Structure ComputationScene Planes and

Homographies

Slides modified from Marc Pollefeys’ slides

Problem Statement

• Given P, P’ or F with great accuracy

• Given x, x’• Compute X

)HP' ,PH , x'x,(H)P' P, , x'x,(

)P' P, , x'x,(1-1-1-

X

Invariant to Projective transformations

Point reconstruction

PXx XP'x'

linear triangulation

XP'x' PXx

0XP'x

0XpXp0XpXp0XpXp

1T2T

2T3T

1T3T

yxyx

2T3T

1T3T

2T3T

1T3T

p'p''p'p''pppp

A

yxyx

0AX

homogeneous

1X

)1,,,( ZYX

inhomogeneous

invariance?

e)(HX)(AH -1

algebraic error yes, constraint no (except for affine)

geometric error

0x̂F'x̂ subject to )'x̂,(x')x̂(x, T22 dd

X̂P''x̂ and X̂Px̂ subject toly equivalentor

possibility to compute using LM (for 2 or more points)

or directly (for 2 points)

Geometric error

Reconstruct matches in projective frame by minimizing the reprojection error

(see Hartley&Sturm,CVIU´97)Non-iterative optimal solution

Optimal 3D point in epipolar plane

Given an epipolar plane, find best 3D point for (x1,x2)

x1

x2

l1 l2

l1x1

x2l2

x1´

x2´

Select closest points (x1´,x2´) on epipolar lines

Obtain 3D point through exact triangulationGuarantees minimal reprojection error (given this epipolar plane)

Optimal epipolar plane

• Reconstruct matches in projective frame by minimizing the reprojection error

• Non-iterative methodDetermine the epipolar plane for reconstruction

Reconstruct optimal point from selected epipolar plane

222

211 XP,xXP,x dd

(Hartley and Sturm, CVIU´97)

222

211 αl,xαl,x DD

(polynomial of degree 6check all minima, incl ∞)

m1

m2

l1 l2

3DOF

1DOF

Reconstruction uncertainty

consider angle between rays

Line reconstruction

P'l'Pl

T

T

L

doesn‘t work for epipolar plane

Scenes and Homographies

Homography given plane

a]|[AP' 0]|[IP

0XπT TT ,1vπ

Hxx' TavAH

0]X|[IPXx

TT ρ,xX

point on plane

TTT x,-vxproject in second view

xavAXP'x' T

Calibrated stereo rig

0]|K[IPE t]|[RK'P'E

-1T K/tnRK'H d

dn/v

TT ) ,n(π d

homographies and epipolar geometry

points on plane also have to satisfy epipolar geometry!

x 0,FxHxFxHx TTT

HTF has to be skew-symmetric

0HFFH TT x ,x'e'Fx

x'x 0,'x'eHx TT

H'eF

l’

Homography also maps epipole

Hee'

Homography also maps epipolar lines

eT

e l'Hl

Compatibility constraint

Hxx'Fxl'e

plane homography given F and 3 points correspondences

Method 1: reconstruct explicitly, compute plane through 3 points derive homography

Method 2: use epipoles as 4th correspondence to compute homography

degenerate geometry for an implicit computation of the homography

Estimastion from 3 noisy points (+F)

Consistency constraint: points have to be in exact epipolar correspodence

Determine MLE points given F and xi↔xi’

Use implicit 3D approach (no derivation here)

T1bMe'AH

2T e'x'/e'x'Ax'x iiiiib

Fe'A

M is a 3x3 matrix with rows xiT

plane homography given F, a point and a line

Tlμe'Fl'H

xle'x'

l'Fxx'e'x'μ

T2

T

application: matching lines(Schmid and Zisserman, CVPR’97)

epipolar geometry induces point homography on lines

x]'l[)xle']'l[(x)(x' T FμFH

Degenerate homographies

Fxl''x

plane induced parallax

Hx'xl

6-point algorithm

6655 Hx'xHx'xe'

x1,x2,x3,x4 in plane, x5,x6 out of plane

Compute H from x1,x2,x3,x4

He'F

Projective depth

ρe'Hxx'

TT ρ,xX

=0 on planesign of determines on which side of plane

Binary space partition

Next class: The Trifocal Tensor