Lecture 33
Multiresolution analysis: A general treatment (contd)
Scaling functions with compact support
In Lecture 28 (Set 10 of Course Notes), we discussed very briey the property that scaling functi
Lecture 24
Wavelets and multiresolution analysis (contd)
The resolutions Vj are nested (contd)
Lets summarize some major recent results:
1. The Haar scaling function (t) = I[0,1) (t) satises the scali
Lecture 15
Discrete Fourier transforms (conclusion)
DFTs of two-dimensional data sets
The DFT is easily extended to handle two-dimensional data sets, e.g., images, by using a tensor
product form of th
Lecture 30
Multiresolution analysis: A general treatment (contd)
Some of the material appearing at the end of the last set of lecture notes (Set 10, Lecture 29) was
actually presented in this lecture
Lecture 27
Wavelets and multiresolution analysis (contd)
Analysis and synthesis algorithms (contd)
In the previous lecture, we showed how the two expansions of a function f1 V1 are related to each
oth
Lecture 21
Sampling Theorem (contd)
A generalized Sampling Theorem for samples obtained by convolutions
In practice, it is generally impossible to obtain perfect samples f (kT ) of a signal f (t). Sam
Lecture 18
Linear Filters
Relevant section from book by Boggess and Narcowich:
Section 2.3, p.
108
The idea of a lter is very important in signal and image processing. A lter F takes a signal f
and pr
Lecture 6
Inner product spaces (contd)
The following idea was introduced in the previous lecture.
Orthogonal/orthonormal sets of a Hilbert space
Let H denote a Hilbert space. A set cfw_u1 , u2 , un H
Lecture 12
Discrete Fourier Transforms (contd)
Thresholding of DFT coecients as a method of data compression
Data compression is a fundamental area of research and development in signal and image proc
Lecture 9
Fourier series on the interval [a, a], even and odd extensions
In a previous lecture, it was mentioned that the following functions comprise an orthonormal set on
the interval [a, a], where
University of Waterloo
Department of Applied Mathematics
AMATH 391: From Fourier to Wavelets
Winter 2010
Lecture Notes
E.R. Vrscay
Department of Applied Mathematics
1
Lecture 1
Introduction: An overvi
Lecture 3
Important concepts from mathematical analysis (contd)
Convergence of sequences in metric spaces
In the previous lecture, we introduced the idea of a metric space (X, d) a set X along with a