Lecture (4 SWS):
every Monday 11:45 - 13:15 in 48-438 and every Wednesday 10:00 - 11:30 in 48-438 (as noted in KIS), first lecture on 25.10.2010
The tutorials take place on every Thursday 17:15 - 18:45 in 48-582.
Lecturer: Martin Gutting
The lecture will be given in English unless unanimously desired otherwise.News
If you want to attend the lecture, but have a scheduling conflict, please contact me.
Illustrations of spherical scaling functions and wavelets: download as
PDF
Lecture handout "Spherical Notation": download as
PDF
Lecture handout "Discrete Wavelet Transform": download as
PDF
Lecture handout "1D-splines and B-splines": download as
PDF
Illustrations of B-splines: download as
PDF
Fundamentals in Functional Analysis: download as
PDF
Illustrations of the introduction: download as
PDF
An introduction to MATLAB: download as
PDF
Lecture Notes as PDF:
lecture notes (9.2.2011)
Exercises for the tutorials
(bonus) sheet 14
sheet 13
data set 1 for Exercise 13.4 data set 2 for Exercise 13.4 data set 3 for Exercise 13.4
sheet 12
sheet 11
sheet 10
Please note: coefficients for the db3-wavelet and coefficients for the Meyer-wavelet.
sheet 9
sheet 8
sheet 7
sheet 6
program for Weierstrass function in Exercise 6.4
sheet 5
data for Exercise 5.4
data for Exercise 5.4
data for Exercise 5.4
sheet 4
sheet 3
sheet 2
sheet 1
data for Exercise 1.1
Solutions to the programming exercises (MATLAB files, sometimes combined as a ZIP-file):
Exercise 13.4
Exercise 12.4 (programs)
Exercise 11.4 (programs)
Exercise 10.4 (programs)
Exercise 7.4 (programs)
Exercise 7.4 (resulting image)
Exercise 6.4
Exercise 5.4
Exercise 3.4
Exercise 1.4
Contents
In practice, a function of interest is only given by its finite number of samples. For this reason, tools for interpolating or approximating the unknown function (based on the finite knowledge about it) are needed. Such problems do not only occur for functions depending on one variable (e.g. time-dependent functions), but also e.g. for functions on spheres in R^3 (e.g. in the geosciences or in medical imaging).
In this lecture, classical (expansion in orthogonal polynomials) and new methods (splines and wavelets) will be taught. Whereas some of the principles are first explained for the less complicated univariate case (i.e. the case where the domain is one-dimensional), the main focus of the lecture is on the treatment of functions on spheres and balls.
Some of the treated topics are: orthogonal polynomials on intervals (in particular Jacobi polynomials and trigonometric polynomials), spheres (spherical harmonics), and balls; particular topics of differentiation and integration on a sphere (surface gradient, surface curl gradient, Beltrami operator); addition theorem for spherical harmonics; Fourier expansions; Funk-Hecke formula; reproducing kernels and spherical Sobolev spaces; reproducing kernel based splines; scaling functions, wavelets, and multiresolution analysis.
Inhalt
In Anwendungen ist die Funktion, für die man sich interessiert nur durch eine endliche Anzahl ihrer Funktionswerte gegeben. Aus diesem Grund benötigt man Verfahren, um die unbekannte Funktion (aus den endlich vielen, gegebenen Daten) zu interpolieren oder zu approximieren. Solche Problemstellungen treten nicht nur bei Funktionen, die von einer Variable (z.B. die Zeit) abhängen, sondern auch z.B. bei Funktionen auf der Sphäre im R^3 auf, d.h. in den Geowissenschaften und in der Medizintechnik.
In dieser Vorlesung werden klassische (Entwicklung in orthogonale Polynome) und insbesondere aktuelle Verfahren (Splines und Wavelets) vorgestellt. Während einige der Gesetzmässigkeiten im einfacheren eindimensionalen Fall vorgestellt werden, liegt der Hauptfokus auf der Sphäre und der Kugel.
Eine Auswahl der Themen, die behandlet werden: orthogonale Polynome auf Intervallen (insbesondere Jacobi- und trigonometrische Polynome), auf der Sphäre (Kugelfunktionen) und auf der Kugel; spezielle Themen zur Differentiation und Integration auf der Sphäre (Oberflächengradient, Oberflächenrotation, Beltrami Operator); das Additionstheorem für Kugelfunktionen; Fourierentwicklungen; Funk-Hecke Formel; reproduzierende Kerne und sphärische Sobolevräume; Reprokern-basierte Splines; Skalierungsfunktionen, Wavelets und Multiresolutionsanalyse.
