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Numerical Algorithms for Visual Computing

Summer Term 2023

Numerical Algorithms for Visual Computing

Examiner: Prof. Dr. Joachim Weickert
Lecturer: Vassillen Chizhov
Office hour: Please contact me via email.

Summer Term 2023

Lectures (4h);
(6 ETCS points)

Lectures: Online and in Person Sessions with Q&A and Tutorial Sections
Tuesday, 14:15-16:00
Friday, 14:15-16:00 (E1.3 107)

First session: Friday, April 14, 2023
Organisational information would be communicated over e-mail.

Description – Prerequisites – Course Design – Registration – Exam – Contents – Assignments – Complementary Reading Literature

Target group: Students in the Master Programme Visual Computing
Lecture aim: To provide some mathematical foundations necessary for visual computing courses such as "Realistic Image Synthesis", "Image Processing and Computer Vision", and "Differential Equations in Image Processing and Computer Vision". The course will consist of two parts, the first will focus on light transport simulation, while the latter would focus on solving partial differential equations (e.g. with applications to heat diffusion, fluid simulation, polyharmonic inpainting). The following topics will be covered:

• a basic pathtracer with multiple importance sampling
• the rendering equation
• Monte Carlo integration
• (short) introduction to partial differential equations (PDEs)
• heat diffusion, fluid simulation, polyharmonic reconstruction
• finite difference, finite volume, and finite element discretisations of PDEs
• iterative solvers for linear systems of equations

This course is suitable for students of visual computing, mathematics, and computer science.
Students attending this course should be familiar with basic concepts of (multi-dimensional) calculus and linear algebra as covered in introductory math courses (such as Mathematik für Informatiker I-III). Nevertheless, the course will provide a refresher of the necessary concepts from linear algebra and calculus.
All material will be in English. Knowledge from image processing may be helpful, but is not required.

There will be reading or/and programming material provided for most lectures. The lectures will be used to reinforce the knowledge of the assigned reading material and will also serve as Q&A sessions.

Throughout the course there will be homeworks meant to reinforce what has been covered in the lectures, or to prepare you for the next lecture. The homeworks should be submitted to Vassillen Chizhov at the latest one week after their publication, and you can work in groups of up to three people on those.

In order to qualify for the exam, it is necessary to achieve 50% of all points from the homeworks. As an alternative one can receive 50% of the points for a specific homework by solving the homework exercises (with help from the tutor if needed) in the tutorial scheduled after the homeworks have been graded.

In order to register for the lecture, write an e-mail to Vassillen Chizhov.
The subject line must begin with the tag [NAVC23].
Please use the following template for the e-mail:

First name: myFirstName
Last name: myLastName
Date of birth: dd.mm.yyyy
Student ID number: ...
Course of study: Bachelor/Master/...
Subject: Computer Science/Visual Computing/Mathematics/...

Note that the e-mail address from which you send this information will be used to provide you with urgent information concerning the lecture.
Such information may include further regulations or urgent additional remarks regarding assignment.

The registration is completely independent of LSF/HISPOS. They require a separate registration.

According to the regulations concerning storage and processing of personal data (Art. 6 (1) Datenschutzgrundverordnung (DSGVO)) we store and process your personal data for the purpose of lecture and tutorial organisation only. I.e. we may use them to contact you, to inform you about your grade, and to transmit your grades to the examination office.

In order to qualify for the final exam, it is necessary to achieve 50% of the points of all homeworks in total.

There will be two written exams, one at the beginning and one at the end of the semester break. The exams allow one A4 "cheat sheet" handwritten by you. In case of qualification, you are allowed to take part in both exams. The better grade counts, but each exam will count as an attempt individually. Please remember that you have to register online for the exam in the HISPOS system of the Saarland University for each attempt separately.

The first exam takes place on
Friday, July 28, 2023, in E1.3 HS002 from 09:00 to 12:00.

The second exam takes place on
Tuesday, October 17, 2023, in E1.3 HS002 from 14:00 to 17:00.

Organizational slides from the first session.

Date	Topic
14.04	Introduction and Organization, Brief History, Camera and Rays
18.04	Normals, Scene, Camera Movement, Shading
21.04	Reflection, Refraction, Lambertian Scattering
25.04	Progressive Rendering
28.04	Tutorial Session for Homework 1
02.05	Progressive Rendering Code and Rendering Equation
05.05	Integration on Manifolds, Monte Carlo, Normals
09.05	Inverse Transform Sampling
12.05	Tutorial Session for HW2
16.05	ITS and Area Formulation of RE
19.05	Tutorial Session for HW3
23.05	Part 1:Inverse Transform Sampling Part 2:Direct Light Sampling
13.06	Part 1:Light Transport Review Part 2:Light Transport Review
16.06	Tutorial Session for Light Transport (in person)
20.06	Part 1:Finite Differences Part 2:Finite Differences
23.06	Tutorial Session (completing the exercises from the previous session)
26.07	Exercises from previous tutorials + extras

Complementary reading material for the course (non-mandatory reading).

Date	Topic
14.04	Appel68, Whitted79, Cook84, Kajiya86, Georgiev's and Lessig's Theses
21.04	Reflection and Refraction
02.05	Georgiev's Thesis Chapter 3, Veach's Thesis Chapter 3
09.05	Lambert's Cosine Law and its formalisation
21.05	The Fundamental Theorem of LA Neumann Series for Generalised Inverses
23.05	Global Illumination Compendium More detailed notes of mine: LT1, LT2, LT3, LT4, LT5
16.06	Veach's Thesis: Chapter 8 and 9
20.06	The books below more or less overlap on the mentioned chapters so you are free to choose between those. I have ordered them in terms of complexity/details (from low to high). Chapter 2.1 and 2.2 of Saad's book Chapter 3 and 4 of Augustin's notes Chapter 1 and 2.1-2.3 of LeVeque's book on FDM for ODEs and PDEs

Date	Assignment
18.04	Homework 1: Triangle Intersection, Normals, Snell's Law
02.05	Homework 2: Intersections, Neumann Series, RE Liouville-Neumann
10.05	Homework 3: Integration on Manifolds, Inverse Transform Sampling

Date	Solutions
28.04	Solution to Homework 1
16.05	Solution to Homework 2 detailed and short
22.05	Solution to Homework 3

There is no specific text book for this class, but here is a selection of some books covering many of the topics in this course, giving background material and providing further reading:

• Ray Tracing in One Weekend
  P. Shirley, 2018.
• Robust Monte Carlo Methods for Light Transport Simulation
  E. Veach, Stanford University, 1997.
• Iterative Methods for Sparse Linear Systems (Second Edition)
  Y. Saad, SIAM, 2003.
• A gentle introduction to the Finite Element Method
  F. Sayas, 2015.
• Numerical Algorithms for Visual Computing 2021 Notes
  M. Augustin, 2021.
• Essential Partial Differential Equations
  D. Griffiths, J. Dold, D. Silvester, Springer Cham, 2015.
• Finite Difference Methods for Ordinary and Partial Differential Equations
  R. J. LeVeque, SIAM, 2007.
• Numerical Solution of Partial Differential Equations in Science and Engineering
  L. Lapidus, G. F. Pinder, Wiley, 1999.
• Numerical Solution of Partial Differential Equations
  K. W. Morton, D. Mayers, Cambridge University Press, 2005.
• Numerical Partial Differential Equations
  Volume I: Finite Difference Methods
  Volume II: Conservation Laws and Elliptic Equations
  J. W. Thomas, Springer, 1995 (Vol. I) / 1999 (Vol. II).
• Numerical Methods for Conservation Laws
  R. J. LeVeque, Birkhäuser, 1992.
• The Finite Difference Method in Partial Differential Equations
  A. R. Mitchell, D. F. Griffiths, Wiley, 1985.
• Numerical Solution of Differential Equations
  W. E. Milne, Wiley, 1960.
• Introduction to Numerical Analysis
  J. Stoer, R. Bulirsch, Springer, 1993.
  English translation of the originally german version.

Most of these and further books can be found in the mathematics and computer science library.

Further references will be provided during the lecture as needed.

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