More on single-view geometry class 10 Multiple View Geometry Comp 290-089 Marc Pollefeys

Multiple View Geometry course schedule (subject to change) Jan. 7, 9 Intro & motivation Projective 2D Geometry Jan. 14, 16 (no class) Jan. 21, 23 Projective 3D Geometry Jan. 28, 30 Parameter Estimation Feb. 4, 6 Algorithm Evaluation Camera Models Feb. 11, 13 Camera Calibration Single View Geometry Feb. 18, 20 Epipolar Geometry 3D reconstruction Feb. 25, 27 Fund. Matrix Comp. Structure Comp. Mar. 4, 6 Planes & Homographies Trifocal Tensor Mar. 18, 20 Three View Reconstruction Multiple View Geometry Mar. 25, 27 MultipleView Reconstruction Bundle adjustment Apr. 1, 3 Auto-Calibration Papers Apr. 8, 10 Dynamic SfM Apr. 15, 17 Cheirality Apr. 22, 24 Duality Project Demos

Single view geometry Camera model Camera calibration Single view geom.

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: ~ ~ ~

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 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

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 the degenerate Quadric:

The importance of the camera center

Moving the image plane (zooming)

Camera rotation conjugate rotation

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

Planar homography mosaicing

close-up: interlacing can be important problem!

Planar homography mosaicing more examples

Projective (reduced) notation

Moving the camera center motion parallax epipolar line

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

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

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)

Orthogonality = pole-polar w.r.t. IAC

The calibrating conic

Vanishing points

ML estimate of a vanishing point from imaged parallel scene lines

Vanishing lines

Orthogonality relation

Five constraints gives us five equations and can determine w

Assumes zero skew, square pixels and 3 orthogonal vanishing points Calibration from vanishing points and lines Principal point is the orthocenter of the trinagle made of 3 orthogonol vanishing lines Assumes zero skew, square pixels and 3 orthogonal vanishing points

Assume zero skew, square pixels,  calibrating conic is a circle; How to find it, so that we can get K?

Assume zero skew , square pixels, and principal point is at the image center Then IAC is diagonal{1/f^2, 1/f^2,1) i.e. one degree of freedom need one more Constraint to determine f, the focal length  two vanishing points corresponding To orthogonal directions.

Next class: Two-view geometry Epipolar geometry