1 Projective 3D Geometry 1 2008-1 Multi View Geometry (Spring '08) Prof. Kyoung Mu Lee SoEECS, Seoul National University Projective 3D Geometry Projective 3D Geometry 2 Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU Points, Planes, Lines and Quadrics in 3D • 3D points: • Projective transform in : Collinear Lines are mapped to lines 15 (4x4-1) DOF 0 , ) 1 , , , ( ) , , , ( 4 4 3 4 2 4 1 3 4 3 2 1 ≠ ⇔ = X X X X X X X in X X X X T T P X 3 4 3 4 2 4 1 ) , , ( R in X X X X X X T X H X 4 4× = ′ Homogeneous representation Inhomogeneous representation 3 P
12
Embed
Multi View Geometry (Spring '08)ocw.snu.ac.kr/sites/default/files/NOTE/2765.pdf · 2018-01-30 · 1 Projective 3D Geometry 1 2008-1 Multi View Geometry (Spring '08) Prof. Kyoung Mu
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Projective 3D Geometry 1
2008-1
Multi View Geometry (Spring '08)
Prof. Kyoung Mu LeeSoEECS, Seoul National University
Projective 3D Geometry
Projective 3D Geometry 2
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
two lines are intersect iff the 4 points are coplanar
Projective 3D Geometry 14
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
Lines - Plücker line coordinates representation
• Results:
( ) [ ] 0ˆˆdetˆ == B,AA,B,|LL
( ) [ ] 0ˆˆdetˆ == Q,PP,Q,|LL
( ) ( )( ) ( )( ) 0ˆ =−= BPAQBQAP| TTTTLL
( ) 0=LL|
i) Plücker internal constraint:
ii) two lines intersect or coplanar:
4 points
4 planes
2 planes and 2 points
8
Projective 3D Geometry 15
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
Quadrics and dual quadrics
• Quadric:
• Dual quadric:
i) Q is a 4x4 symmetric matrixii) 9 DOFiii) in general 9 points define quadriciv) det Q=0 ↔ degenerate quadricv) pole – polar vi) (plane ∩ quadric)=conicvii) transformation
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
Classification of quadrics
•• D represents Q up to projective equivalence.• Signature:
UDUQ ~T= DHHQ T=Scale normalization of D
)()( DQ σσ = Independent of H
Single planeX2=0(1,0,0,0)11
Two planesX2= Y2(1,-1,0,0)0
Single lineX2+ Y2= 0(1,1,0,0)22
Cone X2+ Y2= Z2(1,1,-1,0)1
Single pointX2+ Y2+ Z2=0(1,1,1,0)33
Hyperboloid (1S)X2+ Y2= Z2+1(1,1,-1,-1)0
SphereX2+ Y2+ Z2=1(1,1,1,-1)2
No real pointsX2+ Y2+ Z2+1=0(1,1,1,1)44
RealizationEquationDiagonalSign.Rank
9
Projective 3D Geometry 17
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
Classification of quadrics•
•
•
Nun-ruled quadrics: projectively equivalent to sphere:
Ruled quadrics: contains straight lines
hyperboloids of one sheet
hyperboloid of two sheets
paraboloidsphere ellipsoid
Degenerate ruled quadrics:
cone two planes
Projective 3D Geometry 18
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
Twisted cubics
• A conic in Π2:
• A twisted cubic inΠ3:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
++++++
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
2333231
2232221
2131211
23
2
1 1A
θθθθθθ
θθ
aaaaaaaaa
xxx
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
++++++++++++
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
344
2434241
334
2333231
324
2232221
314
2131211
3
2
4
3
2
1 1
A
θθθθθθθθθθθθ
θθθ
aaaaaaaaaaaaaaaa
xxxx
10
Projective 3D Geometry 19
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
The hierarchy of transformations
⎥⎦
⎤⎢⎣
⎡vTvtAProjective
15dof
Affine12dof
Similarity7dof
Euclidean6dof
Intersection and tangency
Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞
The absolute conic Ω∞
Volume
⎥⎦
⎤⎢⎣
⎡10tA
T
⎥⎦
⎤⎢⎣
⎡10tR
T
s
⎥⎦
⎤⎢⎣
⎡10tR
T
Group Matrix Distortion Invariant properties
Projective 3D Geometry 20
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
The plane at infinity
• The plane at infinity:• Contains directions (points)• two planes are parallel ⇔ line of intersection in • line // line (or plane) ⇔ point of intersection in
• The plane at infinity is a fixed plane under a projective transformation H iff H is an affinity
TXXX )0,,,( 321=D
T)1,0,0,0(=∞π
∞
−
∞−
∞ =
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡
−==′ π
1000
1t0
ππA
AH
TT
A
∞π
∞π
∞π
11
Projective 3D Geometry 21
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
The absolute conic
• the absolute conic Ω∞ is defined on π∞ s.t.
• Thus, a conic with C = I3x3
• The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity
04
23
22
21 =
⎭⎬⎫++
XXXX ( ) ( )T321321 ,,I,, XXXXXX
i) Ω∞ is only fixed as a setii) Circle intersect Ω∞ in two pointsiii) Spheres intersect π∞ in Ω∞
Projective 3D Geometry 22
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
Metric properties
• Once is identified in projective 3-space, angles and relative lengths can be measured.
• orthogonal (conjugacy)
∞Ω
Invariant to projective transform
d1, d2 : directions of two lines
(Intersection points of lines on π∞)
021 =Ω∞dd T
( )( )( )2211
21cosdddd
ddTT
T
=θ
( )( )( )2211
21cosdddd
dd
∞∞
∞
ΩΩ
Ω=
TT
T
θ
Euclidean:
Projective:
plane
normal
12
Projective 3D Geometry 23
Multi View Geometry (Spring '08) K. M. Lee, EECS, SNU
The absolute dual quadric
• The Absolute dual quadric:
• The absolute dual quadric Q*∞ is a fixed quadric under the
projective transformation H iff H is a similarity
⎥⎦
⎤⎢⎣
⎡=∞ 00
0I*TQ
i) 8 DOFii) plane at infinity is the null-vector of Q∞