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Wavelet packets and de-noising based on higher-order-statistics for transient detection
Philippe Ravier, Pierre-Olivier Amblard
To cite this version:
Philippe Ravier, Pierre-Olivier Amblard. Wavelet packets and de-noising based on higher- order-statistics for transient detection. Signal Processing, Elsevier, 2001, 81 (9), pp.1909-1926.
�10.1016/S0165-1684(01)00088-3�. �hal-00595903�
Higher-Order-Statistis for transient Detetion
Philippe Ravier
and Pierre-Olivier Amblard
Laboratoire d'
Eletronique, Signaux, ImagesLESI-ESPEO
BP6744 -45067 Orleans Cedex 2 - Frane
e-mail: Philippe.Ravieruniv-orleans.fr
Tel:(33)-2-38-49-48-63
Fax: (33)-2-38-41-72-45
Laboratoire desImages et desSignaux LIS-ENSIEG
UMR 5083 Groupe NonLineaire
BP46 - 38402 St-Martin d'Heres Cedex- Frane
e-mail: Bidou.Amblardlis.inpg.fr
Tel:(33)-4-76-82-71-06
Fax: (33)-4-76-82-63-84
powerfuldetetiontools:Aloalwaveletanalysisandhigher-orderstatistialproperties
of the signals. Using bothtehniques makes detetion possible in low signal-to-noise
ratioonditions,whenothermeansofdetetion arenolongersuÆient.Theproposed
algorithm uses the adapted wavelet paket transform. It leads to a partition of the
signalwhihis`optimal'aordingto ariterionthatteststheGaussiannature ofthe
frequeny bands. To get a time dependent detetion urve, we perform a de-noising
proedure on the wavelet oeÆients: The Gaussian oeÆients are set to zero. We
thenapplyalassialmethod ofdetetion on thetime reonstruted denoisedsignal.
We study the performane of the detetor interms of experimental ROC urves. We
showthatthedetetorperformsbetterthandeompositionsusingotherlassialsplit-
ting riteria. In a last part, we present an appliation of the algorithm on real ow
reordings of nulear plant pipings. The detetor indiates the presene of a missing
bodyinthepipingat some instantsnotseenwith alassialenergydetetor.
Nouspresentons dans et artileun nouveau deteteur de signaux transitoires aous-
tiques. Ce deteteur tire parti de deux tehniques puissantes utilisees en detetion :
une analyse loale par ondelettes ombinee a une exploitation des proprietes statis-
tiques aux ordres superieurs des signaux. Cette approhe rend possible la detetion
dans des onditions de rapport signal sur bruit diÆilesalors que les methodes las-
siques demeurent insuÆsantes. L'algorithme propose utilise les paquets d'ondelettes
quirealisentunpavagedu plantemps-frequeneadapteal'observation.Ilaboutitala
onstrutiond'unebasededeompositionquiest\optimale"selonunriterequiteste
lanature gaussienne des bandes frequentielles. Pour obtenir une ourbe de detetion
temporelle,oneetueundebruitagedusignalparmiseazerodesoeÆientsdeban-
desgaussiennes.Unemethode lassiquededetetionestensuiteappliqueesurlesignal
debruite reonstruiten temps.
On etudie les performanes du deteteur en terme de ourbes COR experimentales.
Nous montrons que le deteteur donne de meilleurs resultats que les deompositions
quiutilisentd'autresritereslassiquesdesegmentation.Dansunedernierepartie,une
appliation de l'algorithme a un enregistrement de irulation hydraulique dans une
entrale nuleaire est proposee. Le deteteur indiquela presene d'un orps errant a
desinstantsnon reveles parundeteteur lassiqued'energie.
Keywords: transientdetetion, wavelet pakets, multiresolution analysis,adapted
segmentation, de-noising,higher-orderstatistis, ROCperformaneurves.
1 Introdution
1.1 Presentation of the problem
Wavelets andHigher-Order-Statistisare twoof themostsuessfuladvanesinthe eldofsignal
proessing of the twenty last years. Both tehniques are powerful tools that an eÆiently be used
for detetionappliations.But rare are theappliations whereresearhers simultaneously havetried
totake advantage of thetwoapproahes by ombining them.Wepropose todoso inthis paperand
showhow suh aombinationan improvethe quality ofdetetion.
The problem addressed here onerns the detetion of short time impulsive signals that we all
the detetion and exat loalizationof transients inECG orEEG reordings is of great importane
[18,19,5℄. In [1℄, the authors detet wide-band transients that are pressure drop signals originating
fromadeveloped turbuleneexperiment.Inpassivesonarproblems,thedetetion oftransientaous-
tisignalsonstitutesanalternativetootherlassialsonardetetion meansbeauseof the progress
realizedinreduingaoustinoisegeneratedbynavalvessels.Thusthetraditionalmeansofdetetion
are insuÆiently reliable[6,17℄.
In the problem addressed here, the signals are supposed to be embedded in a suÆiently strong
additive noise so that phenomena annot be visually deteted onsidering the observed temporal
shape. The detetor proposed aims to perform better than any energy detetor so that false alarm
ormissed detetion situationsare mitigated.In additionto severe signal-to-noiseratio ontexts, the
signals of interest suer from a lak of information: This makes the problemvery diÆult to solve.
The signals todetet are poorlydened for the following reasons:
{ Theirtemporalsignal shapes are not exatlyknown,
{ Theirarrivaltimes are not known and generallyneed to beestimated,
{ Theirbrief existene bringredued informationfor their haraterization.
However, we need to dene the objets more preisely. Many of them have an osillatory shape
yielding high peaks in their spetrum. This is espeially true for the signals we onsidered suh as
transients generated by metallishoks, raks or slams.
Transient signals an therefore be desribed as models with various parameters. Friedlander and
Porat in[7℄haveproposedatheoretialsolutionofthe problemassumingthe signalwasomposedof
damped parameterized sines orrupted by awhite Gaussiannoise. Detetors investigated are based
onalassoflineardatatransforms.Atually,modelsareratherfarfromrepresentingallthevarietyof
transientsignals.Soweintentionallyhoseanonparametriapproahtopreventfromrestritingthe
problemtoalassofpartiularsignals.Transientswillthenbeharaterizedthroughtheirstatistial
propertieswhih ontrast with the propertiesof the noise.
Wedisuss inthe next setionhowwe alreadyproposed adetetor basedonMalvarwavelets and
its working limitationsfor assumed white Gaussian stationary noise. Wavelet pakets are proposed
instead inmore realistisituations whenthe noise isolored and stationary.
1.2 From Malvar wavelets to wavelet pakets
Ina previous work,we have shown that Malvarwavelets an suessfullybeused when the noise
is stationary white and Gaussian [16℄. The basi idea onsists in disriminating the noise from the
transientsignalbysegmentingthe observationinnetemporalslieswhenatransientispresentand
inlarge oneswhenthereisnoiseonly.Thepurpose istoget the`best' segmentationwhihisadapted
to a desired riterion. Atually, the segmentation orresponds to seleting a basis among the set of
allpossiblebases whih isrelated toa orresponding partitioninthe time-frequenyplane.
Theriteriontohoosethe best basisisbasedonthe GaussianityoeÆients:Whentwoadjaent
segments have Gaussian oeÆients, they are merged, otherwise they are kept separated. Determi-
nationof theGaussianity ofthe wavelet oeÆientsisdone usingHigher-Order Statistis.When the
by replaing mathematialexpetationswith averageomputing. Whenthe noise isolored,wavelet
oeÆientsinthesamesegmentannotbeonsideredstationarysothattheestimationisnotorret.
Forthat reason,the `split and mergetemporalslies'riterionthat isused for the best basis searh,
fails togive good results. To irumvent this problem,we propose to work in the frequeny domain
usingwaveletpakets.Thebestbasisishoseninthesameway:Merging`Gaussianfrequeny bands'.
Inthis ase,the estimationproessisorretbeausetheoeÆientsinthe samefrequeny bandare
stationary. This point is desribed further in the paper. g.
1
The proedure used to searh for the best basis is shematially desribed inthe gure 1 for the
two ases desribed below. In the rst ase depited in [16℄, the best basis is initializedwith a set
of Malvar wavelets supported on minimal equally temporal slies. The merging riterion is applied
on pairs of adjaent slies. It leads to the merging of the slies aording to the Gaussianity of
the wavelet oeÆients in eah slie. When merging, onsidered wavelets are replaed by the same
number of twie longer supported wavelets. In the example of gure 1, the third slie is supposed
tohave nonGaussian oeÆients. The rst step merges the slies 1-2, 5-6and 7-8. In aseond step,
the proedure leads tothe mergingof the lasttwo slies obtained in the rst step. The best basis is
reahed.
In the seond ase (gure 1b), the proess is exatly the same exept the signal is initially seg-
mented in equal frequeny bands. The merging riterion is applied to adjaent frequeny bands.
The time-frequenyplaneevolves aordingto thefrequeny variablewhihis\ative"in thetiling.
Inversely,the timevariablepassivelyreatsbeausefrequeny segmentationimposesthetilinginthe
temporaldomain.
Considering the detetion problem, the purpose, however, is to get a time dependent detetion
statisti.Theobtainedtimefrequenyplanetilingatuallyorrespondstoasetofadmissiblewavelets
onstituting a basis. The signal is projeted on eah element of this basis produingdeomposition
oeÆients.Toreturntoasingletemporalurve,ade-noisingproeduresettingGaussianoeÆients
tozero is performed.A denoised signal is reonstruted from the retained oeÆients. After that, a
standard detetion algorithmis applied.
The exat proedureof detetion isdetailedin setion 3.Studies of performane are desribed.
The method is illustrated insetion 4 with the detetion of a srew boltthat has been forgotten
innulearplant pipings.
In the setion hereafter, we briey introdue wavelet pakets dened as anextension of the mul-
tiresolutionanalysispriniples.WewanttoexplaintheironstrutionaswellastheoeÆientompu-
tation.Beforethat,wegivea justiationintheuse ofadaptedwavelets forthe problemoftransient
detetion.
2.1 Interest of adapted wavelet transforms
Inadetetionproblem,time-frequenyrepresentationsareusedtodistributetheenergyofasignal
on the time-frequeny plane, in suh a way that relevant information an be extrated to make a
good detetion. The results generally depend on the retained parameters and on the method itself
used as atime-frequeny representation.
Forexample,disretewindowedFourier`bases'tilethetime-frequenyplaneinregularellswhih
allhavethe same unertainties. Disrete wavelet bases tilethe time-frequenyplanemore naturally,
aordingtotheevolutionveloityofphenomena.Alowfrequenyneedstobeobservedonalongtime
tobe orretlyestimatedwhereas a highfrequeny an rapidly hangeatany time.Therefore, time-
frequeny loalizationnaturally depends on the `observation sale'. However, these deompositions
impose a xed tiling shape in the time-frequeny plane, independent of the observation. Moreover,
hoosing the analysis window size or the deomposition depth is more or less done empirially. On
the other hand, it is possible, using adapted wavelet transforms, to obtain adapted tilings in the
time-frequenyplane, automatially aording tothe observation. g.
2
Inadetetionproblem,theideaistoadaptthetime-frequenytilingtothesignatureofthesignals
ofinterest.Consideringgure2,thepatternrepresentedontheleftorrespondstoatransientsignalin
aontinuousshort timeFourierrepresentation. Anadapted wavelet transformusingwavelet pakets
asdepitedinpanel() allows thetime-frequenypatterntobeinsulatedwith anappropriatetiling.
In pratie, the signal is of ourse strongly polluted and seeking for an adapted wavelet transform
willperform a better detetion, if ahievable.
Thekeypointistoobtainatilingshowinga`pre-detetion'oftheevents.Atime-frequenytilingis
assoiatedtoaretained wavelet basis amongallthe possible wavelet deompositionbases.Basially,
a basis is onstituted by some admissibleelements in a set of wavelet pakets. Wavelet pakets and
their propertiesare nowpresented.
2.2 Denitions and properties
We introduewavelet pakets as anextensionof the multiresolutionanalysis (MRA). This math-
ematial formalism has been developed for fast omputation of the disrete wavelet transform o-
eÆients [14℄. The MRA is built on two time series lters h
n
and g
n
. The onstrution of wavelet
familiesas wellas the oeÆient omputationrely onthese single sequenes.
From h
n and g
n
, we need todene the two operators H and G aslters on the digitalsignal x
n :
(H x)
n
= P
k h
k x
n k
(Gx)
n
= P
k g
k x
n k :
(H x)(t) = P
k h
k
x(t k)
(Gx)(t) = P
k g
k
x(t k):
Withthese onventions, the lassial sale and wavelet funtions read:
= H
= G:
where is aompression operator applied on aontinuous funtion f(t) suh as: f(t)= p
2f(2t).
Wavelet pakets are reated by further applying the operators either on oron . It produes the
innity
0
;
1
;
2
;:::of wavelets indexed by the arrangement f 2N dened as:
2f
=H
f
2f+1
=G
f
with
0
=H
0 :
Notethat
0
= and
1
= .The familyf
f
(t p);p2Z;f 2Ng onstitutes anorthonormalbasis
of L 2
(R) [20℄ (Thesequenes fh
n
g and fg
n
gare alledquadrature mirrorlters).
Inthe frequeny domain,the family f
f
g, where
f
() standsfor the Fouriertransformof
f (t),
ats as aolletionof subband lters (see gure 3bfor f =4). g.
3
We nowshift and sale the
f
(t) toobtain multisalewavelet pakets:
s;f;p
(t)=2 s
2
f (2
s
t p); s;p2Z; f 2N:
TheoperatorsHandGanagainbeusedtoexpresswaveletpaketsfromonesaletothefollowing
one.Wenotethedeimationoperatoras(#x)
n
=x
2n
andH /
theoperatorHfortheinversesequene
h
n
.The reursive expressions are derived:
s+1;2f;p
=(#H /
s;f;:
)
p
s+1;2f+1;p
=(#G /
s;f;:
)
p :
Note that inrementing the sale modies the `frequeny index' f beause inreasing the sale
orresponds to a ner frequeny analysis and authorizes a greater number of frequenies. In the
time/frequeny plane, applying # H /
and # G /
`splits' the initial band into two subbands whih
doublesthefrequeny resolutionasillustratedingure4.The inverse`merging'operatormergestwo
adjaent frequeny bands ina singlelarger one (gure5). g.
4
g.
5 In this kind of diagram, the entire time-frequeny ell information is supposed to be arried by
the wavelet oeÆient, whih isthe innerprodut of
s;f;p
(t) with the signalx(t):
WP
s;f;p
=hx;
s;f;p
i inthe L 2
(R) sense.
Remarkably, the wavelet oeÆients an be reursively evaluated from the sale s to the next one
WP
s+1;2f;p
= (#H /
WP
s;f;:
)
p
WP
s+1;2f+1;p
= (#G /
WP
s;f;:
)
p :
In the mergingproess, oeÆientsare reonstruted by the following relation:
WP
s;f;p
=(H"WP
s+1;2f;:
)
p
+(G "WP
s+1;2f+1;:
)
p
where the interpolation operator ats like (" x)
2n
= x
n
and (" x)
2n+1
= 0. Sine wavelets are
orthonormal, it is possible to easily reonstrut a signal from the oeÆients and to obtain ltered
versionsof the originalsignal at frequeny f and sale s, suh that x
sf (t)=
P
p WP
s;f;p s;f;p (t).
A wavelet paket basis of L 2
(R) an be onstruted by appropriately seleting a set of wavelets
by their triplets (s;f;p)among the whole olletionf
s;f;p
(t); s;p2Z and f 2Ng. The indies s;f
must beseleted suh thatdyadidisjointintervalsI
sf
= h
f
2 s
; f+1
2 s
i
f 2Z;s2Z over theentire real
positive frequeny axis. Then the funtions
f (2
s
t p); p 2 Z suh that [
s;f I
sf
= R +
onstitute
anorthonormalbasis ofL 2
(R). Letus denoteB allthepossibleonstrutiblebases derived fromthe
admissiblesets in T =f(s;f;p)=[
s;f [2
s
f;2 s
(f +1)℄=R +
; p2Zg. Then the signal x(t) reads:
x(t)= X
(s;f;p)2T WP
s;f;p s;f;p (t):
This relation not only stands for an expansion of the signal on the f
s;f;p
(t); (s;f;p)2 Tg family
but alsoexpresses a signal reonstrution froma set of wavelet oeÆients.
Pratially,foranN =2 L
pointsdigital signalx
n
,the numberofoeÆientsatagiven frequeny
band is sale dependent. At a sale depth s, we have f = 0;:::;2 s
1 frequeny indies and p =
0;:::;2 L s
ell positions equally distributed on the time axis. Globally, the number of oeÆients
in the time-frequeny plane is always the same (i.e. N the sample number of the original signal)
whatever the adapted tilingonsidered.
We nowexplain how toobtain the best adapted wavelet basis.
2.3 Towards the `best' adapted wavelet basis
Thekeypointinthesearhforthebestbasisresultsintheabilityofmakingabasisevolvetowards
the `best' one by iteratively substituting elements of the basis with other admissible elements. For
example, the f
s;f
(p); p=0::2 L s
g wavelets an be substituted by the funtions f
s+1;2f+
(p); =
0;1and p=0::
2 L s
2
g.In the manipulation, the number of wavelets is unhanged for the basis.
This operation aets the time-frequeny tiling by splitting or merging frequeny bands when
hangingthe saleparameter. Thedeisionof modifyingornot the initialbasis ateah stepistaken
aordingtoariterion.Pratially,weneedtostartwithagiveninitialbasis.Thebasisisomposed
of aset ofwavelets withthe samelargest given sale.In the time-frequeny plane,it orresponds to
sliingtheNyquist bandinequalfrequeny bandswiththedesirednest width.The splitand merge
prinipleis only appliedin the merge sense and selets the frequeny bands tobe merged, from the
nest ones to the largestones.
frequeny bands. Traditionalfuntions suh asShannon entropy, onentration in l p
orlogarithmof
energy are used. These funtions stand for an information measure and the aim is to onstrut a
basis with the minimum ost, i.e.to minimizethe total informationmeasure.
Letusall Man informationost funtionalomputed on the wavelet paketoeÆients:
MfWP
s;f;p g=
X
(s;f)2B
IfjWP
s;f (p)jg
forahosenf(s;f;p)gfamily.HereI isarealvaluedadditiveinformationmeasure.Thebestbasis is
obtained byminimizingMfWPg.Thisorresponds tohoosingthe appropriatesequene of(s;f;p)
indies inB neessary to over the time-frequeny plane.The minimization ismade reursively.
Considering two sequenes of oeÆients fWP
s;2f
(p)g and fWP
s;2f+1
(p)g at a xed sale s and
frequeny f, and the sequene overing the same frequeny band at sale s 1 fWP
s 1;f
(p)g, the
riterionmust deide whih oeÆient sets must bekept for the `best' deomposition.The hoie of
minimum informationost isgenerally onsidered whihprovides the following deisionriterion:
8
>
>
>
>
>
<
>
>
>
>
>
:
If IfWP
s;2f
(p)g+IfWP
s;2f+1
(p)g<IfWP
s 1;f (p)g
then keep the oeÆientsWP
s;2f
(p) and WP
s;2f+1
(p)for the wavelet deomposition
Otherwise replae the oeÆients WP
s;2f
(p) and WP
s;2f+1
(p)by the oeÆients WP
s 1;f (p):
Foreah deision,the minimum ost must be kept.
Thisdeisionriterionisbuilt undertheidea ofodingorompressionappliations,wherethe in-
formationmustbearriedbytheminimumnumberofoeÆients.Rememberthatwewanttodetet
transient signals whih is not neessarily ompatible with a onstrution in a minimum ost sense.
The purpose in the ase of detetion is quite dierent: The riterionmust realize the segmentation
that makes the desired signalslearly appear inthe time-frequeny representation.
The next part desribes the riterion wepropose for that purpose.
3 Proedure of detetion and performane
Theproedureof detetionisbased onabest basis searh proedure whihisable toinrease the
signal tonoise ratio by anappropriate time frequeny tiling.
3.1 Searhingfor the best basis and de-noising
Thenoiseissupposedtobestationary.Transientsare oftenosillatingproduingsomelargeoef-
ients in awavelet deomposition.Therefore, the amplitudes of the oeÆients are rather far from
being smoothly distributed.By ontrast, the noise samples are Gaussiandistributed when observed
for a suÆiently long time [3℄. The disrimination between transients and the noise an therefore
be reated using a Gaussianity measure. The presene of a transient will generate nonGaussian
