Projective Geometry and Camera model Class 2

1. Projective Geometry and Camera model Class 2 points, lines, planes conics and quadrics transformations camera model Read tutorial chapter 2 and 3.1 http://www.cs.unc.edu/~marc/tutorial/ Chapter 1, 2 and 5 in Hartley and Zisserman

2. Homogeneous coordinates but only 2DOF Inhomogeneous coordinates 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

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

4. tangent vector normal direction Example Ideal points Line at infinity Ideal points and the line at infinity Intersections of parallel lines • Line at infinity is a collection of all the • points at infinity as x1/x2 varies, i.e. all • Possible directions 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 πif and only if The plane πgoes through the point X if and only if

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

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

8. Dual representation: P and Q are planes; line is span of row space of 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) 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. or homogenized or in matrix form with Conics Curve described by 2nd-degree equation in the plane 5DOF:

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

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

13. In general (C full rank): Dual conics A line tangent to the conic C satisfies 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

14. Note that for degenerate conics 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)

15. 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 • Dual Quadric: defines equation on planes • relation to quadric (non-degenerate)

16. Theorem: A mapping h:P2P2is 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 Definition: Projective transformation or 8DOF 2D projective transformations Definition: A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3lie on the same line if and only if h(x1),h(x2),h(x3) do. projectivity=collineation=projective transformation=homography

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

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

19. 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.

20. 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

21. 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

22. 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

23. 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

24. 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

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

26. The 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

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

28. 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 Conic dual to the circular points l∞

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

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

31. 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

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

33. 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 • Ω∞is only fixed as a set • Circle intersect Ω∞ in two circular points • Spheres intersect π∞ in Ω∞

34. 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:

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

36. Pinhole camera

37. Pinhole camera model linear projection in homogeneous coordinates!

38. Pinhole camera model

39. Principal point offset principal point

40. Principal point offset calibration matrix

41. Camera rotation and translation ~

42. CCD camera

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

44. Radial distortion • Due to spherical lenses (cheap) • Model: R R straight lines are not straight anymore http://foto.hut.fi/opetus/260/luennot/11/atkinson_6-11_radial_distortion_zoom_lenses.jpg

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

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

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

48. Affine cameras

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

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