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Title: Wavelet and multiresolution based signalimage processing


1
Wavelet and multiresolution based signal-image
processing
  • By Fred Truchetet
  • Le2i, UMR 5158 CNRS-Université de Bourgogne,
    France
  • f.truchetet_at_u-bourgogne.fr

2
Overview
3
Wavelet play field signal and image processing
3
7
32 45 3 4 5 67 7 8
  • Signal or image quantitative information
  • Process Analyze
  • Transform
  • Synthesize

4
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

5
A musical sound a function of time, a signal
Separate notes and chord
6
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

7
Distinguish the frequencies
In a chord
8
Distinguish the times and the frequencies
for series of notes
9
A sound a wave
10
A sound a function of time and frequency
11
Wave and impulse
12
A wavelet ?
  • Oscillating mother function, well localized both
    in time and frequency
  • y(t)

13
Wavelet ?
  • A family built by dilation
  • y(t) y(t/2)
    y(t/4)

14
Wavelets ?
  • and translation
  • y(t) y(t-20)
    y(t-40)

15
Waves or wavelets ?
  • Wave
  • frequency
  • Infinite duration
  • No temporal localization
  • Wavelet
  • scale
  • Duration (window size)
  • Temporal localization
  • then
  • Wavelet Note ?

16
Why the wavelets?
17
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

18
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 

19
The wavelets, why?
  • This transform can be seen as the projection over
    the sliding window functions
  • With the inner product

20
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

21
The wavelets, why?
Example of gaboret for two frequencies w (real
part)
The window size does not depend on the frequency
22
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

23
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?
24
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
25
A wavelet, what is it?
  • A mother function oscillating, localized
  • y(t)

26
A wavelet, what is it?
  • A family built by dilation
  • y(t) y(t/2)
    y(t/4)

27
A wavelet, what is it?
  • and translation
  • y(t) y(t-20)
    y(t-40)

28
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
29
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
30
The wavelets, why?
  • Time-frequency plane tiling

The wavelet transform produces a constant Q
analysis Uncertainty principle Df. Dt constant
31
Continuous wavelet transform
32
Wavelet Transform
  • Analysis
  • Searching for the weight of each wavelet (atom of
    signal) in a function f(t)

33
Continuous wavelet transform CWT
  • Continuous wavelet transform
  • In the Fourier space
  • Inverse transform

34
A wavelet has to be admissible
  • Admissibility condition
  • For ordinary localized functions
  • Or, more generally

35
Wavelet Transform
  • Synthesis
  • Add the wavelets weighted by their respective
    weights

36
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
37
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.
38
Wavelet Transform as time-frequency analysis
39
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.
40
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

41
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)

42
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

43
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)

44
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)...

45
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.

46
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

47
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

48
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

49
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.

50
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

51
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

52
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

53
Discrete wavelet transform
  • Multiresolution Analysis orthogonal basis

54
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).

55
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
56
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

57
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
58
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
59
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
60
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.
61
Multi Resolution Analysis of L2(R)
  • Set of axioms for dyadic MRA (S. Mallat, Y.
    Meyer)

62
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)
63
Multiresolution analysis
Detail signal and approximation signal are
characterized by the discrete sequences of
wavelet and scale coefficients
64
Discrete Wavelet Transform Mallats algorithm
  • Recursive algorithm MRA
  • Approximation Detail
  • (wavelet coefficients)

65
Wavelet Transform
66
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.
67
MRA example of Haar analysis
68
MRA general case
69
MRA general case
  • Example of approximations and details of f

f
70
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
71
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
72
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
73
Mallats algorithm
  • Analysis recursive algorithm
  • Linear and invariant digital filtering.
  • Two filters hn (low pass) and gn (high pass) 

74
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
75
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
76
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
77
Dyadic DiscreteWavelet Transform
  • Fast Transform Mallats algorithm
  • Recursive algorithm driving through scales from
    scale j to scale j-1

78
Example of DWT Haar basis
Find the filters h and g for the Haar analysis
Verify the algorithm of Mallat for f(x)x
79
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
80
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
81
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
82
Projection on V0
83
Example of synthesis of a detail signal
84
Projected transform example
d1
d2
d3
85
Example of approximations of the scale function
for the basis Daubechies with N2
86
Orthogonal MRA
  • Properties and building

87
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

88
Orthogonality of the functions and of the
associated filters
For the scale function
Therefore ?
For n0
89
For the wavelets
Between Wj and Vj
Between wavelets within the same scale
Generally
as
Therefore
and
90
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
91
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
92
Scale functions j and associated filters h in the
Fourier domain
Separating odd and even terms
as h is 2p-periodic
or
93
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
94
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
95
Wavelet functions y and associated filters g in
the Fourier domain
Wavelet-scaling function orthogonality
For w0
and
Therefore
96
Wavelet functions y and associated filters g in
the Fourier domain
From
Show that
or
Therefore
97
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?
98
Relationship between h and g in orthogonal bases
From
?
with
The simplest solution with phase linear
For example
99
Relationship between h and g in orthogonal bases
From
Show that
These pairs of filters are called QMF Quadrature
Mirror Filters
Or more generally
100
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
101
(No Transcript)
102
Mallats algorithm
High frequencies
Low frequencies
0
103
Examples of wavelets for orthogonal MRA
  • Haar, Spline, Daubechies

104
Examples of orthogonal MRA
105
Extension of MRA for 2D
106
MRA of multidimensional signal
107
A wavelet can also be an image
108
Line filtering
  • Separable dyadic Image transform 2 stages

Row filtering
- Line by line,
- Then row by row
109
Image transform example of dyadic separable
transform

110
Biorthogonal Bases
111
Biorthogonal bases fro MRA
112
Frames of wavelet
113
Applications
114
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

115
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

116
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
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