epipolar geometry class 7 read notes 3.2.1. feature tracking run iterative l-k warp &...
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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