Multi-View Geometry Part II (Ch7 New book. Ch 10/11 old book)gerig/CS6320-S2015/Materials/CS6320-CV... · 2015. 2. 11. · Fundamental Matrix Fundamental Matrix is singular with rank

Multi-View GeometryPart II

(Ch7 New book.Ch 10/11 old book)

Credits: M. Shah, UCF CAP5415, lecture 23 http://www.cs.ucf.edu/courses/cap6411/cap5415/, Trevor Darrell, Berkeley, C280, Marc Pollefeys

Guido GerigCS 6320 Spring 2015

Multi-View Geometry

Relates

Multi-View Geometry

Relates

• 3D World Points

Multi-View Geometry

Relates

• 3D World Points

• Camera Centers

Multi-View Geometry

Relates

• 3D World Points

• Camera Centers

• Camera Orientations

Multi-View Geometry

Relates

• 3D World Points

• Camera Centers


• Camera Intrinsic Parameters

Multi-View Geometry

Relates

• 3D World Points

• Camera Centers


• Camera Intrinsic Parameters

• Image Points

Stereo

scene point

optical center

image plane

Stereo

Basic Principle: Triangulation• Gives reconstruction as intersection of two rays

Stereo


Stereo


Stereo


Stereo

Basic Principle: Triangulation• Gives reconstruction as intersection of two rays• Requires

– calibration– point correspondence

Stereo Constraints

p p’?

Given p in left image, where can the corresponding point p’in right image be?

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p

Y2X2

Z2O2

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Epipolar Line

Stereo Constraints

X1

Y1

Z1O1

Image plane

Focal plane

M

p p’Y2

X2

Z2O2

Epipolar Line

Epipole

Demo Epipolar Geometry

Java Applet credit to:Quang-Tuan LuongSRI Int.Sylvain Bougnoux

Source: M. Pollefeys

Epipolar constraint

http://www.ai.sri.com/~luong/research/Meta3DViewer/EpipolarGeo.html

• Potential matches for p have to lie on the corresponding epipolar line l’.


Epipolar constraint




Epipolar constraint




Epipolar constraint




Epipolar constraint



• Potential matches for p’ have to lie on the corresponding epipolar line l.


Epipolar constraint


Finding Correspondences

Andrea Fusiello, CVonline

Strong constraints for searching for corresponding points!

Example

Parallel Cameras: Corresponding points on horizontal lines.

Epipolar Constraint

From Geometry to Algebra


O O’

P

p p’


O O’

P

p p’


O O’

P

p p’


O O’

P

p p’


O O’

P

p p’


O O’

P

p p’


O O’

P

p p’


O O’

P

p p’

Linear Constraint:Should be able to express as matrix multiplication.

Review: Matrix Form of Cross Product

Review: Matrix Form of Cross Product

Matrix Form

The Essential Matrix


• Based on the Relative Geometry of the Cameras

• Assumes Cameras are calibrated (i.e., intrinsic parameters are known)

• Relates image of point in one camera to a second camera (points in camera coordinate system).

• Is defined up to scale• 5 independent parameters


What is εp’ ?


Similarly p is the epipolar line corresponding to p in theright camera.

T

p’


What is εe’ ?

• line εp’ converges to epipole e• e’ (center of camera C2 expressed

in frame C1)

eTεe’ ?

e’εe’

Pencil of epipolar lines

Pencil of epipolar lines

http://www.cse.psu.edu/~rcollins/CSE486/lecture19_6pp.pdf

Epipoles

• The epipolar line l’ = εx to each point x (except e) intersects the epipole e0.

Thus e’ satisfies e’T(εx) = (e’T ε)x = 0 for all x.

• This implies that e’T ε = 0T or εTe’ = 0. The epipole e’ is thus a null vector

to εT (in the left null-space of ε).

• Similarly, εe = 0, i.e. e is a null-vector

to ε (in the right null-space of ε).

MVG Hartley & Zisserman



Epipoles


0eR te

The essential matrix ε = [t]×R has 5 degrees of freedom; 3 rotation angles in R, 3 elements in t, but arbitrary scale.Essential Matrix is singular with rank 2.

Similarly, 0 etRetRe TTTT

What if Camera Calibration is not known

Review: Intrinsic Camera Parameters

X

Y

Z C

Image plane

Focal plane

M

m

CCC ZYX ,,

u

vi

j

I

J

II vu ,

101000000

0

0

C

C

C

v

uI

I

ZYX

vfuf

Svu

90

vv

uu

fkffkf

Қ

Fundamental Matrix

Tp 0p system coordinate camerain are and pp

If u and u’ are corresponding image coordinates then we have:

pKupKu

2

1 uKp

KuuKpuKp TTTT

12

11

11

1

TT Ku 1 01

2 uK

0 uFuT TKF 11

2K

Fundamental Matrix

Fundamental Matrix is singular with rank 2.

TKF 11

2K0uFuT

The fundamental matrix F has 7 degrees of freedom: A 3 × 3 homogenous matrix has 8 degrees of freedom. The constraint rank(F) = 2 or det(F) = 0 reduces the number to 7.In principal F has 7 parameters up to scale and can be estimated from 7 point correspondences.Direct Simpler Method requires 8 correspondences (Olivier Faugeras, Computer Vision textbook).

Example II: compute F for a forward translating camera

f

f

X YZ

f

f

X YZ

first image second image

71

Example: forward motion

courtesy of Andrew Zisserman

72

Example: forward motion

courtesy of Andrew Zisserman

e

e’

Estimating Fundamental Matrix0uFuT

Each point correspondence can be expressed as a linear equation:

01

1

333231

232221

131211

vu

FFFFFFFFF

vu

01

33

32

31

23

22

21

13

12

11

FFFFFFFFF

vuvvvvuuvuuu

The 8-point algorithm (Faugeras)

The 8-point AlgorithmScaling: Set F33 to 1 -> Solve for 8 parameters.

Example


Example ctd.


**refers to normal form of line: rho = x cos(phi) + y sin(phi)

uTFu’ = 0 → Fu’=l’, where l’ is epipolar line associated to u.

Example: Left to Right


Example: Right to Left


Example: Epipoles?


Example: Epipoles?


e

Summary: Properties of the Fundamental matrix

Documents

Multi-View Geometry Part II (Ch7 New book. Ch 10/11 old book)gerig/CS6320-S2015/Materials/CS6320-CV... · 2015. 2. 11. · Fundamental Matrix Fundamental Matrix is singular with rank