Wavelet and multiresolution based signal-image processing
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Transcript of Wavelet and multiresolution based signal-image processing
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Wavelet and multiresolution based signal-image processing
By Fred TruchetetLe2i, UMR 5158 CNRS-Université de Bourgogne, France
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Overview
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Wavelet play field: signal and image processing
• Signal or image: quantitative information • Process: Analyze
TransformSynthesize
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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
0 0.5 1 1.5 2 2.5 3
x 105
-0.8
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Notes Accord
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
-0.4
-0.2
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Par
tie r
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Signal temporel
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|STFT|2, Lh=50, Nf=256, Ech. lin., Seuil=5%
Temps
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ACCORD
In a chord
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Distinguish the times and the frequencies
-0.2
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Signal temporel
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Temps
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DOREMI
for series of notes
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A sound: a wave
0 2000 4000 6000 8000 10000 12000 14000 16000 18000-0.5
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0.4Représentation temporelle du son correspondant au mot GABOR
temps
GABOR
0 500 1000 15000
50
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350
400Représentation fréquentielle du son GABOR
fréquence
GABOR
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A sound: a function of time and frequency
-0.2
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Signal temporel
Echelle lin.
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|STFT|2, Lh=50, Nf=256, Ech. Log., Seuil=5%
Temps
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GABOR
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Wave and impulse
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Onde sinusoïdale
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x 104
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Impulsion sinusoïdale
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x 104
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Impulsion sinusoïdale
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A wavelet ?
• Oscillating mother function, well localized both in time and frequency :
(t)
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Wavelet ?
• A family built by dilation
(t) (t/2) (t/4)
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Wavelets ?
• and translation :
(t) (t-20) (t-40)
)(a
bt
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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:
Tfourierf
fte j tdt #
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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 vary It constitutes a family which can be considered as a kind of « basis »
dsetsgsftfT sjslwin
)()(),(
t, s gs te j s #
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The wavelets, why?
• This transform can be seen as the projection over the sliding window functions :
• With the inner product:
,,),( tslwin ftfT
f,g
functions) real(for )()(, dxxgxfgf
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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:
gx 14 e
x2
2 #
f 2
fxf xdx #
Tgaborft, 14
fse
s t2
2 e j sds #
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The wavelets, why?
Example of “gaboret” for two frequencies (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 x=t or x=f for time and frequency* resolution respectively
• For a “gaboret”:
• As
• Then whatever the frequency:
21
x2
x2| x |2dx #
2
2
4
1
)(x
ex
2
1t
2xe e f222
1
f
21 tShow that
d
222 )(ˆ2
1
*
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The wavelets, why?
f
t
t e m p s
f r é q u e n c e
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 is the wavelet function (basis window function).
The scale factor a is as 1/ 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.is called the mother of a family of functions built by dilation and
translation following:
dta
bttf
abafT wav )()(
1),(
a,bt 1a t b
a #
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A wavelet, what is it?
• A mother function oscillating, localized:
(t)
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A wavelet, what is it?
• A family built by dilation
(t) (t/2) (t/4)
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A wavelet, what is it?
• and translation:
(t) (t-20) (t-40)
)(a
bt
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The wavelets, why?
a,bt 1a t b
a #
The norm does not depend on a:
a,b 2
1a t b
a 2dt
1a
| x |2adx
2
#
bawav fbafT ,,),(
The wavelet transform (WT) can be denoted as:
t2 t2| a,0t |2dt
t2 1a t
a 2dt
a2x2 1a | x |2adx
If the temporal resolution of the mother wavelet is taken as unit, then
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The wavelets, why?
1t
Then
And for the frequency resolution, taking in the same way the frequency variance of the mother wavelet as unit
t a
a1
And finally
Show that
0 ~ 1/a then Q=constant10
Q
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The wavelets, why?
Time-frequency plane tiling
a
a
temps
fréquence:1/a
The wavelet transform produces a constant Q analysis
Uncertainty principle: f. t = 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)
)(1
)(, a
bt
atba
baba fC ,, ,
ba,
f
dtttf ba )()( ,*
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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
SynthesisAdd the wavelets weighted by their respective weights
2,, )()(
a
dadbtCctf baba
baC ,
)(, tba
)(tf 2,, )(a
dadbtCc baba
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Wavelets for CWT
• Some examples of admissible wavelets– Haar (this example is presented further)– Mexican hat
– Morletti
t
eet 0
2
24
1
)(
224
12
)1(3
2)(
t
ett
Show that the Morlet wavelet is only close to admissible
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Wavelets for CWT
omega
psi
-10 -5 0 5 10
1
2
omega
psi
Wavelets in the Fourier domain
Morlet for a=1 and a=2 Mexican hat
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
a
a
temps
fréquence:1/a
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:
Zbaanbbaa ooioo
io , with and
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)
• a=a0i with i integer b=nb0a0
i with n integer
• Dyadic analysis: a0=2 b0=1
• Discrete tiling of the scale-time space
dtnttxxnixT iiniod )2(2)(,),( *2/
,
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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 engineerand physicist, Hungarian born English,Nobel price of physics in 1971for inventing holography
Decomposition into constant duration « wave pulses »:
Short Time Fourier Transform (1946)
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The fatherJean 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)...
Who invented wavelets?
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referencesI. Daubechies, «Ten Lectures on Wavelets», SIAM, Philadelphia, PA,
1992.II. S. Mallat, «A theory for multiresolution signal decomposition : the
wavelet representation», IEEE, PAMI, vol. 11, N° 7, pp. 674-693, july 1989.
III. S. Mallat, “Wavelet Tour of Signal Processing”, Academic Press, Chestnut Hill MA, 1999
IV. G. Strang, T. Nguyen, «Wavelets and filter banks», Wellesley-Cambridge Press, Wellesley MA, 1996.
V. F. Truchetet, “Ondelettes pour le signal numérique”, Hermès, Paris, 1998.
VI. 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
VII. 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
.10 with is regularity then the
aroundlocally resembles)( If )(
rrm
tttt o
r
om
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Vanishing Moment
• Moment: j’s moment of the function
• When the wavelet’s k+1 moments are zero
i.e.
the number of Vanishing Moments of the wavelet is k.
Weakly linked to the number of oscillations.
)(t
dtttm j
j )(
kjdttt j ,...,0for0)(
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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 )(ˆ)()( )( kkk jtt
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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)
R=NA R=0.5 R=1.59 R=2.90 VM=1 VM=2 VM=5 VM=10 SS=2 SS=4 SS=10 SS=20
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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 V j , 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 Vj+1 are coarser than in subspace Vj and
– If j goes to - infinity, the subspace must tend toward L2(R).
Aj Ajf Vj #
fx Vj f x2 Vj 1 #
Vj 1 Vj #
limj
Vj L2R #
)(,2
dxxfff
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Multi Resolution Analysis of L2(R)
• Set of axioms for dyadic MRA (S. Mallat, Y. Meyer):
ZkVktxVtx
ZiVtxVtx
V
LV
VVVV
ii
Zii
Zii
ii
,)()(
,)2()(
0
)(
.........
00
1
2
101
R
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 (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 (x)
• The basis is orthogonal if
,)(0Vx
Z nnxx jnj
j
with)2(2)( 2,
Z nnxxn with)()(,0
x x ndx n n Z #
j,n, j,k n k n,k, j Z #
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Multi Resolution Analysis of L2(R)
Ajf n
f, j,n j,n #
The approximation at scale j of the function f is given by:
anj f, j,n #
The approximation coefficients constitutes a discrete signal.
If the basis is orthogonal, then
Ajf 2 n
anj 2
#
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Multi Resolution Analysis of L2(R)
Vj 1 Vj Wj
L2R jZ Wj
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 Wj+1 which is in Vj. Therefore, all the Wj are orthogonal
kj WWjkj then,
V0
V1 W1
V2 W2
V3 W3
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Multi Resolution Analysis of L2(R)
,)(0Wx
In these conditions there exists a function (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 (x)
The basis is orthogonal if
j,nx 2j2 2 jx n avec n Z #
j,n, i,k j i n k j, i,n,k Z #
Z nnxxn with)()(,0
Aj 1f Ajf n
f, j,n j,n #
And the complement of the approximation at scale j can be computed by
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Multi Resolution Analysis of L2(R)
Djf n
f, j,n j,n #
dnj f, j,n #
Aj 1f Ajf Djf #
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):
ZiWV
WVV
ZkVktxVtx
ZiVtxVtx
V
LV
VVVV
ii
iii
ii
Zi
i
Zi
i
ii
,
,)()(
,)2()(
0
)(
.........
1
00
1
2
101
R
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MRA and orthogonal wavelet basis
)2(2)( 2/, ntt iini
with n integer, constitutes an orthogonal basis of Vi, the scaling functions
are not admissible wavelets!
)2(2)( 2/, ntt iini
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):
ni, for i and n integers constitutes an orthogonal basis of L2(R)
Scaling function family:
Wavelet family:
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Multiresolution analysis
nin
nii xxA ,,,
nin
nii xxD ,,,
niin xa ,, ),(, , nixTxd odni
in
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: Mallat’s algorithm
• Recursive algorithm: MRA
AApproximation + DDetail
(wavelet coefficients)
)()()( txDtxAtxA ii1i
Question: initialization?
What are the first approximation coefficients?
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Wavelet Transform
Coarse
Coarse
Coarse
Detail
Detail
Multiresolution analysis
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Example of MRA: Haar basis
x 21.510.50-0.5
1
0.8
0.6
0.4
0.2
0
x 21.510.50-0.5
1
0.5
0
-0.5
-1
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
x
A0x
A1x D1x
A2x D2x
constant has one that such )()1(2,2
2 kkj jjfkLfV ZR
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Transformée projetée 68
MRA: general caseMRA: general case
V 3
V 2
W 3
W 2
V 1 W 2
V 0
2 2
1
1Scale function wavelet
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Transformée projetée 69
MRA: general caseMRA: general case
Example of approximations and details of f
PV0 f
PV1 f PW1
f
PV2 f PW2
f
PV3 f PW3
f
f
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Mallat’s algorithm: analysis
By definition, (x) is a function of V0 and as , (x) can be decomposed on the basis of V-1 and a discrete sequence
with can be found such that
V0 V 1
][nhn Z x
n
hn 1,nx #
With and
or
1,nx 212 2x n nnh ,10,0 ,][
x n
hn21/2 2x n #
Show that nkjk
nj kh 2,1, ][
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Mallat’s algorithm: analysis
The approximation coefficients aj : can be computed following a recursive algorithm:
anj f, j,n
anj
k
hkf, j 1,k 2n #
anj
l
hl 2nf, j 1,l # hn h n # an
j l
h2n lf, j 1,l #
then
If h is considered as the impulse response of a discrete filter, we have a convolution followed by a downsampling by two:
2
anj
l
h2n lal
j 1 #
h~
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Mallat’s algorithm: analysis
n
gn 1,n #
x n
gn 2 2x n #
gn , 1,n #
In the same way, W0 is in V-1 and a discrete sequence g[n] can be found by projecting the wavelet function on the basis of V-1:
or
dnj
l
g2n la lj 1 # 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:
2g~
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Mallat’s algorithm
• Analysis: recursive algorithm• Linear and invariant digital filtering.• Two filters h[n] (low pass) and g[n] (high pass)
nnh ,10,0 ,][ nng ,10,0 ,][
anj-1 an
j
dnj
h
g
2
2
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Mallat’s algorithm: synthesis
Aj 1f n
anj j,n
n
dnj j,n #
anj f, j,n dn
j f, j,n
fAfAA jjj 111 )( Aj 1f n
anj1
Aj 1f, j 1,n j 1,n #
anj 1
k
akj j,k , j 1,n
k
dkj j,k , j 1,n #
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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Mallat’s algorithm: synthesis
j,k l
hl j 1,l 2k #
j,k , j 1,n l
hl
n l 2k
j 1,l 2k , j 1,n #
j,k , j 1,n hn 2k #
j,k , j 1,n gn 2k #
anj 1
k
akj hn 2k
k
dkj gn 2k #
We have seen that
As the basis of Vj-1 is orthogonal
then
and
Therefore from anj 1
k
akj j,k , j 1,n
k
dkj j,k , j 1,n #
a synthesis equation can be written:
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Mallat’s algorithm: synthesis
anj 1
k
akj hn 2k
k
dkj gn 2k #
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:
akj , , , , ,
alj , 0, , 0, , 0, , 0, , 0, , 0
#
anj 1
l
aljhn l
l
d ljgn l #
anj-1
anj
dnj
h
g
2
2
+
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Dyadic Discrete Wavelet Transform
Fast Transform: Mallat’s algorithmRecursive algorithm driving through scales; from scale j to scale j-1
anj 1 ~
han
j 2
~gdn
j 2
ANALYSE
anj 1an
j
dnj
SYNTHESE
2
2
h
g
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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
2
1,
2
1][nh
2
1,
2
1][ng
aon . . . , 0 , 1,2,3,4,5,6,7, . . .
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Mallat’s algorithm:building recursively the basis functions
, o,n n #
Z kjnj and 00, ,
0
and
)(
0
0
n
n
d
na
The mother scale function belongs to V0 and the basis is orthogonal:
and
Then for the mother scale function :
Then an approximation at scale j of can be obtained by cranking the machine up to scale j with a Dirac as approximation coefficient at scale 0 as only input
an-j
h2 h2(n)
j cellules identiques
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Mallat’s algorithm:building recursively the basis functions: the
cascade algorithm
Verify this result for the Haar basis
Z kjnjnj and 0),(, ,
)(
and
0
0
0
nd
a
n
n
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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Transformée projetée 82
Projection on Projection on VV00
a0
a1 d1
a2 d2
a3 d3
Coefficients of the analysis Example of coefficients projections on V0
for some approximations and details
A0 a2
. 0
a2 0 . 0
A0 d3
. 0
0 d3
A0 d2
. 0
0 d2
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Example of synthesis of a detail signal
Analysis Synthesis
Approximation
Détail
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Transformée projetée 84
Projected transform: exampleProjected transform: example
Wavelet coefficients
d1
d2
d3
Detail approximation
A0 d1
A0 d2
A0 d3
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Example of approximations of the scale function for the basis Daubechies with N=2
x 543210
0.8
0.6
0.4
0.2
0 x 1086420
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
x 20151050
0.4
0.3
0.2
0.1
0
-0.1
x 403020100
0.3
0.2
0.1
0
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Orthogonal MRA
Properties and building
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DWT:Properties of the basis functions and of the
associated filters
1. Orthogonality of the scale function and of the associated filter
2. Orthogonality of the wavelet function and of the associated filter
3. Scale functions and filters associated h in the Fourier domain
4. Wavelet functions and filters g associated in the Fourier domain
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Orthogonality of the functions and of the associated filters
Znnntt )()(),(
t n
hn 2 2t n #
k
hkh2n k n #
For the scale function:
Therefore ?
For n=0 12 kh
k
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For the wavelets
Z nntt ,0(),(
Znnntt )()(),(
Zjiknjiknnikj ,,,)()(, ,,
Between Wj and Vj
Between wavelets within the same scale
Generally
as x n
gn 2 2x n #
Therefore nkngkgk
2 02 knhkgk
12 kg
k
0 khkgk
and
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Scale functions and associated filters h in the Fourier domain
h[n] is considered as the impulse response of a discrete linear filter:
h
n
hne in #
n
nznhzH ][)(Transfer function:
Frequency response:
x n
hn 2 2x n #
12
2
n
hne in 2 #
and
therefore
12
h
2
2 #
j 1
12
h
2 j # or
(2-periodic)
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Scale functions and associated filters h in the Fourier domain
Orthogonality in the Fourier domain
n
| 2 n |2 1 #
Show that using autocorrelation in the Fourier domain and the Poisson
formula
xdx 0 0 0 #
Analyzing a function with a non zero mean value shows that we must have:
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n
t nTfourier 2
T n
n 2T #
r
t t dt t, t #
r | |2 #
re r k
k #
re 12r 2
n
n2 #
re n
r 2 n #
n
| 2 n |2 1 #
Poisson equation
autocorrelation
In Fourier
sampling
In Fourier
or
As Fourier transform of Dirac is 1
)()()()( nndtnttnn
and
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Scale functions and associated filters h in Fourier domain
2)()()()( 11 zHzHzHzH
h
2h
2 2 #
n
| 2 2 n |2 n
12
| n |2 h n
2 1 #
12 n
h 2n
2| 2n |2
12 n
h 2n 1
2| 2n 1 |2 1
#
h
2 n
12
| 2 n |2 h
2 n
12
| 2n |2 1 #
as is 2-periodic
Separating odd and even terms:
or
)(ˆ h
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Scale functions and associated filters h in the Fourier domain
h
2 2
2
#
h0
nhn 2
h 0
#
as
For =0 in this equation and in the previous one, it comes
Therefore, h is a low pass filter giving a low resolution version of the signal
0
xdx 1 # and
2n 0 n 0 #
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Wavelet functions and associated filters g in the Fourier domain
n
nzngzG ][)(
n
2 n2 1 #
g[n] is considered as the impulse response of a discrete linear filter:
Transfer function:
Frequency response:
)2(2][)( nxngxn
and
therefore 1
2g
2
2 #
Intra scale wavelet orthogonality
n
jnengg ][)(ˆ
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Wavelet functions and associated filters g in the Fourier domain
|g |2 |g |2 2 #
n
2n 2n 0 # Wavelet-scaling function orthogonality
0
xdx 0 # For =0
g0 n
gn 0
|g | 2 #
and
Therefore
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Wavelet functions and associated filters g in the Fourier domain
From n
2n 2n 0 #
Show that g h g
h 0 #
or 0)()()()( 11 zHzGzHzG
|g |2h
2 |g |2
h
2 #
|g |2 h
2 2 #
Therefore
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Wavelet functions and associated filters g in the Fourier domain
g0 n
gn 0
|g | 2 #
0
xdx 0 # 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
g h g
h 0 # From
g g h
h #
g
h # with
1)(ˆclearly and
1)(ˆ0)(ˆ)(ˆ
periodic2 is )(ˆ
The simplest solution with linear phase : jke 12)(ˆ
g e i h # For example
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Relationship between h and g in orthogonal bases
)()( 112 zGzzH k
g e i h # From
Show that gn 1nh1 n #
Or more generally
Such a pair of filters is called QMF: Quadrature Mirror Filters
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Building an MRA
1. Begin with the scaling function or the approximation subspaces
2. Determine h filters
3. Deduce g filters
4. Finally deduce the wavelet functions
e i 2 1
2
h
2
2
1 and 2 can be switched round
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Résumé des principales propriétés des fonctions et des filtres associés
à une analyse multirésolution orthogonale
t, t n n n Z t, t n 0 n Z
t, t n n n Z
12
h 2
2 12
g 2 2
n| 2 n |2 1
n 2 n
2 1
| 0 | 1 0 0
n 2n 2n 0 g
h g
h 0
k
hkh2n k n k|hk|2 k
|gk|2 1
h
2h
2 2 |g |2 |g |2 2
h
2 |g |2 2
h0
nhn 2
h 0
g0 n
gn 0
|g | 2
g
h
est 2 périodique
0
1
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Low frequencies
High frequencies
x(n)
0% 25% 50%12.5%
g(n) 2 a
h(n) 2
h(n) 2
g(n) 2 b
h(n) 2 d
g(n) 2 c
Mallat’s algorithm
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Examples of wavelets for orthogonal MRA
Haar, Littlewood-Paley, Spline, Daubechies
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Examples of orthogonal MRA: Haar
constant has one that such )()1(2,2
2 kkj jjfkLfV ZR
not if0
101)(
tt
)12()2()( ttt 2 1 e i
2
2 #
h 121 e i #
Hz 121 z 1 #
Gz 121 z 1 #
Mother scaling function:
Approximation subspaces:
Projection on a finer subspace
From x n
hn 2 2x n # It comes
or
and with the QMF property
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Examples of orthogonal MRA: Haar
121 e i
2 2 #
t 2t 2t 1 #
not if0
11
01
)( 21
21
t
t
t
e i 2
sin /2 /2
et 4i e i
2 sin2 4 #
From
We have
Therefore
Scaling and wavelet functions in the Fourier domain:
12g
2
2 #
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Examples of orthogonal MRA: Haar
x 1.41.210.80.60.40.2
1
0.8
0.6
0.4
0.2
0x 100806040200
1
0.8
0.6
0.4
0.2
0
x 1.41.210.80.60.40.20-0.2
1
0.5
0
-0.5
-1 x 100806040200
1
0.8
0.6
0.4
0.2
0
Very compact in space, very bad localized in frequency
Symmetric, no regularity, 1 vanishing moment
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Examples of orthogonal MRA:Littlewood-Paley
It comes from the same idea:
the approximation subspaces in Fourier domain are piecewise constant.
Kind of dual basis to the Haar’s
| | 1 si 0 si
#
the orthogonality property in Fourier is clearly verified: n
| 2n |2 1
If
To have symmetry: a zero-phase condition is set, show that:
t sin t t #
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Examples of orthogonal MRA:Littlewood-Paley
t 1050-5-10
1
0.8
0.6
0.4
0.2
0
-0.2x 6420-2-4-6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
| | 1 si 0 si
# t sin t t #
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Examples of orthogonal MRA:Littlewood-Paley
h 2
2
#
not if0
if2)(ˆ
h
2/
2/sin
2
1][
n
nnh
g e i h
The associated filters
from It comes
Therefore
and with the QMF relationship
2/2
2/2/0
2/2
)(ˆ
ife
if
ife
gi
i
21
21
sin
2
)1(][
n
n
ngn
These filters are IIR
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Examples of orthogonal MRA:Littlewood-Paley
x 1050-5-10
1.4
1.2
1
0.8
0.6
0.4
0.2
0
The wavelet
12g
2
2
not if0
2 if)(ˆ
ie
From
t cos t sin2 t t 1
2 #
t 1050-5-10
1
0.8
0.6
0.4
0.2
0-0.2
-0.4
-0.6
-0.8
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Example of MRA:Spline 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
j
12
h2 j #
L2R
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Example of MRA:Spline bases (Battle-Lemarié)
The approximation subspace Vj is defined as the set of piecewise polynomial functions on 2j width segments.
kn
nfZkRLfV
jkk
j
jj
2 in sderivative continuous 1 with
degree of polynomiala is , that such )()1(2,2
2
not if0
2/12/1if1)(
ttI
1
21
21
21)(
1
)(
)()()()( : even is if
)()()()( : odd is if
n
nn
n
tItItItn
tItItItn
The B-spline basis of order n is built by autoconvolution of a box function:
Therefore
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Example of MRA:Spline bases (Battle-Lemarié)
x 43210
1
0.8
0.6
0.4
0.2
0
Examples of B-spline with n=1 and n=2
Compact support but not orthogonal
x 1.510.50-0.5-1-1.5
1.4
1.2
1
0.8
0.6
0.4
0.2
0
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Example of MRA:Spline bases (Battle-Lemarié)
2)1(
)1(
)(
)1(
)(
2
2sin
)(ˆeven is n if
2
2sin
)(ˆodd is n if
ni
n
n
n
n
e
Therefore
The orthogonalization process is based on the following property of orthogonal bases:
k
f 2k
2 1 #
f
k
f 2k
2 #
It can be shown that if f(t) is a basis, an orthogonal basis is
obtained by:
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Example of MRA:Spline bases (Battle-Lemarié)
n
k
n 2k
2 #
k
n 2k
2b2n 1
#
b2n 1k 2n 1x |x k
#
n
b2n 1
#
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 MRA:Spline bases (Battle-Lemarié)
x 420-2-4
1.4
1.2
1
0.8
0.6
0.4
0.2
0
x 420-2-4
1.4
1.2
1
0.8
0.6
0.4
0.2
0-0.2
sin
22
2 123
13 cos
#
sin
22
3 11120
1330 cos 1
60 cos 2e i3
2 #
Infinitely supported but orthogonal
n=1
n=2
20100-10-20
1.4
1.2
1
0.8
0.6
0.4
0.2
0
In Fourier
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Example of MRA:Spline bases (Battle-Lemarié)
|k| hk (n=1) 0 0.81765 1 0.3973 2 0.06910 3 0.05195 4 0.01697 5 0.00999 6 0.00388 7 0.00202 8 0.00092
Filters
S. quad. :h 2
81 3e i 3e 2i e i3 33 26cos cos 2
33 26cos 2 cos 4 #
Spline linéaire :h 2 1
2 1
2cos 2 cos
2 cos 2 #
k hk (n=2) 2 0.68037 3 0.13796 4 0.12484 5 0.02075 6 0.04198 7 0.00424 8 0.01511 9 0.00112 10 0.00571
|k | hk (cubique)
0 0. 76612
1 0. 43392
2 0.05020
3 0. 11004
4 0.03208
5 0.04207
6 0.01718
7 0.01798
8 0.00869
k hk (cubique)
9 0.00820
10 0.00435
11 0.00388
12 0.00219
13 0.00188
14 0.00110
15 0.00093
16 0.00056
17 0.00046
Cubic spline basis: Battle-Lemarié
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Example of MRA:Spline bases (Battle-Lemarié)
x 43210-1-2
1
0.5
0
-0.5
-1
-1.5
-2
x 420-2-4
1
0.5
0
-0.5
-1
Wavelets
n=1 n=220100-10-20
1.4
1.2
1
0.8
0.6
0.4
0.2
0
In Fourier
Compute an approximation of the Battle-Lemarié wavelet with the matlab wavelet
toolbox