Göm meny

Geometry for Computer Vision 2014 (6hp)

The course topics are representations for various types of geometric objects in geometry (such as points, lines, transformations, constraints) and how to estimatimate these representations from measurments in images, typically of point coordinates.

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.


Some of the material in can be found in either of these two books:
Other parts are covered in articles or conference papers that are linked in the lecture plan.


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


The first lecture is on Tuesday April 29, 10:15 - 12:00 in Seminar room Signalen (Buidling B, map). The subsequent lectures will be one per week, on Tuesday 10:15 - 12:00, in Signalen, except on Tuesday June 3 when the lecture is in Algoritmen.

Preliminary plan:
Lecture Content Literature
1 Introduction to projective spaces, homogeneous coordinates in 2D+3D, points, lines, planes, transformations, homographies, camera matrices. Klas IREG: ch 1 - 7
HZ: ch 2, 3, 4, 6
2 Epipolar geometry: fundamental matrix, triangulation, rectification, degeneracies from planes, transfer. Klas IREG: ch 9
HZ: ch 9
3 Estimation theory: DLT, data normalization, algebraic & geometric errors, Maximum-likelihood, RANSAC, Generalized Hough Transform, Mean-shift. Klas IREG: ch 10 - 15
HZ: ch 4, 11
4 Camera calibration: Zhang's calibration plane, self-calibration. Oriented epipolar geometry. Per-Erik
IREG: ch 16
Chum, Werner, Matas
Affine geometry, multi-body factorization, LCV. Klas Costeira&Kanade
5 Mutli-view and multi-point geometry. Trifocal tensor. 6-point geometry. Klas
HZ: ch 15 - 7
Calibrated multi-view geometry: 5-point problem, bundle adjustment. Per-Erik IREG: ch 14.3, 17
HZ: ch 9
Nistér CVPR03
Triggs et al.
6 Relative pose, estimation of rigid transformations, Procrustes. PnP, P3P. Klas
IREG: ch 13.4
Kneip, Scaramuzza, Siegwart
Sample consensus strategies: LO-RANSAC, preemptive RANSAC, PROSAC & DEGENSAC. Per-Erik Chum&Matas
7 Representations of 3D rotations: exp of so(3), quaternions, Rodrigues. Klas
IREG: ch 8
HZ: ch A4.3
Smoothing of SO(3) and SE(3). SLeRP, and splines on SO(3). Per-Erik Shoemake
Kim, Kim, Shin
8 Rolling shutter and push-broom cameras: geometry and calibration. Per-Erik Ringaby&Forssén


  • Short question seminars on the literature after lectures 4-8.
  • Short (<2h) written examination on concepts from the lectures. See this old exam for an example of what type of questions to expect.
  • Small programming project (2hp) at the end of the course. Own or proposed projects.
Seminar participation plus exam pass is worth 4hp. An additional 2hp can be obtained by doing a programming project.


If you want to participate in the course, notify us by sending an email to klas.nordberg at liu.se, before March 21. The email should contain (1) your full name, (2) your personal number, (3) your department or equivalent, (4) the name of your supervisor.
Klas and Per-Erik

Senast uppdaterad: 2014-10-22