Description:

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. The theory will be demonstrated on practical task of creating mosaics from images, measuring the geometry of objects by a camera, and reconstructing geometrical properties of objects from their projections. We will build on linear algebra 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.

Contents:

1. Geometry of computer vision and graphics and how to study it.
2. Linear and affine spaces.
3. Position and its representation.
4. Mathematical model for perspective camera.
5. Perspective camera calibration and pose computatation.
6. Homography.
7. Invariance and covariant constructions.
8. Projective plane, ideal points and ideal line, vanishing points and horizon.
9. Camera calibration from vanishing points and from planar homography.
10. Projective space. Points, lines, planes.
11. Angle and distace in the projective space.
12. Auticalibration of perspective camera.
13. Epipolar geometry.
14. 3D reconstruction from images.

Seminar contents:

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

Recommended literature:

[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

Keywords:

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

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar