1 / 29

From previous lectures …

From previous lectures …. Transformation groups. Affine transformation. Not a linear transformation! Can be made linear in HOMOGENEOUS COORDINATES. MEMENTO! will appear everywhere. Affine group (contd.). Composition of affine transformations. What is the inverse transformation?.

zev
Download Presentation

From previous lectures …

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. From previous lectures … UCLA Vision Lab

  2. Transformation groups UCLA Vision Lab

  3. Affine transformation • Not a linear transformation! • Can be made linear in HOMOGENEOUS COORDINATES MEMENTO! will appear everywhere UCLA Vision Lab

  4. Affine group (contd.) • Composition of affine transformations. • What is the inverse transformation? UCLA Vision Lab

  5. Orthogonal group • What is the set of transformations that preserve the inner product? • Remember inner product under a transformation? • More on this later … UCLA Vision Lab

  6. Euclidean group UCLA Vision Lab

  7. Gram-Schmidt orthogonalization MEMENTO! will appear in calibration (aka Q-R) Structure of the Parameter matrix UCLA Vision Lab

  8. Symmetric matrices UCLA Vision Lab

  9. Symmetric matrices (contd.) UCLA Vision Lab

  10. Skew-symmetric matrices UCLA Vision Lab

  11. Skew-symmetric matrices (contd.) UCLA Vision Lab

  12. Lecture 3: modeling the geometry of the 3-D world UCLA Vision Lab

  13. Representation of scene motion UCLA Vision Lab

  14. Notation • Points and vectors are different! • Source of confusion! UCLA Vision Lab

  15. Inner product and cross product UCLA Vision Lab

  16. Cross product, skew-symmetry and vee UCLA Vision Lab

  17. Rigid body motion, or Euclidean transformations • A transformation that preserves distances, angles and orientation • Preserves norm, inner product, cross product • Polarization identity • Preserve norm and cross product UCLA Vision Lab

  18. Representation of rigid motions UCLA Vision Lab

  19. Rotations Orthogonal change of coordinates Collect coordinates of one reference frame relative to the other into a matrix R UCLA Vision Lab

  20. Rotations (contd.) • How does a rotation transform points? UCLA Vision Lab

  21. Canonical exponential coordinates • SO(3) has 3 DOF, but is represented with 9 numbers? • i.e. it is skew-symmetric! UCLA Vision Lab

  22. Canonical exponential coordinates (contd.) UCLA Vision Lab

  23. Rodrigues’ formula • Beautiful! • Toss Euler angles, Y-P-R, quaternions etc. • Check Matlab code in laboratory exercises UCLA Vision Lab

  24. General rigid motion (R, T) UCLA Vision Lab

  25. Homogeneous representation • Points • Vectors • Transformation • representation UCLA Vision Lab

  26. Questions: • What is the action of a rigid body motion on a vector? • What is the representation of the inverse action? UCLA Vision Lab

  27. Exponential coordinates for rigid body motions UCLA Vision Lab

  28. Rodrigues’ formula for rigid body motion UCLA Vision Lab

  29. Summary UCLA Vision Lab

More Related