1. Camera: optical system Y da=a -a 2 1 curvature radius r r Z 2 1 thin lens small angles:

2. Y incident light beam lens refraction index: n deviated beam Z deviation angle ? Dq =q’’-q

3. Thin lens rules a) Y=0  Dq = 0 beams through lens center: undeviated b) f Dq = Y  independent of y parallel rays converge onto a focal plane Y f Dq

4. r  f Hp: Z >> a the image of a point P belongs to the line (P,O) P image plane O p p = image of P = image plane ∩ line(O,P) interpretation line of p: line(O,p) = locus of the scene points projecting onto image point p

5. Projective 2D geometry Notes based on di R.Hartley e A.Zisserman “Multiple view geometry”

6. Projective 2D Geometry • Points, lines & conics • Transformations & invariants • 1D projective geometry and the Cross-ratio

7. Homogeneous representation of points on if and only if Homogeneous coordinates but only 2DOF Inhomogeneous coordinates Homogeneous coordinates Homogeneous representation of lines equivalence class of vectors, any vector is representative Set of all equivalence classes in R3(0,0,0)T forms P2 The point x lies on the line l if and only if xTl=lTx=0

8. 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 Line joining two points: parametric equation A point on the line through two points x and x’ is y = x + q x’ Example

9. tangent vector normal direction Example Ideal points Line at infinity Ideal points and the line at infinity Intersections of parallel lines Note that in P2 there is no distinction between ideal points and others

10. Example • The linear combination z = ax + by of two points x and y is a point zcolinear with them (i.e., on the line through x and y)  • The linear combination n = al + bm of two lines l and m is a line nconcurrent with them (i.e., through the point on l and m)

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

12. or stacking constraints yields Five points define a conic For each point the conic passes through

13. Polarity: cross ratio Cross ratioof 4 colinear points y = x + q x’ (with i=1,..,4) ratio of ratios i i Harmonic 4-tuple of colinear points: such that CR=-1

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

15. In general (C full rank): in fact Dual conics A line tangent to the conic C satisfies -1 T Linelis thepolar line of y : y = Cl , but since y Cy = 0  l CCC l = 0  C = C = C -T -1 T -T -1 * Dual conics = line conics = conic envelopes

16. 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)

17. 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 represented by a vector x it is true that h(x)=Hx Definition: Projective transformation or 8DOF 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

18. Mapping between planes central projection may be expressed by x’=Hx (application of theorem)

19. Removing projective distortion select four points in a plane with known coordinates (linear in hij) (2 constraints/point, 8DOF  4 points needed) Remark: no calibration at all necessary, better ways to compute (see later)

20. More examples

21. Transformation for conics Transformation for dual conics Transformation of lines and conics For a point transformation Transformation for lines

22. A hierarchy of transformations Projective linear group 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 leave distances unchanged

23. orientation preserving: orientation reversing: Class I: Isometries (iso=same, metric=measure) 3DOF (1 rotation, 2 translation) special cases: pure rotation, pure translation Invariants: length, angle, area

24. Class II: Similarities (isometry + scale) 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

25. Class III: Affine transformations where 6DOF (2 scale, 2 rotation, 2 translation) non-isotropic scaling! (2DOF: scale ratio and orientation) Invariants: parallel lines, ratios of parallel segment lengths, ratios of areas

26. Action of affinities and projectivitieson line at infinity Line at infinity stays at infinity, but points move along line Line at infinity becomes finite, allows to observe vanishing points, horizon,

27. Class VI: Projective transformations 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 ratios)

28. Projective geometry of 1D 3DOF (2x2-1) The cross ratio Invariant under projective transformations

29. Overview transformations 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.

30. Number of invariants? The number of functional invariants is equal to, or greater than, the number of degrees of freedom of the configuration less the number of degrees of freedom of the transformation e.g. configuration of 4 points in general position has 8 dof (2/pt) and so 4 similarity, 2 affinity and zero projective invariants

31. Recovering metric and affine properties from images • Parallelism • Parallel length ratios • Angles • Length ratios

32. 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

33. Affine properties from images projection (affine) rectification in fact, any point x on l’ is mapped to a point at the ∞ ∞

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

35. The circular points The circular points I, J are fixed points under the projective transformation H iff H is a similarity

36. l∞ Algebraically, encodes orthogonal directions The circular points “circular points” Intersection points between any circle and l∞

37. Circular points invariance • {I,J} = l∞∩any circumference • Similarity: circ’  circ” • Similarity: circ’ ∩l∞ circ’’ ∩l∞ • Similarity: {I,J}  {I,J} •  circular points: invariant under similarity

38. The dual conic is fixed conic under the projective transformation H iff H is a similarity Note: has 4DOF l∞ is the null vector Conic dual to the circular points : line conic = set of lines through any of the circular points

39. Projective: (orthogonal) Angles Euclidean:

40. Metric properties from images Rectifying transformation from SVD

41. Why ? Normally: SVD (Singular Value Decomposition) with U and V orthogonal But is symmetric  and SVD is unique  Observation : H=U orthogonal (3x3): not a P2 isometry

42. Metric from affine Once the image has been affinely rectified

43. Metric from affine

44. Metric from projective

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

46. Projective 3D geometry

47. Singular Value Decomposition

48. Singular Value Decomposition • Homogeneous least-squares • Span and null-space • Closest rank r approximation • Pseudo inverse

49. Projective 3D Geometry • Points, lines, planes and quadrics • Transformations • П∞, ω∞and Ω ∞

50. 3D points 3D point in R3 in P3 projective transformation (4x4-1=15 dof)