Content

1. ROBOT VISIONLesson 3: Projective GeometryMatthias RütherSlides courtesy of Marc Pollefeys Department of Computer ScienceUniversity of North Carolina, Chapel Hill

2. Content • Projective geometry in 3D • Points • Planes • Lines • Points from planes, planes from points • Hierarchy of transformations / invariants • Parameter Estimation • Problem: intersecting lines -> SVD • Problem: rectification • Normalization • Robust Estimation

3. Projective 3D Geometry • Points, lines, planes and quadrics • Transformations

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

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

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

7. (solve as right nullspace of ) Points from planes Representing a plane by its span

8. Lines (4dof) Example: X-axis

9. Points, lines and planes

10. Plücker matrices Plücker matrix (4x4 skew-symmetric homogeneous matrix) • L has rank 2 • 4dof • generalization of • L independent of choice A and B • Transformation Example: x-axis

11. Plücker matrices Dual Plücker matrix L* Correspondence Join and incidence (plane through point and line) (point on line) (intersection point of plane and line) (line in plane) (coplanar lines)

12. Estimation • 2D Problem: Lines l1, l2, l3 intersect in a point l1 ?? Point of intersection x x l2 l3 • Solve equation system: How many solutions exist? When does a solution exist? Geometric interpretation?

13. Estimation • Practical 2D Problem: measured lines l1, l2, l3, … lmapproximately intersect in a point x l1 ?? Point of intersection x x l2 l3 lm • Find an approximate solution to the system: Find a solution that minimizes |Lx|.

14. Singular Value Decomposition (SVD) • SVD: algorithm that decomposes Amxn (mn) to • Properties: • U has orthogonal columns: UTU = I • U is norm-preserving: |Ux| = |x| • D is diagonal, diagonal elements are the singular values • V is orthogonal: VTV = I

15. Singular Value Decomposition (SVD) • Solve Ax=0  minimize |Ax| subject to |x|=1: • minimize |DVTx| subject to |VTx|=1 1 is in the same row as the smallest singular value in D. X is the column of V corresponding to the smallest Singular Value.

16. Robust Estimation • Practical 2D Problem: SVD obtains an optimal least squares solution: l1 x l2 l3 • Problem with outliers: l4 l1 x l2 l3

17. RANdom SAmple Consensus • RANSAC algorithm: given a set of lines l1,…,lm, find an intersection point x such that |lkx| is minimal for the largest subset of k=1..m, subject to the condition that none of the valid lines deviates from x by more than t. lm l4 l1 t x l2 l3

18. RANSAC • Objective • Robust fit of model to data set S which contains outliers • Algorithm • Randomly select a sample of s data points from S and instantiate the model from this subset. • Determine the set of data points Si which are within a distance threshold t of the model. The set Si is the consensus set of samples and defines the inliers of S. • If the subset of Si is greater than some threshold T, re-estimate the model using all the points in Si and terminate • If the size of Si is less than T, select a new subset and repeat the above. • After N trials the largest consensus set Si is selected, and the model is re-estimated using all the points in the subset Si

19. Parameter estimation • 2D homography Given a set of (xi,xi’), compute H (xi’=Hxi) • 3D to 2D camera projection Given a set of (Xi,xi), compute P (xi=PXi) • Fundamental matrix Given a set of (xi,xi’), compute F (xi’TFxi=0) x3 x3‘ x4‘ x4 ?H? x2 x1 x1‘ x2‘

20. Number of measurements required • At least as many independent equations as degrees of freedom required • Example: 2 independent equations / point 8 degrees of freedom 4x2≥8

21. Approximate solutions • Minimal solution 4 points yield an exact solution for H • More points • No exact solution, because measurements are inexact (“noise”) • Search for “best” according to some cost function • Algebraic or geometric/statistical cost

22. Gold Standard algorithm • Cost function that is optimal for some assumptions • Computational algorithm that minimizes it is called “Gold Standard” algorithm • Other algorithms can then be compared to it

23. Direct Linear Transformation(DLT)

24. Direct Linear Transformation(DLT) • Equations are linear in h • Only 2 out of 3 rows are linearly independent (indeed, 2 eq/pt) (only drop third row if wi’≠0) • Holds for any homogeneous representation, e.g. (xi’,yi’,1)

25. Direct Linear Transformation(DLT) • Solving for H size A is 8x9 or 12x9, but rank 8 Trivial solution is h=09T is not interesting 1-D null-space yields solution of interest pick for example the one with

26. Direct Linear Transformation(DLT) • Over-determined solution No exact solution because of inexact measurement i.e. “noise” • Find approximate solution • Additional constraint needed to avoid 0, e.g. • not possible, so minimize

27. DLT algorithm • Objective • Given n≥4 2D to 2D point correspondences {xi↔xi’}, determine the 2D homography matrix H such that xi’=Hxi • Algorithm • For each correspondence xi ↔xi’ compute Ai. Usually only two first rows needed. • Assemble n 2x9 matrices Ai into a single 2nx9 matrix A • Obtain SVD of A. Solution for h is last column of V • Determine H from h

28. Inhomogeneous solution Since h can only be computed up to scale, pick hj=1, e.g. h9=1, and solve for 8-vector Solve using Gaussian elimination (4 points) or using linear least-squares (more than 4 points) However, if h9=0 this approach fails also poor results if h9 close to zero Therefore, not recommended Note h9=H33=0 if origin is mapped to infinity