Structure Computation Scene Planes and Homographies
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Structure Computation Scene Planes and Homographies Slides modified from Marc Pollefeys’ slides
description
Structure Computation Scene Planes and Homographies. Slides modified from Marc Pollefeys’ slides. Problem Statement. Given P, P’ or F with great accuracy Given x, x’ Compute X. Invariant to Projective transformations. Point reconstruction. linear triangulation. homogeneous. invariance?. - PowerPoint PPT Presentation
Transcript of Structure Computation Scene Planes and Homographies
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
