epipolar geometry

The geometric entities involved in epipolar geometry

Fig 8. 3

Converging cameras

Fig 8. 4

Motion parallel to the image plane

Fig. 8. 5 Geometric derivation

Point transfer via a plane

The fundamental matrix F • x l’ • Geometric Derivation • Step 1: Point transfer via a plane There is a 2 D homography Hp mapping each xi to xi’ Step 2: Constructing the epipolar line

Constructing the epipolar line

Cross products • If a = ( a 1, a 2 , a 3)T is a 3 -vector, then one define a corresponding skew-sysmmetric matrix as follows:

Cross products 2 • Matrix [a]x is singular and a is its null vector • a x b = ( a 2 b 3 - a 3 b 2, a 3 b 1 - a 1 b 3 , a 1 b 2 – a 2 b 1)T • a x b = [a]x b =( a. T [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

Fig. 8. 6 a Epipolar line homography 1

Fig. 8. 6 b Epipolar line homography 2

Epipolar line homography

The epipolar line homography

A pure camera motion

Pure translation

Fig. 8. 8

Pure translation motion

Example of pure translation

Fig. 8. 9 General camera motion

General camera motion

Example of general motion

Pure planar motion

Retrieving the camera matrices Using 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 F 2

Projective ambiguity of cameras given F 3

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