More on single-view geometry class 10 Multiple View Geometry Comp 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 (i)Linear solution: (a)Normalization: (b)DLT (ii)Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error: (iii)Denormalization: ~~ ~
More Single-View Geometry Projective cameras and planes, lines, conics and quadrics. Camera calibration and vanishing points, calibrating conic and the IAC
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)
Action of projective camera on lines forward projection back-projection
Action of projective camera on conics back-projection to cone example:
Images of smooth surfaces The contour generator is the set of points X on S at which rays are tangent to the surface. The corresponding apparent contour is the set of points x which are the image of X, i.e. is the image of The contour generator depends only on position of projection center, depends also on rest of P
Action of projective camera on quadrics back-projection to cone The plane of for a quadric Q is camera center C is given by =QC (follows from pole-polar relation) The cone with vertex V and tangent to the quadric Q is
The importance of the camera center
Moving the image plane (zooming)
Camera rotation conjugate rotation
Synthetic view (i)Compute the homography that warps some a rectangle to the correct aspect ratio (ii)warp the image
Planar homography mosaicing
more examples
Projective (reduced) notation
Moving the camera center motion parallax epipolar line
What does calibration give? An image line l defines a plane through the camera center with normal n=K T l measured in the camera’s Euclidean frame
The image of the absolute conic mapping between ∞ to an image is given by the planar homogaphy x=Hd, with H=KR image of the absolute conic (IAC) (i)IAC depends only on intrinsics (ii)angle between two rays (iii)DIAC= * =KK T (iv) K (cholesky factorisation) (v)image of circular points
A simple calibration device (i)compute H for each square (corners (0,0),(1,0),(0,1),(1,1)) (ii)compute the imaged circular points H(1,±i,0) T (iii)fit a conic to 6 circular points (iv)compute K from through cholesky factorization (= Zhang’s calibration method)
Orthogonality =conjuacy and pole-polar w.r.t. IAC
The calibrating conic
Vanishing points
Vanishing lines
Orthogonality relation
Calibration from vanishing points and lines
Next class: Two-view geometry Epipolar geometry