Epipolar GeometryClass 7

Read notes 3.2.1

Feature tracking

run iterative L-K

run iterative L-K

warp & upsample

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• Tracking

• Good features

• Multi-scale

Transl.

Affine transf.

• Feature monitoring

(i) Correspondence geometry: Given an image point x in the first image, how does this constrain the position of the

corresponding point x’ in the second image?

(ii) Camera geometry (motion): Given a set of corresponding image points {xi ↔x’i}, i=1,…,n, what are the cameras P and P’ for the two views?

(iii) Scene geometry (structure): Given corresponding image points xi ↔x’i and cameras P, P’, what is the position of (their pre-image) X in space?

Three questions:

The epipolar geometry

C,C’,x,x’ and X are coplanar

The epipolar geometry

What if only C,C’,x are known?

The epipolar geometry

All points on project on l and l’

The epipolar geometry

Family of planes and lines l and l’ Intersection in e and e’

The epipolar geometry

epipoles e,e’= intersection of baseline with image plane = projection of projection center in other image= vanishing point of camera motion direction

an epipolar plane = plane containing baseline (1-D family)

an epipolar line = intersection of epipolar plane with image(always come in corresponding pairs)

Example: converging cameras

Example: motion parallel with image plane

(simple for stereo rectification)

Example: forward motion

e

e’

The fundamental matrix F

algebraic representation of epipolar geometry

l'x

we will see that mapping is (singular) correlation (i.e. projective mapping from points to lines) represented by the fundamental matrix F

The fundamental matrix F

geometric derivation

xHx' π

x'e'l' FxxHe' π

mapping from 2-D to 1-D family (rank 2)

The fundamental matrix F

algebraic derivation

λCxPλX IPP

PP'e'F

xPP'CP'l

(note: doesn’t work for C=C’ F=0)

xP

λX

The fundamental matrix F

correspondence condition

0Fxx'T

The fundamental matrix satisfies the condition that for any pair of corresponding points x↔x’ in the two images 0l'x'T

The fundamental matrix F

F is the unique 3x3 rank 2 matrix that satisfies x’TFx=0 for all x↔x’

(i) Transpose: if F is fundamental matrix for (P,P’), then FT is fundamental matrix for (P’,P)

(ii) Epipolar lines: l’=Fx & l=FTx’(iii) Epipoles: on all epipolar lines, thus e’TFx=0, x

e’TF=0, similarly Fe=0(iv) F has 7 d.o.f. , i.e. 3x3-1(homogeneous)-1(rank2)(v) F is a correlation, projective mapping from a point x to

a line l’=Fx (not a proper correlation, i.e. not invertible)

Fundamental matrix for pure translation

Fundamental matrix for pure translation

Fundamental matrix for pure translation

PP'e'F

0]|K[IP t]|K[IP'

0KP

-1

00

0e'F

xy

xz

yz

eeeeee

General motion

Pure translation

for pure translation F only has 2 degrees of freedom

The fundamental matrix F

relation to homographies

lHl' -T

π FHe'

π

valid for all plane homographies

eHe'π

The fundamental matrix F

relation to homographies

FxlxH'xππ

requires

πl

πx

x x

Fe'H e.g. 0e'e'T 0e'lT

π

Projective transformation and invariance

-1-T FHH'F̂ x'H''x̂ Hx,x̂

Derivation based purely on projective concepts

X̂P̂XHPHPXx -1

F invariant to transformations of projective 3-space

X̂'P̂XHHP'XP'x' -1

FP'P,

P'P,F

unique

not unique

canonical form

m]|[MP'0]|[IP

MmF

PP'e'F

Projective ambiguity of cameras given Fprevious slide: at least projective ambiguitythis slide: not more!

Show that if F is same for (P,P’) and (P,P’), there exists a projective transformation H so that P=HP and P’=HP’

~ ~

~ ~

]a~|A~

['P~

0]|[IP~

a]|[AP' 0]|[IP

A

~a~AaF

T1 avAA~

kaa~ kandlemma:

kaa~Fa~0AaaaF2rank

TavA-A~

k0A-A~

kaA~

a~Aa

kkIkT1

1

v0H

'P~

]a|av-A[

v0a]|[AHP'

T1

T1

1

kk

kkIk

(22-15=7, ok)

Canonical cameras given F

Possible choice:

]e'|F][[e'P' 0]|[IP

Canonical representation:

]λe'|ve'F][[e'P' 0]|[IP T

0I]e'|F][[e'][e'PP'][e'F

I.e'e'e'.e'][e'][e' TT

λFF.e'e'e'.e' TT

Epipolar geometry?

courtesy Frank Dellaert

Triangulation

C1m1

L1

m2

L2

M

C2

Triangulation

- calibration

- correspondences

Triangulation• Backprojection

• Triangulation

Iterative least-squares

• Maximum Likelihood Triangulation

Backprojection

• Represent point as intersection of row and column

Useful presentation for deriving and understanding multiple view geometry(notice 3D planes are linear in 2D point coordinates)

• Condition for solution?

Next class: computing F