Geometry of two cameras
G.Georgiev1, N.Danailova2, V.Radulov3
Key words: projective coordinates, projection matrix.
Scientific area: mathematics (projective geometry)
ABSTRACT
The investigation of image formation and object modeling is widely
developed in the last thirty years. The formation of planar images of our three
dimensional world plays an important role in our century of communications through
computers. The geometry of multiple images provides us the description of the
geometric laws for creation of different views of a scene. The inverse problem of
reconstruction of the object by its images is studied also for the needs of
photogrametry.
The central projection is may be the most used projection because of its
closeness to the human perceptions. Although the laws of the central projection and
perspective are known for many years, the mathematicians investigate them now in
the framework of the projective, affine and Euclidian geometry to obtain more
efficient and elegant results.
In this context we present here the main law for central projection in
projective coordinate system and applications. The system of two cameras is studied
also, which is the subject of the so called epipolar geometry. Some applications are
given in the context of the projective and Euclidean geometry from algebraic
viewpoint.
1. Introduction
The planar image creation of the three dimensional world is investigated
lately in the framework of the projective, affine or Euclidian geometry (see [1], [2]
and [3]). Some correspondences can be established between two or three and more
images, which properties can be applied.
We introduce projective coordinate space system as well as projective
homogeneous and non-homogeneous coordinates, which are the most common case
of coordinates. The projective coordinates have a nice property - they do not change
1
G.Georgiev, Asso.Prof., PhD,Shumen University, partially supported under grant
RD-05-289/11.03.2009, g.georgiev@fmi.shu-bg.net
2
N.Danailova, Asso. Prof., PhD, UACEG-Sofia, dichevan@yahoo.com
3
V.Radulov, Assis. Prof., UACEG-Sofia, vradulov@yahoo.com
after central projection. The affine and Euclidian case will be considered as particular
cases.
We define a well-known cross ratio of 4 points on a line as follows:
 ABCD  
AC AD  ABC 
:

BC BD  ABD 
Let five points be fixed in the space (Fig.1), called basic points, so that no
four of them lie on a plane. We call U, V, W – excluded points, О – a zero point and
Е – a unit point. Let denote the intersection points by EU  OU  (VEW ) ,
EV  OV  (UEW ) and EW  OW  (UEV ). If А is an arbitrary point, not
lying in the plane (UVW), then we can get by analogy points A U, AV, AW. The
numbers U A  (UOEU AU ), V A  (VOEV AV ), W A  (WOE W AW ) are called
non-homogeneous projective coordinates of the point А. We compare to the basic
points the following coordinates; U(∞; 0; 0), V(0; ∞; 0), W(0; 0; ∞), О(0; 0; 0), Е(1;
1; 1).
The defined by this way coordinate system К(U, V, W, О, Е) compares to
each point (except those lying in the plane (UVW)) an ordered triple of real numbers
and conversely - for each ordered triple of numbers corresponds a point. In order of
points on (UVW) have coordinates, we introduce homogeneous projective
coordinates: Homogeneous projective coordinates of the point А is called the
ordered four of numbers (u A , v A , wA , t A ), as follows:
- if А is a point, not lying in (UVW), then
where t A
u A  U A t A , v A  V A t A , wA  WA t A ,
 0 is an arbitrary real number.
- if А is a point in the plane (UVW), then t A  0 , u A  0, v A  0, w A  0 .
The geometric sense of the homogeneous projective coordinates of a point A
is, that they are proportional to the ratios of the distances from the points A and E to
the walls of the coordinate tetrahedron, namely:
uA 
a
a1
a
a
, vA  2 , w A  3 , t A  4 .
e1
e2
e3
e4
2. Analytical model of central projection
Let 5 points U, V, W, O, E be given, so that no 4 of them lie in a plane
(fig.2), and let consider the projective coordinate system K(U, V, W, O, E). If А is an
arbitrary point, then
uA 
A, (OVW )
a1
. The first projective coordinate is

e1
E, (OVW )
equal to the relation of the distances from А and Е to the plane (OVW). If we draw
the lines АА1 and ЕЕ1, orthogonal to (VWO), and A1 , E1  (VWO) , then
'
uA 
A, (VWO)
E , (VWO)

AA1
EE1'
. Let take an arbitrary plane π as a projective one,
containing the point Е, and the projection center be the point О. Denoting by
U ' , V 'W ' , A' the central projections of the corresponding points, and E  E'. The
A' A1' is orthogonal to the plane (VWO). The projecting ray OAA' is projected
'
orthogonally on (VWO) in the line OA1 A1 . From the similarity of the triangles OAA1
AA1 OA
'

  A , or AA1   A . A' A1' . Let draw in
and OA' A1 it follows that
A' A1' OA'
the plane π the perpendiculars
A'a' and Ee' to V 'W '. From
'
'
A' A1  (VWO), a' A1  (VWO) it follows, that the triangle A' A1' a' is
'
'
rectangular with right angle A1 . By analogy we consider the triangle EE1e'. The
'
'
rectangular triangles A' A1 a' and EE1e' are similar, since A' a' // Ee'
'
'
(perpendiculars to V 'W ' ) and A' A1 \ \ EE1 (perpendiculars to OVW). Hence,
line
A' A1' A' a'

. In this way we have
EE1'
Ee'
uA 
where
AA1  A . A' A1'  A . A' a


  A u A' ,
'
'
Ee'
EE1
EE1
u A' is the first projective coordinate of’ A' in the projective coordinate system
K ' U ' ,V ' ,W ' , E'. By analogy v A   A .v A' , w A   A .wA' . If the projection
plane does not pass through Е, then the coefficient of the proportion between the
space projective homogenous coordinates of А and the plane projective homogenous
coordinates of its central projection A' is
A
,
E
where
A 
OA
is the ratio of the
OA'
distances from the projection center to the given point and to its central projection..
We obtain, that if A(u A , v A , wA , t A ) is an arbitrary point with projective
homogenous coordinates in projective coordinate system К(U, V, W, O, E)), and
A' (u A' , v A' , wA' ) is the central projection of А, so that we project on arbitrary
projection plane with projection center O, and the projective homogenous coordinates
of А’ are in K ' U ' ,V ' ,W ' , E' – the central projections of the points U, V, W and
E, then the following relations are true:


A '
uA
E

v A  A v A'
E
A '
wA 
wA .
E
uA 
E
uA
A

v' A  E v A
A
E
w' A 
wA .
A
u' A 
, which is equivalent to
Using matrix equation, this correspondence can be given by
PA  A' ,
1


P0
0



0 0 0

1
0 0 ,
 1 
0  0


A
.
E
(1)
The matrix P is called projection matrix of size (3 x 4) and rang (P) = 3.
3. Projection matrix and arising correspondences
Let us consider an arbitrary projective coordinate system with origin – the
0
 
0
point C   , which is the projection centre also, and an arbitrary projection plane ω.
0
 
1
 
u
 
v
To each point in the space M   there corresponds a point – its central projection
w
 
t
 
u
'
 
 
m v '   CM   . This correspondence is given by PM = m, where the projection
 w' 
 
matrix is given by (1).
1 0 0


 1 0
T
Let consider the matrix P   0  1 . It arises a correspondence
0 0 


0 0 0


between the set of lines g ' in ω and the set of planes  ' , containing the center С.
a
 
Really, let g '≡ au ' + bv ' + cw '= 0, or g '  b . Then
c
 
a
 

0 0

1  a  b 
0  
 1 . b      . This vector corresponds to the plane
0  c  c 
   
0 0


0
 
a
b
c
 '  u  v  w  0t  0, or  '  au  bv  cw  0t  0. Evidently
1


T
P g'   0
0

0




that C(0,0,0,1) belongs to it.
In this way to each line g' in ω there corresponds a plane π, passing through
the projection center.
Let Р+ be a matrix of size (4 x 3), so that Р.Р+ = Е3. It is easily seen that
 0

0 

P 
00

a b

0

0
, where a, b, c are arbitrary real numbers. The matrix Р+ is related


c 
with the plane   au  bv  cw  t  0, which does not pass through the
projection center, since   0. So, let consider this correspondence
 a 


 b 

P   
(2)
c 


  



Theorem 1. To each point m  the matrix P compares the point
M  .
 u' 
 
Proof: Let m v '    . It follows, that
 w' 
 
u '
 0 0

 u

  u'  
  


0

0

v
'



 v
P m  
. v'   

   w   M . We check that
00  
w'



  
 a b c   w'   au 'bv'cw'   t 



  
au  bv  cw  t  au 'bv'cw'  (au 'bv'cw' )  0.  M   .
  0 0 a


Now let consider the matrix P   0  0 b .
 0 0  c


T
Тheorem 2. The matrix P determines a correspondence between the set
T
of planes, different from π, and the set of lines in ω, comparing to each plane α ≠ π a
line l '  , which is the image (central projection) of the line l     .
 A
 
B
Proof Let     be an arbitrary plane, α ≠ π, and let l     . We have
C
 
 D
 
 A
  0 0 a    A  aD 

 B  

PT    0  0 b     B  bD   l '
 0 0  c  C   C  cD 

 D  

 
l '  ( A  aD)u  ( B  bD)v  ( C  cD) w  0
(3)
is a line in ω.
We shall show, that it is the image of
l     . Really, if
 m1 
 
 m2 
 l     , then these coordinates satisfy the equations
М
m3 
 
m 
 4
Am1  Bm 2  Cm3  Dm 4  0
am1  bm2  cm3  m4  0
The central projection is
 m1 


m m2   CM   . These coordinates satisfy the equation (3), and m  l '.
 m 
 3
Theorem 3: The above matrices have the following properties:
1
1.
PP T 
2.
 T P  0, P.C  0 .
3.
4.
5.
6.
2
E3 .
  ( E4  P  P)  C T .
1
P T P  2 ( E 4  C T ) .

1
P T P T  E 4  C T .

T
P P  E3
T
4. Double central projection and geometry of two cameras.
Let С and С1 be two different projection centers, ω and ω1 be the
corresponding projection planes, and let P, Q be the projection matrices (fig .3). Take
М to be an arbitrary point with central projections m  CM   and
m1  C1M  1 - the two images. The corresponding projection matrices are P and

Q: PМ = m and QМ = m1. With the matrix P the plane π is related so that
P m  M , M  . It follows QP m  QM  m1 . Let us denote the matrix
QP  H  .
Theorem 4. The matrix Нπ, related to the plane π, arises one-to-one and
invertible correspondence between ω and ω1. The converse correspondence is
1

determined by the matrix H   P.Q .
 0 0


0  0

Proof: Let m be a point in ω. Consider the matrix P  
, where a, b, c
0 0 


a b c


are arbitrary real numbers. The plane π is defined   au  bv  cw  t  0,

There exists a point М in π, for which P m  M . But the point M is projected on
1
by
QM  m1  1. A point in ω1 is compared to each point in ω:
 0 0
 u '   qu ' 
 u' 
 q 0 0 0 
 



0

0

  

If m v ' , then m1  QP m   0 q 0 0 
 v'    qv'  .

 w' 
 0 0 q 0  0 0   w'   qw' 
 

 a b c   



 u' ' 
 q'  ' u ' ' 
 


'
If m'  v' '  is another point in ω, by analogy m1   q '  ' v' ' .
 w' ' 
 q'  ' w' ' 
 


qu '  q'  ' u ' '
If we assume that
m1  m1' , then qv'  q'  ' v' ' or
qw'  q'  ' w' '
u ' v' w'


 q'  ' / q  k . It means that u '  ku' ' , v'  kv' ' , w'  kw' ' , or
u ' ' v' ' w' '
m  m' . This contradiction shows, that different points in ω are compared to
different points in ω1.

The correspondence QP  H 
is one-to-one correspondence. It is
determined by matrix of size (3 x 3) and rang H  = 3, therefore it is convertible, or
1
the matrix H 

Property 3 of Th. 3 confirms, that
It follows, that

there exists. From QP  H  we obtain QP P  H  P.
H  P  Q( E 4 
H  PQ  QQ 
1

P  P  E4 
1

1

C T .
C T ) . Multiply to the right:
QC T Q . Since QQ  E3 (Property 1 Th. 3) and
 Q  0 (Property 2 Th. 3), we have H  PQ  E3
T

1

or H   PQ .
This is the matrix of the converse correspondence from ω1 to ω.
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