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Interpolation and Approximation for Visual Computing

Winter Term 2020

Interpolation and Approximation for Visual Computing

Dr. Matthias Augustin

Winter Term 2020

Lectures (3h) with exercises (1h);
(6 ETCS points)

Lectures: Online Sessions with Q&A and Tutorial Sections
Monday, 16:15-18:00
Thursday, 12:15-14:00

First online session: Thursday, November 05th, 2020

Announcements – Description – Prerequisites – Tutorials – Registration – Exam – Contents – Assignments – Literature

• 2021-02-02: Sample solution to Assignment 05 online.
• 2021-01-25: Assignment 05 is online.
  You can submit this assignment until 2021-02-01, 18:00 via e-mail to Matthias Augustin.
  Lecture notes updated: Including a missing factor in the third line of Equation (5.1.10).
• 2020-11-23: Exam guidelines online. Please note that you have submit a declaration of consent for the exam to be held in digital form via video conference.
  Quiz 05 online.
• 2020-10-29: Schedule online. Reference version of lecture notes online. Introductory slides online.
• 2020-10-05: In order to protect your health and inhibit the spread of Sars-Cov2, this year's iteration of IAVC will be fully digital. We will use Microsoft Teams for communication and distributing lecture content. Regular teaching will begin on Thursday, November 5, 2020, but you can already register for the course.

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 computer science.
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.

Due to the current sitation caused by SARS-Cov2, this lecture will be closer to an inverted classroom / blended learning setting. This means that for each live online sessions, students will be required to prepare using the available material. A schedule detailing the amount of content that you need to prepare will be made available at the start of the lecturing period. Live sessions themselves are intended to follow a Q&A structures and might present further exercise material.

Due to a shorter lecture period, there will be a total of 5 homework assignments which will be graded. Assignments will be published on the Teams file repository and this website. Students are expected to submit their solutions to these assignments via email to Matthias Augustin within one week after publications. Working together in groups of up to 3 students is permitted and encouraged. Some assignments contain programming exercises.

In order to qualify for the final exam, it is necessary to achieve 50% of the points of all assignment sheets in total. All exams will be oral.

In order to register for the lecture, write an e-mail to Matthias Augustin.
Registration is open until Monday, November 16, 12 am.
The subject line must begin with the tag [IAVC20].
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 information that you provide in your registration will be used to add you to the course in MS Teams. As this addition needs to be done manually, it might be delayed.
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 assignment sheets in total.

There will be two oral exams, one at the beginning and one at the end of the semester break. 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
Monday, February 22, 2021.
The second exam takes place on
Wednesday, March 31, 2021.

Due to the ongoing pandemic of Sars-Cov2, exams will be conducted via MS Teams. Further information is provided by these guidelines
Note that in order to take the exam in this format, you have to submit a declaration of consent. This declaration of consent is in German and you have to submit the German form. A translation to English for your convenience is available here.

Introductory slides

Lecture notes
reference version
newest version
Topics for online sessions are according to the following schedule.
Links to additional exercises will be added to the schedule after each live session.

Date	Please prepare	Pages
11/05	Chapter 1 up to Lemma 1.5	1--7
11/09	Theorem 1.6 to Example 1.15	7--15
11/12	Section 1.2	15--21
11/16	Chapter 2 up to Remark 2.11	23--30
11/19	Lemma 2.12 to Theorem 2.20	30--36
11/23	Lemma 2.21 to Corollary 2.26	36--43
11/26	Assignment 01	--
11/30	After Corollary 2.26 to end of Chapter 2	43--51
12/03	Chapter 3 up to before Section 3.2.1	53--60
12/07	Sections 3.2.1 and 3.2.2	60--65
12/10	Assignment 02	--
12/14	Section 3.3	65--72
12/17	Chapter 4 up to Example 4.5	73--78
01/04	Theorem 4.6 to Example 4.15	78--85
01/07	Assignment 03	--
01/11	Remark 4.16 to Remark 4.23	86--92
01/14	Lemma 4.24 to end of Section 4.3.2	92--99
01/18	Chapter 5 up to Remark 5.8	109--115
01/21	Assignment 04	--
01/25	After Remark 5.8 to Figure 5.2.4	115--122
01/28	Section 5.3 up to Example 5.19	122--131
02/01	Remark 5.20 to end of Section 5.4	131--139
02/04	Assignment 05	--

No.	Title	Date		Submit until
Assignment 01	Newton Interpolation	11/16	[download]	11/23
Assignment 02	Interpolation with B-Splines	11/30	[download]	12/07
Assignment 05	Interpolation with Synthesis Functions	01/25	[download]	02/01

No.	Title	Date		Submit until
Assignment 01	Interpolation and Approximation with Polynomials	11/16	[download]	11/23
Assignment 02	Splines and B-Splines	11/30	[download]	12/07
Assignment 03	Splines in 2D	12/14	[download]	01/04
Assignment 04	Trigonometric Polynomials	01/11	[download]	01/18
Assignment 05	Sampling Theorem and Synthesis Functions	01/25	[download]	02/01

No.	Title	Date
Assignment 01	Interpolation and Approximation with Polynomials	11/24	[download]
Assignment 01	Sample Programme: Newton Interpolation	11/24	[download]
Assignment 02	Splines and B-Splines	12/08	[download]
Assignment 02	Sample Programme: Interpolation with B-Splines	12/08	[download]
Assignment 03	Splines in 2D	01/05	[download]
Assignment 04	Trigonometric Polynomials	01/19	[download]
Assignment 05	Sampling Theorem and Synthesis Functions	02/02	[download]
Assignment 05	Sample Programme: Interpolation with Synthesis Functions	11/24	[download]

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.
• Numerical Methods
  W. Boehm, H. Prautzsch, CRC Press, 1993.
  
• Interpolation and Approximation
  P. Davis, Blaisdell, 1963.
  Reprinted by Dover, 2014.
• Approximation Theory
  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.
• Multivariate Splines
  C. Chui, SIAM, 1991.
• Scattered Data Interpolation
  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.

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