In todays show warm up for next weeks torture

1
The Story of WaveletsTheory and Engineering
Applications
Jamboree 7 February 28, 2001

• In todays show (warm up for next weeks torture)
• Recall vector spaces and orthogonal projection
• An introduction to multiresolution approximation
• The scaling functions and the wonders it can
  do
• Scaling functions close friend the wavelet
  function and the Haar wavelet
• Filter implementation of the Haar wavelet
• I would do more, but by this time you will be
  brain dead

2
Recall Orthogonal Projection
V2
V3
Coarse approximation of X3 at level 2
3
Vector Spaces

• VN-1is the next coarser vector space, it is a
  subspace of VN.
• WN-1 is orthogonal to VN-1, and
• In general
• and

4
Intro. To MRA

• A function f(t) can be approximated using
  piecewise linear functions of interval 1. Lets
  call this approx. fo(t) of space V0
• The best approximation is simply the average of
  the function in that interval
• If we use functions of interval 2, we get a
  poorer accuracy. This approximation is coarser
  than the previous. Lets call this approx. f1(t)
  of space V1
• If we use functions of interval ½, we get a
  better accuracy. Lets call this approx. f-1(t)
  of space V-1
• Note that

5
DetailsLiterally

• If we approximate function f(t) at level k using
  fk(t), then

Detail function at level k. If we add this detail
to fk then we obtain the better resolution
(finer) approximation at level k-1
6
Scaling Function

• The piecewise linear function though which we
  obtain various approximations have various
  properties
• This is the Scaling Function
• Haar scaling function is an example of such
  functions

?(t)
1
1
7
Approximations

• Using the Haar scaling function, we can
  approximate any function in L2 using
• where
• Note that the scaling functions are orthonormal
  to their own translates.

8
Details

• The detail functions g(t) can also be represented
  using a set of orthonormal functions, called
  wavelets.
• where d(k,l) detail coefficients can be obtained
  from
• and where the wavelet is obtained from

or
?(t)
1
1
t
1/2
-1
9
The Reconstruction

• Since
  then

This is the discrete wavelet transform (DWT)
synthesis, where d(k,l) constitute the DWT
coefficients at resolution level k.