1 / 31

More on single-view geometry class 10

More on single-view geometry class 10. Multiple View Geometry Comp 290-089 Marc Pollefeys. Gold Standard algorithm. Objective Given n≥6 2D to 2D point correspondences {X i ↔x i ’}, determine the Maximum Likelyhood Estimation of P Algorithm Linear solution: Normalization: DLT

driscoll-short
Download Presentation

More on single-view geometry class 10

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. More on single-view geometryclass 10 Multiple View Geometry Comp 290-089 Marc Pollefeys

  2. Gold Standard algorithm • Objective • Given n≥6 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P • Algorithm • Linear solution: • Normalization: • DLT • Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error: • Denormalization: ~ ~ ~

  3. More Single-View Geometry • Projective cameras and planes, lines, conics and quadrics. • Camera calibration and vanishing points, calibrating conic and the IAC

  4. Action of projective camera on planes The most general transformation that can occur between a scene plane and an image plane under perspective imaging is a plane projective transformation (affine camera-affine transformation)

  5. Action of projective camera on lines forward projection back-projection

  6. Action of projective camera on conics back-projection to cone example:

  7. Images of smooth surfaces The contour generator G is the set of points X on S at which rays are tangent to the surface. The corresponding apparent contour g is the set of points x which are the image of X, i.e. g is the image of G The contour generator G depends only on position of projection center, g depends also on rest of P

  8. Action of projective camera on quadrics back-projection to cone The plane of G for a quadric Q is camera center C is given by P=QC (follows from pole-polar relation) The cone with vertex V and tangent to the quadric Q is

  9. The importance of the camera center

  10. Moving the image plane (zooming)

  11. Camera rotation conjugate rotation

  12. Synthetic view • Compute the homography that warps some a rectangle to the correct aspect ratio • warp the image

  13. Planar homography mosaicing

  14. Planar homography mosaicing more examples

  15. Projective (reduced) notation

  16. Moving the camera center motion parallax epipolar line

  17. What does calibration give? An image line l defines a plane through the camera center with normal n=KTl measured in the camera’s Euclidean frame

  18. The image of the absolute conic mapping between p∞ to an image is given by the planar homogaphy x=Hd, with H=KR image of the absolute conic (IAC) • IAC depends only on intrinsics • angle between two rays • DIAC=w*=KKT • w  K (cholesky factorisation) • image of circular points

  19. A simple calibration device • compute H for each square • (corners  (0,0),(1,0),(0,1),(1,1)) • compute the imaged circular points H(1,±i,0)T • fit a conic to 6 circular points • compute K from w through cholesky factorization (= Zhang’s calibration method)

  20. Orthogonality =conjuacy and pole-polar w.r.t. IAC

  21. The calibrating conic

  22. Vanishing points

  23. Vanishing lines

  24. Orthogonality relation

  25. Calibration from vanishing points and lines

  26. Calibration from vanishing points and lines

  27. Next class: Two-view geometryEpipolar geometry

More Related