The document discusses geometric transformations, including translation, rotation, scale, similarity, and affine transformations, highlighting their degrees of freedom. It introduces projective transformations and outlines the process for computing them using homogeneous coordinates. Additionally, it raises questions about the number of corresponding points needed to determine these transformations.

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?