3D Projective Geometry and Applications for Computer Vision

1. Computer Visioncmput 428/615 Lecture 8: 3D projective geometry and it’s applications Martin Jagersand

2. First 1D Projective lineProjective Coordinates ac ac bd bd • Basic projective invariant in P1: the cross ratio b a d c = inhomogeneous d b c Requires 4 points: a 3 to create a coordinate system The fourth is positioned within that system

3. Cross ratio HZ CH 2.5 • Basic projective invariant in P1: the cross ratio • Properties: • Defines coordinates along a 1d projective line • Independent of the homogeneous representation of x • Valid for ideal points • Invariant under homographies homogeneous

4. Projective 3D space • Points Points and planes are dual in 3d projective space. • Planes • Lines: 5DOF, various parameterizations • Projective transformation: • 4x4 nonsingular matrix H • Point transformation • Plane transformation • Quadrics:Q Dual: Q* • 4x4 symmetric matrix Q • 9 DOF (defined by 9 points in general pose)

5. 3D points 3D point in R3 in P3 projective transformation (4x4-1=15 dof)

6. Planes Transformation Euclidean representation 3D plane Dual: points ↔ planes, lines ↔ lines

7. Planes from points (solve as right nullspace of ) Or implicitly from coplanarity condition

8. Representing a plane byby lin comb of 3 points Representing a plane by its nullspace span M All points lin comb of basis Canonical form: Given a plane nullspace span M is

9. Points from planes (solve as right nullspace of )

10. Lines Join of two points: A, B (4dof) Intersection of two planes: P, Q Example: X-axis

11. Points, lines and planes Join of point X and line W is plane p Intersection of line W with plane p is point X

12. Affine space Difficulties with a projective space: • Nonintuitive notion of direction: • Parallelism is not represented • Infinity not distinguished • No notion of “inbetweenness”: • Projective lines are topologically circular • Only cross-ratios are available • Ratios are required for many practical tasks A +/- B • Solution:find the plane at infinity! • Transform the model to give ∞ its canonical coordinates • 2D analogy: fix the horizon line Determining the plane at infinity upgrades the geometry from projective to affine

13. From projective to affine 3D  2 1x = 3 2(1+x) 2D l • Finding the plane (line, point) at infinity • 2 or 3 sets of parallel lines (meeting at “infinite” points) • a known ratio can also determine infinite points 1D  cross ratio p 1 x 2 1 1 1 2

14. From projective to affine Projective Affine HZ 3D 2D

15. Affine space • Affine transformation • 12 DOF • Leaves  unchanged • Invariants • Ratio of lengths on a line • Ratios of angles • Parallelism Kutulakos and Vallino, Calibration-Free Augmented Reality, 1998

16. Metric space • Metric transformation (similarity) • 7 DOF • Maps absolute conic to itself • Invariants • Length ratios • Angles • The absolute conic Without a yard stick, this is the highest level of geometric structure that can be retrieved from images

17. The absolute conic  • Absolute conic is an imaginary circle on • It is the intersection of every sphere with • In a metric frame On : absolute dual quadric

18. From affine to metric 3D 2D • Identify on OR identify • via angles, ratios of lengths • e.g. perpendicular lines • Upgrade the geometry by bringing to its canonical form via an affine transformation 1D ?

19. Examples 2D Two pairs of perpendicular lines 3D Affine Metric 5 known points

20. What good is a projective model? f3 f1 f2 Tcol(f1,f2,f3) = f1 - f2f3 Represents fundamental feature interactions Used in rendering with an unconventional engine: Physical Objects! Visual Servoing Achieving 3d tasks via 2d image control collinearity task

21. What good is a projective model? 1 y1 1 y3 1 y2 2 y3 2 Y1 2 y2 Represents fundamental feature interactions Used in rendering with an unconventional engine: Physical Objects! Visual Servoing Achieving 3d tasks via 2d image control image collinearity constraint

22. Collinearity? A point meets a line in each of two images...

23. Collinearity? But it doesn’t guarantee task achievement !

24. 1 1 1 1 1 0 0 1 1 0 0 0 1 1 0  0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 All achievable tasks... 4pt. coincidence 3pt. coincidence Coincidence 2 2pt. coincidence 2pt. coincidence Cross Ratio One point Collinearity Noncoincidence Coplanarity are ALL projectively distinguishable coordinate systems General Position General Position

25. Cinj Cwk Composing possible tasks Tpp point-coincidence Injective Task primitives Tcol collinearity Tpp Tcopl coplanarity Tcr Projective cross ratios (a) AND OR NOT T1 0 if T = 0 T1 T2 = T1•T2 T1 T2 = T = T2 1 otherwise Task operators

26. 5 6 2 1 8 7 4 3 Result: Task toolkit wrench: view 1 wrench: view 2 Tpp(x1,x5)  Tpp(x3,x7)  Twrench(x1..8) = Tcol(x4,x7,x8)  Tcol(x2,x5,x6)

27. Geometric strata: 3d overview 3DOF 5DOF Faugeras ‘95

28. Perspective and projection • Euclidean geometry is not the only representation - for building models from images - for building images from models • But when 3d realism is the goal, • how can we effectively build and use projective affine metric representations ?