3dphoto02_projectiveeewGeometry

3D photography

Kevin Kser, Marc Pollefeys

Spring 2012

http://www.cvg.ethz.ch/teaching/2012spring/3dphoto/

Feb 20 Introduction

Feb 27 Lecture: Geometry, Camera Model, Calibration

Mar 5 Lecture: Features, Tracking/Matching

Mar 12 Project Proposals by Students

Mar 19 Lecture: Epipolar Geometry

Mar 26 Short lecture Stereo

+ 2 papers (2 teams of 2 students)

Apr 2 Short lecture Active Ranging, Structured Light + 2 papers

Apr 9 Easter

Apr 16 Project Updates (Sechseluten in afternoon)

Apr 23 Short lecture Volumetric Modeling +2 papers

Apr 30 Short lecture Single View + 2 papers

May 7 Short lecture SfM from Photo Collections + 2 papers

May 14 Short lecture Silhouettes + 2 papers

May 21 Short lecture + 2 papers

May 28 Pentecost / White Monday

May 29-June 1 Final Demos

Schedule (tentative)

Projective Geometry and Camera model

Class 2

points, lines, planes conics and quadrics

transformations camera model

See http://www.cs.unc.edu/~marc/tutorial/

Book: Hartley/Zisserman: Multiple View Geometry in Computer Vision

Homogeneous coordinates

0=++ cbyax ( ) ( ) 0=1x,y,a,b,c T

( ) ( ) 0,~ ka,b,cka,b,c TT

Homogeneous representation of 2D points and lines

equivalence class of vectors, any vector is representative

Set of all equivalence classes in R3(0,0,0)T forms P2

( ) ( ) 0,1,,~1,, kyxkyx TT

The point x lies on the line l if and only if

Homogeneous coordinates

Inhomogeneous coordinates ( )Tyx,( )T

321 ,, xxx but only 2DOF

Note that scale is unimportant for incidence relation

0=xlT

Points from lines and vice-versa

l'l=x

Intersections of lines

The intersection of two lines and is l l'

Line joining two points

The line through two points and is x'x=lx x'

Example

1=x

1=y

1

)1-,1,0( y

x

1

)1-,0,1( y

x

Note:

[ ] 'xx='xx

with

0-

0-

-0

x

xy

xz

yz

Ideal points and the line at infinity

T0,,l'l ab

Intersections of parallel lines

( ) ( )TT and ',,=l' ,,=l cbacba

Example

1=x 2=x

Ideal points ( )T0,, 21 xxLine at infinity ( )T1,0,0=l

22 l= RP

tangent vector

normal direction

-ab,( )ba,

Note that in P2 there is no distinction

between ideal points and others

3D points and planes

Homogeneous representation of 3D points and planes

The point X lies on the plane if and only if

0=+++ 44332211 XXXX

0=XT

The plane goes through the point X if and only if

0=XT

Planes from points

0

X

X

X

3

2

1

T

T

T

0=X 0=X 0,=X 321TTT

andfromSolve

(solve as right nullspace of )

T

T

T

3

2

1

X

X

X

Points from planes

0X

3

2

1

T

T

T

x=X M 321 XXXM

0= MT

0=X 0=X 0,=X X 321TTT

andfromSolve

(solve as right nullspace of ) X

T

T

T

3

2

1

Representing a plane by its span

Lines

T

T

B

AW BA

22

** 0=WW=WWTT

0001

1000W

0010

0100W*

Example: X-axis

(4dof)

Representing a line by its span

T

T

Q

PW* QP

Dual representation (planes)

(Alternative: Plcker representation, details see e.g. H&Z)

Points, lines and planes

TX

WM 0=M

T

W*M 0X M

W

X

*W

Common plane for point X and line W

Intersection line W and plane

Plcker coordinates

Elegant representation for 3D lines

jijiij ABBAl -

TL 344223141312 ,,,,, llllll5P

0B,AB,A,det| LL( ) 0=|LL (Plcker internal constraint)

(two lines intersect)

( )123413421423231442133412+++++=| llllllllllllLL

(for more details see e.g. H&Z)

(with A and B points)

Conics

Curve described by 2nd-degree equation in the plane

0=+++++ 22 feydxcybxyax

0=+++++ 2332312

221

2

1 fxxexxdxcxxbxax

3

2

3

1 ,x

xy

xx

x or homogenized

0=xx CT

or in matrix form

fed

ecb

dba

2/2/

2/2/

2/2/

Cwith

{ }fedcba :::::5DOF:

Five points define a conic

For each point the conic passes through

0=+++++ 22 feydxcyybxax iiiiii

or

( ) 0=.1,,,,, 22 ciiiiii yxyyxx ( )Tfedcba ,,,,,=c

0

1

1

1

1

1

55

2

555

2

5

44

2

444

2

4

33

2

333

2

3

22

2

222

2

2

11

2

111

2

1

c

yxyyxx

yxyyxx

yxyyxx

yxyyxx

yxyyxx

stacking constraints yields

Tangent lines to conics

The line l tangent to C at point x on C is given by l=Cx

l x

C

Dual conics

0=ll *CTA line tangent to the conic C satisfies

Dual conics = line conics = conic envelopes

1-* CC In general (C full rank):

Degenerate conics

A conic is degenerate if matrix C is not of full rank

TT ml+lm=C

e.g. two lines (rank 2)

e.g. repeated line (rank 1)

Tll=C

l

l

m

Degenerate line conics: 2 points (rank 2), double point (rank1)

( ) CC **Note that for degenerate conics

Quadrics and dual quadrics

(Q : 4x4 symmetric matrix) 0=QXXT

9 d.o.f.

in general 9 points define quadric

det Q=0 degenerate quadric

tangent plane

Q

QX=

0=Q *T

-1* Q=Q relation to quadric (non-degenerate)

2D projective transformations

A projectivity is an invertible mapping h from P2 to itself

such that three points x1,x2,x3 lie on the same line if and

only if h(x1),h(x2),h(x3) do.

Definition:

A mapping h:P2P2 is a projectivity if and only if there

exist a non-singular 3x3 matrix H such that for any point

in P2 reprented by a vector x it is true that h(x)=Hx

Theorem:

Definition: Projective transformation

3

2

1

333231

232221

131211

3

2

1

'

'

'

x

x

x

hhh

hhh

hhh

x

x

x

x=x' Hor

8DOF

projectivity=collineation=projective transformation=homography

Transformation of 2D points, lines and conics

Transformation for lines

l=l' -TH

Transformation for conics

-1-TCHHC ='

Transformation for dual conics

THHCC

** ='

x=x' HFor a point transformation

Fixed points and lines

e=e H

(eigenvectors H =fixed points)

l=l TH

(eigenvectors H-T =fixed lines)

(1=2 pointwise fixed line)

Hierarchy of 2D transformations

100

2221

1211

y

x

taa

taa

100

2221

1211

y

x

tsrsr

tsrsr

333231

232221

131211

hhh

hhh

hhh

100

2221

1211

y

x

trr

trr

Projective

8dof

Affine

6dof

Similarity

4dof

Euclidean

3dof

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.

invariants transformed

squares

The line at infinity

3

2

1

1

-

1t-

0ll

l

l

l

AA

AH

T-T

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

110

t2

1

x

x

xx AA

H

10

t111- AAHA

An affinity maps points

Lines map with an inverse, transposed H:

1-

l

1

0

0

1t-

0ll

A

AH

T-T

A

The line at infinity thus is mapped as:

Affine properties from images

projection rectification

APA

lll

HH

321

010

001

0,l 3321 llllT

Affine rectification v1 v2

l1

l2 l4

l3

l

21 vvl

211 llv

432 ll=v

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

0=++++ 2332312

2

2

1 fxxexxdxxx 0=+ 222

1 xx

l

T

T

0,-,1J

0,,1I

i

i

( ) ( )TT 0,1,0+0,0,1=I i

Algebraically, encodes orthogonal directions

03 x

Conic dual to the circular points

TT JIIJ* C

000

010

001*

C

T

SS HCHC*

*

The dual conic is fixed conic under the

projective transformation H iff H is a similarity

*

C

l

I

J

Angles

( )( )22

2

1

2

2

2

1

2211

++

+=cos

mmll

mlml

( )T321 ,,=l lll ( )

T

321 ,,=m mmm

Euclidean:

Projective: mmll

mlcos

**

*

CC

C

TT

T

0=ml *CT

(orthogonal)

Transformation of 3D points, planes and quadrics

Transformation for lines

( )l=l' -TH

Transformation for conics

( )-1-TCHHC ='

Transformation for dual conics

( )THHCC ** ='

( )x=x' H

For a point transformation

X=X' H

=' -TH

-1-TQHH=Q'

THHQ=Q' **

(cfr. 2D equivalent)

Hierarchy of 3D transformations

vTv

tAProjective 15dof

Affine

12dof

Similarity

7dof

Euclidean

6dof

Intersection and tangency

Parallellism of planes,

Volume ratios, centroids,

The plane at infinity

Angles, ratios of length

The absolute conic

Volume

10

tAT

10

tRT

s

10

tRT

The plane at infinity

1

0

0

0

1t-

0

A

AH

TT

A

The plane at infinity is a fixed plane under a projective transformation H iff H is an affinity

1. canonical position

2. contains directions

3. two planes are parallel line of intersection in 4. line // line (or plane) point of intersection in

( )T1,0,0,0=( )T0,,,=D 321 XXX

The absolute conic

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

04

2

3

2

2

2

1

X

XXX

The absolute conic is a (point) conic on . In a metric frame:

T321321 ,,I,, XXXXXXor conic for directions: (with no real points)

1. is only fixed as a set 2. Circle intersect in two circular points 3. Spheres intersect in

The absolute dual quadric

00

0I*

T

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

1. plane at infinity is the nullvector of 2. Angles:

( )( )2

*

21*

1

2

*

1

=cos

TT

T

Camera model

Relation between pixels and rays in space

?

Pinhole camera

Pinhole camera model

TT ZfYZfXZYX )/,/(),,(

101

0

0

1

Z

Y

X

f

f

Z

fY

fX

Z

Y

X

linear projection in homogeneous coordinates!

Pinhole camera model

101

0

0

Z

Y

X

f

f

Z

fY

fX

101

01

01

1Z

Y

X

f

f

Z

fY

fX

PXx

0|I)1,,(diagP ff

Principal point offset

T

yx

T pZfYpZfXZYX )+/,+/(),,(

principal point T

yx pp ),(

101

0

0

1

Z

Y

X

pf

pf

Z

ZpfY

ZpfX

Z

Y

X

y

x

x

x

Principal point offset

101

0

0

Z

Y

X

pf

pf

Z

ZpfY

ZpfX

y

x

x

x

camX0|IKx

1

y

x

pf

pf

K calibration matrix

Camera rotation and translation

( )C~-X~R=X~ cam

X10

RC-R

1

10

C~

R-RXcam

Z

Y

X

camX0|IKx XC~

-|IKRx

t|RKP C~R-t PX=x

~

CCD camera

1

yy

xx

p

p

K

11

y

x

y

x

pf

pf

m

m

K

General projective camera

1

yx

xx

p

ps

K

1

yx

xx

p

p

K

C~|IKRP non-singular

11 dof (5+3+3)

t|RKP

intrinsic camera parameters

extrinsic camera parameters

Radial distortion

Due to spherical lenses (cheap)

Model:

R

y

xyxKyxKyx ...))()(1(),( 442

22

1R

http://foto.hut.fi/opetus/260/luennot/11/atkinson_6-11_radial_distortion_zoom_lenses.jpg straight lines are not straight anymore

Camera model


?

Projector model


(dual of camera)

(main geometric difference is vertical principal point offset

to reduce keystone effect)

?

Affine cameras

Action of projective camera on points

and lines

forward projection of line

( ) b+a=PB+PA=B)+P(A=X

back-projection of line

lP= T

PXlX TT ( )PX= x0;=xlT

PX=x

projection of point

Action of projective camera on conics

and quadrics

back-projection to cone

CPPQ Tco 0=CPXPX=CxxTTT

( )PX=x

projection of quadric

TPPQC ** 0lPPQlQT*T*T

( )lP= T

Image of absolute conic

Practical Camera Calibration

Unknown: constant camera intrinsics K

(varying) camera pose R,t

Known: 3D coordinates of chessboard corners

=> Define to be the z=0 plane ( X=[X1 X2 0 1] )

Point is mapped as x = K (r1 r2 r3 t) X x = K (r1 r2 t) [X1 X2 1]

1

yy

xx

pf

pf

K

Homography H between image and chess coordinates,

estimate from known Xi and measured xi

Method and Pictures from Zhang (ICCV99): Flexible Camera Calibration By Viewing a

Plane From Unknown Orientations

Direct Linear Transformation (DLT)

ii Hxx 0Hxx ii

i

i

i

i

xh

xh

xh

Hx3

2

1

T

T

T

iiii

iiii

iiii

ii

yx

xw

wy

xhxh

xhxh

xhxh

Hxx12

31

23

TT

TT

TT

0

h

h

h

0xx

x0x

xx0

3

2

1

TTT

TTT

TTT

iiii

iiii

iiii

xy

xw

yw

Tiiii wyx ,,x

0hA i


Equations are linear in h

0

h

h

h

0xx

x0x

xx0

3

2

1

TTT

TTT

TTT

iiii

iiii

iiii

xy

xw

yw

0AAA 321 iiiiii wyx

0hA i Only 2 out of 3 are linearly independent

(indeed, 2 eq/pt)

0

h

h

h

x0x

xx0

3

2

1

TTT

TTT

iiii

iiii

xw

yw

(only drop third row if wi0)

Holds for any homogeneous

representation, e.g. (xi,yi,1)


Solving for H

0Ah 0h

A

A

A

A

4

3

2

1

size A is 8x9 or 12x9, but rank 8

Trivial solution is h=09T is not interesting

1-D null-space yields solution of interest

pick for example the one with 1h


Over-determined solution

No exact solution because of inexact measurement

i.e. noise

0Ah 0h

A

A

A

n

2

1

Find approximate solution

- Additional constraint needed to avoid 0, e.g.

- not possible, so minimize

1h Ah0Ah

DLT algorithm

Objective

Given n4 2D to 2D point correspondences {xixi}, determine the 2D homography matrix H such that xi=Hxi

Algorithm

(i) For each correspondence xi xi compute Ai. Usually only two first rows needed.

(ii) Assemble n 2x9 matrices Ai into a single 2nx9 matrix A

(iii) Obtain SVD of A. Solution for h is last column of V

(iv) Determine H from h

Importance of normalization

0

h

h

h

0001

1000

3

2

1

iiiiiii

iiiiiii

xyxxxyx

yyyxyyx

~102 ~102 ~102 ~102 ~104 ~104 ~102 1 1

orders of magnitude difference!

Monte Carlo simulation

for identity computation based on 5 points

(not normalized normalized)

Normalized DLT algorithm

Objective

Given n4 2D to 2D point correspondences {xixi}, determine the 2D homography matrix H such that xi=Hxi

Algorithm

(i) Normalize points

(ii) Apply DLT algorithm to

(iii) Denormalize solution

,x~x~ ii inormiinormi xTx

~,xTx~

norm

-1

norm TH~

TH

1

norm

100

2/0

2/0

T

hhw

whw

Geometric distance measured coordinates

estimated coordinates

true coordinates

xxx

2H

xH,xargminH iii

d Error in one image

e.g. calibration pattern

221-H

Hx,xxH,xargminH iiiii

dd

Symmetric transfer error

d(.,.) Euclidean distance (in image)

ii

iiii

i

ii ddii

xHx subject to

x,xx,xargminx,x,H22

x,xH,

Reprojection error

Reprojection error

221- Hx,xxHx, dd

22 x,xxx, dd

Statistical cost function and Maximum Likelihood Estimation

Optimal cost function related to noise model

Assume zero-mean isotropic Gaussian noise (assume outliers removed)

22 2/xx,

22

1xPr de

22

i 2xH,x /2i

2

1H|xPr

idei

constantxH,xH|xPrlog2

i2i2

1id

Error in one image

Maximum Likelihood Estimate

2

i xH,x id

Statistical cost function and Maximum Likelihood Estimation

Optimal cost function related to noise model

Assume zero-mean isotropic Gaussian noise (assume outliers removed)

22 2/xx,

22

1xPr de

22i2i 2xH,xx,x /

2i2

1H|xPr

iidd

ei

Error in both images

Maximum Likelihood Estimate

2i2

i x,xx,x ii dd

Gold Standard algorithm

Objective

Given n4 2D to 2D point correspondences {xixi}, determine the Maximum Likelyhood Estimation of H

(this also implies computing optimal xi=Hxi)

Algorithm

(i) Initialization: compute an initial estimate using

normalized DLT or RANSAC

(ii) Geometric minimization of reprojection error:

Minimize using Levenberg-Marquardt over 9 entries of h

or Gold Standard error:

compute initial estimate for optimal {xi}

minimize cost over {H,x1,x2,,xn}

if many points, use sparse method

2i2

i x,xx,x ii dd


Unknown: constant camera intrinsics K

(varying) camera pose R,t

Known: 3D coordinates of chessboard corners

=> Define to be the z=0 plane ( X=[X1 X2 0 1] )

Point is mapped as x = K (r1 r2 r3 t) X x = K (r1 r2 t) [X1 X2 1]

1

yy

xx

pf

pf

K

Homography H between image and chess coordinates,

estimate from known Xi and measured xi

Method and Pictures from Zhang (ICCV99): Flexible Camera Calibration By Viewing a

Plane From Unknown Orientations


1

yy

xx

pf

pf

K

H has the shape:

H = K(r1 r2 t) K1H = (r1 r2 t)

With rotation matrix constraints

r1Tr2 = 0

r1Tr1=r2

Tr2=1

Squaring yields 2 constraints on IAC (KKT)-1

h1T KTK1h2 = 0

hT1KTK1h1 = h

T2K

TK1h2

n images provide 2n constraints on IAC

More measurements => better estimate of IAC !


1

yy

xx

pf

pf

K

Given an estimate for the IAC, factorize (KKT)-1

Factorize

[Obtain Poses]

Clamp

Optimize

Reminder

Project presentation in 2 weeks

Find team mate / decide topic soon

Limited number of kinects: first come, first served !

3dphoto02_projectiveeewGeometry

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