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projective-geometry

# projective-geometry - Projective geometry Ames Room Slides...

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Projective geometry Ames Room Slides from Steve Seitz and Daniel DeMenthon

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Projective geometry—what’s it good for? Uses of projective geometry Drawing Measurements Mathematics for projection Undistorting images Focus of expansion Camera pose estimation, match move Object recognition

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Applications of projective geometry Vermeer’s Music Lesson
1 2 3 4 1 2 3 4 Measurements on planes Approach: unwarp then measure What kind of warp is this?

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Image rectification To unwarp (rectify) an image solve for homography H given p and p’ solve equations of the form: w p’ = Hp – linear in unknowns: w and coefficients of H H is defined up to an arbitrary scale factor how many points are necessary to solve for H ? p p’
Solving for homographies

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Solving for homographies A h 0 Linear least squares Since h is only defined up to scale, solve for unit vector ĥ Minimize 2n × 9 9 2n Solution: ĥ = eigenvector of A T A with smallest eigenvalue Works with 4 or more points
(0,0,0) The projective plane Why do we need homogeneous coordinates? represent points at infinity, homographies, perspective projection, multi-view relationships What is the geometric intuition? a point in the image is a ray in projective space (sx,sy,s) Each point (x,y) on the plane is represented by a ray (sx,sy,s) all points on the ray are equivalent: (x, y, 1) 2245 (sx, sy, s) image plane (x,y,1) y x z

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Projective lines What does a line in the image correspond to in projective space? A line is a plane of rays through origin all rays (x,y,z) satisfying: ax + by + cz = 0 [ ] = z y x c b a 0 : notation vector in A line is also represented as a homogeneous 3-vector l l p
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projective-geometry - Projective geometry Ames Room Slides...

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