Projective 3D geometry class 4 Multiple View Geometry Slides modified from Marc Pollefeys Comp...
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Transcript of Projective 3D geometry class 4 Multiple View Geometry Slides modified from Marc Pollefeys Comp...
Projective 3D geometryclass 4
Multiple View GeometrySlides modified from Marc Pollefeys Comp 290-089
Last week …
T1,0,0l
TT JIIJ* C
000
010
001*C
0ml * CT
(orthogonality)
circular points (similarities)
line at infinity (affinities)
Last week …pole-polar relation
0xy CT
0222 wyx
conjugate points & lines
projective conic classification
affine conic classification
xl C lx *C
0lm * CT
A B
C
DX
Chasles’ theorem
cross-ratio
Fixed points and lines
λee H (eigenvectors H =fixed points)
lλl TH (eigenvectors H-T =fixed lines)
(1=2 pointwise fixed line)
Singular Value Decomposition
Tnnnmmmnm VΣUA
IUU T
021 n
IVV T
nm
XXVT XVΣ T XVUΣ T
TTTnnn VUVUVUA 222111
000
00
00
00
Σ
2
1
n
Singular Value Decomposition
• Homogeneous least-squares
• Span and null-space
• Closest rank r approximation
• Pseudo inverse
1X AXmin subject to nVX solution
nrrdiag ,,,,,, 121 0 ,, 0 ,,,,~
21 rdiag TVΣ~
UA~
TVUΣA
0000
0000
000
000
Σ 2
1
4321 UU;UU LL NS 4321 VV;VV RR NS
TUVΣA 0 ,, 0 ,,,, 112
11 rdiag
TVUΣA
Projective 3D Geometry
• Points, lines, planes and quadrics
• Transformations
• П∞, ω∞ and Ω ∞
3D points
TT
1 ,,,1,,,X4
3
4
2
4
1 ZYXX
X
X
X
X
X
in R3
04 X
TZYX ,,
in P3
XX' H (4x4-1=15 dof)
projective transformation
3D point
T4321 ,,,X XXXX
Planes
0ππππ 4321 ZYX
0ππππ 44332211 XXXX
0Xπ T
Dual: points ↔ planes, lines ↔ lines
3D plane
0X~
.n d T321 π,π,πn TZYX ,,X~
14 Xd4π
Euclidean representation
n/d
XX' Hππ' -TH
Transformation
Planes from points
0π
X
X
X
3
2
1
T
T
T
2341DX T123124134234 ,,,π DDDD
0det
4342414
3332313
2322212
1312111
XXXX
XXXX
XXXX
XXXX
0πX 0πX 0,πX π 321 TTT andfromSolve
(solve as right nullspace of )π
T
T
T
3
2
1
X
X
X
0XXX Xdet 321
Or implicitly from coplanarity condition
124313422341 DXDXDX 01234124313422341 DXDXDXDX 13422341 DXDX
Points from planes
0X
π
π
π
3
2
1
T
T
T
xX M 321 XXXM
0π MT
I
pM
Tdcba ,,,π T
a
d
a
c
a
bp ,,
0Xπ 0Xπ 0,Xπ X 321 TTT andfromSolve
(solve as right nullspace of )X
T
T
T
3
2
1
π
π
π
Parameterizing points on a plan by representing a plane by its span
Lines
T
T
B
AW μBλA
T
T
Q
PW* μQλP
22** 0WWWW TT
0001
1000W
0010
0100W*
Example: X-axis
(4dof)
Points, lines and planes
TX
WM 0π M
Tπ
W*
M 0X M
W
X
*W
π
Plücker matrices
jijiij ABBAl TT BAABL
Plücker matrix (4x4 skew-symmetric homogeneous matrix)
1. L has rank 22. 4dof3. generalization of
4. L independent of choice A and B5. Transformation
24* 0LW T
yxl
THLHL'
0001
0000
0000
1000
1000
0
0
0
1
0001
1
0
0
0
L T
Example: x-axis
Plücker matrices
TT QPPQL*
Dual Plücker matrix L*
-1TLHHL -'*
*12
*13
*14
*23
*42
*34344223141312 :::::::::: llllllllllll
XLπ *
LπX
Correspondence
Join and incidence
0XL*
(plane through point and line)
(point on line)
(intersection point of plane and line)
(line in plane)0Lπ
0π,L,L 21 (coplanar lines)
Plücker line coordinates
TL 344223141312 ,,,,, llllll 5P
0
000001
000010
000100
001000
010000
100000
34
42
23
14
13
12
344223141312
l
l
l
l
l
l
llllll
B,A,BA,ˆ, LL
LLLLT ˆ|Kˆ
ˆˆˆˆˆˆB,AB,A,det 123413421423231442133412
llllllllllll
0231442133412 llllll on Klein quadric 0K LLT
Plücker line coordinates
0B,AB,A,detˆ| LL
0Q,PQ,P,detˆ| LL
0BPAQBQAPˆ| TTTTLL
0| LL (Plücker internal constraint)
(two lines intersect)
(two lines intersect)
(two lines intersect)
Quadrics and dual quadrics
(Q : 4x4 symmetric matrix)0QXX T
1. 9 d.o.f.
2. in general 9 points define quadric
3. det Q=0 ↔ degenerate quadric
4. pole – polar
5. (plane ∩ quadric)=conic
6. transformation
Q
QXπ QMMC T MxX:π
-1-TQHHQ'
0πQπ * T
-1* QQ 1. relation to quadric (non-degenerate)
2. transformation THHQQ' **
Quadric classification
Rank Sign. Diagonal Equation Realization
4 4 (1,1,1,1) X2+ Y2+ Z2+1=0 No real points
2 (1,1,1,-1) X2+ Y2+ Z2=1 Sphere
0 (1,1,-1,-1) X2+ Y2= Z2+1 Hyperboloid (1S)
3 3 (1,1,1,0) X2+ Y2+ Z2=0 Single point
1 (1,1,-1,0) X2+ Y2= Z2 Cone
2 2 (1,1,0,0) X2+ Y2= 0 Single line
0 (1,-1,0,0) X2= Y2 Two planes
1 1 (1,0,0,0) X2=0 Single plane
Quadric classificationProjectively equivalent to sphere:
Ruled quadrics:
hyperboloids of one sheet
hyperboloid of two sheets
paraboloidsphere ellipsoid
Degenerate ruled quadrics:
cone two planes
Twisted cubic
2333231
2232221
2131211
23
2
1 1
A
aaa
aaa
aaa
x
x
x
conic
344
2434241
334
2333231
324
2232221
314
2131211
3
2
4
3
2
1 1
A
aaaa
aaaa
aaaa
aaaa
x
x
x
x
twisted cubic
1. 3 intersection with plane (in general)
2. 12 dof (15 for A – 3 for reparametrisation (1 θ θ2θ3)
3. 2 constraints per point on cubic, defined by 6 points
4. projectively equivalent to (1 θ θ2θ3)
5. Horopter & degenerate case for reconstruction
Hierarchy of transformations
vTv
tAProjective15dof
Affine12dof
Similarity7dof
Euclidean6dof
Intersection and tangency
Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞
The absolute conic Ω∞
Volume
10
tAT
10
tRT
s
10
tRT
Screw decompositionAny particular translation and rotation is
equivalent to a rotation about a screw axis and a translation along the screw axis.
ttt //
screw axis // rotation axis
The plane at infinity
π
1
0
0
0
1t
0ππ
A
AH
TT
A
The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity
1. canical position2. contains directions 3. two planes are parallel line of intersection in π∞
4. line // line (or plane) point of intersection in π∞
T1,0,0,0π
T0,,,D 321 XXX
The absolute conic
The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity
04
23
22
21
X
XXX
The absolute conic Ω∞ is a (point) conic on π.
In a metric frame:
T321321 ,,I,, XXXXXXor conic for directions:(with no real points)
1. Ω∞ is only fixed as a set2. Circle intersect Ω∞ in two points3. Spheres intersect π∞ in Ω∞
The absolute conic
2211
21
dddd
ddcos
TT
T
2211
21
dddd
ddcos
TT
T
0dd 21 T
Euclidean:
Projective:
(orthogonality=conjugacy)
plane
normal
The absolute dual quadric
00
0I*T
The absolute conic Ω*∞ is a fixed conic under the
projective transformation H iff H is a similarity
1. 8 dof2. plane at infinity π∞ is the nullvector of Ω∞
3. Angles:
2*
21*
1
2*
1
ππππ
ππcos
TT
T
Next classes:Parameter estimation
Direct Linear TransformIterative EstimationMaximum Likelihood Est.Robust Estimation