Projective 2D geometry (cont’)course 3

Multiple View GeometryComp 290-089Marc Pollefeys

Content

• Background: Projective geometry (2D, 3D), Parameter estimation, Algorithm evaluation.

• Single View: Camera model, Calibration, Single View Geometry.

• Two Views: Epipolar Geometry, 3D reconstruction, Computing F, Computing structure, Plane and homographies.

• Three Views: Trifocal Tensor, Computing T.• More Views: N-Linearities, Multiple view

reconstruction, Bundle adjustment, auto-calibration, Dynamic SfM, Cheirality, Duality

Multiple View Geometry course schedule(subject to change)

Jan. 7, 9 Intro & motivation Projective 2D Geometry

Jan. 14, 16

(no course) Projective 2D Geometry

Jan. 21, 23

Projective 3D Geometry Parameter Estimation

Jan. 28, 30

Parameter Estimation Algorithm Evaluation

Feb. 4, 6 Camera Models Camera Calibration

Feb. 11, 13

Single View Geometry Epipolar Geometry

Feb. 18, 20

3D reconstruction Fund. Matrix Comp.

Feb. 25, 27

Structure Comp. Planes & Homographies

Mar. 4, 6 Trifocal Tensor Three View Reconstruction

Mar. 18, 20

Multiple View Geometry MultipleView Reconstruction

Mar. 25, 27

Bundle adjustment Papers

Apr. 1, 3 Auto-Calibration Papers

Apr. 8, 10

Dynamic SfM Papers

Apr. 15, 17

Cheirality Papers

Apr. 22, 24

Duality Project Demos

Last week …

l'lx 0xl T x'xl T1,0,0l

Points and lines

0xx CT xl C0ll * CT 1* CC

Conics and dual conics

ll' -TH

-1-TCHHC ' THHCC **'

xx' HProjective transformations

Last week …

1002221

1211

y

x

taa

taa

1002221

1211

y

x

tsrsr

tsrsr

333231

232221

131211

hhh

hhh

hhh

1002221

1211

y

x

trr

trr

Projective8dof

Affine6dof

Similarity4dof

Euclidean3dof

Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio

Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids). The line at infinity l∞

Ratios of lengths, angles.The circular points I,J

lengths, areas.

Projective geometry of 1D

x'x 22H

The cross ratio

Invariant under projective transformations

T21, xx

3DOF (2x2-1)

02 x

4231

43214321 x,xx,x

x,xx,xx,x,x,x Cross

22

11detx,x

ji

ji

ji xx

xx

Recovering metric and affine properties from images

• Parallelism• Parallel length ratios

• Angles • Length ratios

The line at infinity

l

1

0

0

1t

0ll

A

AH

TTA

The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity

Note: not fixed pointwise

Affine properties from images

projection rectification

APA

lll

HH

321

010

001

0,l 3321 llll T

Affine rectificationv1 v2

l1

l2 l4

l3

l∞

21 vvl

211 llv 432 llv

Distance ratios

badd :c,b:b,a

T0,1v' H

TTT 1,,1,,1,0 baa

c,b,a H

The circular points

0

1

I i

0

1

J i

I

0

1

0

1

100

cossin

sincos

II

iseitss

tssi

y

x

S

H

The circular points I, J are fixed points under the projective transformation H iff H is a similarity

The circular points

“circular points”

0233231

22

21 fxxexxdxxx 02

221 xx

l∞

T

T

0,,1J

0,,1I

i

i

TT 0,1,00,0,1I iAlgebraically, encodes orthogonal directions

03 x

Conic dual to the circular points

TT JIIJ* C

000

010

001*C

TSS HCHC **

The dual conic is fixed conic under the projective transformation H iff H is a similarity

*C

Note: has 4DOF

l∞ is the nullvector

*C

Angles

22

21

22

21

2211cosmmll

mlml

T321 ,,l lll T321 ,,m mmmEuclidean:

Projective: mmll

mlcos

**

*

CC

CTT

T

0ml * CT (orthogonal)

Length ratios

sin

sin

),(

),(

cad

cbd

Metric properties from images

vvv

v

'

*

*

**

TT

TT

T

TT

T

K

KKK

HHCHH

HHHCHHH

HHHCHHHC

APAP

APSSAP

SAPSAP

TUUC

000

010

001

'*

Rectifying transformation from SVD

UH

Metric from affine

000

0

3

2

1

321

m

m

m

lllTKK

0,,,, 2221211

212

21122122111 T

kkkkkmlmlmlml

Metric from projective

0c,,,,, 332332133122122111 5.05.05.0 mlmlmlmlmlmlmlmlml

0vvv

v

3

2

1

321

m

m

m

lllTT

TT

K

KKK

Pole-polar relationship

The polar line l=Cx of the point x with respect to the conic C intersects the conic in two points. The two

lines tangent to C at these points intersect at x

Correlations and conjugate points

A correlation is an invertible mapping from points of P2 to lines of P2. It is represented by a 3x3 non-singular matrix A as l=Ax

Conjugate points with respect to C(on each others polar)

0xy CT

Conjugate points with respect to C*

(through each others pole)

0ml * CT

Projective conic classificationTUDUC -TCUUD 1

321321321 ,,diag,,diag,,diag ssseeesssD0or 1ie

Diagonal Equation Conic type

(1,1,1) improper conic

(1,1,-1) circle

(1,1,0) single real point

(1,-1,0) two lines

(1,0,0) single line

0222 wyx

0222 wyx

022 yx

022 yx

02 x

Affine conic classification

ellipse parabola hyperbola

Chasles’ theorem

A B

C

DX

Conic = locus of constant cross-ratio towards 4 ref. points

Iso-disparity curves

X0X1

C1

X∞

C2

Xi Xj

ii

i

i

1

1:

1

01

1

01

11

:

11

10

11

0

Fixed points and lines

ee H (eigenvectors H =fixed points)

ll TH (eigenvectors H-T =fixed lines)

(1=2 pointwise fixed line)

Next course:Projective 3D Geometry

• Points, lines, planes and quadrics

• Transformations

• П∞, ω∞ and Ω ∞