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 functions
(x) and their associated wavelet functions (x) can
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 scaling relation
(t) = (2t) + (2t 1),
(1)
which is a special
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 the orthogonal basis elements. In what follows, we assume
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 in particular, Property No. 3 on p. 235, i.e., that the
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
other. We can generalize this result to the decomposition
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). Samples are
generally obtained by some kind of convolution
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 processes it to produce a modied signal f . Examples of p
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 is said to form an orthogonal set in H if
ui , uj = 0
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 processing.
Practically speaking, youd like to get as many
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 a > 0:
1
1
1
1
x
x
e0 = , e1 = cos
, e2 = sin
, e3 = co
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 overview of the course
The rst sentence in the calendar descr
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 metric
d that assigns distances between any two element