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Unformatted text preview: Spring 2006 Projective Geometry 2D Projective geometry 2D 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, &quot; Multiple View Geometry in Computer Vision &quot; Wednesday, February 3, 2010 ..to map one 3D plane to another 3D plane Homography Wednesday, February 3, 2010 y [pixels] x [pixels] ! 253 ! 55 143 342 540 738 936 1135 1333 1531 ! 94 ! 15 64 144 223 302 381 461 540 619 Wednesday, February 3, 2010 ! 178 ! 105 ! 32 41 114 188 261 334 407 480 Wednesday, February 3, 2010 Wednesday, February 3, 2010 http://thebrain.mcgill.ca / Mapping from retina to visual cortex ..in perspective Wednesday, February 3, 2010 http://thebrain.mcgill.ca / left eye Wednesday, February 3, 2010 Projective Geometry 2D How? Compute the keypoints Construct a description of these keypoints (including location information). E.g., a SIFT descriptor Match a set of keypoints from one image to a set of points from the other image. you need a matching criterion. Example: use Euclidean distance in the 128dimensional feature space. prune the matches based on a model. Example. use the homography model. Using the matched points, compute the model parameters reproject, align, and blend the images together to create a larger panorama. 8 Wednesday, February 3, 2010 Projective Geometry 2D This lecture we learn about such geometric projection models begin with homogeneous representation of points in space projective space and the homography model .. and more. 9 Wednesday, February 3, 2010 Projective Geometry 2D 10 Homogeneous coordinates Homogeneous representation of lines equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3 (0,0,0) T forms P 2 Wednesday, February 3, 2010 Projective Geometry 2D 10 Homogeneous coordinates Homogeneous representation of lines equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3 (0,0,0) T forms P 2 Homogeneous representation of points on if and only if Wednesday, February 3, 2010 Projective Geometry 2D...
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 Fall '08
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