Lecture 6 Metti 5 Spring School - Roscoff - June 13-18

Transcription

1 ecure 6 Me 5 Sprng School - Roscoff - June 3-8 me/space nose and «hermal» processng of emperaure sgnal JC Basale, C Pradere REFE-IM, Esplanade des Ars e Méers, 3345 alence, France E-mal: Jean-Chrsophe.basale@ensam.eu Absrac. hs e proposes smple mehods for he processng of me and space emperaure felds such as he felds provded by nfrared hermography devces, for he non-desrucve evaluaon of heerogeneous samples. A frs par s devoed o nose consderaons and he possble errors comng from he nsrumen and he epermenal suaon. he second par s concernng he esmaon of felds of hermophyscal properes from space-me felds of emperaures. As llusraon eamples, he space dervave of he sgnal and he local esmaon of hermal dffusvy felds are analyzed n he case of D ransverse or n-plane dffuson. he man sraeges conss n projecng he space-me sgnal n a suable bass of funcons, neverheless oher sraeges such as consderng he correlaon beween space laplacan and me dervave of he observable feld, can be useful even f he sgnal s very nosy..inroducon he processng of space and me emperaure felds s more and more necessary because devces are now currenly avalable n order o quckly measure, sore and process hermal nformaon. Infrared hermography devces are he mos usual. Bu a lo of oher possbles wh conac or non-conac sensors (wh opcal or mechancal scans) wll appear n he fuure and a lo of quesons abou how o process and how o esmae hermophyscal properes from such felds are now posed. I s here proposed o revew some of he dffcules occurrng wh such nsrumens and how o overcome hem. he frs par wll be devoed o he analyss of he nose and sgnal perurbaon from such knd of space-me sensors. A D emperaure feld can be he sgnaure of a lo of hea ransfer phenomena a he surface of a sold or a lqud (hea conducon or dffuson hrough homogeneous or heerogeneous meda, convecve ranspor n comple sysems, ). One of he man nuve way o process he sgnal s o sudy he me or space dervaves of such felds, n order o lnk he emperaure observaon o a hea ransfer model. he second par of hs e wll llusrae on smple eamples, some ways and dffcules relaed o he dervaons of he sgnals and he esmaon of hermophyscal properes wh regards o he general hea ransfer equaon. o whom any correspondence should be addressed.

2 . Nose and sgnal perurbaon from emperaure sensors. Characerzaon of me/space nose, dependng on he measuremen devce: -- Monosensor In he pas (only 3 years ago!), only one measuremen generally a seady sae was relaed o a very epensve and echnologcal epermen. he guarded ho plae was he emblemac eample of such devces, n order o esmae he hermal conducvy of a homogeneous sample. In such devces, only one or wo emperaure measuremens were mplemened and hand-conrolled for mananng he seady-sae regme. Now, mllons of hermal daa relaed o smple epermens are avalable from nfrared cameras or opcal and mechancal devces. I s necessary o eamne he characerscs of such devces and o analyze he valdy of he sgnals, bu frs keep n mnd ha he enormous amoun of daa s advanageous snce he compuaonal effor relaed o such daa s reduced. Generally a monosensor (hermocouple/ elecrcal ressance, quanum deecor, phoomulpler ) s gvng a regularly me-spaced nformaon comng from a comple unknown elecronc chan (amplfer, analogcal/numercal converer, flers,.). I s nsrucve o observe (as far as possble) a saonary sgnal (ryng o avod any perurbaon) comng from a measuremen chan, n order o nuvely se ou some characerscs. he fgure s llusrang several suaons ofen encounered (gaussan nose, perodc parase, correlaed sgnal, dgzaon nose ). Such nose, n he dfferen suaons of fgure, has a zero mean and a que unform sandard devaon, even f he fundamenal naure of he nose s no he same A B C- D- Fgure : several nose llusraons from a monosensor (A-random sgnal, B-hsogram of A- wh me seps (nsead of n fg A-), C- parasc perodc nose superposed o he sgnal, D-Dgzaon nose from A)

3 For praccal reasons, he nose assocaed o he real sgnal vecor Y of a monosensor wll be consdered as an undesred random flucuaon (random varable: e Y ) whch s added o a sgnal Y (even dsored and based) comng he physcal phenomenon of neres, such as: Y Y + e Y he man characerscs of a monosensor s generally he nose amplude (or he sandard devaon of e Y ) and he mean value whch s assumed o be zero ( E( Y ˆ ) E(Y)---> E(e Y ) ).(wh E( ): epeced value operaor). he sgnal o nose rao s generally he rao beween he sandard devaon and he mean value. Generally, s assumed ha he error on Y has a consan sandard devaon σ such as : cov ( e Y ) σ I.. Sensor array Snce he years, he nfrared cameras are offerng sgnals form a focal plane array of deecors whch gves a he same me no only one sgnal bu a mar of sgnals comng from all he deecors wh dfferen characerscs. he consderaon of he whole mar of deecors as unform s a dangerous assumpon and nsead of one mean value and one sandard devaon, s suable o consder he covarance mar of he array (very ofen and because s smpler, he covarance of he array wll be consdered as unform as he covarance of a monosensor, such as: cov ( e Y ) σ I ) A B C- D- Fgure ; Eamples of nose occurrng wh a D array of sensors, A- random Gaussan, B- non-soropc spaal correlaon, C- Dgzed nose D-Non-unformy dsoron

4 . Eamples of sysemac errors wh hermocouples and IR cameras.. Inera and poson errors wh conac monosensors Even f he prevous random nose s undesred, he properes of he mean value and he knowledge of he varance, sandard devaon or covarance mar allow o process he sgnal. Unforunaely, sysemac errors can also occur and nroduce a non-deecable error. A hermocouple s a sold sensor whch s perurbang he emperaure feld especally when hgh emperaure gradens are mplemened. Even f several precauons are aken such as pung he sensor and he connecon wre along he assumed soherm curve or plane, he nera n he case of ransen epermen canno be avoded (see [] ). Generally he oupu sgnal of he sensor s consdered as a convoluon produc whch ake no accoun he hermal conac ressance or an echange coeffcen h and an apparen hea capacy c for he sysem such as: Y () K ep h c c τ U( τ )dτ H (τ )U( τ )dτ H ( τ)u(τ )dτ K s an arbrary mulplcave consan and U() s he dealzed sgnal whou nera and he neral and ressance effecs can be represened under an mpulsonal response: H(). he mplemenaon of he prevous epresson needs ofen a dscrezaon and can be presened as: Y j H j U j or Y H Y H. H.. Ym H 3 m H H H. H. H U U.. H U m m One way o avod such dffcules s o esmae he ransfer funcon beween he measured emperaure response and a surface hea flu whch causes he ransen graden. A prerequse condon s o be able o ece he sysem wh a calbraed hea flu and o esmae from a model he mpulse response or he ransfer funcon (see [])... Some eamples of sysemac errors wh sensor-arrays,(calbraon, non unformy of a deecors array, bad pels, dead me...) -Calbraon, emsson and refleon In he case of nfrared cameras, he sysemac errors are no comng from he nera or he hermal ressances (only wh conac sold sensors) bu from he calbraon of he grea amoun of radave sensors and he esmaon of he radave balance beween he sensor (proper emsson, refleon and nfluence of he envronmen) (see [3], [4]). he umnance comng from he observed surface s a funcon of he emperaure of he objec (Planck s law) and mus be calbraed prevously wh a black body source. Such calbraon s a source of sysemac error (non lneares, emssvy,..) whch wll be here assumed o be overcome. If he nfluence of he amosphere beween he camera and he measured surface s consdered as perfecly ransparen, he measured lumnance s hen consdered as only dependng on he proper emsson of he surface and he reflecon of he envronmen. In order o avod he parasc effecs of he refleon, s convenen o sudy he ransen emperaure response of he surface of a sysem o a calbraed even localsed heang (see [5]).

5 -Non-Unformy Correcon he nfrared cameras are generally sold wh a pre-calbraon sysem of he pels, A dsrbuon of gan and offse for each pel mus be regularly re-esmaed (Non Unformy Correcon). he esmaon s generally obaned from he measuremen of he emsson of an sohermal surface a wo dfferen emperaures (generally wh an eended blackbody). Somemes, hese dsrbuons are correced by he nernal emperaure of he camera n order o ake no accoun he me derve of he envronmen or he elecronc sysem. hen he relaonshp beween he measured hea flu (measured lumnance) s relaed o he emperaure of he observed surface by a global calbraon (esmaon of Planck law parameers or of polynomal parameers fng he Planck aw). -Bad or dead Pels Ou of he non-unformy correcon, he deecor array can presen defecs such as bad or dead pels (less han.5%). Generally, hese pels are recognzed nally by he devce provder and correced by a sgnal averaged from he neghbourng pels (Bad Pel Replacemen). -me recordng, dead me sep One oher mporan defec relaed o he nfrared cameras consss n he dead me deecons. Even f he recordng of frames s assumed o be a regularly spaced me seps s necessary o eamne he real me recordng seps provded by he las generaon of hermographc devces (see [5]). In fac, a lo of delays have o be consdered n a hermographc devce. he negraon me s he delay for he recordng of he radaon emed by he consdered surface, by he deecor. he deecors of he array are recordng smulaneously he hermal scene (snapsho mode) and herefore he negraon me s he me resoluon lm of he devce. he elecronc ransfer of he nformaons from he deecors o he sorage memory s hen nsured wh a delay (elecronc ransfer delay) generally much longer han he negraon me (several ms nsead of several µs). hs characersc s mporan because for fas phenomena observaon (or fas apparen scannng) he elecronc ransfer can fal and gve rregular recordng me seps. I s hen more advanageous o mplemen a hgh frequency perodc phenomenon and o record he mages wh a rgger conroled a sroboscopc frequences (see [6]). In all cases, f he recordng me s no perfecly conrolled, he errors nduced, for nsance, wh a me dervaon of he sgnal wll be srongly dfferen from he classcal random nose assumpon snce he me s consdered as an eplcave varable (known whou errors ). Generally hs phenomenon s no affecng he vsualzaon of he hermal phenomenon, bu he processng (see [7]). -hermal sably of he nsrumen he sably of he sgnal delvered by he nfrared camera s ofen relaed o he nernal emperaure of he deecor array whch s abou 8 K. Unforunaely he freezng of he deecor array and he hermal regulaon s no always sable ( o 5 mk). hs nsably s nfluencng he nomnal properes of he deecor array and consequenly he Non Unformy Correcon. By he same way, he deecor array s an energec sysem nfluenced by he nernal hea pump and also by he eernal amben emperaure condons. -Space resoluon he apparen number of pels of an mage s no a sure ndcaon abou he space

6 resoluon. Wh he ancen hermographc devces, he mage was bul wh an opcal scannng and a sngle deecor. Very ofen he scannng was nducng an overlappng of he pels and a srong spaal correlaon acng as a space convoluon effec (or blurrng effec) of he mage dramacally damaged by he absence of possbly of snapsho mode (me lag for each pel). he effecs of such pel correlaon are aenuaed wh a focal plane array of deecors and hen mprovng he mage resoluon (or he apparen mage qualy). Neverheless, he eamnaon of he correlaon of he pels wh he neghbors remans necessary. A classcal es s he sl response funcon measuremen. A cool sl (a sl) wh a varable wdh s placed n fron of a ho plae a plae. he wdh of he sl s vared n order o oban he Sl Response Funcon (SRF()) such as : ( ) sl SRF( ) plae sl, Such a funcon s correspondng wh symmery consderaons o a sep response see fgure 3. I means ha he mage sgnal s convolued wh he dervave of he SRF. (see fgure 3). Generally he characersc wdh of he s abou or 3 pels bu can vary n he mage. he measuremen of such a response n he cener of he mage gves ofen slghly dfferen resuls on he boundary of he mage. he dervave of he SRF can be consdered as a weghng funcon n a spaal convoluon produc, such as: Or under dscree form : Y () p(χ)u( χ)dχ p( χ)u(χ)dχ Y Y p Y p. p U.. U. 3 N p ju j j or.... p. Y m p p U m p p p p p p 3.. hs space convoluon s dfferen from he me convoluon prevously mplemened wh sold sensors. Generally hs weghng funcon s cenered and symmercal. he correspondng mar s a band mar.

7 real sgnal.6.5 SRF() Measured sgnal p()dervave of SRF() symercal par wdh of he sl A- B- Fgure 3: A-Eample of Sl Response Funcon and correspondng pel posons, B- Dervave of he prevous SRF funcon: p() whch s he kernel of he movng average appled o he sgnal..3 Concluson/ Synhess of he frs par he prevous ls of possble errors and unwaned noses relaed o emperaure feld esmaon s ofen frghenng for he begnner. Maybe a rough synhess can help he neperenced user o sar and pracce he processng of such nosy and plenful daa. ree caegores of nose can be globally consdered: -he random nose : wh zero mean value s an unwaned perurbang nose bu able o be processed wh smples asumpons (relaed o he unform covarance mar). -he sysemac errors: (NUC, me derve, radave parasc effecs, sensor posons...) whch mus be fough, deeced or bypassed by he epermener. -he space and me convoluons and correlaons of he sgnal acng on he real me and space resoluon lm. Such convoluons are consdered as flers on he space and me sgnal. I wll be generally dffcul o oban good esmaons when he resoluon lm s passed (even f deconvoluon s a class of nverse problems). Such remark s remanng ha he processng of a large amoun of daa mus no gve he lluson o dspose of he complee nformaon abou he phenomenon o be suded. Oher flers or convoluons wll be consdered n he second par of hs e, devoed o he sgnal processng.. hermal» processng of a D ransen (,y,) feld From a D ransen (,y,) feld, and a hea ransfer model (even smplfed) s ofen possble o esmae a resulan hermophyscal properes feld. he dffuson and convecon ranspor modes can be consdered n order o se ou an denfcaon model.he smple observaon of he evoluon of he emperaure feld can gve he nuon of such phenomena (see he fgure 4).

8 Fgure 4. Illusraon of he evoluon of convecve, n plane dffusve feld, capacve ransverse feld... he processng of such felds needs of course o be aware of he prevous nose and perurbaons consderaons abou he emperaure feld recordng. he random nose wh a zero mean and a unform and dagonal covarance mar wll generally be consdered, bu several aspecs relaed o he bad pels, he space or me correlaon and he non regular me seps wll appear on he followng eamples. Maybe, he frs naural processng of such feld s o ry o se ou he dervave (versus space or me) of he feld. he space-dervave s ofen a mean o analyse he sgnal by consderng he graden of he sgnal (from dsplacemen o sress n sold mechancs see [8]).. Sraeges for he esmaon of he me and space dervave of he sgnal : he space or me dervaon of nosy felds s a dffcul ask, because such operaor s amplfyng he random measuremen nose, f no precauon s aken (see fgure 5). A flerng s hen necessary, bu he rsk s o lose a par of he orgnal nformaon. Several sraeges wll be here eamned (he fne dfferences, he polynomal f, he orhogonal bass decomposon and he convoluon), n order o numercally mplemen he dervaon of a dcree sgnal. In order o es hese sraeges he relaaon of an nal sep such as: (,) f <<b/ and (,) f b/<< Afer a me, he emperaure feld s relaed (or flered) by dffuson, and an appromaon of he feld s hen: N sn( αnb) (, ) b / + / ep( aαn )cos( αn) α n n

9 wh: α n nπ / and n,n fne negers. Rgourously, he prevous epresson s a sere wh N endng o nfny. A physcal fler (due o he eponenal erm) s acng here and lmng he space-frequency conen of he sgnal. he parameer whch allows o conrol he fler s here he observaon me. hs observaon me wll be fed n hs secon and only he space feld wll be consdered. he observed emperaure s a vecor obaned from he prevous epresson a regularly spaced space seps and a whe gaussan nose s added. () Dervave of () A- B- Fgure 5: A- emperaure feld from he prevous analycal epresson a me.5s; a -5 m s - ; b/;.m; (connuous lne: real sgnal, o : dscree nosy sgnal); B-Dervave of he prevous nosy sgnal by fne dfferences. he nal gaussan nose s wh zero mean and σ.. he resulng nose s amplfed by he dervaon operaon... Fne dfferences he fne dfference approach used on fgure 5 s generally presened a each space sep, as: ˆ ' ˆ + ˆ If s assumed, ha he relaon beween he observed emperaure ˆ and he real emperaure s: ˆ + e, wh e represenng he random varable relaed o he gaussan nose, unform whaever he poson. he asympoc epanson around s such as: Wh: ˆ ' ( + ) ( ) +ε( + ) + e + e lm ε ( ) wo knds of errors have hen o be consdered: he appromaon error ε ( ) relaed o he res of he asympoc epanson and he random error relaed o he random varable e. Unforunaely, when he space sep s endng o zero, he appromaon error s effecvely endng o zero, bu he random error s endng o nfny!

10 he dfference of wo random varables s a lnear operaon whch amplfy he nal nose. In order o avod such a dffculy, s necessary o fler he sgnal or o projec he dscree observed nformaon n a bass of funcons. One of he smples bass of funcon can be a polynomal bass... Polynomal fng From he same prevous measuremens, a polynomal fng can be mplemened such as: N ) n ( β n n he esmaon of he βn parameers n vecor: a lnear leas-square relaon such as : B ( X X) X ˆ Wh: X. m m... In order o mnmze he dsance: ˆ XB ˆ XB ˆ XB ( ) ( ) B [, β, β... β β 3 n ] can hen be obaned by he consrucon of he dervave wll hen conss n consderng he dervave of he polynomal funcon. he Malab sofware s convenen n order o mplemen such calculaons because he X X mar can be bad condoned (Wandermonde mar) and mus be nversed wh specal precauons. One eample of such processng s shown on he fgure ().6.4. Dervave of () A- B- Fgure 6: A-polynomal fng of he sgnal shown on fgure 5. B- Dervaon. he chosen degree of he polynomal fng s here: N3. he number of observaon pons s. Is shown here ha he drec dervaon of he esmaed polynomal epresson s gvng a suable connuous appromaon n he consdered doman. Appromaon errors are occurrng a he boundary of he doman f no precauons or assumpons are aken. One oher sraegy s o chose a bass near from he consdered physcal phenomenon. Here, he deal bass f he bass made of he Fourer cosne funcons, because he cosne vecors are here verfyng he boundary condons (null dervave a and ) and are also he

11 egenvecors of he dffuson phenomenon (egenvecors of he aplacan operaor n Caresan coordnaes and adabac boundares)...3 Fourer cosne bass he same processng as he polynomal fng can be consdered wh such an epresson: M ( ) βn cos( αn) wh: α n nπ / n he esmaon of he parameer vecor: [ β, β, β3... β M ] B s hen obaned by he same epressons as prevously (n secon..), eceped ha he X X mar s orhogonal, and hen very easy o be nvered. he covarance mar relaed o he parameers s hen as dagonal as he observable vecor. he resul of he esmaon s hen, wh 3 erms for he sere, llusraed on fgure 7. () dervave of () A- B- Fgure 7: A-Fourer fng of he sgnal shown on fgure 5. B- Dervaon. he number of erms s N3. he number of observaon pons s...4 Flerng wh a convoluon kernel he flerng s a usual operaon n sgnal processng, whch conss n weghng he sgnal wh a movng average. I mus be noced ha he observable sgnal hmself s maybe prevously flered by he proper nsrumen. Such as eplaned n par. A new ~ appromaon of he sgnal can hen be consdered by ( ) such as : ~ ( ) p( χ) ( χ) dχ p() mus be normed such as : p( χ) dχ he dervave of ~ ( ) s hen convenenly consdered by he commuably of he convoluon produc such as : ~ d ( ) d d( χ) p( χ) dχ d dp( χ) ( χ) dχ d he dscree appromaon of he dervave s hen convenenly consdered by a convoluon wh a «derved» kernel.

12 One very smple llusraon s gven on fgure 8. he dscree convoluon kernel of he fler s for eample [ ]/4 and an appromaon of he convoluon kernel for he dervaon s hen [/ -/]. I can be noced ha hs slgh convoluon (affecng only a few number of neghbors and very smlar o he fne dfference mehod) allows o oban good resuls wh less effor.. 5 () dervave of () A- B- Fgure 8: A-Flerng of he sgnal shown on fgure 5, wh a convoluon kernel: [ ]/4 B- Dervaon wh a convoluon kernel: [/ -/] (plo o ) and comparson wh he fne dfference dervave of he prevously flered sgnal (plo + )...5 Sngular value decomposon of he whole space and me sgnal he prevous mehods consss n fndng a compromse beween he appromaon error and he flerng. he number of erms of he sere (or he rank of bass), or he wdh of he convoluon kernel are basng he sgnal f hey are used oo far. A lo of oher mehods can be consdered (for eample more sophscaed regularsaon echncs, see [9]). When a large feld mus be processed he choce of he compromse beween he flerng and he bas s made by ral and error. he key pon s he knowledge of he random nose (a he mnmum he sandard devaon). Ofen, he eperencer does no know he characersc of he nose n hs proper eperence. I s hen dffcul o mplemen an opmal flerng of he sgnal. One way o separae he avalable sgnal from he random nose, when he eperencer has a grea amoun of space and me nformaon, consss n mplemenng he sngular value decomposon (SVD) of he (, ) feld, or he ˆ (, j ) dscree observable mar. Applyng he SVD o emperaures mar ˆ yelds ˆ U nn Σ nn V nm Where Σ s a sparse dagonal mar as descrbed below Σ Σ ( ) n m n n n m n So ha fnally he followng runcaed formulaon of he SVD s commonly acceped ˆ U nn Σ nn V nm n k γ k ( U k.v k )

13 Where he modesu k and V k are he column vecors of marces U and V respecvely, and he sngular values γ k are he dagonal elemens of Σ nn arranged n descendng order. Such a decomposon of he global space/me feld s offerng a lo of advanages. Frs, he eamnaon of he sngular values γ k allows o selec he really avalable sgnal. hermal phenomena are ofen relaed o dffuson problems (naurally flered), wh only a few avalable sngular values. he decomposon obaned wh he reduced number of sngular values s hen offerng a reduced represenaon of he feld whch allows a lo of possbles (reduced compuaonal effor for he furher parameers esmaons, low memory sorage, new orhogonal bass and projecon possbles, see []). In he case llusraed on fgure 9, he SVD s allowng an opmal flerng whou any prevous knowledge abou he random nose a any me. he fne dfferences dervave s hen avalable whou precauons. (,).5.5 Sngular values A B values Uk() -. Vk().. V() V3() U() -.4 U() U3() C D- V() (,.) Space dervave o (,.) E F Fgure 9: A- (,) nosy emperaure feld (me eenon from fgure 5), B-3 frs sngular values,c-d-frs U and V vecors, comng from he SVD decomposon, E- Reconsrucon of he prevous sgnal a me.s, wh 3 erm of he SVD decomposon.f- Nosy dervaon of he prevous sgnal (nal nosy observaon) compared he fne dfferences dervave of he SVD reconsrucon)

14 he prevous eample s showng ha he ransen suaon even non-saonnary, s gvng a greaer amoun of daa han a unque mage. he SVD decomposon s a mean o squze and o process he grea amoun of daa que wh he same effor as n he saonnarycase. I mus be noced ha he orhogonal bass U s more reduced han he he bass of cosne funcons used n secon Esmaon of a ransverse dffusvy feld from flash epermens (comparson of classcal Non Desrucve Evaluaon mehods):.. Esmaon wh physcal asympoc epanson: Non Desrucve Evaluaon (NDE) wh nfrared cameras consss generally o apply a hea pulse on a non homogeneous paralleleppedc slab and o process he emperaure response (,y,) from one face of he sample (he fron face or he rear face). he am of such processng s o esmae some characerscs of he heerogenees (srucure, sze, naure, poson nsde he sample ). he resoluon of a general drec problem of ransen hea ransfer n a 3D heerogeneous geomery s ofen heavy and no convenen o mplemen such mehods. Here, hn samples wh small heerogenees such as he ransfer s locally D (versus z drecon), wll be consdered. One asympoc epanson assumng ha he heerogenees flucuaons are small compared o he mean value of one hermophyscal propery of he sample yelds a lnear relaonshp, such as: f (, y, z, ) q(, y) f ( z,, β + ) β (, y) β ( z,, β ) Where he funcons q(,y) s he spaal dsrbuon of energy comng from a flash ecaon and β (, y) s he spaal hermophyscal propery varaon(dffusvy, conducvy, hckness ). q(,y) and β (, y) have o be specfed wh a fne number of parameers whch wll be he objec of he esmaon procedure. Such procedure wll hen consss n processng he weghed sum of he mages (,y,z,) where he weghng funcons are he sensvy funcons f(z,,β ) and f β ( z,, β ). he emperaure response of he fron or he rear face of he sample wll be recorded by a camera n order o esmae a map or a feld of hermophyscal properes. In he deal case, he sample (a plane plae of small hckness ) s assumed o be hermally nsulaed and wh a emperaure feld nally unform ((,y,z,)). If he hea ransfer s supposed -D, hen, he emperaure response relaed o a unque locaon (,y) correspondng o a pel, o an nsananeous hermal pulse, s gven (See []) on he fron face (z) : Q n² π ² a Q ( z, ) + ep f ( a / ) c n Such epresson s raher ncomplee because he smplfed assumpons (D ransfer, adabacy ) can nroduce a bas beween modelng and epermen. herefore, hs epresson s convenen o undersand he dfferen esmaon procedure sraeges. From he prevous epresson, he esmaon problem of several parameers can be consdered. Insead of he hermal dffusvy, he esmaon problem of he sample hckness, he hermal conducvy λ and he volumc hea capacy c can be consdered. In each case, a reference appromaed value of he parameer mus be known (and noed:, λ and c ). c

15 In many cases such reference values can be obaned by a prevous global measuremen. he followng frs order asympoc epansons can be for eample wren for he fron face, a each me : -If a hckness varaon s o be esmaed: ( ) ( ) + ) / ( / / ) (, c f c f c Q c f c Q λ λ λ -If a volumerc capacy varaon s o be esmaed: ( ) ( ) + ) / ( / / ) (, c f c f c c c Q c f c Q λ λ λ -If a hermal conducvy varaon s o be esmaed: ( ) ) / ( / ) (, c f c Q c f c Q λ λ λ λ + I s very mporan o noce ha n all prevous cases, s possble o replace he sensvy funcons f(z,,β ) and ),, ( β β z f by a lnear combnaon of f(z,,β ) and he me logarhmc dervave ),, ( β z f. I s hen possble o mplemen a lnear relaonshp such as: ) ( ) ( ) (, X X β β β β + he funcons ) ( X j β are he sensvy funcons of ), ( o parameers j β. Such funcons are shown on fgure. If oher parameers combnaon esmaon (such as hermal dffusvy esmaon) s consdered, he resulng sensvy funcon wll be a lnear combnaon of funcons f and ( f/ ).

16 Fgure Sensvy curves relaed o he -- secon. he parameers β j are defned n each esmaon suaon such as: -If a hckness varaon s o be esmaed Q β and c β Q c -If a volumerc capacy varaon s o be esmaed Q β and c β Q c c c -If a hermal conducvy varaon s o be esmaed Q β and c Q λ β c λ ha s o say wh mar noaon, consderng he vecors and marces: [ (, ) (, )], N X β ( )... X ( N ) β X X X, β ( )... ( N ) β I yelds under mar noaons: β X β

17 If he measuremen nose on each componen of he epermenal emperaure vecor ˆ s assumed wh a consan sandard devaon and no correlaed, n he doman of valdy of he prevous asympoc epansons, hen, he opmal esmaor of he parameers vecor ˆ β ˆ β [ ] s obaned by : ˆ β ˆ β ( X X) X ˆ he lnear appromaon allows no only he esmaon of he parameers, bu also he confdence nerval of hs esmaon o be suded. An nermedae sage s he covarance mar of he esmaon on he vecor ˆ β ˆ [ β ], gven dependng on he sandard devaon of he emperaure measuremen nose σ : ˆ β cov ˆ β ( X X) σ he mehod can be used whaever he lengh of he vecor ˆ. hese epressons can be mplemened smulaneously wh all of he pels of he mage. hus, he mar produc X ˆ can be ncremened and consss n a real me weghng. he choce of he weghng s lnked o he choce of he esmaon sraegy (esmaon of, c or λ). he erms of he sensvy mar are heorecally calculaed wh he references values or he averaged mages. hs sequenal mehod consderably eases he problems of sorage and mages manpulaon. I s very suable for a smple Non Desrucve Evaluaon. Moreover, f() can be measured on he daa, because a each me sep, he average of he mage s a flered appromaon of f(). he compuaon of he logarhmc dervave of f() would hen gve a suable mehod applcable whou any dea abou he knowledge of he nomnal values of he hermophyscal properes of he consdered sample. Unforunaely, he logarhmc dervave of hs epermenal sgnal s no easy wh smple fne dfferences mehods. In order o overcome such dffcules Shepard [] proposed nuvely a logarhmc me-fng each pel sgnal and Rajc[3] o consder he SVD of he global daa cube... he logarhmc polynomal me-fng [] In order o convenenly process he grea amoun of daa provded by he flash NDE epermen, Shepard proposed o decompose he sgnal wh a polynomal fng, such as: n((,y,z,))β (,y)+ β (,y)n()+ β (,y)n ()+ Such decomposon has no physcal meanng because he new parameer vecor s no relaed o a physcal model, bu he me-logarhmc dervave (consdered by Shepard) appears o be very well correlaed wh he deph or hermophyscal properes changes of he esed samples. he calculaon of he logarhmc dervave s hen akng he advanages relaed n secon --. I s also a very effcen and convenen mehod n order o reduce and manpulae he grea amoun of daa (only N mages correspondng o he degree of he polynomal epressons are o be manpulaed).

18 ..3 he SVD decomposon ([3]-[]) An oher way o reduce he amoun of daa consss n consderng he SVD of he nformaon cube. Rajc [3] proposed he SVD decomposon (eplaned n..5) such as: (, y, z, ) u ( ) γ v K k k k k ( ) Such daa obaned from NDE epermens appears o be ncely reduced by or 3 erms of he prevous sere. Generally, he U vecor (or u ( ) funcon) s gvng a good appromaon of he spaal energy dsrbuon. he V vecor (or v ( ) funcon) s relaed o f(). he U vecor (or u ( ) funcon) s gvng a good appromaon of he defecs localsaon. Bamford [] proposed o compare he asympoc epanson eplaned n he.. secon o he prevous SVD decomposon. he slgh dfferences are comng from he nonorhogonaly of he funcons n... In fac all he mehods presened n secon. are very smlar. hey conss n projecng he daa cube n a suable bass of funcons (physcal or no) and hen o ry o process he sgnal wh a physcal model. In he ne secon, an oher sraegy s proposed. I consss n usng he physcal model n order o reduce or elmnae he non useful daa (because nohng physcally happens or because he sensor s provdng a wrong sgnal)..3 Esmaon of n-plane dffusvy feld-me-space correlaon and elmnaon of he non useful daa Inally, n-plane characerzaon mehods were relaed o modal mehods (usng cosne Fourer ransform or projecon on cos(α n ) funcons bass) allowng o esmae he hermal dffusvy of homogeneous ansoropc samples (see [4],[5],[6]). he man drawback of hese mehods s o be non adaped o heerogeneous samples, and also o consder only hea pulse or hea sep heang responses. When he sample s heerogeneous and when hea source erms can occur whaever he me or he space localzaon, s dffcul o se ou analycal Fourer soluons, even polynomal fng nor SVD decomposon (especally when he heang s random n me). Nodal mehods are hen suable and allow o consder oher esmaon sraeges (see [7]). he local energy balance s hen dscree and consdered such as : k k k k k k wh: ( ) Fo, j k k k, j + Φ, j δ, j, j, j, j, j, j 4 s proporonal o he local aplacan of he + +, j emperaure feld, a, j Fo, j Is he Fourer nondmensonal local parameer relaed o he hermal dffusvy a, ; he pel sze ; and he me sep. j k, j Φ s a nodal source erm whch can occur randomly n space or me. δ k k + k, j, j, j s relaed o he dscree me-dervave.

19 I s hen proposed o process he emperaure feld by lookng only for he zone where here s a pure dffuson phenomenon (verfyng only Fo, j k, j δ k, j ). A creron suable o deec such zones s o consder he local correlaon beween he laplacan and he me dervave: j F, j F F k k, j δ, j k, j F k δ, j where F s a emporal wndow such as: F [ k, k + l] wh k [, N l], k s he me sep number and l he wdh of he me wndow. If such a coeffcen s near from, a dffusvy parameer s hen esmable: F Fo, j F k k, j δ, j F k δ, j F, j F F k, j k δ, j One eample of emperaure feld processng s gven on he fgure o 4. A- B- Fgure. Source erm A- Poson of he ho spo B- me evoluon for 3 pels n he cener of he mage A B Fgure. emperaure response A- emperaure feld a 55 wu (whou un), B- Evoluon of several pels n he cener of he mage

20 A B Fgure.3 Correlaon coeffcens calculaed for F 4. A- Feld a 55 wu (jus afer he hea sep) B- me-evoluon for3 pels e he cener of he mage. he feld of correlaon coeffcen llusraed on fgure 3 are showng he zones where he hermal dffusvy os able o be esmaed (correlaon near from ) and he zone where no esmaon s possble. hs knowledge of he correlaon feld allow o esmae he dffusvy only n he suable area (see fgure 4). A- B- Fgure 4: Esmaon of he reduced dffusvy A- Feld obaned from he whole nformaon cube B- Quas nsananeous esmaon (wndow of 4 me seps). From fgure 5 s neresng o remark ha he physcal correlaon process allow o dscrmnae he zone where here s a pure dffuson phenomenon from he zone where here s a source erm from also he zones where nohng appear or maybe he pel are dead. hs processng can hen somemes allow o parly overcome he processng and he consderaons eplaned n he par. Especally when he sgnal s very nosy, he smple consderaon of correlaons can help o dscrmnae he areas where a source erm or a purely dffusve relaaon appear.

21 Hea source on.8 Hea source on dervave() Pure dffuson No dffuson or bad pel aplacan() dervave().6.4. Pure dffuson No dffuson or bad pel aplacan() A- B- Fgure 5 Sudy of he correlaon beween aplacan and me dervave for several pels ( o pel wh or whou source erm wh dffuson, + pel where no dffuson appear or bad pel. A- From nosy sgnal, B-From less nosy sgnal. 3 Concluson: he me-space hermal sgnal s offerng a grea amoun of daa whch mus be processed from several pons of vews. Frs, he dfferen knds of nose and bas (random, sysemac and space and me correlaons) comng from he nsrumen mus be analysed and undersood. hen several sraeges are avalable n order o process he sgnal n relaon wh a physcal model. Ofen he man sraegy wll conss n projecng he sgnal n a funcon bass whch can come from nuon, sascal processng or physcal analyss. hs projecon s advanageous n order o fler he random nose, o reduce he grea amoun of daa and o convenenly manpulae and esmae he parameers. Bu he projecon s no always he mos suable sraegy. he drec condderaon of a physcal model wll allow o elmnae or dscrmnae he daa correlaed or no wh a chosen physcal phenomenon (one eample has been evocaed wh he esmaon of a hermal dffusvy feld). References: [] B. Bourouga, V. Goze, J.-P. Bardon, es aspecs héorques régssan l nsrumenaon d un capeur hermque paréal à fable nere, Inernaonal Journal of hermal Scences 39 () (January ) [] J.. Baagla, J.C. Basale, Esmaon of hea flu and emperaure n a ool durng urnng, Inverse Probl. Eng. 8 () [3] Gaussorgue G., Infrared hermography, 994, Chapman e al, ondon. [4] Maldague X.P.V., heory and Pracce of Infrared echnology for Nondesrucve esng, John Wley & sons, Inc., 684 p,. [5] Balageas D., Delpech P., Boscher D., Deom A., New developmens n smulaed nfrared hermography appled o non desrucve evaluaon of lamnaes, Revew on

22 Progress n Quanave Non-Desrucve esng, ED hompson and Chmen (Plenum Press, New York, 99, Vol A, pp [6] Basale JC, Chrysochoos A, Pron, H., Warsse B.,, Analyse hermographque du compormen des maérau, n Mesure de champs e denfcaon en mécanque des soldes, hermes, avoser, [7] Pradere C., Clerjaud., Dhlare S.,JC Basale,, Hgh speed heerodyne nfrared hermography appled o hermal dffusvy denfcaon, Rev Sc. Insrum, 8, 549 [8] Fessel P..,, Du déplacemen à la déformaon, n Mesure de champs e denfcaon en mécanque des soldes, hermes, avoser, [9] hkonov A. and Arsenne V., 977 Soluons for ll posed problems, Wnson and Sons, Washngon. [] Bamford M., Basale J.C., Fudym O., Nodal and Modal Sraeges for longudnal hermal Dffusvy Profle Esmaon. Applcaon o he non-desrucve Evaluaon of SC/SC composes under unaal ensle ess, Infrared Physcs and echnology (8), Volume 5, Issue, January 9, Pages -3. [] Parker WJ, Jenkns, W., Abo J.., 96 Flash mehod of deermnng hermal dffusvy, Hea capacy and hermal conducvy, Journal Appl. Phys., vol 3, No 9, pp , [] S.M. Shepard, Y.. Hou,. Ahmed and J.R. hoa, Reference-free Inerpreaon of Flashhermography Daa, Insgh, Volume 48, No. 5, Brsh Insue of ND, May 6, pp [3] Rajc N., Prncpal componen hermography for flaw conras enhancemen and flaw deph characersaon n compose srucures, Compose Srucures Volume 58, Issue 4, December, Pages 5-58 [4] Phlpp I., Basale J.C., Malle D., Degovann A., Measuremen of hermal dffusvy hrough processng of nfrared mages, Rev. Sc. Insrum. 66() 995, 8-9. [5] Krapez JC, Spagnolo., FreB M., Maer H.P., Neuer, Measuremen of n-plane dffusvy n non-homogeneous slabs by applyng flash hermography, Inernaonal Journal of hermal Scences 43 (4) [6] Souhar Y.,, Caracérsaon hermque de maérau ansoropes à haues empéraures, INP Docorae hess, Nancy. [7] Pradère C., Morkawa J., ouan J., Basale J.C., Hayakawa E., Hashmoo., Mcroscale hermography of freezng bologcal cells n vew of cryopreservaon, QIR Journal, VO 6/ pp do:.366/qr