He who loves practice without theory is like the sailor who boards a ship without a rudder and compass and never knows where he may cast. — Leonardo Da Vinci (1452-1519)

And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art. — Albrecht Durer (1471-1528)

As for everything else, so for a mathematical theory: beauty can be perceived but not explained. — Arthur Cayley (1821–1895)

We will explain Euclidean, Affine, and Projective geometry basics, introduce a model of the perspective camera, and explain how images change when moving a camera. We will show how to compute camera poses and the 3D scene geometry from images. We will demonstrate the theory in practical panorama construction tasks, finding the camera pose, adding a virtual object to a real scene, and reconstructing a 3D model of a scene from its images. We will build on our previous knowledge of linear algebra and provide fundamentals of geometry for computer vision, computer graphics, augmented reality, image processing, and object recognition.

Tomas Pajdla, Torsten Sattler

Martin Matoušek, Viktor Korotynskiy , Vojtěch Pánek, Diana Sungatullina

See Labs for more details.

1. All homework must be submitted via BRUTE and accepted.
2. At least 50% of points in total for the homework.
3. At least 50% of points in total from the tests.
4. Regular submission of homework ends on May 23, 2021. Later submissions are possible only by an agreement with the assistants.
5. All the above conditions have to be fulfilled, and the results have to be recorded in the Submission system before the exam.

The exam consists of a written and an oral part. It is required to achieve at least 50% of points from the written exam to be admitted to the oral exam. The grade depends on the exam (40%), tests (30%), and homework (30%). You may skip the oral exam if you are satisfied with the result after the written exam.

Exam content:

1. Linear algebra [4,5,6,7]: linear space, basis, coordinates, linear dependence/independence, matrices, rank, determinant, eigenvalues, and eigenvectors, solving systems of linear equations, Frobenius theorem and linear independence, linear function, affine function, linear mapping, and its matrix, computing roots of a polynomial via eigenvalues of its companion matrix, dual space, dual basis, change of the dual basis corresponding to a change of a basis, vector product and derived linear mappings, SVD, dual space, and dual basis.
2. Course material

Written exam terms

1. 23.5.2022 11:00-13:00 JP:B671
2. 30.5.2022 11:00-13:00 JP:B671
3. 06.6.2022 11:00-13:00 JP:B671

Written exam organization:

1. The written exam will be done online via MS Teams. The written exam is in person in room KN:E112 JP:B671.

Oral exam terms:

1. 27.5.2022 10:00-12:00 MS Teams (4 slots)
2. 06.6.2022 11:00-16:00 MS Teams (10 slots)

Oral exam organization:

1. You may skip the oral exam if you are satisfied with the result after the written exam.
2. The oral face-to-face exam will be done online via MS Teams and will take about 30 mins.

1. Lecture: It is very difficult to pass the course without attending the lectures.
2. Labs: It is impossible to pass the course without attending the labs.
3. Homework: Homework is assigned at a lab where it can be discussed with teaching assistants. Students work out homework individually (rulesin Czech). The deadline for submitting homework via BRUTE is on Monday at 6:00 in the morning two weeks after the assignment. Late submissions are penalized (10% for each commenced day of delay but not more than 50% of points).
4. Assessment: see above.
5. Tests: Students take tests individually.

1. T Pajdla Elements of Geometry for Computer Vision
2. R Hartley, A Zisserman Multiple View Geometry in Computer Vision Cambridge University Press, 2003
3. K Daniilidis, J Shi. Robotics: Perception - Coursera Online Course
4. G Sanderson Essence of Linear Algebra from 3Blue1Brown
5. J Strom, K Astrom, T Akenine-Moller Interactive Linear Algebra Course
6. P Olšák Úvod do algebry, zejména lineární ČVUT 2007
7. P Pták Introduction to Linear Algebra ČVUT 2007
8. Maple A0B01MVM Matematika v Maple Installation