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Homography

  • Projective – mapping between any two PPs with

the same center of projection

– rectangle should map to arbitrary quadrilateral – parallel lines aren’t – but must preserve straight lines – same as: project, rotate, reproject called Homography = 1 y x * * * * * * * * * w y ' w x ' w

To apply a homography H

p’

H

p

PP1

PP2

  • Compute

p

= Hp

(regular matrix multiply)

  • Convert p’ from homogeneous to image

coordinates

1D homography

  • Reproject to different line

w

w=1

x

1D homography

  • Reproject to different line

  • Equivalent to rotating 2D points

Î reprojection is linear in homogeneous coordinates w

w=1

x

1D homogeneous coordinates

  • Add one dimension to make life simpler

  • (x, w) represent point x/w

w

x

1D homography

  • Reproject to different line

w

x

Same in 2D

  • Reprojection = homography

  • 3x3 matrix

p’ w = * * * w y ' w x '

H

* * *

1 y x * * * p

w=1

w=1

PP1

PP2

3

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