1. Geometric Transformations CSE P 576 Larry Zitnick (larryz@microsoft.com)

2. What are geometric transformations?

3. Translation

4. Translation and rotation

5. Scale

6. Similarity transformations Similarity transform (4 DoF) = translation + rotation + scale

7. Aspect ratio

8. Shear

9. Affine transformations Affine transform (6 DoF) = translation + rotation + scale + aspect ratio + shear

10. What is missing? Are there any other planar transformations? Canaletto

11. General affine We already used these How do we compute projective transformations?

12. Homogeneous coordinates One extra step:

13. Projective transformations a.k.a. Homographies “keystone” distortions

14. Finding the transformation How can we find the transformation between these images?

15. Finding the transformation Translation = 2 degrees of freedom Similarity = 4 degrees of freedom Affine = 6 degrees of freedom Homography = 8 degrees of freedom How many corresponding points do we need to solve?