Interpolation and Approximation for Visual Computing
Lecturer: Vassillen Chizhov
Dr. Joachim Weickert
Winter Term 2022
Lectures (3h) with exercises (1h);
(6 ETCS points)
Lectures: Sessions with Q&A and Tutorial Sections
The lectures will be held online (Zoom link shared over e-mail)
Target group: Students in the Master Programme Visual Computing
Lecture aim: Give an introduction to the concepts of interpolation
and (function) approximation. This includes
- interpolation with polynomials,
- least-square fitting,
- polynomial splines,
- some Fourier theory,
- radial basis functions, and
- applications in image processing.
This course is suitable for students of visual computing, mathematics, and
Students attending this course should be familiar with basic concepts of
(multi-dimensional) calculus and linear algebra as covered in introductory
maths course (such as Mathematik für Informatiker I-III). Mathematical
prerequisites which exceed the basic mathematics courses are provided within
the lecture notes. All material will be in English. Knowledge from image
processing may be helpful, but is not required.
In order to register for the lecture, write an e-mail to
The subject line must begin with the tag [IAVC22].
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
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
There will be two written exams: the first on 21.02, 14-17 (E1 3, HS002), and the second on
You are allowed to take part in both exams.
The better grade counts, but each exam will count as an attempt
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 will be made available on this homepage.
Additional organizational information, examples and explanations
that may be relevant for your understanding and the exam are provided
in the online sessions and tutorials.
Here you can find: Dr. Augustin's notes.
We will follow the above notes for some topics (e.g. polynomials).
However, I will occasionally provide more details on specific topics, and
less on others. This will be made clear during the lectures.
For instance, I plan to cover some more applications for
interpolation and approximation for higher dimensions
(e.g. finite elements, PDE inpainting).
Polynomial Interpolation, Lagrange Basis
Polynomial Interpolation, Newton Interpolation
Interpolation Error, Polynomial Approximation
Splines II: B-Splines
Splines III: Approximation and Smoothing Splines
Solving Mock Exam I
Solving Mock Exam I
Higher Dimensions I: Mairhuber-Curtis Theorem and Multivariate Polynomials
Higher Dimensions II: Tensor Product Splines, Affine Transformations
Higher Dimensions III: Voronoi, Delaunay, and Multiharmonic Reconstruction
Fourier Theory I: Approximation in Orthonormal Bases, Fourier Series
Fourier Theory II: Fourier Transform, Convolution
Fourier Theory III: Discrete Fourier and Cosine Transforms
Fourier Theory IV: Degradation Phenomena in Reconstruction
Synthesis Functions I: Sampling Theorem
Synthesis Functions II: Inteprolation Using Synthesis Functions
Synthesis Functions III: Transfer to Higher Dimensions
Synthesis Functions IV: From Energy Minimizing Ideas
Interpolation and Approximation in Inner Product Spaces
Trigonometric Polynomial Approximation Exercises
Interpolation in Reproducing Kernel Hilbert Spaces
Fourier Transform, Convolution, Interpolation Exercises
Convolution, Bochner's Theorem Exercises
Mock Exam: Chebyshev Polynomials
Mock Exam: Monomial, Newton, Lagrange, Hermite Bases
Mock Exam: Hermite, De Casteljau, Approximation, Splines
Mock Exam: B-Splines, Fourier, Theory
There is no specific text book for this class as it touches on many topics
for which specialized books exist.
Introduction to Numerical Analysis
J. Stoer, R. Bulirsch, Springer, 1993.
English translation of the originally german version.
W. Boehm, H. Prautzsch, CRC Press, 1993.
Interpolation and Approximation
P. Davis, Blaisdell, 1963.
Reprinted by Dover, 2014.
O. Christensen, K. Christensen, Springer, 2005.
Mathematics of Approximation
J. de Villiers, Springer, 2012.
Fourier Analysis and Applications
C. Gasquet, P. Witomski, Springer, 1999.
The Fourier Transform and its Applications
R. N. Bracwell, McGraw Hill, 1999.
A Practical Guide to Splines
C. de Boor, Springer, 2001.
C. Chui, SIAM, 1991.
Scattered Data Approximation
H. Wendland, Cambridge University Press, 2005.
Most of these and further books can be found in the
mathematics and computer science library.
Further references will be provided during the lecture.