1. Geometric Transformations
CSE 455
Linda Shapiro

2. What are geometric transformations?
Why do we need them?

3. Translation
Preserves: Orientation

4. Translation and rotation

5. Scale

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

7. Aspect ratio

8. Shear

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

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

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
Preserves: Straight Lines

14. 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?

15. Finding the transformation
How can we find the transformation between these images?
How many corresponding points do we need to solve?

16. What can I use homographies for?

17. For one thing: Panoramas