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Projective Geometry and Camera model
Class 2
points, lines, planesconics and quadrics
transformationscamera model
Read tutorial chapter 2 and 3.1http://www.cs.unc.edu/~marc/tutorial/
Chapter 1, 2 and 5 in Hartley and Zisserman
Homogeneous coordinates
0=++ cbyax ( ) ( ) 0=1x,y,a,b,c T
( ) ( ) 0≠∀,~ ka,b,cka,b,c TT
Homogeneous representation of 2D points and lines
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3(0,0,0)T forms P2
( ) ( ) 0≠∀,1,,~1,, kyxkyx TT
The point x lies on the line l if and only if
Homogeneous coordinates
Inhomogeneous coordinates ( )Tyx,
( )T321 ,, xxx but only 2DOF
Note that scale is unimportant for incidence relation
0=xlT
Points from lines and vice-versa
l'×l=x
Intersections of lines
The intersection of two lines and is l l'
Line joining two points
The line through two points and is x'×x=lx x'
Example
1=x
1=y
1
)1-,1,0( y
x
1
)1-,0,1( y
x
Note:[ ] 'xx='x×x ×
with
0-
0-
-0
x
xy
xz
yz
Ideal points and the line at infinity
T0,,l'l ab
Intersections of parallel lines
( ) ( )TTand ',,=l' ,,=l cbacba
Example
1=x 2=x
Ideal points ( )T0,, 21 xx
Line at infinity ( )T1,0,0=l∞
∞22 l∪= RP
tangent vector
normal direction
-ab,( )ba,
Note that in P2 there is no distinction
between ideal points and others
• Line at infinity is a collection of all the points at infinity as x1/x2 varies, i.e. all Possible directions
3D points and planes
Homogeneous representation of 3D points and planes
The point X lies on the plane π if and only if
0=π+π+π+π 44332211 XXXX
0=XπT
The plane π goes through the point X if and only if
0=XπT
Planes from points
0π
X
X
X
3
2
1
T
T
T
0=πX 0=πX 0,=πX π 321TTT andfromSolve
(solve as right nullspace of )π
T
T
T
3
2
1
X
X
X
Points from planes
0X
π
π
π
3
2
1
T
T
T
x=X M 321 XXXM
0π MT
0=Xπ 0=Xπ 0,=Xπ X 321TTT andfromSolve
(solve as right nullspace of )X
T
T
T
3
2
1
π
π
π
Representing a plane by its span
• M is 4x3 matrix. Columns of M are null space of Tπ
Lines
T
T
B
AW μBλA
2×2** 0=WW=WWTT
0001
1000W
0010
0100W*
Example: X-axis
(4dof)
Representing a line by its span: two vectors A, B for two space points
T
T
Q
PW* μQλP
Dual representation: P and Q are planes; line is span of row space of
(Alternative: Plücker representation, details see e.g. H&Z)
• Line is either joint of two points or intersection of two planes
*W
Points, lines and planes
TX
WM 0π M
Tπ
W*
M 0X M
W
X
*W
π
• Plane defined by join of point X and line W is the null space of M
π
• Point X defined by the intersection of line W with plane is the null space of Mπ
Conics
Curve described by 2nd-degree equation in the plane
0=+++++ 22 feydxcybxyax
0=+++++ 233231
2221
21 fxxexxdxcxxbxax
3
2
3
1 , xxyx
xx or homogenized
0=xx CT
or in matrix form
fed
ecb
dba
2/2/
2/2/
2/2/
Cwith
{ }fedcba :::::5DOF:
Five points define a conic
For each point the conic passes through
0=+++++ 22 feydxcyybxax iiiiii
or
( ) 0=.1,,,,, 22 ciiiiii yxyyxx ( )Tfedcba ,,,,,=c
0
1
1
1
1
1
552555
25
442444
24
332333
23
222222
22
112111
21
c
yxyyxx
yxyyxx
yxyyxx
yxyyxx
yxyyxx
stacking constraints yields
Conic c is a null vector of the above matrix
Dual conics
0=ll *CTA line tangent to the conic C satisfies
Dual conics = line conics = conic envelopes
1-* CC In general (C full rank):
Points lie on a point lines are tangent conic to the point conic C; conic C is the envelope of lines l
0CxxT
0* lClT
Degenerate conics
A conic is degenerate if matrix C is not of full rank
TT ml+lm=C
e.g. two lines (rank 2)
e.g. repeated line (rank 1)
Tll=C
l
l
m
Degenerate line conics: 2 points (rank 2), double point (rank1)
( ) CC ≠**Note that for degenerate conics
Quadrics and dual quadrics
(Q : 4x4 symmetric matrix)0=QXXT
• 9 d.o.f.• in general 9 points define quadric • det Q=0 ↔ degenerate quadric• tangent plane
Q
QX=π
0=πQπ *T
-1* Q=Q• relation to quadric (non-degenerate)
• Dual Quadric: defines equation on planes
2D projective transformations
A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3 lie on the same line if and
only if h(x1),h(x2),h(x3) do.
Definition:
A mapping h:P2P2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2 reprented by a vector x it is true that h(x)=Hx
Theorem:
Definition: Projective transformation
3
2
1
333231
232221
131211
3
2
1
'
'
'
x
x
x
hhh
hhh
hhh
x
x
x
x=x' Hor
8DOF
projectivity=collineation=projective transformation=homography
Transformation of 2D points, lines and conics
Transformation for lines
l=l' -TH
Transformation for conics-1-TCHHC ='
Transformation for dual conicsTHHCC ** ='
x=x' HFor a point transformation
Fixed points and lines
e=e λH (eigenvectors H =fixed points)
l=l λTH (eigenvectors H-T =fixed lines)
(1=2 pointwise fixed line)
Hierarchy of 2D transformations
1002221
1211
y
x
taa
taa
1002221
1211
y
x
tsrsr
tsrsr
333231
232221
131211
hhh
hhh
hhh
1002221
1211
y
x
trr
trr
Projective8dof
Affine6dof
Similarity4dof
Euclidean3dof
Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio
Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids). The line at infinity l∞
Ratios of lengths, angles.The circular points I,J
lengths, areas.
invariantstransformed squares
Projective geometry of 1D
The cross ratio
Is invariant under projective transformations in P^1.
4231
43214321 x,xx,x
x,xx,xx,x,x,x Cross
22
11detx,x
ji
ji
ji xx
xx
Four sets of four collinear points; each set is related to the others by a line projectivitySince cross ratio is an invariant under projectivity, the cross ratio has the same value for all the sets shown
Concurrent Lines
Four concurrent lines l_i intersect the line l in the four points x_i; The cross ratio of these points is an invariant to the projective transformation of the plane
Coplanar points x_i are imaged onto a line by a projection with center C. The cross ratioof the image points x_i is invariant to the position of the image line l
A hierarchy of transformations
• Euclidean transformations (rotation and translation) leave distances unchanged
• Similarity: circle imaged as circle; square as square; parallel or perpendicular lines have same relative orientation
• Affine: circle becomes ellipse; orthogonal world lines not imaged as orthogonoal; But, parallel lines in the square remain parallel
• Projective: parallel world lines imaged as converging lines; tiles closer to camera larger image than those further away.
Similarity Affine projective
The line at infinity
00
l l 0 lt 1
1A
AH
A
TT
T T
The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity
Note: not fixed pointwise
Affine properties from images
projection rectification
APA
lll
HH
321
010
001
0≠,l 3321∞ llll T
Two step process:1.Find l the image of line at infinity in plane 22. Plug into H_pa
Image of line at infinity in plane
The circular points
0
1
I i
0
1
J i
I
0
1
0
1
100
cossin
sincos
II
iseitss
tssi
y
x
S
H
The circular points I, J are fixed points under the projective transformation H iff H is a similarity
Two points on l_inf which are fixed under any similarity transformation
Canonical coordinates of circular points
The circular points
“circular points”
0=++++ 233231
22
21 fxxexxdxxx 0=+ 2
221 xx
l∞
T
T
0,-,1J
0,,1I
i
i
( ) ( )TT 0,1,0+0,0,1=I i
Algebraically, encodes orthogonal directions
03 x
Two points on l_inf: Every circle intersects l_inf at circular points
Line at infinity
Circle:
Conic dual to the circular points
TT JIIJ*∞ C
000
010
001*∞C
TSS HCHC *
∞*∞
The dual conic is fixed conic under the projective transformation H iff H is a similarity
*∞C
Note: has 4DOF (3x3 ho mogeneous; symmetric, determinant is zero)
l∞ is the nullvector
*∞C
l∞
I
J
Angles
( )( )22
21
22
21
2211
++
+=cos
mmll
mlmlθ
( )T321 ,,=l lll ( )T
321 ,,=m mmmEuclidean:
Projective: mmll
mlcos
**
*
CC
CTT
T
0ml * CT (l and m orthogonal)
Once is identified in the projective plane, then Euclidean angles may be measured by equation (1)
(1)
*C
Transformation of 3D points, planes and quadrics
Transformation for lines
( )l=l' -TH
Transformation for conics
( )-1-TCHHC ='
Transformation for dual conics
( )THHCC ** ='
( )x=x' H
For a point transformation
X=X' H
π=π' -TH
-1-TQHH=Q'
THHQ=Q' **
(cfr. 2D equivalent)
Hierarchy of 3D transformations
vTv
tAProjective15dof
Affine12dof
Similarity7dof
Euclidean6dof
Intersection and tangency
Parallellism of planes,Volume ratios, centroids,The plane at infinity π∞
Angles, ratios of length
The absolute conic Ω∞
Volume
10
tAT
10
tRT
s
10
tRT
The plane at infinity0
00π π π
0- t 1
1
A
AH
A
TT
T T
The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity
1. canonical position2. contains directions 3. two planes are parallel line of intersection in π∞
4. line // line (or plane) point of intersection in π∞
( )T1,0,0,0=π∞
( )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 circular points3. Spheres intersect π∞ in Ω∞
The absolute dual quadric
00
0I*∞ T
The absolute dual quadric Ω*∞ is a fixed conic under
the projective transformation H iff H is a similarity
1. 8 dof ( symmetric matrix, det is zero)2. plane at infinity π∞ is the nullvector of Ω∞
3. Angles:
( )( )2
*∞21
*∞1
2*∞1
πΩππΩπ
πΩπ=cos
TT
T
θ
Set of planes tangent to an ellipsoid, then squash to a pancake
Pinhole camera model
TT ZfYZfXZYX )/,/(),,(
101
0
0
1
Z
Y
X
f
f
Z
fY
fX
Z
Y
X
linear projection in homogeneous coordinates!
Principal point offset
Tyx
T pZfYpZfXZYX )+/,+/(),,(
principal pointT
yx pp ),(
101
0
0
1
Z
Y
X
pf
pf
Z
ZpfY
ZpfX
Z
Y
X
y
x
x
x
Principal point offset
101
0
0
Z
Y
X
pf
pf
Z
ZpfY
ZpfX
y
x
x
x
camX0|IKx
1y
x
pf
pf
K calibration matrix
Camera rotation and translation
( )C~
-X~
R=X~
cam
X10
RC-R
1
10
C~
R-RXcam
Z
Y
X
camX0|IKx XC~
-|IKRx
t|RKP C~
R-t PX=x
~
General projective camera
1yx
xx
p
ps
K
1yx
xx
p
p
K
C~
|IKRP
non-singular
11 dof (5+3+3)
t|RKP
intrinsic camera parametersextrinsic camera parameters
Radial distortion
• Due to spherical lenses (cheap)• Model:
R
y
xyxKyxKyx ...))()(1(),( 44
222
1R
http://foto.hut.fi/opetus/260/luennot/11/atkinson_6-11_radial_distortion_zoom_lenses.jpgstraight lines are not straight anymore
Projector model
Relation between pixels and rays in space
(dual of camera)
(main geometric difference is vertical principal point offset
to reduce keystone effect)
?
Action of projective camera on points and lines
forward projection of line
( ) μb+a=μPB+PA=μB)+P(A=μX
back-projection of line
lP=Π T
PXlX TT ( )PX= x0;=xlT
PX=x
projection of point
Action of projective camera on conics and quadricsback-projection to cone
CPP=Q Tco 0=CPXPX=Cxx TTT
( )PX=x
projection of quadric
TPPQC ** 0lPPQlQ T*T*T
( )lP=Π T
A simple calibration device
(i) compute H for each square (corners @ (0,0),(1,0),(0,1),(1,1))
(ii) compute the imaged circular points H(1,±i,0)T
(iii) fit a conic to 6 circular points(iv) compute K from w through cholesky factorization
(≈ Zhang’s calibration method)
A hierarchy of transformations• Group of invertible nxn matrices with real elements general linear group on n
dimensionsGL(n);• Projective linear group: matrices related by a scalar multiplier PL(n); three
subgroups:• Affine group (last row (0,0,1))• Euclidean group (upper left 2x2 orthogonal)• Oriented Euclidean group (upper left 2x2 det 1)
• Alternative, characterize transformation in terms of elements or quantities that are preserved or invariant
• e.g. Euclidean transformations (rotation and translation) leave distances unchanged
• Similarity: circle imaged as circle; square as square; parallel or perpendicular lines have same relative orientation
• Affine: circle becomes ellipse; orthogonal world lines not imaged as orthogonoal; But, parallel lines in the square remain parallel
• Projective: parallel world lines imaged as converging lines; tiles closer to camera larger image than those further away.
Similarity Affine projective