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
  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.
• or detection of higher degree components
  singularities
• If a wavelet is k times differentiable, it has at
  least k vanishing moments

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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 scale 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
Scale function family
with n integer, constitutes an orthogonal basis
of Vi, the scale 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
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Discrete Wavelet Transform Mallats algorithm

• Recursive algorithm MRA
• Approximation Detail
• (wavelet 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, f(x) is a function of V0 and as
, f(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 subsampling 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 subsampling 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 we know that
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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 upsampled versions of a and d are
introduced
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Dyadic DiscreteWavelet 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
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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 only input at scale 0 as
approximation coefficient
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Mallats algorithmbuilding recursively the
basis functions the cascad 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
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
Orthogonality in the Fourier domain
Show that using autocorrelation in the Fourier
domain and the Poisson formula
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Scale functions j and associated filters h in the
Fourier domain
Separating odd and even terms
as h 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 phase linear
For example
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Relationship between h and g in orthogonal bases
From
Show that
These pairs of filters are 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, Spline, Daubechies

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Examples of orthogonal MRA
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Extension of MRA for 2D
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MRA of multidimensional signal
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A wavelet can also be an image
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Line filtering

• Separable dyadic Image transform 2 stages

Row filtering
- Line by line,
- Then row by row
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Image transform example of dyadic separable
transform

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Biorthogonal Bases
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Biorthogonal bases fro MRA
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Frames of wavelet
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Applications
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Which Wavelet Transform?

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

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DWT variations and extensions

• Wavelet packets
• Multi-wavelets
• Rational wavelets
• Multi-dimensional wavelets
• Separables
• Non-separables
• Geometric wavelets
• Ridgelets
• Curvelets
• Contourlets
• Multi-valued wavelets
• Gaborets
• Pseudo, semi, wavelets

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Wavelet applicationsmain contributing properties

• Time-scale analysis
• Scalograms
• Transient detection and characterization
• Feature extraction
• Multiscale analysis
• Characterization of fractal behaviour
• Texture analysis
• Organizing information in signal
• Compression
• Reversibility
• Filtering
• De-noising