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Geometric Transformations ECE/CSE 576 Linda Shapiro
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Page 1: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Geometric Transformations

ECE/CSE 576Linda Shapiro

Page 2: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

What are geometric transformations?

Why do we need them?

Page 3: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Translation

Preserves: Orientation

Page 4: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Translation and rotation

Page 5: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Scale

Page 6: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Similarity transformations

Similarity transform (4 DoF) = translation + rotation + scale

Preserves: Angles

Page 7: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Aspect ratio

Page 8: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Shear

Page 9: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Affine transformations

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

Preserves: Parallelism

Page 10: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

What is missing?

Are there any other planar transformations?

Canaletto

Page 11: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

General affine

We already used these

How do we compute projective transformations?

Page 12: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Homogeneous coordinates

One extra step:

Page 13: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Projective transformationsa.k.a. Homographies

“keystone” distortions

Preserves: Straight Lines

Page 14: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Finding the transformation

Translation = 2 degrees of freedomSimilarity = 4 degrees of freedomAffine = 6 degrees of freedomHomography = 8 degrees of freedom

How many corresponding points do we need to solve?

Page 15: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

Finding the transformation

- How can we find the transformation between these images?

- How many corresponding points do we need to solve?

Page 16: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

What can I use homographies for?

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Page 17: Geometric Transformations - University of Washington · 2019-04-02 · Geometric Transformations ECE/CSE 576 Linda Shapiro. What are geometric transformations? Why do we need them?

For one thing: Panoramas

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