1. Projective Geometry and Camera model Class 2
points, lines, planes
conics and quadrics
transformations
camera model
Read tutorial chapter 2 and 3.1
Chapter 1, 2 and 5 in Hartley and Zisserman

2. Homogeneous coordinates
Homogeneous representation of 2D points and lines
The point x lies on the line l if and only if
Note that scale is unimportant for incidence relation
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3(0,0,0)T forms P2
Homogeneous coordinates
Inhomogeneous coordinates
but only 2DOF

3. Points from lines and vice-versa
Intersections of lines
The intersection of two lines and is
Line joining two points
The line through two points and is
Example
Note:
with

4. Ideal points and the line at infinity
Intersections of parallel lines
Example
tangent vector
normal direction
Ideal points
Line at infinity is a collection of all the
points at infinity as x1/x2 varies, i.e. all
Possible directions
Line at infinity
Note that in P2 there is no distinction
between ideal points and others

5. 3D points and planes
Homogeneous representation of 3D points and planes
The point X lies on the plane π if and only if
The plane π goes through the point X if and only if

6. Planes from points
(solve as right nullspace of )

7. Points from planes M is 4x3 matrix. Columns of M are null space of
(solve as right nullspace of )
Representing a plane by its span
M is 4x3 matrix. Columns of M are null space of

8. Lines Line is either joint of two points or intersection of two planes
Representing a line by its span: two vectors A, B for two space points
(4dof)
Dual representation: P and Q are planes; line is span of row space of
Example: X-axis
(Alternative: Plücker representation, details see e.g. H&Z)

9. Points, lines and planes
Plane defined by join of point X and line W is the
null space of M
Point X defined by the intersection of line W with
plane is the null space of M

10. Plücker coordinates Elegant representation for 3D lines
(with A and B points)
(Plücker internal constraint)
(two lines intersect)
(for more details see e.g. H&Z)

11. Conics Curve described by 2nd-degree equation in the plane
or homogenized
or in matrix form
with
5DOF:

12. Five points define a conic
For each point the conic passes through or
stacking constraints yields
Conic c is a null vector of the above matrix

13. Tangent lines to conics
The line l tangent to C at point x on C is given by l=Cx l x C

14. Dual conics Points lie on a point lines are tangent
A line tangent to the conic C satisfies
In general (C full rank):
Dual conics = line conics = conic envelopes
Points lie on a point lines are tangent
conic to the point conic C; conic
C is the envelope of lines l

15. Degenerate conics
A conic is degenerate if matrix C is not of full rank
e.g. two lines (rank 2)
e.g. repeated line (rank 1)
Degenerate line conics: 2 points (rank 2), double point (rank1)
Note that for degenerate conics

16. Quadrics and dual quadrics
(Q : 4x4 symmetric matrix)
9 d.o.f.
in general 9 points define quadric
det Q=0 ↔ degenerate quadric
tangent plane
3. and thus defined by less points
4. On quadric=> tangent, outside quadric=> plane through tangent point
5. Derive X’QX=x’M’QMx=0
Dual Quadric: defines equation on planes
relation to quadric (non-degenerate)

17. 2D 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.
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 reprented by a vector x it is true that h(x)=Hx
Theorem:
Definition: Projective transformation or
8DOF
projectivity=collineation=projective transformation=homography

18. Transformation of 2D points, lines and conics
For a point transformation
Transformation for lines
Transformation for conics
Transformation for dual conics

19. Fixed points and lines (eigenvectors H =fixed points)
(1=2  pointwise fixed line)
(eigenvectors H-T =fixed lines)

20. Hierarchy of 2D transformations
transformed squares
invariants
Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio
Projective
8dof
Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids).
The line at infinity l∞
Affine
6dof
Ratios of lengths, angles.
The circular points I,J
Similarity
4dof
Euclidean
3dof
lengths, areas.

21. Projective geometry of 1D
The cross ratio
Is invariant under projective transformations in P^1.
Four sets of four collinear points; each set is related to the others by a line projectivity
Since cross ratio is an invariant under projectivity, the cross ratio has the same value for all the sets shown

22. 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 ratio
of the image points x_i is invariant to the position of
the image line l

23. 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

24. Note: not fixed pointwise
The line at infinity
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

25. Affine properties from images
projection
rectification
Two step process:
1.Find l the image of line
at infinity in plane 2
2. Plug into H_pa
Image of line at infinity in plane

26. Affine rectification v1 l∞ v2 l1 l3 l2 l4

27. The circular points Canonical coordinates of circular points
Two points on l_inf which are fixed under any similarity transformation
Canonical coordinates
of circular points
The circular points I, J are fixed points under the projective transformation H iff H is a similarity

28. The circular points “circular points” l∞ Circle: Line at infinity
Two points on l_inf: Every circle intersects l_inf at circular points
“circular points” l∞
Circle:
Line at infinity
Algebraically, encodes orthogonal directions

29. Conic dual to the circular points
The dual conic is fixed conic under the
projective transformation H iff H is a similarity
Note: has 4DOF (3x3 ho mogeneous; symmetric, determinant is zero)
l∞ is the nullvector

30. Angles Euclidean: Projective: (1) (l and m orthogonal)
Once is identified in the projective plane, then Euclidean angles
may be measured by equation (1)

31. Transformation of 3D points, planes and quadrics
For a point transformation
(cfr. 2D equivalent)
Transformation for lines
Transformation for conics
Transformation for dual conics

32. Hierarchy of 3D transformations
Projective
15dof
Intersection and tangency
Parallellism of planes,
Volume ratios, centroids,
The plane at infinity π∞
Affine
12dof
Similarity
7dof
Angles, ratios of length
The absolute conic Ω∞
Euclidean
6dof
Volume

33. The plane at infinity
The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity
Represents 3DOF between projective and affine
canonical position
contains directions
two planes are parallel  line of intersection in π∞
line // line (or plane)  point of intersection in π∞

34. The absolute conic
The absolute conic Ω∞ is a (point) conic on π.
In a metric frame:
or conic for directions:
(with no real points)
The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity
Represent 5 DOF between affine and similarity
Ω∞ is only fixed as a set
Circle intersect Ω∞ in two circular points
Spheres intersect π∞ in Ω∞

35. The absolute dual quadric
Set of planes tangent to an ellipsoid, then squash to a pancake
The absolute dual quadric Ω*∞ is a fixed conic under the projective transformation H iff H is a similarity
8 dof ( symmetric matrix, det is zero)
plane at infinity π∞ is the nullvector of Ω∞
Angles:

36. Camera model
Relation between pixels and rays in space ?

37. Pinhole camera

38. Pinhole camera model
linear projection in homogeneous coordinates!

39. Pinhole camera model

40. Principal point offset

41. Principal point offset
calibration matrix

42. Camera rotation and translation~

43. CCD camera

44. General projective camera
11 dof (5+3+3)
non-singular
intrinsic camera parameters
extrinsic camera parameters

45. Radial distortion Due to spherical lenses (cheap) Model: R R
straight lines are not straight anymore

46. Camera model
Relation between pixels and rays in space ?

47. Projector model Relation between pixels and rays in space
(dual of camera)
(main geometric difference is vertical principal point offset
to reduce keystone effect) ?

48. Meydenbauer camera vertical lens shift to allow direct
ortho-photographs

49. Affine cameras

50. Action of projective camera on points and lines
projection of point
forward projection of line
back-projection of line

51. Action of projective camera on conics and quadrics
back-projection to cone
projection of quadric

52. Image of absolute conic

53. A simple calibration device
compute H for each square
(0,0),(1,0),(0,1),(1,1))
compute the imaged circular points H(1,±i,0)T
fit a conic to 6 circular points
compute K from w through cholesky factorization
(≈ Zhang’s calibration method)

54. Exercises: Camera calibration

55. Next class: Single View Metrology
Antonio Criminisi

56. 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 Affine projective
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.