Lecture 1

C OMPUTER V ISION : P ROJECTIVE G EOMETRY

IIT Kharagpur

Computer Science and Engineering,
Indian Institute of Technology
Kharagpur.

(IIT Kharagpur) Projective Geometry Jan ’10 1 / 40

Planar Geometry
Geometry is the study of points and lines and their relationships.
Geometry can be studied in terms of properties of geometric
primitives.
An algebraic approach to studying geometry involves establishing
a coordinate system.

Algebraic geometry
Points and lines are represented as vectors.
A conic section is represented by a symmetric matrix.
Results derived using algebraic geometry are very useful for
developing practical computation methods.


The 2D projective plane Notation
A point (x, y ) can be considered as a vector in the vector space
IR2
Geometric entities can be represented by a column vector.
Generally x represents a column vector and xT represents a row
vector.
A point x gets represented by a column vector:

x
x= = (x, y )T
y


Homogeneous representation of lines
Equation of a line: ax + by + c = 0
Line as a vector: (a, b, c)T
Vectors (a, b, c)T and k (a, b, c)T represent the same line.
For different values of scalar k , we get an equivalence class of
vectors.
Any particular vector (a, b, c)T is a representative of the
equivalence class.

Projective space
The set of equivalence classes of vectors in IR3 forms the
projective space IP2 .
The vector (0, 0, 0)T is excluded from the projective space since it
does not correspond to any line.


Homogeneous representation of points
A point x = (x, y )T lies on the line l = (a, b, c)T if
 
 a
 

(x, y , 1)  b
 
ax + by + c = 0 


=0



c
 

The point (x, y )T in IR2 is represented as a 3-vector by adding a
ﬁnal coordinate of 1.
Since (kx, ky , k )l = 0, the set of vectors (kx, ky , k )T for varying
values of k would represent the same point (x, y T ) in IR2 .
A homogeneous vector of general form x = (x1 , x2 , x3 )T
represents the point (x1 /x3 , x2 /x3 )T in IR2 .

Points as homogeneous vectors are also elements of IP2 .


Homogeneous coordinate
Point x lies on line l if and only if xT l = 0
xT l = lT x = x.l
Homogeneous coordinate (3-vector) x = (x1 , y1 , z1 )T .
Inhomogeneous coordinate (2-vector) x = (x, y )T .

Two lines l = (a, b, c)T and l = (a , b , c )T intersect at a point x.

x=l×l

The line through two points x and x is

l=x×x


Ideal Points Line at ∞
Consider two parallel lines l = (a, b, c)T and l = (a, b, c )T .
Intersection of l and l is given by l × l .
 
 b
 

l × l = (c − c )  −a
 


 

 
0
 

The inhomogeneous representation of the point of intersection

b/0
−a/0

Parallel lines meet at inﬁnity.


Ideal points and the line at inﬁnity
All homogeneous 3-vectors form the projective space IP2 .
The points for which the last coordinate x3 = 0 are the ideal points
 
 x1 
 
 x 

 2 

 

0
 

The set of all ideal points (x1 , x2 , 0)T lie on a single line, the Line
at Inﬁnity, denoted l∞ = (0, 0, 1)T
A line l = (a, b, c)T intersects l∞ in the ideal point (b, −a, 0)T .
The vector (b, −a)T is tangent to the line and orthogonal to the
line normal (a, b) and so represents the line’s direction.


Advantage of projective geometry
Projective plane IP2
In IP2 , two distinct lines meet in a single point and two points lie on
a single line.
In the standard Euclidean geometry of IR2 , parallel lines form a
special case.

The study of the geometry of IP2 is known as projective geometry.
In the purely geometric study of projective geometry, one does not
make any distinction between points at inﬁnity (ideal points) and
ordinary points.


A model for projective plane
Points in IP2 correspond to rays in IR3 .
The set of all vectors k (x1 , x2 , x3 )T as k varies forms a ray through
origin.
The lines in IP2 are planes passing through origin in IR3
Getting inhomogeneous representation: Points and lines may be
obtained by intersecting this set of of rays and planes with the
plane x3 = 1


Duality
The role of points and lines can be interchanged in statements
concerning the properties of lines and points.
E.g. lT x = 0 also implies xT l = 0

To any theorem of 2-dimensional projective geometry there
corresponds a dual theorem, which may be derived by interchanging
the roles of points and lines in the original theorem.
A line through 2 points is dual to the point of intersection of the
two lines.


Conics and Dual Conics
A conic is a curve described by a second-degree equation in the
plane.
E.g. hyperbola, ellipse, parabola.
Inhomogeneous coordinates → equation of a conic:

ax 2 + bxy + cy 2 + dx + ey + f = 0

Homogenizing this by replacements x1 → x1 /x3 , y → x2 /x3
2 2 2
ax1 + bx1 x2 + cx2 + dx1 x2 + ex2 x3 + fx3


Conic in matrix form
 
 a b/2 d/2 
 
xT Cx = 0 C =  b/2 c e/2 
 

 

 
d/2 e/2 f
 

The matrix C is a homogeneous representation of the conic.
The conic has 5 degrees of freedom, i.e. the ratios:
{a : b : c : d : e : f }
Five points are required to deﬁne a conic.

Tangent to the conic
The line l tangent to the conic C is given by l = Cx


Conic in matrix form
 
 a b/2 d/2 
 
xT Cx = 0 C =  b/2 c e/2 
 

 

 
d/2 e/2 f
 

The matrix C is a homogeneous representation of the conic.
The conic has 5 degrees of freedom, i.e. the ratios:
{a : b : c : d : e : f }


Projective Transformations
2D projective geometry is the study of properties of the projective
plane IP2 that are invariant under a group of transformations
known as projectivities.
A projectivity is an invertible mapping from points in IP2 to points in
IP2 .
A projectivity is an invertible mapping h from IP2 to itself such that
three points x1 , x2 and x3 lie on the same line if and only if h(x1 ),
h(x2 ) and h(x3 ) do.
Also called as: collineation, projective transformation or a
homography.


Homography Projective Transformation
Algebraic deﬁnition:
A mapping h : P2 → IP2 is a projectivity if and only if there exists a
non-singular 3 × 3 matrix H such that for any point in P2 represented by
vector x it is true that h(x) = Hx.

H is a linear transformation
    
 x1   h11 h12 h13
     x1 
 
x = Hx  x = h
 2   21 h22 h23

   x 

 2  

  
  
 

x3 h31 h32 h33 x3
    

H is a homogeneous matrix
Only ratios of the matrix elements is signiﬁcant.
There are 8 degrees of freedom.


Projective Transformation
A projective transformation leaves the projective properties
invariant.
A projective transformation in P2 is simply a linear transformation
of R3 .


Transformation of Lines
Points xi get transformed as xi = Hxi
If these points xi lie on a line l, then lT xi = 0
The transformed points xi would lie on a line l .

l = H−T l


Transformation of Conics
Points x get transformed as x = Hx
If the point x lies on a conic C, then xT Cx = 0

xT Cx = x T [H−1 ]T C H−1 x
= x T H−T C H−1 x

Under a point transformation x = Hx, a conic C transforms to

C = H−T C H−1


Hierarchy of Transformations
General linear group: GL(n) −→ Group of invertible n × n matrices
with real elements.
Projective linear group: PL(n) −→ Matrices are related by a scalar
multiplier. Quotient group of GL(n).

Projective Linear Group Subgroups
Afﬁne group: −→ Matrices for which the last row is (0, 0, 1)
Euclidean group: −→ Additionally, the upper left hand 2 × 2 matrix
is orthogonal.
Oriented Euclidean group: PL(n) −→ Additionally, the upper left
hand 2 × 2 matrix has determinant 1.


Invariants
A transformation can be described in terms of those elements or
quantities that are preserved or invariant.
A (scalar) invariant of a geometric conﬁguration is a function of the
conﬁguration whose value is unchanged by a particular
transformation.

Euclidean invariants Similarity invariants
Distance between two points. Distance
Angle between two lines. Angle between two lines.


Examples of Projective transformations


Example of Projective Correction


Isometries
     
 x 
  
 cosθ − sinθ tx   x 
  
 y  = 



 
 sinθ cosθ ty   y 
 
 

 where = ±1

 
 
  
  

1 0 0 1 1
     

Isometries are transformations of the plane R2 that preserve
Euclidean distance.
If = 1, the isometry is orientation-preserving and is a Euclidean
transformation. Euclidean transformation is a composition of
translation and rotation.
If = −1, the isometry reverses orientation.


Isometries In short form
R t
x = HE x = x
0T 1

R is a 2 × 2 rotation matrix. RT R = RRT = I
t is a translation 2-vector.
0 is a null 2-vector.
It has 3 degrees of freedom: 1 for rotation, 2 for translation.

Invariants Isometry
Length
Angle
Area


Similarity Transformation
     
 x 
   s cosθ −s sinθ tx   x 
   
 y  =  s sinθ s cosθ t   y 
      where s = scaling








 y  
 
 



1 0 0 1 1
     

It is an isometry composed with an isotropic scaling.
Preserves the shape.
Has 4 degrees of freedom −→ scaling(1), rotation(1),
translation(2).


Similarity Transformation In short form
sR t
x = HS x = x
0T 1

R is a 2 × 2 rotation matrix. RT R = RRT = I
t is a translation 2-vector.
0 is a null 2-vector.

Invariants Isometry
Angle
Parallel lines remain as parallel.
Length: Ratio of two lengths is preserved.
Area: Ratio of two areas is preserved.


Metric Structure
Metric Structure implies that the structure is deﬁned up to a similarity.


Afﬁne Transformation (Afﬁnity)
     
 x 
   a11 a12 tx   x 
   
 y  =  a21 a22 ty   y 

     
 
 
  
  

     
1 0 0 1 1
     

A t
x = HA x = x
0T 1

A is a 2 × 2 non-singular matrix.
Has 6 degrees of freedom −→ 6 matrix elements.
The transformation can be computed using 3 point
correspondences.


Decomposition of an Afﬁne transform
A = R(θ) R(−φ) D R(φ)

λ1 0
D is a diagonal matrix. D =
0 λ2
R(θ) and R(φ) are rotations by θ and φ respectively.


Affine transform Non-isotropic scaling
Non-isotropic scaling means there is a scaling direction (angle φ),
and a ratio of scaling parameters λ1 : λ2 in orthogonal directions.
It has 2 extra degrees of freedom compared to a similarity
transform.

Invariants Affine Transform
Angle
Parallel lines remain as parallel.
Length: Ratio of two lengths is preserved for parallel lines.
Area: Ratio of two areas is preserved. In fact areas are scaled by
factor λ1 λ2 .
There can be orientation preserving and orientation reversing affinities
depending on the sign of detA


Projective Transformation
A t
x = HP x = x where v = (v1 , v2 )T
vT v

Has 8 degrees of freedom −→ 9 elements with only ratio
signiﬁcant.
The transformation can be computed using 4 point
correspondences, with no 3 collinear on either plane.

Invariants
A ratio of ratios (cross ratio) of lengths on a line is a projective
invariant.


Similarity (4 dof)
↓
Afﬁnity (6 dof) Afﬁnity: Scaling of area is the same all over the
↓ plane. Orientation of a transformed line does not
Projectivity (8 dof) depend on its position on the plane.
Projectivity: Area scaling varies with position.
Orientation of a transformed line depends on its
initial orientation and position.
The vector v is responsible for non-linear effects.
   
 x1   A x1 
A t    
 x2  =  x2 

 
 
 

0T 1 
 
 
 

0 0
   

   
 x1  x1
A t
 
 x  =  A x

   

 2   
2
vT
 
v 
 
 
 

0 v1 x1 + v2 x2
   


Decomposition of a Projective Transform

sR t K 0 I 0 A t
H = HS HA HP = =
0T 1 0T 1 vT v vT v

A = sRK + tvT
K is an upper-triangular matrix normalized as det K = 1
The decomposition is valid if v 0, is unique if s is positive.

 
 1.707 0.586 1.0 
 
H =  2.707 8.242 2.0 
 

 

 
1.0 2.0 1.0
 

 2cos45o −2sin45o 1   0.5 1 0   1 0 0 
   
   
=  2sin45o 2cos45o 2   0 2 0   0 1 0 

   
 
 
 

   
0 0 1 0 0 1 1 2 1
   

Rectifying a Projective Transform
Affine rectification
Similarity rectification (Metric structure)
Euclidean rectification


Number of Invariants
The number of functionally independent invariants
≥
the number of degrees of freedom of the conﬁguration
−
the number of degrees of freedom of the transformation


Projective (8 dof) Invariants
 
 h11 h12 h13
Concurrency, collinearity
 

 h21 h22 h23

 


Order of contact

 

h31 h32 h33
 
Cross ratios

Afﬁne (6 dof) Invariants
 
 a11 a12 tx 
  Parallelism
 a21 a22 ty 

 

Ratios of areas, ratio of

 

0 0 1
 
lengths on parallel lines
The line at ∞


Similarity (4 dof) Invariants
 
 sr11 sr12 tx 

 sr
 Ratio of lengths,
 21 sr22 ty 
 

Angles

 

0 0 1
 
The circular points

Euclidean (3 dof) Invariants
 
 r11 r12 tx 

 r
 Lengths,
 21 r22 ty 
 

Area

 

0 0 1
 


The projective geometry of 1D


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