1 / 26

Recovery of affine and metric properties from images in 2D Projective space

Recovery of affine and metric properties from images in 2D Projective space. 2013-03-20 Ko Dae -Won. Recovery of affine and metric properties from images in 2D Projective space. Affine properties(line at infinity) Parallelism Parallel length ratios Metric properties(circular points)

efrem
Download Presentation

Recovery of affine and metric properties from images in 2D Projective space

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Recovery of affine and metric properties from images in 2D Projective space 2013-03-20 KoDae-Won

  2. Recovery of affine and metric properties from images in 2D Projective space Affine properties(line at infinity) • Parallelism • Parallel length ratios Metric properties(circular points) • Angles • Length ratios Recover the original shape

  3. Homogeneous coordinates but only 2DOF Inhomogeneous coordinates Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Homogeneous coordinates equivalence class of vectors, any vector is representative Set of all equivalence classes in R3(0,0,0)T forms P2 Homogeneous representation of points on if and only if The point x lies on the line l if and only if xTl=lTx=0

  4. Line joining two points The line through two points and is Ideal points Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Points from lines and vice-versa Intersections of lines The intersection of two lines and is Intersections of parallel lines Line at infinity

  5. Duality principle: To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Duality

  6. or homogenized or in matrix form with Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Conics Curve described by 2nd-degree equation in the plane

  7. Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Tangent lines to conics The line l tangent to C at point x on C is given by l=Cx l x C

  8. In general : Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Dual conics A line tangent to the conic C satisfies Dual conics = line conics = conic envelopes

  9. Theorem: A mapping h:P2P2is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2represented by a vector x it is true that h(x)=Hx Definition: Projective transformation or Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Projective transformations Definition: A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3lie on the same line if and only if h(x1),h(x2),h(x3) do. projectivity=collineation=projective transformation=homography

  10. Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties The line at infinity

  11. Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Affine properties from images

  12. Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties

  13. Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties

  14. Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Distance ratio

  15. Recovery of affine and metric properties from images in 2D Projective space 1. Recovery of affine properties Distance ratio

  16. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties

  17. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties The circular points

  18. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties The circular points

  19. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties Conic dual to the circular points

  20. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties Angles

  21. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties Length ratios

  22. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties

  23. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties Length ratios

  24. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties Metric from affine

  25. Recovery of affine and metric properties from images in 2D Projective space 2. Recovery of metric properties Metric from projective

More Related