2. Reconnaissance d’objets et vision artificielle Jean Ponce (ponce@di.ens.fr) http://www.di.ens.fr/~ponce Equipe-projet WILLOW ENS/INRIA/CNRS UMR 8548 Laboratoire d’Informatique Ecole Normale Supérieure, Paris

3. Outline Motivation: Making mosaics Perspetive and weak perspective Coordinates changes Intrinsic and extrensic parameters Affine registration Projective registration

4. Feature-based alignment outline

5. Extract features

6. Feature-based alignment outline Extract features Compute putative matches

7. Feature-based alignment outline Extract features Compute putative matches Loop: Hypothesize transformation T (small group of putative matches that are related by T)

8. Feature-based alignment outline Extract features Compute putative matches Loop: Hypothesize transformation T (small group of putative matches that are related by T) Verify transformation (search for other matches consistent with T)

9. Feature-based alignment outline Extract features Compute putative matches Loop: Hypothesize transformation T (small group of putative matches that are related by T) Verify transformation (search for other matches consistent with T)

10. 2D transformation models Similarity (translation, scale, rotation) Affine Projective (homography) Why these transformations ???

11. Pinhole perspective equation NOTE: z is always negative..

12. Affine models: Weak perspective projection is the magnification. When the scene relief is small compared its distance from the Camera, m can be taken constant: weak perspective projection.

13. Affine models: Orthographic projection When the camera is at a (roughly constant) distance from the scene, take m=1.

14. Analytical camera geometry

15. Coordinate Changes: Pure Translations O B P = O B O A + O A P, B P = A P + B O A

16. Coordinate Changes: Pure Rotations

17. Coordinate Changes: Rotations about the z Axis

18. A rotation matrix is characterized by the following properties: Its inverse is equal to its transpose, and its determinant is equal to 1. Or equivalently: Its rows (or columns) form a right-handed orthonormal coordinate system.

19. Coordinate changes: pure rotations

20. Coordinate Changes: Rigid Transformations

21. Pinhole perspective equation NOTE: z is always negative..

22. The intrinsic parameters of a camera Normalized image coordinates Physical image coordinates Units: k,l : pixel/m f : m  : pixel

23. The intrinsic parameters of a camera Calibration matrix The perspective projection equation

24. The extrinsic parameters of a camera

25. Perspective projections induce projective transformations between planes

26. Weak-perspective projection Paraperspective projection Affine cameras

27. Orthographic projection Parallel projection More affine cameras

28. Weak-perspective projection model r (p and P are in homogeneous coordinates) p = A P + b (neither p nor P is in hom. coordinates) p = M P (P is in homogeneous coordinates)

29. Affine projections induce affine transformations from planes onto their images.

30. Affine transformations An affine transformation maps a parallelogram onto another parallelogram

31. Fitting an affine transformation Assume we know the correspondences, how do we get the transformation?

32. Fitting an affine transformation Linear system with six unknowns Each match gives us two linearly independent equations: need at least three to solve for the transformation parameters

33. Beyond affine transformations What is the transformation between two views of a planar surface? What is the transformation between images from two cameras that share the same center?

34. Perspective projections induce projective transformations between planes

35. Beyond affine transformations Homography: plane projective transformation (transformation taking a quad to another arbitrary quad)

36. Fitting a homography Recall: homogenenous coordinates Converting to homogenenous image coordinates Converting from homogenenous image coordinates

37. Fitting a homography Recall: homogenenous coordinates Equation for homography: Converting to homogenenous image coordinates Converting from homogenenous image coordinates

38. Fitting a homography Equation for homography: 3 equations, only 2 linearly independent 9 entries, 8 degrees of freedom (scale is arbitrary)

39. Direct linear transform H has 8 degrees of freedom (9 parameters, but scale is arbitrary) One match gives us two linearly independent equations Four matches needed for a minimal solution (null space of 8x9 matrix) More than four: homogeneous least squares

40. Application: Panorama stitching Images courtesy of A. Zisserman.