几何变换

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陈成

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是指从具有几何结构之集合至其自身或其他此类集合的一种对射。几何变换是一种数学解题的方法思路。在几何的解题中，当题目给出的条件显得不够或者不明显时，我们可以将图形作一定的变换，这样将有利于发现问题的隐含条件，使问题得以突破。

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1 .Geometric Transformations EE/CSE 576 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 D oF ) = translation + rotation + scale + aspect ratio + shear Preserves: Parallelism

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 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? 16

17 .For one thing: Panoramas 17

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