AE4M33UGR - Uncertain Geometric Reasoning

March, 21-25 2011

Times: lectures: 13:00-14:30 in G205 and exercises: 16:15-17:45 in G3 (daily)

Location: Department of Cybernetics, Czech Technical University in Prague, Karlovo namesti 13, buiding G

Course Description

Lectures of this intensive course provide an introduction into the concepts of uncertain geometric reasoning using projective entities with applications in Computer Vision. They cover aspects such as the representation of uncertain projective entities, the uncertainty propagation, performing statistical testing of geometric relations and optimal estimation of geometric entities and transformations.

Five lectures per 90 minutes will be given during a single week of 21-25. March 2011. Software, short exercises parallel to the lectures and a project will be provided. The lectures will be given by a leading expert in the field, Prof. Wolfgang Förstner, University of Bonn, Germany.


Geometric computations in Computer Vision cover a large range of applications, such as calibration, orientation, reconstruction and grouping. They all are based on image features, mostly points or line segments. These are uncertain e. g. due to image noise, imperfection of their definition and possibly due to the suboptimality of the image analysis procedures. If this uncertainty is modelled statistically one can track the uncertainty of the basic features through the geometric reasoning chain, which consists of the derivation of new geometric entities and decisions based on expected geometric relations. As the tools from projective geometry ease geometric reasoning integrating projective geometry and statistics appears to be of great advantage. This integration conceptually is not straight forward: e. g. homogeneous entities have a free scale, fixing the scale leads to singular covariance matrices, the propagation of uncertainty of non-linear functions leads to distributions which, even when starting from a Gaussian distribution, are non-trivial, large estimation problems using homogeneous entities have to introduce constraints, directly or indirectly.


  • Linear algebra (eigenvalues, singular values, determinants, nullspace, Kronecker product and vec-operator)
  • Probability theory and statistics (continuous pdf’s (Gauss, chi-square), variance propagation, classical hypothesis testing)
  • Projective geometry (homogeneous representation of 2D points and lines, 3D points and planes, spatial relations, constructions and transformations, algebraic solutions for transformations from correspondences)
  • Multi view geometry (projection matrix and its properties, epipolar geometry, fundamental and essential matrix)
  • Image analysis (least squares matching (Lucas-Kanade/Ackermann), key point extraction, structure tensor (Harris), line extraction)


All lectures will take place during the week of March, 21-25 2011. Location: G205 Time: 13:00-14:30 daily

Lecturer: Wolfgang Förstner

Day Date Location Materials Contents
1. 21.3. G205 lecture1
Gauss-Markov model and in a Gauss-Helmert model with constraints
2. 22.3. G205 lecture2
Maximum likelihood estimation in GM/GH model, projective geometric entities
3. 23.3. G205 lecture3
Representation of uncertain geometric entities, uncertain geometric constructions
4. 24.3. G102 lecture4
Estimation of geometric entities and testing geometric relations, uncertain transformations
5. 25.3. G205 lecture5
Representation of uncertain transformations, estimation of transformations, orientation of a camera, pitfalls in uncertain reasoning, uncertainty of minimal solutions

All lectures in a single PDF, with corrections (last update: March 28, 2011)

All *annotated* lectures in a single PDF (last update: April 1, 2011)


The exercises will consist of exercises as a part of the intensive course. Answers to exercises are due the following date at the lecture. The student should collect at least 50% of points from each exercise to qualify for the exam.


Day Date Time Location Handout Due
1. 21.3. 16:15-17:45 G3 exercise01 22.3. 13:00
2. 22.3. 16:15-17:45 G205 exercise02 23.3. 13:00
3. 23.3. 16:15-17:45 G3 exercise03 24.3. 13:00
4. 24.3. 16:15-17:45 G3 exercise04 25.3. 13:00


Write a sugr-routine for ML-estimation of the essential matrix from point pairs similar ot the routine for estimating a homography in SUGR (copy and modify, including a test routine for checking the resultant covariance matrix of the five parameters)

The project is due by the end of the semester.

SUGR software (last update Mar 25, 2011)


  1. Heuel, Stephan: Uncertain reasoning in Projective Geometry, Springer, LNCS 3008, 2004 See at publisher
  2. Förstner, Wolfgang: Minimal Representations for Uncertainty and Estimation in Projective Spaces, In: Proceedings of the Asian Conference on Computer Vision. Queenstown, New Zealand 2010. download
  3. Förstner, Wolfgang: Optimal Vanishing Point Detection and Rotation Estimation of Single Images of a Legolandscene, In: Proceedings of the ISPRS Symposium Commision III PCV. Paris 2010. download
  4. Meidow, Jochen; Beder, Christian; Förstner, Wolfgang: Reasoning with uncertain points, straight lines, and straight line segments in 2D In: ISPRS Journal of Photogrammetry and Remote Sensing, 64. Jg. 2009, Vol: 2, pp. 125-139. download
  5. Förstner, Wolfgang: Uncertainty and Projective Geometry, In: Bayro Corrochano, Eduardo (Hg.): Handbook of Geometric Computing. 2005, pp. 493-535. download
  6. McGlone, Chris, Bethel, Jim, Mikhail, Ed M. (Eds): Manual of Photogrammetry, ASPRS, 2004
  7. Koch, Karl-Rudolf: Parameter Estimation and Hypothesis Testing in Linear Models, 2. Edition, Bonn (1999) (download the German Version from 1997)
  8. Förstner, Wolfgang: Reliability Analysis of Parameter Estimation in Linear Models with Applications to Mensuration Problems in Computer Vision, In: CVGIP - Computer Vision, Graphics, and Image Processing. 1987, pp. 273-310. download
  9. Bishop, C.M: The Gaussian Distribution, Chap 2.3 In Pattern Recognition and Machine Learning. Springer 2006 download scan (internal access only)


Lecturer Local contact
Wolfgang Förstner Radim Šára
KN, room G 103
tel. (22435) 7203
courses/ae4m33ugr/start.txt · Last modified: 2013/10/04 13:02 (external edit)