A Multiresolution Formulation of Wavelet Systems

A Multiresolution Formulation of Wavelet Systems
Discrete Wavelet transforms
Prerequisites
Wavelets: A wavelet is a wave-like oscillation with an amplitude that starts out at zero,
increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation"
like one might see recorded by a seismograph or heart monitor.
Discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are
discretely sampled
Concepts of Multiresolution Analysis
the first component to multiresolution analysis is vector spaces. For each vector space, there is
another vector space of higher resolution until you get to the final image. Also, each vector
space contains all vector spaces that are of lower resolution. The basis of each of these vector
spaces is the scale function for the wavelet.
Nested function spaces spanned by the wavelet and scaling functions.
Subspace of L 2 (R) spanned by function k
(t) is
Span { ( t)}
0 k
Subspace w0
k
1
0
w
0
2
0
w
0
w
1
The Wavelet system expansion is a tool used to decompose a function into a set of weighted
basis functions that are localized in time and frequency
For an orthogonal wavelet system, the coefficients can be calculated by their inner products

Display of the Discrete Wavelet Transform and the Wavelet Expansion
Samples of the signal – for a high starting scale j 0 samples of the signal are the DWT at that
scale.
A 3D plot of the expansion coefficients of the DWT values c(k) and d j (k)
Components of the signal are generated using time functions at each scale:
f ( t ) f
where
f
j0
j0
c ( k ) ( t k )
j
f
j
( t )
and
f
j
( t )
k
d
j
( k )2
j / 2
(2
jt k
)

By generating time functions f k (t) at each translation the time localization of the wavelet
expansion is obtain
f ( t)
c(
k ) ( t k ) d ( k )2 (2 t k )
k
j
j
Tilling time frequency plane: we use the term time frequency tile of a particular basic function to
designate the region in the plane which contains most of the function`s energy
j / 2
j
Generalized tiling which adapts in time as well as in frequency
The tiling shows the sampling in time and frequency
Examples of wavelet expansions
Objective: show the way a wavelet expansion decomposes a signal and what the components look like
at different scale
Projection of the Houston Skyline Signal onto ν spaceA characteristic of Daubechies systems is
that low order polynomials are completely contained in the scaling function space.
DWT used for edge detection

Discrete Wavelet Transform of a Doppler signal
Illustrates coefficients of the DWT directly as a function of j and k

Scaling functions approximations and the wavelet decomposition of the chirp signal
Daubechies
Daubechies wavelets
the Daubechies wavelets are a family of orthogonal wavelets defining a DWT and characterized
by a maximal number of vanishing moments for some given support. With each wavelet type of
this class, there is a scaling function (also called father wavelet) which generates an orthogonal
multiresolution analysis.
Ingrid Daubechies