5B1308 Wavelets

Course contents

Lecture 1

Introduction,
Splitting of time-frequency space:    Standard representation by samling
                                                            Fourier tranforms, harmonic oscillations
                                                             Wave packets.  wavelets

Infinite sequences, local rotations, invariance under even translations
Exampel: rotations 45 degree on a constant sequence.
One moment condition. Haar functions.
Application of wavelet on compression of images.
Home exercize:  Do consequtive Haar transform  on data (16 points)  sampled from your
favorite  function  f- which is constant on some subintervals

Lecture 2

Construction of orthogonal sequences  of length 4 by
means of two local rotation operators. Two moment conditions.
Daubechies orthogonal sequeneces h= {h_n} and g={g_n} of length 4.
The space of square summabel sequences. Orthogonal translation invariant system
{T_2k h}  U {T_2k g}.
Aplication: med with noise reductions on reccording by Caruso on vax rolls
Home exercise: Do consecutive filtering with Daubechies orthogonal filter of length 4
on data samples from your favorite funtion f - which is linear on some subintervals
 

Lecture 3

Representation by orthogonal traslationinvariant basis.  Decomposition by low and high
pass filter  and reconstruction. The wavelet filter tree.  (Wavelet packet filter tree was also
mentioned)
Construction of function by iteration of the formula using the filter h (the Cascade algoritm
see http://cm.bell-labs.com/who/wim/cascade/
Exercize: Play with the Cascade algorithm.

Lecture 4

The limiting function in the Cascade algothitm. The scaling equation.
The orthogonality properties. Convergence questions.
Z- and Fourier transform of h and g.
Orthogonality in subspaces of functions  <-->  orthogonality in sequence spaces.
Multi -scale analysis.

Lecture 5

Multiscale analysis (cont.) . Construction of the orhonormal wavelet basis for L2(R) generated
by  the "mother wavelet" function
 

Lecture 6

Orthonormal wavelet basis in dimension 2:
Full tensor basis (we will not use that) .
The nonstandard tensor basis (which usually is used) .
The wavelet filter tree in dimension 2.
Lowpass and highpass filtering  and their relation to convolution. Peparation to
Home Exercise 1 in Matlab (wavelet decomposition).
 

Lecture 7

The reconstruction of the original date from the wavelet coefficients, relation between decomposition and reconstruction filter.
Biorthorthogonal basis and dual basis. Symmectric biorthogonal filters and their construction by
lifting.
Preparation for Home Exercise 2 in Matlab ( reconstruction of data from its wavelet coefficients)

Lecture 8
Estimate of size of wavelet coefficients. Moment conditions on wavelet functions and scaling
functions. Lowpass and higpass filter of polynomial sequences.
Tresholding of wavelet coefficientents
Home Exercise 3: Thesholding of coefficients - apply  to signals in  'http://math.kth.se:~janolov/5B1308/signals1.mat'

Lecture 9
Wavelet packets.  Cost functionions. Entropy. Best basis. Best basis algorithm.
Home Exercisee 4:  Reconstruction from wavelet coefficients in dimension 2.

Lecture 10
Local descriminant basis. Feature extraction for classification.
Home Exercise 5:  Compression of an image: "Birds"  and and noise
reduction of a noise images  "BirdsN"  using thresholding of wavelet
coefficients. Signals in : 'http://math.kth.se:~janolov/5B1308/Birds'
and 'http://math.kth.se:~janolov/5B1308/BirdsN'

Lecture 11
Wavelets have not so successful in representation of music. Try the
music signals in: 'http://math.kth.se:~janolov/5B1308/guitar.wav'
and 'http://math.kth.se:~janolov/5B1308/noisyguitar.wav'
Local trigonometrical function basis.
Daubechies contstructions of compact orthogonal wavelets
using  Fourier transform characterization of the ON-conditions
of the wavelet functions.

Lecture 12

Short summary of the content of the course. Most parts of
on wavelet theory from chapters 1-7 coverered in the course.
Wavelet  applications in chapters 8,9 and some part of
chapter 10  and 14 was alse covered in the course.

The course ended by a short overview of wavelet projects
we are involved in at the Department of Mathematics at
KTH.