Wavelets 5B1308 ht 2005

Second meeting  (September 5)

Topics / keywords:

Example application representation of data.: Music CD

Standard(sampling ) basis <---------> Fourier basis

Time frequency plane .

We want good description both in time and frequency.

Heisenberg's uncertainty principle:      product of the spread of time and the spread of
                                                             frequency is bounded from below by a constant times the
                                                             square of the norm of a function.

Gauss function  is the extremal function for which the equality with the best constant is obtained.

Gabor basis:   a basis with translation of a window function + local fourier series in each window.

Can we choose the window function such that both its spread of time and the spread of its
frequency is fine  such that the corresponding Gabor basis an Orthonormal:

Balian-Low theorem:  Answer is No.

'Classical' orthonormal wavelet basis., generated by a 'mother wavelet

by scale and translation parameters.

First example: Haar basis (1910) - is not continuous

Shanon basis not well localized in time

Sketching on blackboard : approximation by Haar basis

Topics / keywords
Back to the discrete world:
An orthonormal basis gives an isometri between L^2(R) and l^2(Z
Subspaces V of L^2(R) with a translation  invariant basis.
Example : piesewise konstant functions on integer intervals.
The Haar basis and its disrete version.
but first orthog0nal rotation of vectors in R^2.

    s= x cos a + y sin a

    d= - x sin a + y cos a

    x= s cos a - d sin a

    y = s sin a + d cos a

Norm equivalence : s^2+d^2 = x^2 +y^2

Pairwise local rotation on l^2.
given (x_n)_n
organize into pairs (x,y) = (x_0,x_1)  ,    (x,y)= (x_2, x_3) and so on.
Inverse local rotation and norm equivalence as as above.

Example:
Discrete Haar basis : set angel a = 45 degree (= Pi/4 ).
Do a hierarchy of successive local rotations all with angle 45 degree:

        first s^0 =(x_n)_n into s^1 and d^1
        then s^1 into s^2 and d^2
        then s^2 int s^2 and d^3
        and so on.
        finally stop with s^m and d^m

What is the connection with the Haar basis above?

Construction of a wavelet filter of length 4 by two different local rotations.

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4th Meeting  September 22
Longer ON -filter of finite length
       factorized to  a group of local rotations
       the comlementary ON filter
       The union of   ON wavelet filter h  with its  translates  T^2k h
        and its complementary ON-filter g with its translates T^2k g
        constitutes an ON basis for  l^2.
       The wavelet filters and moment conditions.

5th Meeting  September  27  
Multi-scale analysis

6th Meeting  October 5

the  scaling equation, the cascade algorithm
biorthogonal filters, interpolating filters

7th Meeting  October 12

Biorthogonal filters, dual filters to spline filters, factorization and construction
of ON wavelet filters.
Some matlab hints to the homework assignment.
Multi-scale analysis and ON wavelet bases in dimension = 2.

8th Meeting  October 18

Listing course contents in J.Bergh et al: Chapters 1,2,3,4,5,6,7,8,,9,10,14.
Wavelets in dim 2 (cont.)   2-dim wavelet  ON wavelet filters.
 Two dimensional filtertree.
 ON wavelets conditions given in the Fourier transform language.

9th Meeting  October 25

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Homework 1a

Calculation by hand (calulator) of Haar wavelets coefficien of a short sequence

Let the sequence a in l^2(Z) be given by the 16 terms given below, the remaining tern

are all equal to zero.

( We may also think of it as a sequence in l^2(Z/16) .

a) Compute all (16) Haar wavelet coefficient of the sequence in 4 levels

( including the the the lowpass term on the last leve.l)

b) Reconstruct the sequence exactly from those coefficients.

c) Find 8 coefficients with largest absolute value and replace the value of the remaining 8 coefficients by the value zero. Repeat now the reconstruction as in exercise b) after this change of the coefficients. .

500. The construction done in c) would be an approximation of the original sequence. Plot both the original sequence and the approximation. Find also the maximum error in the approximation.

Given a=(a_n) n =0,1,2,...,15 where

a_n = n for n=0,1,2,...,6 and a_n=1 for n=7,8,9,...,15.

Homework Ib

Calculation of the Daubechies walelet filters of length 4

The sketch of how the those wavelet filters can be calculated by two local rotations would be

given in the lecture. Complete the details in this calculation. This filters will be used later

in the course.

Homework is supposed to be handed in at next lecture.