Geometric Computer Vision

1. Geometric Computer Vision Marc Pollefeys Fall 2009 http://www.inf.ethz.ch/personal/pomarc/courses/gcv/

2. Geometric Computer Vision course schedule(tentative)

3. From Last Class Varing focus does change the focal length for our image formation (probably very slightly).

4. Single View MetrologyClass 3

5. Single View Metrology

6. Measuring in a plane Need to compute H as well as uncertainty

7. Direct Linear Transformation(DLT)

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

9. 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

10. 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

11. 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

12. Importance of normalization 1 ~104 ~102 ~102 ~102 ~102 ~102 1 ~104 orders of magnitude difference! Monte Carlo simulation for identity computation based on 5 points (not normalized ↔ normalized)

13. Normalized DLT algorithm • Objective • Given n≥4 2D to 2D point correspondences {xi↔xi’}, determine the 2D homography matrix H such that xi’=Hxi • Algorithm • Normalize points • Apply DLT algorithm to • Denormalize solution

14. measured coordinates estimated coordinates true coordinates Error in one image Symmetric transfer error Reprojection error Geometric distance d(.,.) Euclidean distance (in image) e.g. calibration pattern

15. Reprojection error

16. Maximum Likelihood Estimate Statistical cost function and Maximum Likelihood Estimation • Optimal cost function related to noise model • Assume zero-mean isotropic Gaussian noise (assume outliers removed) Error in one image

17. Maximum Likelihood Estimate Statistical cost function and Maximum Likelihood Estimation • Optimal cost function related to noise model • Assume zero-mean isotropic Gaussian noise (assume outliers removed) Error in both images

18. Gold Standard algorithm • Objective • Given n≥4 2D to 2D point correspondences {xi↔xi’}, determine the Maximum Likelyhood Estimation of H • (this also implies computing optimal xi’=Hxi) • Algorithm • Initialization: compute an initial estimate using normalized DLT or RANSAC • Geometric minimization of reprojection error: • ● Minimize using Levenberg-Marquardt over 9 entries of h • or Gold Standard error: • ● compute initial estimate for optimal {xi} • ● minimize cost over {H,x1,x2,…,xn} • ● if many points, use sparse method

19. Uncertainty: error in one image • Estimate the transformation from the data • Compute Jacobian , evaluated at • The covariance matrix of the estimated is given by

20. Uncertainty: error in both images separate in homography and point parameters

21. Error in two images (if h and x independent, i.e. new points) Using covariance matrix in point transfer Error in one image

22. Example: s=1 pixel S=0.5cm (Criminisi’97)

23. Example: s=1 pixel S=0.5cm (Criminisi’97)

24. Example: (Criminisi’97)

25. Monte Carlo estimation of covariance • To be used when previous assumptions do not hold (e.g. non-flat within variance) or to complicate to compute. • Simple and general, but expensive • Generate samples according to assumed noise distribution, carry out computations, observe distribution of result

26. Single view measurements:3D scene

27. Background: Affine Geometry • Invariant properties: Parallelism, ratio of areas, ratio of lengths on collinear or parallel lines • Invariant properties from projective transformation: cross ratio

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

29. Vanishing points • Under perspective projection points at infinity can have a finite image • The projection of 3D parallel lines intersect at vanishing points in the image

30. Basic geometry

31. Basic geometry • Allows to relate height of point to height of camera

32. Homology mapping between parallel planes • Allows to transfer point from one plane to another

33. Single view measurements

34. Single view measurements

35. Forensic applications 190.6±2.9 cm 190.6±4.1 cm A. Criminisi, I. Reid, and A. Zisserman. Computing 3D euclidean distance from a single view. Technical Report OUEL 2158/98, Dept. Eng. Science, University of Oxford, 1998.

36. Example courtesy of Antonio Criminisi

37. La Flagellazione di Cristo (1460) Galleria Nazionale delle Marche by Piero della Francesca (1416-1492) http://www.robots.ox.ac.uk/~vgg/projects/SingleView/

38. More interesting stuff • Criminisi demo http://www.robots.ox.ac.uk/~vgg/presentations/spie98/criminis/index.html • work by Derek Hoiem on learning single view 3D structure and apps http://www.cs.cmu.edu/~dhoiem/ • similar work by Ashutosh Saxena on learning single view depth http://ai.stanford.edu/~asaxena/learningdepth/

39. Next class • Feature tracking and matching