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Camera: optical system
Y
da=a -a
2
1

2
1
curvature radius
Z
thin lens
small angles:
a1 
Y
1
a2  -
Y
2
Y
incident light beam
lens refraction index: n
deviated beam
q ' '-a 2 sin(q ' '-a 2 )

=n
q '-a 2 sin(q '-a 2 )
q - a1 sin(q - a1 )

=n
q '-a1 sin(q '-a1 )
deviation angle ? Dq = q’’-q
Dq  (n - 1)Y (
1
1

1
2
)
Z
Thin lens rules
a) Y=0  Dq = 0
b) f Dq = Y 
beams through lens center: undeviated
f =
(n - 1)(
1
1
1

1
2
independent of y
)
parallel rays converge onto a focal plane
Y
Dq
f
Hp: Z >> a
rf
the image of a point P belongs to the line (P,O)
P
image plane
O
p
p = image of P = image plane ∩ line(O,P)
interpretation line of p: line(O,p) =
locus of the scene points projecting onto image point p
Projective 2D geometry
Notes based on
di R.Hartley e A.Zisserman “Multiple view geometry”
Projective 2D Geometry
• Points, lines & conics
• Transformations & invariants
• 1D projective geometry and
the Cross-ratio
Homogeneous coordinates
Homogeneous representation of lines
ax  by  c = 0
a,b,cT
(ka) x  (kb) y  kc = 0, k  0
a,b,cT ~ k a,b,cT
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3-(0,0,0)T forms P2
Homogeneous representation of points
T
T
x = x, y  on l = a,b,c if and only if ax  by  c = 0
x,y,1a,b,cT = x,y,1 l = 0
x, y,1T ~ k x, y,1T , k  0
The point x lies on the line l if and only if xTl=lTx=0
Homogeneous coordinates x1, x2 , x3 
T
Inhomogeneous coordinates x, y 
T
but only 2DOF
Points from lines and vice-versa
Intersections of lines
The intersection of two lines l and l' is x = l l'
Line joining two points
The line through two points x and x' is l = x x'
Line joining two points: parametric equation
A point on the line through two points x and x’
is y = x + q x’
Example
y =1
x =1
Ideal points and the line at infinity
Intersections of parallel lines
l = a, b, c and l' = a, b, c'
T
T
Example
l l' = b,-a,0
T
b,-a  tangent vector
a, b  normal direction
x =1 x = 2
Ideal points
Line at infinity
x1, x2 ,0T
T
l = 0,0,1
P 2 = R 2  l
Note that in P2 there is no distinction
between ideal points and others
Example
• The linear combination z = ax + by of two
points x and y is a point z colinear with
them (i.e., on the line through x and y)

• The linear combination n = al + bm of two
lines l and m is a line n concurrent with
them (i.e., through the point on l and m)
Conics
Curve described by 2nd-degree equation in the plane
ax2  bxy  cy2  dx  ey  f = 0
or homogenized x  x1 x , y  x2 x
3
3
ax1  bx1x2  cx2  dx1x3  ex2 x3  fx32 = 0
2
2
or in matrix form
b / 2 d / 2
 a

e / 2 
x T C x = 0 with C =  b / 2 c
d / 2 e / 2
f 
5DOF:
a : b : c : d : e : f 
Five points define a conic
For each point the conic passes through
axi2  bxi yi  cyi2  dxi  eyi  f = 0
or
x , x y , y , x , y , f c = 0
2
i
i
i
2
i
i
i
stacking constraints yields
 x12
 2
 x2
 x32
 2
 x4
x2
 5
x1 y1
y12
x1
x2 y 2
y22
x2
x3 y3
y32
x3
x4 y 4
y42
x4
x5 y5
y52
x5
y1 1

y2 1
y3 1c = 0

y4 1
y5 1
c = a, b, c, d , e, f 
T
Polarity: cross ratio
Cross ratio of 4 colinear points y = x + q x’ (with i=1,..,4)
i
ratio of ratios
θ1 - θ 3
θ1 - θ 4
i
θ 2 - θ3
θ2 - θ4
Harmonic 4-tuple of colinear points: such that CR=-1
Tangent lines to conics
The line l tangent to C at point x on C is given by l=Cx
x
l
C
Dual conics
lT C* l = 0
A line tangent to the conic C satisfies
In general (C full rank):
Line
C* = C-1
in fact
-1
but since yT Cy
l is the polar line of y : y = C l ,
T -T
-1
-T
-1
lC CC l=0 C =C =C
*
Dual conics = line conics = conic envelopes
=0
Degenerate conics
A conic is degenerate if matrix C is not of full rank
m
e.g. two lines (rank 2)
l
C = lmT  mlT
e.g. repeated line (rank 1)
C = llT
l
Degenerate line conics: 2 points (rank 2), double point (rank1)
Note that for degenerate conics
C   C
* *
Projective transformations
Definition:
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.
Theorem:
A mapping h:P2P2 is a projectivity if and only if there
exist a non-singular 3x3 matrix H such that for any point
in P2 represented by a vector x it is true that h(x)=Hx
Definition: Projective transformation
 x'1   h11
  
 x'2  = h21
 x'  h
 3   31
h12
h22
h32
h13  x1 
 

h23  x2 
h33  x3 
or
x' = H x
8DOF
projectivity=collineation=projective transformation=homography
Mapping between planes
central projection may be expressed by x’=Hx
(application of theorem)
Removing projective distortion
select four points in a plane with known coordinates
x' =
x'1 h11 x  h12 y  h13
=
x'3 h31 x  h32 y  h33
y' =
x'2 h21 x  h22 y  h23
=
x'3 h31 x  h32 y  h33
x' h31 x  h32 y  h33  = h11 x  h12 y  h13
(linear in hij)
y' h31 x  h32 y  h33  = h21 x  h22 y  h23
(2 constraints/point, 8DOF  4 points needed)
Remark: no calibration at all necessary,
better ways to compute (see later)
More examples
Transformation of lines and conics
For a point transformation
x' = H x
Transformation for lines
l' = H -T l
Transformation for conics
C' = H-TC H-1
Transformation for dual conics
C'* = HC*HT
A hierarchy of transformations
Projective linear group
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 leave distances unchanged
Class I: Isometries
(iso=same, metric=measure)
 x'   cosq
  
 y '  =   sin q
1  0
  
- sin q
cosq
0
t x  x 
 

t y  y 
1  1 
 = 1
orientation preserving:  = 1
orientation reversing:  = -1
 R t
x' = H E x =  T  x
0 1
RTR = I
3DOF (1 rotation, 2 translation)
special cases: pure rotation, pure translation
Invariants: length, angle, area
Class II: Similarities
(isometry + scale)
 x'   s cosq
  
 y '  =  s sin q
1  0
  
- s sin q
s cosq
0
t x  x 
 

t y  y 
1  1 
sR t 
x'= H S x =  T  x
 0 1
RTR = I
4DOF (1 scale, 1 rotation, 2 translation)
also know as equi-form (shape preserving)
metric structure = structure up to similarity (in literature)
Invariants: ratios of length, angle, ratios of areas,
parallel lines
Class III: Affine transformations
 x'   a11
  
 y '  = a21
1  0
  
a12
a22
0
t x  x 
 

t y  y 
1  1 
 A t
x' = H A x =  T  x
0 1
A = UDVT = (UVT )(VDVT )
A = Rq R-  DR 
where
1
D=
0
0
2 
6DOF (2 scale, 2 rotation, 2 translation)
non-isotropic scaling! (2DOF: scale ratio and orientation)
Invariants: parallel lines, ratios of parallel
segment lengths, ratios of areas
Action of affinities and
projectivities
on line at infinity
A
0 T

 x1    x1  
t     A  
 x2  =   x2  

v   

0
0

  
Line at infinity stays at infinity,
but points move along line
A
vT

 x1  
x  
t     A 1  
x  
 x2  = 

 2
v   

v
x

v
x
0
1
1
2
2


 
Line at infinity becomes finite,
allows to observe vanishing points, horizon,
Class VI: Projective
transformations
A
x'= H P x =  T
v
t
x

v
v = v1, v2 
T
8DOF (2 scale, 2 rotation, 2 translation, 2 line at infinity)
Action: non-homogeneous over the plane
Invariants: cross-ratio of four points on a line
(ratio of ratios)
Projective geometry of 1D
x1, x2 T
x2 = 0
x' = H 22 x
3DOF (2x2-1)
The cross ratio
Crossx1 , x 2 , x 3 , x 4  =
x1 , x 2 x 3 , x 4
x1 , x 3 x 2 , x 4
Invariant under projective transformations
 xi1
x i , x j = det 
 xi 2
x j1 
x j 2 
Overview transformations
Projective
8dof
Affine
6dof
Similarity
4dof
Euclidean
3dof
 h11 h12
h
 21 h22
 h31 h32
 a11 a12
a
 21 a22
 0
0
h13 
h23 
h33 
tx 
t y 
1 
 sr11 sr12 t x 
 sr

sr
t
22
y
 21
 0
0
1 
 r11 r12 t x 
r

r
t
21
22
y


 0 0 1 
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.
Number of invariants?
The number of functional invariants is equal to, or greater than, the
number of degrees of freedom of the configuration less the number of
degrees of freedom of the transformation
e.g. configuration of 4 points in general position has 8 dof (2/pt)
and so 4 similarity, 2 affinity and zero projective invariants
Recovering metric and affine
properties from images
• Parallelism
• Parallel length ratios
• Angles
• Length ratios
The line at infinity
l = H
 A
l =  T -T
- t A
-T
A 
-T
0
0 
 0  = l 
1 
1
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
1 0
H ' P =  0 1
l1 l2
0
0  H A
l3 
(affine) rectification
l' = l1 l2 l3  , l3  0
T
in fact, any point x on l’∞is mapped to a point at the ∞
Affine rectification
l∞
v1
l1
l2
l3
v2
v 2 = l3  l 4
l4
v1 = l1  l2
l = v1  v2
The circular points
1
 
I=i
0
 
 s cosq
I = H S I =  s sin q
 0
1
 
J = -i
0
 
- s sin q
s cosq
0
t x  1 
1


 

iq
t y  i  = se  i  = I
0
1  0 
 
The circular points I, J are fixed points under the
projective transformation H iff H is a similarity
The circular points
“circular points”
l∞
x12  x22  dx1x3  ex2 x3  fx32 = 0
x3 = 0
x12  x22 = 0
I = 1, i,0 
T
Intersection points between any circle and l∞
J = 1,-i,0 
Algebraically, encodes orthogonal directions
I = 1,0,0  i0,1,0
T
T
T
Circular points invariance
•
•
•
•
•
{I,J} = l∞ ∩ any circumference
Similarity: circ’  circ”
Similarity: circ’ ∩ l∞  circ’’ ∩ l∞
Similarity: {I,J}  {I,J}
 circular points: invariant under similarity
Conic dual to the circular points
C* = IJT  JI T
1 0 0


*
C  = 0 1 0 
0 0 0
C*: line conic = set of lines through any of the circular points
C* = HS C*HTS
The dual conic C* is fixed conic under the

projective transformation H iff H is a similarity
Note: C* has 4DOF
l∞ is the null vector
Angles
Euclidean: l = l1 , l2 , l3 
m = m1, m2 , m3 
l1m1  l2 m2
T
cosq =
Projective: cos q =
l
2
1
l
T

 l22 m12  m22

l T C* m
T

C* l m T C* m
lT C* m = 0 (orthogonal)

Metric properties from images
C* ' = H P H A H S C* H P H A H S 
T
= H P H A H S C* H TS H P H A 
T
= H P H A C* H P H A 
T
KK T
= T
v K
K T v
T 
v v
Rectifying transformation from SVD
1 0 0 
C* ' = U 0 1 0 U T
0 0 0
H=U
1 0 0 
Why C* ' = U 0 1 0 U T ?


0 0 0
Normally: SVD (Singular Value Decomposition)
 a 0 0
C* ' = V 0 b 0 U T
0 0 c 
with U and V orthogonal
But C* ' is symmetric  C* 'T = UDVT = VDUT = C* '
and SVD is unique 
Observation : H=U
U=V
orthogonal (3x3): not a
P2
isometry
Metric from affine
Once the image has been affinely rectified
C ' = HAC HA
*

K
*
C ' =  T
0
*

T
1 0 0 T
t
K

0 1 0  T


1
t

0 0 0
0 KK T
= T
1  0
0

0
Metric from affine
l1
l2
KK
l3 
 0
T
 m1 
0 
 m2  = 0
0 
 m3 
l1m1, l1m2  l2 m1, l2 m2 k
2
11
 k , k11k12 , k
2
12

2 T
22
=0
Metric from projective
l1
l2
KK
l3  T
v K
T
 m1 
K v  
m = 0
T  2 
v v  
 m3 
T
l1m1,0.5l1m2  l2 m1 , l2 m2 ,0.5l1m3  l3m1 ,0.5l2 m3  l3m2 , l3m3 c = 0
Fixed points and lines
He = e
H
-T
(eigenvectors H =fixed points)
(1=2  pointwise fixed line)
l =  l (eigenvectors H-T =fixed lines)
Projective 3D geometry
Singular Value Decomposition
Amn = UmmΣmn VnTn
 1 0  0 
0   0
2



  
Σ=

0
0


n






 0 0  0 
mn
1   2     n  0
UT U = I
VT V = I
A = U1 1 V1T  U2  2 V2T   Un  n VnT
UΣ
Σ VT X
Singular Value Decomposition
A = UΣ V
T
• Homogeneous least-squares
min AX subject to X = 1
• Span and null-space
S L = U1 U2 ; N L = U3U4 
S R = V1V2 ; N R = V3V4 
solution X = Vn
 1 0
0 
2
Σ=
0 0

0 0
0
0
0
0
0
0
0

0
• Closest rank r approximation
~
~ T ~
A = UΣ
UΣ V  = diag1, 2 ,, r , 0 ,, 0 
• Pseudo inverse
A  = VΣ  U T  = diag1-1, 2-1,, r-1, 0 ,, 0 
Projective 3D Geometry
• Points, lines, planes and quadrics
• Transformations
• П∞, ω∞ and Ω ∞
3D points
3D point
 X , Y , Z T in R3
T
X =  X1, X 2 , X 3 , X 4 
in P3
T
 X1 X 2 X 3 
T
X = 
,
,
,1 =  X , Y , Z , 1
 X4 X4 X4 
projective transformation
X' = H X (4x4-1=15 dof)
 X 4  0
Planes
3D plane
Transformation
X' = H X
π'= H -T π
π1 X  π2Y  π3Z  π4 = 0
π1 X1  π2 X 2  π3 X 3  π4 X 4 = 0
πT X = 0
Euclidean representation
~
n . X d = 0
n = π1, π2 , π3 
π4 = d
T
~
T
X = X ,Y , Z 
X4 =1
d/ n
Dual: points ↔ planes, lines ↔ lines
Planes from points
Solveπ from X1T π = 0, XT2 π = 0 andX3T π = 0
X1T 
 T
X 2  π = 0
X 3T 
 
 X 1T 
 
(solve π as right nullspace of  X T2  )
 X 3T 
 
Or implicitly from coplanarity condition
 X 1  X 1 1  X 2 1
 X X  X 
2
1 2
2 2
det 
detX X1X2X3  = 0
 X 3  X 1 3  X 2 3

 X 4  X 1 4  X 2 4
 X 3 1 
 X 3 2 
=0
 X 3 3 
 X 3 4 
X1D234 - X 2 D134  X 3 D124 - X 4 D123 = 0
T


π = D234 ,-D134 , D124 ,-D123
Points from planes
SolveX from π1T X = 0, πT2 X = 0 and π3T X = 0
 π1T 
 T
π 2  X = 0
 π 3T 
 
(solve Xas right nullspace of
 π1T 
 T)
π 2 
 π 3T 
 
Representing a plane by its span
X=Mx
M = X1X2X3 
πT M = 0
p 
M= 
I
π = a, b, c, d 
T
 b c d
p =  - ,- ,- 
 a a a
T
Quadrics and dual quadrics
XT QX = 0
1.
2.
3.
4.
5.
6.
(Q : 4x4 symmetric matrix)


Q=



9 d.o.f.
in general 9 points define quadric
det Q=0 ↔ degenerate quadric
pole – polar π = QX
(plane ∩ quadric)=conic C = MT QM
transformation Q'= H-TQH-1













π : X = Mx
π T Q*π = 0
1. relation to quadric Q = Q
(non-degenerate)
*
* T
2. transformation Q' = HQ H
*
-1
Quadric classification
Rank
Sign.
Diagonal
Equation
4
4
(1,1,1,1)
X2+ Y2+ Z2+1=0
2
(1,1,1,-1)
X2+ Y2+ Z2=1
Sphere
0
(1,1,-1,-1)
X2+ Y2= Z2+1
Hyperboloid (1S)
3
(1,1,1,0)
X2+ Y2+ Z2=0
Single point
1
(1,1,-1,0)
X 2 + Y 2 = Z2
Cone
2
(1,1,0,0)
X2 + Y2 = 0
Single line
0
(1,-1,0,0)
X 2 = Y2
Two planes
1
(1,0,0,0)
X2=0
Single plane
3
2
1
Realization
No real points
Quadric
classification
Projectively equivalent to sphere:
sphere
ellipsoid
Ruled quadrics:
hyperboloid paraboloid
of two sheets
hyperboloids
of one sheet
Degenerate ruled quadrics:
cone
two planes
Hierarchy of transformations
Projective
15dof
A
vT

t
v 
Affine
12dof
 A t
0T 1


Similarity
7dof
s R t 
 0T 1


The absolute conic Ω∞
Euclidean
6dof
 R t
0T 1


Volume
Intersection and tangency
Parallellism of planes,
Volume ratios, centroids,
The plane at infinity π∞
Screw decomposition
Any particular translation and rotation is
equivalent to a rotation about a screw axis
and a translation along the screw axis.
screw axis // rotation axis
t = t //  t 
The plane at infinity
 0
 
-T
A
0  0 
-T

π = H A π = 
  = π 
- A t 1 0 
1
 
The plane at infinity π is a fixed plane under a
projective transformation H iff H is an affinity
1.
2.
3.
4.
canical position π = 0,0,0,1
contains directions D =  X1 , X 2 , X 3 ,0T
two planes are parallel  line of intersection in π∞
line // line (or plane)  point of intersection in π∞
T
The absolute conic
The absolute conic Ω∞ is a (point) conic on π.
In a metric frame:
2
2
2
X1  X 2  X 3 
=0
X4

or conic for directions:
(with no real points)
X1, X 2 , X 3 IX1, X 2 , X 3 T
The absolute conic Ω∞ is a fixed conic under the
projective transformation H iff H is a similarity
1. Ω∞ is only fixed as a set
2. Circle intersect Ω∞ in two points
3. Spheres intersect π∞ in Ω∞
Absolute conic invariance
•
•
•
•
•
Ω∞ = π ∩ any sphere
Similarity: sphere’  sphere”
Similarity: sphere’ ∩ π  sphere’’ ∩ π
Similarity:
Ω∞  Ω∞
 Ω∞ : invariant under similarity
The absolute conic
Euclidean:
Projective:
cos q =
cos q =
d d 
d d d d 
d  d 
d  d d  d 
T
1
2
T
1 1
T
1
T
1
d1T d 2 = 0
 1
T
2
2

2
T
2

2
(orthogonality=conjugacy)
normal
plane
The absolute dual quadric
I
 = T
0
*

0
0
The absolute conic Ω*∞ is a fixed conic under the
projective transformation H iff H is a similarity
1. 8 dof
2. plane at infinity π∞ is the nullvector of Ω∞
3. Angles:
π T * π
cos q =

1
2
π  π π  π 
T
1
*

1
T
2
*

2
P
Y
X
c
f
O
Z
x
p
y
X
x= f
Z
Y
y= f
Z
perspective projection
-nonlinear
-not shape-preserving
-not length-ratio preserving
Homogeneous coordinates
• In 2D: add a third coordinate, w
•Point [x,y]T expanded to [u,v,w]T
•Any two sets of points [u1,v1,w1]T and [u2,v2,w2]T
represent the same point if one is multiple of the other
•[u,v,w]T  [x,y] with x=u/w, and y=v/w
•[u,v,0]T is the point at the infinite along direction (u,v)
Transformations
translation by vector [dx,dy]T
scaling (by different factors in x and y)
rotation by angle q
Homogeneous coordinates
In 3D: add a fourth coordinate, t
•Point [X,Y,Z]T expanded to [x,y,z,t]T
•Any two sets of points [x1,y1,z1,t1]T and [x2,y2,z2,t2]T
represent the same point if one is multiple of the other
•[x,y,z,t]T  [X,Y,Z] with X=x/t, Y=y/t, and Z=z/t
•[x,y,z,0]T is the point at the infinite along direction
(x,y,z)
Transformations
translation
scaling
rotation
Obs: rotation matrix is an orthogonal matrix
i.e.: R-1 = RT
CAMERA GEOMETRY
colinearity is preserved  linear relation among homogeneous coords
X
X
 
 
u
 
Y 
Y 
X =    u =  v  = P3x4   = P3x4 X = M 3x3
Z
Z


 
 
 w
1
1
 
 
invertible
m 3x1 X
SCENE: viewing ray from image point
null-space of camera projection matrix O
a point Y on the line X, O
its image
PO = 0
Y = aX  bO
u = PY = aPX  bPO = PX
For all points Y on (X,O) project on image of X,
 O is camera center
Image of camera center is (0,0,0)T, i.e. undefined
Finite cameras:
 o   - M -1m 

O =   = 

 1  1 
Viewing ray associated to image point u:
image point u is image of X if X is on a certain line through O
viewing ray associated to u
Consider the point at the infinity along this line
u= M m
d
= M d
0
The locus of the points x whose image is u
is a straight line through o having direction
X=
d
0
d = M-1  u
o is the camera viewpoint (perspective projection center)
line(o, d) = viewing ray associated to image point u
CAMERA
dcam = Rcam-world dworld
Ycam
x
c
y
1
d
Xcam
O
Zcam
U
u
V
geometric coordinates
U
X
V = Y  d cam
1
-1
Ycam
x
c
y
1
Xcam
O
Zcam
“1” is equal to fx horizontal pixels
“1” is equal to fy vertical pixels
U
u
d
V
pixel coordinates
u
x
u v  y
w
1
principal point (cartesian pixel coordinates) c =
Uo
Vo
x = u / w = U o  f xU
fx
=
y = v / w = Vo  f yV
0
u
fx
u v  0
w
0
with K =
0 Uo
f y Vo
U
fx
V =
0
1
0 Uo
d cam
f y Vo
0 Uo
f y Vo d cam = Kd cam = KR cam- world d world
0 1
fx
0
Uo
0
fy
Vo
0
0
1
M = KR cam-world
and, from dworld = M-1  u
given M, find K and R by Q-R
matrix decomposition of inv(M)
Radial distortion
Exterior orientation
Calibrated camera, position and orientation unkown
 Pose estimation
6 dof  3 points minimal (4 solutions in general)
Properties of perspective transformations
1) vanishing points
V image of the point at the ∞ along direction d
d
uV = M m 
= M d
0
d = M-1  uV
the interpretation line of V is parallel to d
P
d
O
V
The images of parallel lines are concurrent lines
Properties of perspective transformations ctd.
2) cross ratio invariance
Given four colinear points p1 , p2 , p3 , p4 
let
x1, x2 , x3 , x4 
be their abscissae
x1 - x3
x -x
CRp1 , p 2 , p3 , p 4  = 1 4
x2 - x3
x2 - x4
Cross ratio invariance under perspective transformation
X = [ x, y, z, t ]T = [ x,0,0, t ]T
a point on the line y=0=z
its image
u = [u, v, w]T = P  X
its coordinate u
u = [u, w] =
CRu1 , u2 , u3 , u4  =
det u1 , u2 det u3 , u4
=
det x1 , x 2 det x3 , x 4
det u1 , u3 det u2 , u4
det x1 , x3 det x 2 , x 4
T
=
p1
p4
belongs to a line

x
t
= P14  x
det P14 det x1 , x 2 det P14 det x3 , x 4
det P14 det x1 , x 3 det P14 det x 2 , x 4
= CRx1 , x 2 , x3 , x 4 
What can be told from a single image?
Action of projective camera on planes
X 
X 
Y


x = PX = p1p 2 p3p 4 
= p1p 2 p 4  Y 
0
 1 
 1 
The most general transformation that can occur between
a scene plane and an image plane under perspective
imaging is a plane projective transformation
Action of projective camera on lines
forward projection
Xμ  = P(A μB) = PA  μPB = a  μb
back-projection
l x=0
T
x = PX
l PX = 0 =  X
T
T
with
 = PTl
Interpretation plane of line l
Image of a conic
X 
X 
Y 
x = PX = p1p 2 p3p 4   = p1p 2 p 4  Y  = p1p 2 p 4 x = Px
0
 1 
 
T 
x T Cx = 0 = x T P-TCP -1x
therefore
C' = P -T CP -1
Action of projective camera on conics
back-projection of a conic C to cone
C
Q co
Q co
back-projection of a conic C to cone
x Cx = 0
Q co
T
x = PX
x TCx = XT PTCPX = 0 = XTQcoX
with
Interpretation cone of a conic C
Qco = PTCP
example:
T
T




T
K
K
CK
0
Q co =   C K | 0 = 
0
0 
 0
Images of smooth surfaces
The contour generator G is the set of points X on S at
which rays are tangent to the surface. The corresponding
apparent contour g is the set of points x which are the
image of X, i.e. g is the image of G
The contour generator G depends only on position of
projection center, g depends also on rest of P
Action of projective camera on quadrics
apparent contour of a quadric Q
dual quadric
Q =Q
*
-1
the set of planes tangent to Q
is a plane quadric:
T Q* = 0
Let us consider only those planes that are backprojection of image lines
TQ* = lT PQ*PT l = 0
 = PTl
with
C = PQ P
*
* T
its dual is
*-1
C=C
The plane containing the apparent contour G of a quadric Q
from a camera center O follows from pole-polar relationship
=QO
The cone with vertex V and tangent to the quadric Q is
QCO = (VTQV)Q - (QV)(QV)T
back-projection to cone
QCO V = 0
What does calibration give?
x = K[I | 0]d 
0 
d = K -1x
cosq =
T
T
d1 d 2
d d d
T
1
1
T
2
d2

=
x
x1 (K-T K -1 )x2
T
1

T
(K-T K -1 )x1 x 2 (K-T K -1 )x2

An image line l defines a plane through the camera center
with normal n=KTl measured in the camera’s Euclidean
frame. In fact the backprojection of l is PTl
 n=KTl
The image of the absolute conic  
d
x = PX = KR[I | -O]  = KRd
0
mapping between p∞ to an image is given by the planar
homogaphy x=Hd, with H=KR
absolute conic (IAC), represented by I3 within
its image (IAC)


T -1
ω = KK
= K -T K -1
p∞ (w=0)
C  H
(i) IAC depends only on intrinsics
(ii) angle between two rays cosq =
(iii) DIAC=w*=KKT
(iv) w  K (Cholesky factorization)
(v) image of circular points belong
to w (image of absolute conic)
-T
CH -1

T
x
x1 ωx2
T
1

T
ωx1 x 2 ωx2

A simple calibration device
(i)
compute Hi for each square
(corners  (0,0),(1,0),(0,1),(1,1))
(ii) compute the imaged circular points Hi
(iii) fit a conic w to 6 imaged circular points
[1,±i,0]T
(iv) compute K from w=K-T K-1 through Cholesky
factorization
(= Zhang’s calibration method)
Orthogonality relation
cosq =
T
v
1
v1 ωv2
T

ωv1 v 2 ωv2
v ωv2 = 0
T
1
l ω*l2 = 0
T
1
T

Calibration from vanishing points and
lines
Calibration from vanishing points and
lines