Welcome to the homepage of the lecture

Interpolation and Approximation for Visual Computing

Winter Term 2019

Interpolation and Approximation for Visual Computing

Dr. Matthias Augustin
Office hour: Friday, 13:30 - 14:30.

Winter Term 2019

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

Lectures:
Monday, 12:00-14:00, Building E1.3, Lecture Hall 001
Thursday, 12:00-14:00, Building E1.3, Lecture Hall 001

First lecture: Thursday, October 17, 2019

Onetime alternates:

• Friday, October 18, 2018, 8:00-10:00, Building E1.3, Lecture Hall 001 as substitute for Monday, October 14, 2019

First tutorial: Thursday, November 07, 2019; see also below.

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

• 2020-02-06: Sample solution to homework assignment 07 is online.
  Chapter 05 updated; corrected some minor typos.
• 2020-01-23: Sample solution to homework assignment 06 is online.
  Chapter 05 online. Only the parts of Chapter 05 that will be covered in the lecture will be relevant for the exam.
  Due to the limitied number of participants, exams will be oral.

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,
• Fourier Series, Fourier Transform, and related concepts,
• discrete Fourier transform and FFT,
• splines,
• 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. Lectures and tutorials will be in English. Knowledge from image processing may be helpful, but is not required.

There will be a total of 7 tutorials which will take place instead of regular lectures on

• 2018-11-07,
• 2018-11-21,
• 2018-12-05,
• 2018-12-19,
• 2019-01-09,
• 2019-01-23, and
• 2019-02-06.

The tutorials include homework assignments which have to be submitted before the lecture proceeding the tutorial and which will be graded. Working together in groups of up to 3 students is permitted and encouraged. Some assignments may contain programming exercises.

In order to qualify for the final exam, it is necessary to achieve 50% of the points of all homework assignment sheets in total. There will be either oral exams or written exams, depending on the number of participants.

Registration is closed since the submitting date of the first assignment passed (2018-11-04).

This registration is for internal purposes at our chair only and completely independent of any system like 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.

The first exam takes place on Monday, February 17, 2020.

The second exam takes place on Monday, March 30, 2020.

Further information about the exam (time, written or oral) will be available here after the winter break.

Course material will be made available on this homepage as the lecture proceeds. All provided material is meant to support the classroom teaching and the tutorials, not to replace them. Additional organizational information, examples and explanations that may be relevant for your understanding and the exam are provided in the lectures and tutorials.

Lecture notes in separate chapters

No.	Title	Date
Slides	Organizational Issues, Introduction	10/17	[download]
Chapter 1	Polynomials	10/28	[download]
Chapter 2	Splines	12/06	[download]
Chapter 3	Fourier Theory	01/09	[download]
Chapter 4	Sampling and Synthesis	01/17	[download]
Chapter 5	Techniques in Higher Dimensions	02/06	[download]
Appendix A	Summary of Basic Mathematical Concepts and Notation	11/28	[download]
Bibliography	Literature	10/17	[download]

No.	Title	Date		Submit until
Assignment 01	Interpolation, Estimates, Divided Differences	10/28	[download]	11/04
Assignment 02	Polynomial Interpolation, Hermite Interpolation, Least-Square	11/11	[download]	11/18
Assignment 03	Cubic Splines, B-Splines	11/25	[download]	12/02
Assignment 04	More about B-Splines	12/09	[download]	12/16
Assignment 05	Trigonometric Polynomials, Fourier Series	12/16	[download]	01/06
Assignment 06	Trigonometric Interpolation, Fourier Transforms, Convolution	01/13	[download]	01/20
Assignment 07	Sampling Theorem, Tensor Products, Method of Plates	01/27	[download]	02/03

No.	Title	Date		Submit until
Assignment 02	Newton Interpolation	11/11	[download]	11/18
Assignment 03	Linear and Cubic Spline Interpolation	11/25	[download]	12/02
Assignment 04	Interpolation with B-Splines	12/09	[download]	12/16

No.	Title	Date
Assignment 01	Interpolation, Estimates, Divided Differences	11/07	[download]
Homework Assignment 02	Polynomial Interpolation, Hermite Interpolation, Least-Square	11/25	[download]
Programming Assignment 02	Newton Interpolation	11/25	[download]
Homework Assignment 03	Cubic Splines, B-Splines	12/06	[download]
Programming Assignment 03	Linear and Cubic Spline Interpolation	12/06	[download]
Homework Assignment 04	More about B-Splines	12/19	[download]
Programming Assignment 04	Interpolation with B-Splines	12/19	[download]
Homework Assignment 05	Trigonometric Polynomials, Fourier Series	01/09	[download]
Homework Assignment 06	Trigonometric Interpolation, Fourier Transforms, Convolution	01/23	[download]
Homework Assignment 07	Sampling Theorem, Tensor Products, Method of Plates	02/06	[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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