Welcome to the homepage of the lecture

Numerical Algorithms for Visual Computing

Summer Term 2024

Numerical Algorithms for Visual Computing

You can find the main webpage for the course on CMS

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

Summer Term 2024

Lectures (4h);
(6 ETCS points)

Lectures and Tutorials
Monday, 12:15-14:00 (E1.3 HS001)
Thursday, 14:15-16:00 (E1.3 HS001)

First session: Thursday, April 18, 2024
Organisational information would be communicated over e-mail.

Description – Prerequisites – Registration – Exams – Contents – Complementary Reading – Assignments – Literature

Target group: Students in the Master Programme Visual Computing
Lecture aim: An introduction to numerical methods. This should provide some of the mathematical foundations necessary for courses such as "Image Processing and Computer Vision", "Differential Equations in Image Processing and Computer Vision", and "Realistic Image Synthesis". Some of the following topics will be covered depending on the pace:

• Applications
• path tracing
• wave equation
• ray-marching (intersections)
• continuous-scale morphology
• harmonic inpainting
• heat diffusion
• edge detection
• Linear Equations
• coordinate systems, change of coordinates
• linear and affine transformations: rotation, non-uniform scale, translation
• intersections: plane, triangle, disk
• linear systems of equations and linear least squares (projections)
• iterative and direct linear system solvers
• optimisation problems resulting in linear systems
• Nonlinear Equations
• root-finding: bisection, fixed-point, Newton, secant
• quadrics intersections, ray-marching (potentially fractals)
• optimisation: gradient descent, Newton, quasi-Newton
• nonlinear least squares and constrained optimisation
• Discrete Calculus
• refresher of integration and differentiation
• differential forms and generalised Stokes
• numerical integration and differentiation
• discrete differential calculus
• Integral and Differential Equations
• differential equations: diffusion, wave, transport
• integral equations: the rendering equation
• surface flows, discrete differential geometry
• integro-differential equations: volumetric scattering
• finite differences, finite volumes, finite elements
• Monte Carlo integration, importance sampling

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. Additionally students should have some basic programming skills C/Python (e.g. being able to work with arrays and program flow control).
All material will be in English. Knowledge from image processing and rendering may be helpful, but is not required.

You can register for the lecture in CMS.

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.

Course material and a schedule for online sessions will be made available on this homepage at the start of the lecture period.

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

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:

• Numerical Algorithms for Visual Computing 2021 Notes
  M. Augustin, 2021.
• Scientific Computing: An Introductory Survey (Revised Second Edition)
  M. Heath, SIAM, 2018.
• Numerical Methods in Engineering with Python 3
  J. Kiusalaas, Cambridge University Press, 2013.
• Numerical Methods: An Inquiry-Based Approach with Python
  E. Sullivan, 2021.
• Matrix Computations (4th Edition)
  G. Golub, C. Van Loan, John Hopkins University Press, 2013.
• Numerical Linear Algebra: A Concise Introduction with MATLAB and Julia
  F. Bornemann, Springer, 2018.
• Iterative Methods for Sparse Linear Systems (Second Edition)
  Y. Saad, SIAM, 2003.
• Essential Partial Differential Equations
  D. Griffiths, J. Dold, D. Silvester, Springer Cham, 2015.
• A gentle introduction to the Finite Element Method
  F. Sayas, 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.
• Robust Monte Carlo Methods for Light Transport Simulation
  E. Veach, Stanford University, 1997.
• Ray Tracing in One Weekend
  P. Shirley, 2018.

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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