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08 ProjGeom

# 08 ProjGeom - Projective Transformations Acknowledgements...

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Spring 2006 Projective Geometry 2D Projective Transformations Acknowledgements Marc Pollefeys: for allowing the use of his excellent slides on this topic http://www.cs.unc.edu/~marc/mvg/ Richard Hartley and Andrew Zisserman, " Multiple View Geometry in Computer Vision " Friday, February 5, 2010

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..to map one 3D “plane” to “another 3D plane” Homography Friday, February 5, 2010
x 1 x 2 x 3 = h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 x 1 x 2 x 3 Projective Geometry 2D 3 Projective transformations A projectivity is an invertible mapping h from P 2 to itself such that three points x 1 ,x 2 ,x 3 lie on the same line if and only if h (x 1 ), h (x 2 ), h (x 3 ) do. Definition: Friday, February 5, 2010

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x 1 x 2 x 3 = h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 x 1 x 2 x 3 Projective Geometry 2D 3 Projective transformations A projectivity is an invertible mapping h from P 2 to itself such that three points x 1 ,x 2 ,x 3 lie on the same line if and only if h (x 1 ), h (x 2 ), h (x 3 ) do. Definition: A mapping h : P 2 P 2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P 2 represented by a vector x it is true that h (x)= H x Theorem: Friday, February 5, 2010
x 1 x 2 x 3 = h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 x 1 x 2 x 3 Projective Geometry 2D 3 Projective transformations A projectivity is an invertible mapping h from P 2 to itself such that three points x 1 ,x 2 ,x 3 lie on the same line if and only if h (x 1 ), h (x 2 ), h (x 3 ) do. Definition: A mapping h : P 2 P 2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P 2 represented by a vector x it is true that h (x)= H x Theorem: Definition: Projective transformation or 8DOF Friday, February 5, 2010

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x 1 x 2 x 3 = h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 x 1 x 2 x 3
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08 ProjGeom - Projective Transformations Acknowledgements...

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