Projective geometry- 2D
Acknowledgements
Marc Pollefeys: for allowing the use of his excellent slides on this topic
http://www.cs.unc.edu/~marc/mvg/
Richard Hartley and Andrew Zisserman, "Multiple View Geometry in Computer Vision "
Homogeneous coordinates
Homogeneous representation of lines
ax + by + c = 0
(a,b,c )T
(a,b,c )T ~ k (a,b,c )T
(ka) x + (kb) y + kc = 0, k 0
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
Spring 2006
Projective Geometry 2D
but only 2DOF
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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'
Example
y =1
x =1
Projective Geometry 2D
Spring 2006
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Ideal points and the line at infinity
Intersections of parallel lines
l = (a, b, c ) and l' = (a, b, c')
T
T
l l' = (b,a,0 )
T
Example
x =1 x = 2
Ideal points
Line at infinity
(x1 , x2 ,0)T
T
l = (0,0,1)
P 2 = R 2 l
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Note that this set lies on a single line,
Note that in P2 there is no distinction
between ideal points and others
Projective Geometry 2D
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Summary
The set of ideal points lies on the line at infinity,
intersects the line at infinity in the ideal point
A line
parallel to l also intersects
point, irrespective of the value of c’.
in the same ideal
In inhomogeneous notation,
is a vector tangent to the line.
It is orthogonal to (a, b) -- the line normal.
Thus it represents the line direction.
As the line’s direction varies, the ideal point
varies over
.
--> line at infinity can be thought of as the set of directions of lines in the
plane.
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Projective Geometry 2D
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A model for the projective plane
Points represented by rays through origin
Lines represented by planes through origin
x1x2 plane represents line at infinity
exactly one line through two points
exaclty one point at intersection of two lines
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Projective Geometry 2D
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Duality
x
l
x Tl = 0
lT x = 0
x = l l'
l = x x'
Duality principle:
To any theorem of 2-dimensional projective geometry
there corresponds a dual theorem, which may be
derived by interchanging the role of points and lines in
the original theorem
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Projective Geometry 2D
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Conics
Curve described by 2nd-degree equation in the plane
ax 2 + bxy + cy 2 + dx + ey + f = 0
or homogenized x x1 x , y x2 x
3
2
3
2
ax1 + bx1 x2 + cx2 + dx1 x3 + ex2 x3 + fx32 = 0
or in matrix form
b / 2 d / 2
a
e / 2 x C x = 0 with C = b / 2 c
d / 2 e / 2
f T
5DOF:
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{a : b : c : d : e : f }
Projective Geometry 2D
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4
Conics …
http://ccins.camosun.bc.ca/~jbritton/jbconics.htm
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Projective Geometry 2D
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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
c = (a, b, c, d , e, f )
T
stacking constraints yields
x12
2
x2
x32
2
x4
x2
5
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x1 y1
x2 y 2
x3 y3
x4 y 4
x5 y5
y12
y22
y32
y42
y52
x1
x2
x3
x4
x5
y1
y2
y3
y4
y5
1
1
1c = 0
1
1
Projective Geometry 2D
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Tangent lines to conics
The line l tangent to C at point x on C is given by l=Cx
x
l
C
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Projective Geometry 2D
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Dual conics
A line tangent to the conic C satisfies
In general (C full rank):
l T C* l = 0
C* = C 1
C* : Adjoint matrix of C.
Dual conics = line conics = conic envelopes
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Projective Geometry 2D
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Degenerate conics
A conic is degenerate if matrix C is not of full rank
m
l
e.g. two lines (rank 2)
C = lm T + 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
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(C ) C
* *
Projective Geometry 2D
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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:P2P2 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
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Projective Geometry 2D
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Mapping between planes
central projection may be expressed by x’=Hx
(application of theorem)
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Projective Geometry 2D
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Removing projective distortion
select four points in a plane with know coordinates
h x + h12 y + h13
h x + h22 y + h23
x'
x'
x' = 1 = 11
y ' = 2 = 21
x'3 h31 x + h32 y + h33
x'3 h31 x + h32 y + h33
x' (h31 x + h32 y + h33 ) = h11 x + h12 y + h13
y ' (h31 x + h32 y + h33 ) = h21 x + h22 y + h23
(linear in hij)
(2 constraints/point, 8DOF 4 points needed)
Remark: no calibration at all necessary,
better ways to compute (see later)
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Projective Geometry 2D
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Transformation of lines and conics
For a point transformation
x' = H x
Transformation for lines
l' = H -T l
Transformation for conics
C' = H -T CH -1
Transformation for dual conics
C'* = HC*H T
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Projective Geometry 2D
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Distortions under center projection
Similarity: squares imaged as squares.
Affine: parallel lines remain parallel; circles become ellipses.
Projective: Parallel lines converge.
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Projective Geometry 2D
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Class I: Isometries
(iso=same, metric=measure)
x' cos y ' = sin 1 0
sin cos 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
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Projective Geometry 2D
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Class II: Similarities
x' s cos y ' = s sin 1
0
s sin s cos 0
sR t x' = H S x = T
x
0 1
(isometry + scale)
t x x t y y 1 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
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Projective Geometry 2D
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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 = R ( )R ( )DR ( )
0 D= 1
0 2 6DOF (2 scale, 2 rotation, 2 translation)
non-isotropic scaling! (2DOF: scale ratio and orientation)
Invariants: parallel lines, ratios of parallel lengths,
ratios of areas
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Projective Geometry 2D
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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 ratio)
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Projective Geometry 2D
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Action of affinities and projectivities on line at infinity
A
0 T
x1 x t A 1 x2 = x v 2 0 0 Line at infinity stays at infinity,
but points move along line
A
vT
x1 x t A 1 x x
=
2
2
v 0 v1 x1 + v2 x2 Line at infinity becomes finite,
allows to observe vanishing points, horizon.
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Projective Geometry 2D
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Decomposition of projective transformations
sR t K
H = H S H AH P = T
T
0 1 0
0 I
1 v T
decomposition unique (if chosen s>0)
0 A
=
v v T
t
v A = sRK + tv T
K upper-triangular, det K = 1
Example:
1.707 0.586 1.0 H = 2.707 8.242 2.0
2.0 1.0 1.0
2 cos 45
H = 2 sin 45
0
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2 sin 45
2 cos 45
0
1.0 0.5 1 0 1 0 0
2.0 0 2 0 0 1 0
1 0 0 1 1 2 1
Projective Geometry 2D
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Overview transformations
Projective
8dof
Affine
6dof
Similarity
4dof
Euclidean
3dof
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h11
h
21
h31
h12
h22
h32
h13 h23 h33 a11
a
21
0
a12
tx t y 1 a22
0
sr11 sr12 t x sr
21 sr22 t y 0
0
1 r11 r12 t x r
21 r22 t y 0 0 1 Projective Geometry 2D
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.
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