Göm meny

Geometry for Computer Vision 2010 (6hp)

The course starts with two lectures on basic projective, and epipolar geometry. The remaining six lectures cover a selection of more recent techniques and methods that are widely used in the Computer Vision community.

These six lectures are each accompanied by a scientific paper that should be read by the participants before the lecture.


Where possible the lectures are based on the book:
"Multiple View Geometry in Computer Vision" by Hartley and Zisserman . About half the content is however not covered by the book, and is instead based on a number of research papers that will be made available to the course participants.
If you are on campus you can use this link to browse the electronic version


The lectures are held by Klas Nordberg and Per-Erik Forssén .


The lectures will be held on Fridays 9-11 in Algoritmen .
  1. March 19: Introduction to projective spaces, homogeneous coordinates in 2D+3D, points, lines, planes, homographies, camera matrices Klas 2x45min. Slides
  2. March 26: Epipolar geometry: fundamental matrix, triangulation, rectification, degeneracies from planes, transfer Klas 2x45min. Slides
  3. April 9: Estimation theory: DLT, data normalization, algebraic & geometric errors, Maximum-likelihood, RANSAC, Voting, Mean-shift. Per-Erik 2x45min. Slides
  4. April 16: Camera calibration: Zhang's calibration plane, self-calibration.  Oriented epipolar geometry.  Per-Erik  45min Slides .  Multi-body factorization.  Klas  45 min Slides . Articles: [ Mendoca&Cipolla ] [ Costeira&Kanade ]
  5. April 23: Three view geometry: Trifocal tensor, applications, estimations, internal constraints.  Klas 45 min Slides .  Calibrated multi-view geometry: 5-point problem, P3P, bundle adjustment.  Per-Erik 45 min Slides . Article: [ Nistér ]
  6. April 30: 6-point geometry.  Klas  45 min. Slides . LO-RANSAC, preemptive RANSAC, PROSAC & DEGENSAC. Per-Erik  45 min. Slides . Articles: [ Quan ] [ Chum&Matas ]
  7. May 7: Representations of 3D rotations: exp of so(3), quaternions, Rodrigues.  Estimation of 2D and 3D rigid transformations, Horn+Procrustes.  Klas.  45 min Slides .  SLERP, Smoothing of SO(3) and SE(3).  Per-Erik 45 min Slides . [ Shoemake ]
  8. May 21: Rolling shutter and push-broom cameras: geometry and calibration.  Per-Erik  2x45 min Slides . [ Forssén&Ringaby ]


  • Short question seminar on the literature after lectures 4-8.
  • Small programming project (2hp) at the end of the course.  Own or proposed projects.
  • Short (<2h)  written examination on concepts from the lectures.


If you want to participate in the course, notify us by sending an email to perfo at isy.liu.se, before March 19. The email should contain (1) your full name, (2) your personal number, (3) your department or equivalent, (4) the name of your supervisor.
Per-Erik and Klas

Senast uppdaterad: 2014-03-18