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Reconnaissance d’objets et vision artificielle

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 Sup érieure, Paris. Outline. Motivation: Making mosaics Perspetive and weak perspective

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Reconnaissance d’objets et vision artificielle

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  1. 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

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

  3. Feature-based alignment outline

  4. Feature-based alignment outline • Extract features

  5. Feature-based alignment outline • Extract features • Compute putative matches

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

  7. 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)

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

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

  11. 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.

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

  13. Analytical camera geometry

  14. Coordinate Changes: Pure Translations OBP = OBOA + OAP ,BP = AP + BOA

  15. Coordinate Changes: Pure Rotations

  16. Coordinate Changes: Rotations about the zAxis

  17. 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.

  18. Coordinate changes: pure rotations

  19. Coordinate Changes: Rigid Transformations

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

  21. The intrinsic parameters of a camera Units: k,l :pixel/m f :m a,b : pixel Physical image coordinates Normalized image coordinates

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

  23. The extrinsic parameters of a camera

  24. Perspective projections induce projective transformations between planes

  25. Affine cameras Weak-perspective projection Paraperspectiveprojection

  26. More affine cameras Orthographic projection Parallel projection

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

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

  29. Affine transformations • An affine transformation maps a parallelogram onto • another parallelogram

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

  31. 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

  32. 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?

  33. Perspective projections induce projective transformations between planes

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

  35. Fitting a homography • Recall: homogenenous coordinates Converting from homogenenousimage coordinates Converting to homogenenousimage coordinates

  36. Fitting a homography • Recall: homogenenous coordinates • Equation for homography: Converting from homogenenousimage coordinates Converting to homogenenousimage coordinates

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

  38. 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

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

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