Inst. för Matematik    |   KTH    | 



     
 

SF2702 Wavelets, 6hp 

Kursansvarlig/föreläsare:Jan-Olov Strömberg, 
janolov@math.kth.se
tel. 08-790 6676 
(12 gånger under period 1 -2)
Kursstart Torsdag den 30 augusti kl.14.15.
Föreläsningslokal:   Seminarierum 3721, Lindstedtsv. 25
Därefter är föreläsningar preliminärt ändrat till: Tisdagar kl. 13.15-15.00, Seminarie-
rum 3733 (fr.o.m 4 september)
Kurslitteratur: Bergh/Ekstedt/Lindberg: Wavelets.[Säljes på Kårbokhandel .]
Föreläsningsanteckningar (delas ut) 
Hemuppgifter.  Dataövningar på Matlab
Examensform: Inlämningsuppgifter under kursens gång 
  First meeting  (August 30)
  This is an early meating, in fact one day before the official semester start.  It is not a problem for new student to start at a later occation. We need to get some idea how many will attend the course,
so we get an appropriate size lecture room.  Preliminary we plan lectures on Tuesdays 1-3

In this first couse meeting the were given some introductatary information about the administration
of the course. 
Also ther were given some very general and very short description of wavelets as wave packages  and some application of it to imageprocessing were demonstrated.

In this wavelet course we will assume that the students have some knowledge in two basic
areas in mathematics:  1. Linera algebra,  with basis systems , specially orthormlal basis.
                                    2. Fourier series  and/or Fourier transforms.
The student are assumed to have accress to Matlab during the course

In this course we will jump back and forth between  four scenarious  (or types) of functions:
(2 x 2 = 4 kombinations)
Countinuous  / Discrete                     Non-period  / Periodic
functions and their "fourier" transforms. (Some of which topics most student have already seen and
other topics may be new for most students)

Also, if I  get enough time we plan to describe the system of Haar functions.

Lecture 1 (September 4)

A short repition from first introductary meeting about the four senarious of functions , the
Fourier transform on those, and finaly  expansion of a function in the Haar system.
A more detailed list of headlines is available here.
Students own readin in the text book (Bergh, Ekstedt,Lindberg: Wavelets)
                                      about the Haar system in Ch. 1,
                                      about vector space and fouriertransforms in some different setting in Ch.2
                                      

Homework assignmet 1 was handed out, it is also available here.



Lecture 2 (September 11)
We saw that the expansion of function with the Haar filter corresponds to
filter operations on the sequence space l^2 with two filter  the Lowpass filter
h = (1,1)/sqrt(2)    and the  Highpass filter g=(-1,1)/sqrt(2) , arranging the
iterated filter operation in the Wavelet filter tree .
The filter h  and g  generatate the translation invariant ON-sets {T^2k g}_k
resp.   {T^2k g}_k .  Those two ON-sets are mutually orthogonal and makes
together an ON- basis for l^2.  The  Low-  and High- pass filter operation can
be seen as changing coordinates system to this new ON-basis.
This change of coordinates can be seen as locally doing an 45 degree  in
many copies of  R².   By  repeated local rotations (with specially selected angles)
 one will construct longer filters h and g with will generate similar translation invariant ON-sets and ON- basis.
Usually one selects the rotation angles  so that the constructed filter fullfill some moment conditions.


Lecture 3 (September 18)
Given a filter h of finite length such that {T^2k  h}_k  is a ON-set
we saw how to get the complimentary filter g such that {T^2k  h}_k 
union {T^2k  g}_k will be a ON-basis for l^2.
We saw also how to obtain such filters  h of arbitary length 2m  by  m local rotations
A local rotations  is  defined on  pairvise   coodinates  (a_2k, a_2k+1)  by the matrix
                      cos A sin A
         R_A= (                    )  each local rotation is interwined by a translation T:
                      -sin A cos A    

Thus   h = R_A1 T RA_2......TR_Am Dirac_0 
 and    g = R_A1 T RA_2......TR_Am Dirac_1.
gives ON-filter of length 2m.
The rotation angels A1,...,Am are chose usually so that g satisfies some moment
conditions.
Homwork number 2 is available here.
Inlevering utsatt till 9 oktober pga sen WEB-publisering.

Lecture 4 (September 25)
First some hints about doing the filtration in matlab.
           Matlab start indexing arrays with index one.
           Often Low- and  highpass filters involves filters where
           some non-zero elements have negative index. These implies
           some shifting or other tricks is necessary when using Matlab.
           If  one consider a given finites set of data surrounded by zero
           elements, the Low- and Highpass filtering with result in a
           totally larger set of coefficients then the data points. To get
           around this one often consider the data as a set of periodic data.
           This means that the convolution in Matlab for Low and Highpass
           filter must be followed by a periodization.
We saw how the ON-conditions of the filters h and g can be written in term
             of their Fourier transform och Z-transforms.
Multi-scale analysis. We stated sixth conditions for multi-scale analysis, for a
             family of subspace V_j  of  L^2 (R)  and saw how this leads to a
             a filter h  such that {T^2k  h}_k  is a ON-set.
             We also saw how to use the complimentaty fileter g to construct the
              corresponding  ON- wavelet basis.

Lecture 5 (October 2)
 Here is a list of topics from lecture 5.
 Homework number 3 was handed out.
Lecture 6 (October 9)
Using the scale equation we can construct  the scaling function
by iteration . This is the cascade algorithm for  filters h of length
4 can be found at http://plan9.bell-labs.com/who/wim/cascade/
In this lecture we also saw how an image which is represented
of a matrix with grey-scale pixelvalues may be filtered using
lowpass and highpass filter in the x- and the y directions.
We saw that using this procedure in several levels we with
will get the wavelet coefficients of this  matrix  and that
most of the coefficient will be  very small.  By  setting
the small coefficient to zero, we can represent the image
by a small part of'the coefficients. This leads to compression
algorithms.


Lecture 7 (October 16)
Mathlab hints: use the built in Matlab functions sum and reshape to perform
periodizations without loops.
We look at bi-orthonormal basis,  and biorthonormal wavelet filters.
bi-orthonormal filter can be spit into a product of filters of length 3-1.
a so called lifting. Using this liftings gives a very fast implementation
of the bi-orthonormal filtering procedure.
Convolution of filter related to convolution of scalingfunctions by the cascade
algorithm.
Th spline filter b_n=(1,1) * ....* (1,1)     i.e  iterated convolution with n  (1,1)
filters.   Let   b_2n  ,be the bi-orthogonal wavelet scaling filter (length = 2n+1),
then its dual filter b^*_2n  of length 2n-1, can be found by solving a few linear equations.     
The filter I_n = b_2n * (b^_2n)^  will be a so called interpolet filter of length 4n-1 (whose dual
is the Dirac_0-filter.) With some numerical  efforts  one may split I_n as I_n=h_n * (h_n)^~.
In this way we have constructed an orthonormal wavelet filter h_n of length 2n.
Homework number 4  was handed out.

Lecture 8
Estimate of size of wavelet coefficients. Moment conditions on wavelet functions and scaling
functions. Lowpass and higpass filter of polynomial sequences.  The moments of a wavelet
with m times continuous derivatives.
Tresholding of wavelet coefficientents

Lecture 9
The time-frequency plane, the uncertainty principle.
The continuous wavelet transform.
Homework number 5 was handed out.
No lectures on November 6 and November 13

Here is the last homework ( with a correction of index in a summation formula)
Homework number 6

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Avdelning Matematik Sidansvarig: Jan-Olov Strömberg  
Uppdaterad: 2007-09-17