Wavelet and multiresolution based signalimage processing

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Wavelet and multiresolution based signal-image
processing

• By Fred Truchetet
• Le2i, UMR 5158 CNRS-Université de Bourgogne,
  France
• f.truchetet_at_u-bourgogne.fr

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Overview
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Wavelet play field signal and image processing
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32 45 3 4 5 67 7 8

• Signal or image quantitative information
• Process Analyze
• Transform
• Synthesize

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Wavelets, why?

• Signal processing analysis, transformation,
  characterization, synthesis
• Example
• Analysis of a musical sequence
• For automatic creation of score (music sheet)
• Synthesis of music from score
• For automatic reading and playing score

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A musical sound a function of time, a signal
Separate notes and chord
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Analysis and synthesis ?

• In a score the music stream is segmented into
   atoms  or notes defined by their
• Pitch C, D, E, etc
• Duration (whole note, half note, quarter note,
  etc)
• Position in time (measure bars)
• It provides an analysis of the musical signal
• With the score the musician can play the music as
  it has been originally created
• It is the synthesis stage

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Distinguish the frequencies
In a chord
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Distinguish the times and the frequencies
for series of notes
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A sound a wave
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A sound a function of time and frequency
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Wave and impulse
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A wavelet ?

• Oscillating mother function, well localized both
  in time and frequency
• y(t)

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Wavelet ?

• A family built by dilation
• y(t) y(t/2)
  y(t/4)

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Wavelets ?

• and translation
• y(t) y(t-20)
  y(t-40)

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Waves or wavelets ?

• Wave
• frequency
• Infinite duration
• No temporal localization

• Wavelet
• scale
• Duration (window size)
• Temporal localization
• then
• Wavelet Note ?

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Why the wavelets?
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The wavelets, why?(again but with mathematical
arguments)

• In the real world, a signal is not stationary.
• The information is in the statistical,
  frequential, temporal, spatial varying features
• Examples vocal signal, music, images
• Joseph Fourier, in 1822, proposed a global
  analysis
• Integrals are from - ? to ?
• Spatial or temporal localization is lost
• Fourier Transform

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The wavelets, why?

• A straightforward idea cut the integration
  domain into sliding windows
• Window Fourier transform or Short Time Fourier
  Transform (STFT)
• We denote the window function as
• When t and w vary It constitutes a family which
  can be considered as a kind of  basis 

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The wavelets, why?

• This transform can be seen as the projection over
  the sliding window functions
• With the inner product

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The wavelets, why?

• Many window functions are used Hanning, Hamming,
  and Gauss
• For the Gauss window, the transform is called
  Gabor transform. The basis function is called
  gaboret. These functions are normalized with
• Gabor Transform

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The wavelets, why?
Example of gaboret for two frequencies w (real
part)
The window size does not depend on the frequency
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The wavelets, why?

• The resolution in the frequency-time plane can be
  estimated by the variance of the window function
• With xt or xf for time and frequency
  resolution respectively
• For a gaboret
• As
• Then whatever the frequency

Show that

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The wavelets, why?
Time-frequency plane tiling provided by the Gabor
Transform
Not optimal As some periods are necessary for
frequency measurement a low temporal resolution
comes naturally for low frequencies, for high
frequencies a finer temporal resolution is
possible. Question how to find an automatic
trade-off between time and frequency resolution
for all the frequencies?
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The wavelets, why?

• Answer the Wavelet Transform

a is the scale factor and b the translation
parameter and Y is the wavelet function (basis
window function). The scale factor a is as 1/w,
the greater a the larger the wavelet. If a is
small, the frequency is high and the window is
small allowing a high temporal resolution for the
analysis. Y is called the mother of a family of
functions built by dilation and translation
following
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A wavelet, what is it?

• A mother function oscillating, localized
• y(t)

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A wavelet, what is it?

• A family built by dilation
• y(t) y(t/2)
  y(t/4)

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A wavelet, what is it?

• and translation
• y(t) y(t-20)
  y(t-40)

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The wavelets, why?
The norm does not depend on a
The wavelet transform (WT) can be denoted as
If the temporal resolution of the mother wavelet
is taken as unit, then
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The wavelets, why?
Then
And for the frequency resolution, taking in the
same way the frequency variance of the mother
wavelet as unit
Show that
w0 1/a then
Qconstant
And finally
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The wavelets, why?

• Time-frequency plane tiling

The wavelet transform produces a constant Q
analysis Uncertainty principle Df. Dt constant
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Continuous wavelet transform
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Wavelet Transform

• Analysis
• Searching for the weight of each wavelet (atom of
  signal) in a function f(t)

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Continuous wavelet transform CWT

• Continuous wavelet transform
• In the Fourier space
• Inverse transform

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A wavelet has to be admissible

• Admissibility condition
• For ordinary localized functions
• Or, more generally

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Wavelet Transform

• Synthesis
• Add the wavelets weighted by their respective
  weights

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Wavelets for CWT

• Some examples of admissible wavelets
• Haar (this example is presented further)
• Mexican hat
• Morlet

Show that the Morlet wavelet is only close to
admissible
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Wavelets for CWT
Wavelets in the Fourier domain
Mexican hat
Morlet for a1 and a2
As a is increasing, the frequency size shrinks
while the temporal window enlarges. The original
trade-off is maintained whatever the scale factor.
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Wavelet Transform as time-frequency analysis
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Sampling for discrete wavelet transform
The time-scale plane can be sampled to avoid or
limit the redundancy of the CWT. To respect the
Q-constant analysis principle, the sampling must
be such that
i is the discrete scale factor and n the discrete
translation parameter, both are integer.
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Discrete wavelet transform DWT

• Discrete analysis with continuous wavelet
• Isomorphism between L2(R) and l2(R) (continuous
  functions ? discrete sequences)
• aa0i with i integer bnb0a0i with n integer
• Dyadic analysis a02 b01
• Discrete tiling of the scale-time space

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Which Wavelet Transform?

• Continuous, CWT, for signal analysis, without
  synthesis redundant
• Discrete, DWT, (dyadic or not, Mallat or lifting
  scheme), for signal or image analysis if
  synthesis is required
• Non redundant
• Orthogonal basis
• Non orthogonal basis (biorthogonal)
• Redundant non decimated DWT, Frame
• Wavelet packets (redundant or not)

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Who invented wavelets?

• From Joseph Fourier to Jean Morlet
• and after ...
• almost a French story
• The ancestor
• Joseph FOURIER born in Auxerre
• (Burgundy, France) in 1768,
• amateur mathematician, provost of Isère
• published in 1822 a theory of heat
• Every physical  function can be
• written as a sum of sine-waves
• Fourier Transform

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Who invented wavelets?

• The grandfather
• Dennis GABOR electrical engineer
• and physicist, Hungarian born English,
• Nobel price of physics in 1971
• for inventing holography
• Decomposition into constant duration
• wave pulses
• Short Time Fourier Transform (1946)

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Who invented wavelets?

• The father
• Jean MORLET French engineer from Ecole
  Polytechnique, geologist for petrol company
• Elf Aquitaine
• Decomposition into wavelets with duration in
  inverse proportion to frequency (1982)
• The children
• A.Grossmann (1983), Y.Meyer (1986),
• S.Mallat (1987), I.Daubechies (1988), J.C.Fauveau
  (1990), W. Sweldens (1995)...

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references

• Daubechies, Ten Lectures on Wavelets, SIAM,
  Philadelphia, PA, 1992.
• S. Mallat, A theory for multiresolution signal
  decomposition the wavelet representation,
  IEEE, PAMI, vol. 11, N 7, pp. 674-693, july
  1989.
• S. Mallat, Wavelet Tour of Signal Processing,
  Academic Press, Chestnut Hill MA, 1999
• G. Strang, T. Nguyen, Wavelets and filter
  banks, Wellesley-Cambridge Press, Wellesley MA,
  1996.
• F. Truchetet, Ondelettes pour le signal
  numérique, Hermès, Paris, 1998.
• F. Truchetet, O. Laligant, Industrial
  applications of the wavelet and multiresolution
  based signal-image processing, a review, proc.
  of QCAV 07, SPIE, vol. 6356, may 2007
• M. Vetterli, J. Kovacevic, Wavelets and Subband
  Coding , Prentice Hall, Englewood Cliffs, NJ,
  1995.

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Which wavelet?

• Freedom to choose a wavelet
• Blessing or Curse?
• How much efforts need to be made for finding a
  good wavelet?
• Any wavelet will do?
• What properties of wavelets need to be
  considered?
• Symmetry, regularity, vanishing moments, compacity

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Symmetry

• In some applications the analyzing function needs
  to be symmetric or antisymmetric
• Real world images
• This is related to phase linearity
• Symmetric Haar, Mexican hat, Morlet
• Non symmetric Daubechies, 1D compact support
  orthogonal wavelets

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Regularity

• The order of regularity of a wavelet is the
  number of its continuous derivatives.
• Regularity can be expanded into real numbers.
  (through Fourier Transform equivalent of
  derivative)
• Regularity indicates how smooth a wavelet is

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Vanishing Moment

• Moment js moment of the function
• When the wavelets k1 moments are zero
• i.e.
• the number of Vanishing Moments of the wavelet is
  k.
• Weakly linked to the number of oscillations.

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Vanishing moments

• When a wavelet has k vanishing moments, WT leads
  to suppression of signals that are polynomial of
  degree lower or equal to k. (whatever the scale)
• or detection of higher degree components
  singularities
• If a wavelet is k times differentiable, it has at
  least k vanishing moments

Show that from
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Compacity (size of the support)

• The number of FIR filter coefficients.
• The number of vanishing moments is proportional
  to the size of support.
• Trade-off between computational power required
  and analysis accuracy
• Trade-off between time resolution and frequency
  resolution
• A compact orthogonal wavelet cannot be symmetric
  in 1D

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Which wavelet examples for DWT

• Db1 (Haar) Db2 (D4) Db5 (D10) Db10
  (D20)
• RNA R0.5 R1.59
  R2.90
• VM1 VM2 VM5
  VM10
• SS2 SS4 SS10
  SS20

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Discrete wavelet transform

• Multiresolution Analysis orthogonal basis

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Multi Resolution Analysis of L2(R)

• Approximation spaces
• Working space L2(R), for continuous functions,
  f(x), on R with finite norm (finite energy)
• An analysis at resolution j of f is obtained by a
  linear operator
• Vj is a subspace of L2(R), Aj is a projection
  operator (idempotent)
• A multiresolution analysis (MRA) is obtained with
  a set of embedded subspaces Vj , such that going
  from one to the next one is performed by
  dilation
• In the dyadic case for instance, the dilation
  factor is 2.
• The functions in subspace Vj1 are coarser than
  in subspace Vj and
• If j goes to - infinity, the subspace must tend
  toward L2(R).

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Multi Resolution Analysis of L2(R)

• Set of axioms for dyadic MRA (S. Mallat, Y.
  Meyer)

The last property allows the invariance for
translation by integer steps
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Multi Resolution Analysis of L2(R)

• In these conditions there exists a function f(x)
  called scaling function from which, by integer
  translation, a basis of V0 can be built.
• Then a basis can be obtained for each subspace by
  dilating f(x)
• The basis is orthogonal if

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Multi Resolution Analysis of L2(R)
The approximation at scale j of the function f is
given by
The approximation coefficients constitutes a
discrete signal. If the basis is orthogonal, then
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Multi Resolution Analysis of L2(R)
For each subspace Vj its orthogonal complement Wj
in Vj-1 can be defined. It is called the detail
subspace at scale j
As Wj is orthogonal to Vj, it is also orthogonal
to Wj1 which is in Vj. Therefore, all the Wj are
orthogonal
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Multi Resolution Analysis of L2(R)
In these conditions there exists a function y(x)
called wavelet function from which, by integer
translation, a basis of W0 can be built. Then
a basis can be obtained for each subspace by
dilating y(x) The basis is orthogonal if
And the complement of the approximation at scale
j can be computed by
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Multi Resolution Analysis of L2(R)
The details of f at scale j are obtained by a
projection on Wj as
These coefficients are the wavelet coefficients
or the coefficients of the discrete wavelet
transform DWT associated to this MRA. They
constitute a discrete signal.
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Multi Resolution Analysis of L2(R)

• Set of axioms for dyadic MRA (S. Mallat, Y.
  Meyer)

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MRA and orthogonal wavelet basis
Scaling function family
with n integer, constitutes an orthogonal basis
of Vi, the scaling functions are not admissible
wavelets!
Wavelet family
with n integer, constitutes an orthogonal basis
of Wi
All Wi are orthogonal and the direct sum of all
these subspaces is equal to L2(R)
for i and n integers constitutes an orthogonal
basis of L2(R)
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Multiresolution analysis
Detail signal and approximation signal are
characterized by the discrete sequences of
wavelet and scale coefficients
Sampling is a consequence of MRA
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Discrete Wavelet Transform Mallats algorithm

• Recursive algorithm MRA
• Approximation Detail
• (wavelet coefficients)

Question initialization? What are the first
approximation coefficients?
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Wavelet Transform
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Example of MRA Haar basis
The scale function
The wavelet function
Verify invariance, normality and describe the
functions of Vj and Wj and give the Haar analysis
of f(x)x.
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MRA example of Haar analysis
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MRA general case
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MRA general case

• Example of approximations and details of f

f
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Mallats algorithm analysis
By definition, j(x) is a function of V0 and as
, j(x) can be decomposed on the
basis of V-1 and a discrete sequence with
can be found such that
With
and or
Show that
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Mallats algorithm analysis
The approximation coefficients aj
can be computed following a recursive
algorithm
then
If h is considered as the impulse response of a
discrete filter, we have a convolution followed
by a downsampling by two
2
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Mallats algorithm analysis
In the same way, W0 is in V-1 and a discrete
sequence gn can be found by projecting the
wavelet function on the basis of V-1
or
Show that
If g is considered as the impulse response of a
discrete filter, we have a convolution followed
by a down sampling by two
2
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Mallats algorithm

• Analysis recursive algorithm
• Linear and invariant digital filtering.
• Two filters hn (low pass) and gn (high pass) 

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Mallats algorithm synthesis
The analysis at scale j-1 gives two components,
one in Vj and the other in Wj
with
As Aj is a projection operator (idempotent)
then
and therefore
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Mallats algorithm synthesis
We have seen that
As the basis of Vj-1 is orthogonal
then
and
Therefore from
a synthesis equation can be written
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Mallats algorithm synthesis
This equation can be seen as the sum of two
convolution products (digital linear filtering)
if two up sampled versions of aj and dj are
introduced
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Dyadic Discrete Wavelet Transform

• Fast Transform Mallats algorithm
• Recursive algorithm driving through scales from
  scale j to scale j-1

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Example of DWT Haar basis
Find the filters h and g for the Haar analysis
Verify the algorithm of Mallat for f(x)x and one
scale
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Mallats algorithmbuilding recursively the
basis functions
The mother scale function belongs to V0 and the
basis is orthogonal
and
Then for the mother scale function j
Then an approximation at scale j of j can be
obtained by cranking the machine up to scale j
with a Dirac as approximation coefficient at
scale 0 as only input
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Mallats algorithmbuilding recursively the
basis functions the cascade algorithm
Verify this result for the Haar basis
A similar result can be obtained for the wavelets
therefore
The only detail coefficient sequence is a Dirac
at scale 0
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Synthesis of a projection on Vj or Wj
More generally, an approximation or a detail
function at scale j can be obtained by following
the synthesis algorithm
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Projection on V0
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Example of synthesis of a detail signal
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Projected transform example
d1
d2
d3
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Example of approximations of the scale function
for the basis Daubechies with N2
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Orthogonal MRA

• Properties and building

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DWTProperties of the basis functions and of the
associated filters

• Orthogonality of the scale function and of the
  associated filter
• Orthogonality of the wavelet function and of the
  associated filter
• Scale functions j and filters associated h in the
  Fourier domain
• Wavelet functions y and filters g associated in
  the Fourier domain

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Orthogonality of the functions and of the
associated filters
For the scale function
Therefore ?
For n0
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For the wavelets
Between Wj and Vj
Between wavelets within the same scale
Generally
as
Therefore
and
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Scale functions j and associated filters h in the
Fourier domain
hn is considered as the impulse response of a
discrete linear filter
Transfer function
Frequency response
(2p-periodic)
and
therefore
or
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Scale functions j and associated filters h in the
Fourier domain
Analyzing a function with a non zero mean value
shows that we must have
Orthogonality in the Fourier domain
Show that using autocorrelation in the Fourier
domain and the Poisson formula
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Poisson equation
autocorrelation
In Fourier
sampling
In Fourier
or
and
As Fourier transform of Dirac is 1
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Scale functions j and associated filters h in
Fourier domain
Separating odd and even terms
as is 2p-periodic
or
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Scale functions j and associated filters h in the
Fourier domain
as
For w0 in this equation and in the previous one,
it comes
and
Therefore, h is a low pass filter giving a low
resolution version of the signal
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Wavelet functions y and associated filters g in
the Fourier domain
gn is considered as the impulse response of a
discrete linear filter
Transfer function
Frequency response
and
therefore
Intra scale wavelet orthogonality
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Wavelet functions y and associated filters g in
the Fourier domain
Wavelet-scaling function orthogonality
For w0
and
Therefore
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Wavelet functions y and associated filters g in
the Fourier domain
From
Show that
or
Therefore
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Wavelet functions y and associated filters g in
the Fourier domain
y is an admissible wavelet function
g is a high pass filter keeping the high
frequency components, i.e. the details
How to deduce g from h?
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Relationship between h and g in orthogonal bases
From
?
with
The simplest solution with linear phase
For example
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Relationship between h and g in orthogonal bases
From
Show that
Such a pair of filters is called QMF Quadrature
Mirror Filters
Or more generally
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Building an MRA

• Begin with the scaling function or the
  approximation subspaces
• Determine h filters
• Deduce g filters
• Finally deduce the wavelet functions

1 and 2 can be switched round
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Mallats algorithm
High frequencies
Low frequencies
0
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Examples of wavelets for orthogonal MRA

• Haar, Littlewood-Paley, Spline, Daubechies

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Examples of orthogonal MRA Haar
Mother scaling function
Approximation subspaces
Projection on a finer subspace
From
It comes
or
and with the QMF property
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Examples of orthogonal MRA Haar
We have
From
Therefore
Scaling and wavelet functions in the Fourier
domain
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Examples of orthogonal MRA Haar
Very compact in space, very bad localized in
frequency Symmetric, no regularity, 1 vanishing
moment
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Examples of orthogonal MRALittlewood-Paley
It comes from the same idea the approximation
subspaces in Fourier domain are piecewise
constant. Kind of dual basis to the Haars
If
the orthogonality property in Fourier is clearly
verified
To have symmetry a zero-phase condition is set,
show that
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Examples of orthogonal MRALittlewood-Paley
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Examples of orthogonal MRALittlewood-Paley
The associated filters
from
It comes
Therefore
and with the QMF relationship
These filters are IIR
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Examples of orthogonal MRALittlewood-Paley
The wavelet
From
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Example of MRASpline bases (Battle-Lemarié)

• Improve the Haar basis for a better piecewise
  approximation using polynomial functions
• Keep the symmetry (linear phase)
• Use the B-spline basis properties in connection
  with
• The B-spline functions are a basis for piecewise
  polynomial functions but not an orthogonal basis
  in
• An orthogonalization process is required

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Example of MRASpline bases (Battle-Lemarié)
The approximation subspace Vj is defined as the
set of piecewise polynomial functions on 2j width
segments.
The B-spline basis of order n is built by
autoconvolution of a box function
Therefore
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Example of MRASpline bases (Battle-Lemarié)
Examples of B-spline with n1 and n2
Compact support but not orthogonal
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Example of MRASpline bases (Battle-Lemarié)
Therefore
The orthogonalization process is based on the
following property of orthogonal bases
It can be shown that if f(t) is a basis, an
orthogonal basis is obtained by
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Example of MRASpline bases (Battle-Lemarié)
The orthogonal scaling function basis is given by
It can be shown that the normalization factor can
be computed with discrete B-splines
with
Therefore finally
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Example of MRASpline bases (Battle-Lemarié)
n1
n2
Infinitely supported but orthogonal
In Fourier
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Example of MRASpline bases (Battle-Lemarié)
Filters




















Cubic spline basis Battle-Lemarié
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Example of MRASpline bases (Battle-Lemarié)
Wavelets
n1
n2
In Fourier
Compute an approximation of the Battle-Lemarié
wavelet with the matlab wavelet toolbox