Geometric Transformations

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Presentation transcript:

Geometric Transformations EE/CSE 576 Linda Shapiro

What are geometric transformations? Why do we need them?

Translation Preserves: Orientation

Translation and rotation

Scale

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

Aspect ratio

Shear

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

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

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

Homogeneous coordinates One extra step:

Projective transformations a.k.a. Homographies “keystone” distortions Preserves: Straight Lines

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?

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

What can I use homographies for?

For one thing: Panoramas