Subject description - AE4M33GVG

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AE4M33GVG Geometry of Computer Vision and Graphics
Roles:PO, V Extent of teaching:2P+2C
Department:13133 Language of teaching:EN
Guarantors:  Completion:Z,ZK
Lecturers:  Credits:6
Tutors:  Semester:L


We will explain fundamentals of image and space geometry including Euclidean, affine and projective geometry, the model of a perspective camera, image transformations induced by camera motion, and image normalization for object recognition. Then we will study methods of calculating geometrical objects in images and space, estimating geometrical models from observed data, and for calculating geometric and physical properties of observed objects. The theory will be demonstrated on practical task of creating mosaics from images, measuring the geometry of objects by a camera, and reconstructing geometrical and physical properties of objects from their projections. We will build on linear algebra, probability theory, numerical mathematics and optimization and lay down foundation for other subjects such as computational geometry, computer vision, computer graphics, digital image processing and recognition of objects in images.

Study targets:

The goal is to present the theoretical background for modeling of perspective cameras and solving tasks of measurement in images and scene reconstruction.

Course outlines:

1. Computer vision, graphics, and interaction - the discipline and the subject.
2. Modeling world geometry in the affine space.
3. The mathematical model of the perspective camera.
4. Relationship between images of the world observed by a moving camera.
5. Estimation of geometrical models from image data.
6. Optimal approximation using points and lines in L2 and minimax metric.
7. The projective plane.
8. The projective, affine and Euclidean space.
9. Transformation of the projective space. Invariance and covariance.
10. Random numbers and their generating.
11. Randomized estimation of models from data.
12. Construction of geometric objects from points and planes using linear programming.
13. Calculation of spatial object properties using Monte Carlo simulation.
14. Review.

Exercises outline:

1Introduction, a-test 2-4Linear algebra and optimization tools for computing with geometrical objects 5-6Cameras in affine space - assignment I 7-8Geometry of objects and cameras in projective space - assignment II 9-10Principles of randomized algorithms - assignment III. 11-14Randomized algorithms for computing scene geometry - assignment IV.


[1] P. Ptak. Introduction to Linear Algebra. Vydavatelstvi CVUT, Praha, 2007.
[2] E. Krajnik. Maticovy pocet. Skriptum. Vydavatelstvi CVUT, Praha, 2000.
[3] R. Hartley, A.Zisserman. Multiple View Geometry in Computer Vision.
Cambridge University Press, 2000.
[4] M. Mortenson. Mathematics for Computer Graphics Applications. Industrial Press. 1999


A standard course in Linear Algebra.



Computer vision and graphics, Euclidean, affine, projective geometry, perspective camera, random numbers, randomized algorithms, Monte Carlo simulation, linear programming.

Subject is included into these academic programs:

Program Branch Role Recommended semester
MEKME1 Wireless Communication V 2
MEKME5 Systems of Communication V 2
MEKME4 Networks of Electronic Communication V 2
MEKME3 Electronics V 2
MEKME2 Multimedia Technology V 2
MEOI4 Computer Graphics and Interaction PO 2
MEEEM1 Technological Systems V 2
MEEEM5 Economy and Management of Electrical Engineering V 2
MEEEM4 Economy and Management of Power Engineering V 2
MEEEM3 Electrical Power Engineering V 2
MEEEM2 Electrical Machines, Apparatus and Drives V 2
MEOI3 Computer Vision and Image Processing PO 2
MEKYR4 Aerospace Systems V 2
MEKYR1 Robotics V 2
MEKYR3 Systems and Control V 2
MEKYR2 Sensors and Instrumentation V 2

Page updated 5.3.2021 17:52:09, semester: Z/2020-1, L/2021-2, L/2020-1, Z/2021-2, Send comments about the content to the Administrators of the Academic Programs Proposal and Realization: I. Halaška (K336), J. Novák (K336)