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"
       Spring 2006            Projective Geometry 2D
 Friday, February 5, 2010
                  Homography
                                                 ..to map one 3D 
                                                 “plane” to “another 
                                                 3D plane”
 Friday, February 5, 2010
         Projective transformations
         Definition:     A projectivity is an invertible mapping h from P2 to itself 
                         such that three points x ,x ,x  lie on the same line if and 
                                                  1  2  3
                         only if h(x ),h(x ),h(x ) do.
                                   1      2     3
            ′                          
              x         h      h     h        x
               1          11    12     13      1
            ′                          
              x    = h21 h22 h23              x2
               2
               ′
              x         h31    h32   h33      x3
               3
                                         Projective Geometry 2D                  3
 Friday, February 5, 2010
         Projective transformations
         Definition:     A projectivity is an invertible mapping h from P2 to itself 
                         such that three points x ,x ,x  lie on the same line if and 
                                                  1  2  3
                         only if h(x ),h(x ),h(x ) do.
                                   1      2     3
         Theorem:
                             2    2
             A mapping h:P →P  is a projectivity if and only if there 
             exist a non-singular 3x3 matrix H such that for any point 
             in P2 represented by a vector x it is true that h(x)=Hx
            ′                          
              x         h      h     h        x
               1          11    12     13      1
            ′                          
              x    = h21 h22 h23              x2
               2
               ′
              x         h31    h32   h33      x3
               3
                                         Projective Geometry 2D                  3
 Friday, February 5, 2010