Self-Calibration of PIV Video-Cameras in Scheimpflug Condition

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Self-Clibio of PIV Video-Ces i Scheiflug Codiio T. Fouel J.-M. Lves S. Coude F. Collge LTSI UMR CNRS 556 ue Boui 4 Si-Eiee Fce LASMEA UMR CNRS 66 6377 Aubièe cede Fce Absc I ode o ioduce oe fleibili i clibio he budle djuse och is ioduced fo Seeo PIV sses i Scheiflug codiio. The clibio ge is osiioed b hd fo of he video-ces diffee locios i sce. Fo he ulile views of he clibio ge he ees of he video-ces wih boh he ge locios d he 3D clibio ois e esied b o lie les e sques. I he oosed clibio ocedue he lse le equio is deeied b sechig fo he hoogh bewee he ige les. The bili of he ocedue fo self-clibig wo f5 video-ces d ecoveig he equio of he esuee le is eeiell esed. Ioducio Seeo Picle Ige Velocie echique (SPIV) ebles iseous veloci field esuees i vious siuios. Wheve e he seeoscoic se-u d he ossible oicl iefces he evluio of he hee veloci cooes i secio of he flow is bsed o he fucio ig 3D locio o is ige. The ig fucio odelig oe video-ce is clssicll oloil fucio (V Ood 997) o hooghic oe (Rffel e l. 998) whose ees e ecoveed he clibio sge fo 3D well-kow ois d hei coesodeces i he ige le. Usull kow fl gid is lced i he edi le of he ligh shee d ccuel oved o diffee locios llel o he ligh shee (Soloff e l. 997). A ech locio iges of he gid e ecoded. Such clibio ehod be usble elive o o llel d o equll sced ge locios esecill i iel flows (e.g. ie flow). I is he eso wh (Quéo e l. ) oosed ehod usig sigle ige of do gid ge o clibe PIV video-ce wih he focl legh d seso iel iches s ddiiol ifoio. Nowds uli-level ges o which diffee -osiios e ese e vilble fo he secil cse of SPIV clibio (Nqwi ). Such objec hs o be ccuel defied d cefull hdled duig d fe he eeie. I ddiio seies of 3D clibio ges

39 Sessio 7 is equied o isec lge ge of fields fo icopiv fields o eodics oes. The i of his icle is o obi esie PIV clibio oocol b hdosiioig ied fl clibio ge. I coue visio such selfclibio bsed o he budle djuse echiques (Lves e l. 998 Tiggs e l. ) led woks i licios usig oo (Li d Lves 996 De Agio e l. ) o isecios wih ve sho focl legh. I his ulile d hoogeicl och descibed i secio oe equies eihe eoe cool slio o ccue clibio ge : he video-ce iisic ees d he clibio ge geoe e ecoveed siuleousl. I secio 3 we oose wo iisic odels cosise wih he secific cse of he Scheiflug codiio used i he SPIV gul ehod. The oble of islige is ckled i secio 4 whee he locliio of he lse le is cied ou i he clibed seeo device coodie sse. Coed o ohe wok o he oic (Riou 999 Bjokquis 998 Coude d Scho ) we oose o deeie he hoogh bewee boh ige les whe igig he lse le. I ode o vlide he och d o ssess is ecisio dislcees of he fl do gid used fo he self-clibio of wo f 5 video-ces i Scheiflug codiio wee esued (secio 5). The bili of he lse le locliio ehod hs bee esed o ge wih ied do do e. The budle Adjuse Aoch.. Self-clibio fo ulile iges The budle djuse echique is les e sque ehod used i coue visio fo self-clibig video-ces fo clibio ge. This echique is bsed o uli-ige schee ledig o highl ove-deeied o-lie sse d elible esiio of boh he iisic ees of he videoces d he 3D clibio ois. I such hoogeic och eisic ees givig he locliio of oe video-ce e disiguished fo he iisic ees. Le us coside he i-hole odel deiced i Fig. whee he coveio of he ie hs bee doed (i.e. he ige le is lced fo of he ojecio cee i ode o wok wih osiive vlue fo gificio). This is silified odel of he lef (o he igh) video-ce ssued o wok ccodig o esecive ojecio. To fis ode he elioshi bewee 3D oi give i he efeece coodie sse W XYZ d is ige i he video-ce coodie sse C XYZ is descibed b he followig equios :

Seeoscoic PIV 393 R λ (.) whee is he disce seig he ojecio cee fo he ige le (deoed i Fig. ) ( ) ij R d ( ) T e esecivel he oio i d he slio veco bewee he wold d video-ce coodie sses. Fig.. The i-hole ce odel ige geoe d coodie sses. B eliiig i (.) he scle fco λ which s he 3D oi o is ige oe obis he followig so-clled colliei equios : Z Y X Z Y X Z Y X Z Y X 33 3 3 3 33 3 3 3 (.)

394 Sessio 7 e s- B kig io ccou disoios ( δ δ) he coodies ( ) foed io he iel coodie ( u v) s follows : ( u ε u ) d δ u (.3) δ ( v ε v v )d u v esued i he iel whee ε u ε v e he eos o he ige locio ( ) ( v ) is d he ige le ( d) u e he iel coodies of he iesecio bewee he oicl d he iel sies. Disoios e usull divided i wo s o oe hd he dil disoios give b he ees 3 o he ohe hd he geil oes give b he ees (ASP 984) : 4 6 δ ( u u )( 3 ) d (.4) 4 6 δ v v δ ( )( )d ( u u ) 3 ( u u ) v v d (.5) ( v v ) u u δ ( v v ) d whee d eese he ige disces eessed i iel d d ( u u ) ( ) ui d v v is he dil disce fo he ige cee. Subsiuig (.3) (.4) d (.5) io (.) he esuee eos c be eessed s ( ) X Y 3Z δ δ ε u Φ u u 3X 3Y 33Z d (.6) X Y Z v ( ) v 3 δ δ ε Φ v X Y Z d 3 3 33 whee Φ is he veco of ukow ees. The veco Φ is esied b iiiig he su ( ε ε ) i ui vi X Y Z K fo he ois ( i i i ) i u K of fl clibio gid d hei iges ( i v i ) i loclied i he iel. A his sge he ee veco Φ cosiss of iisic d eisic ees of he video-ce. B lookig (.4) (.5) (.6) d b keeig i

Seeoscoic PIV 395 id h esecive ojecio is lws defied wihi scle fco he ige disces d e s iisic ees. The ige cee coodies ( ) v u d he oicl disoio ees 3 e he ohe oes. The eisic ees e he hee ideede oio gles of he oio R (α oig oud X -is β oud Y -is d γ oud Z -is) d he slio veco ( ) T. Thus he ee veco deoed hee 6 9 Φ is ( ) T v u Φ 3 6 9 γ β α Oe jo souce of clibio eos is he esul of esuee eos. These eos c be loced o he 3D coodies of he clibio ge ois bu lso o he locliio of hei coesodeces i he ige le. Oe w o iove his is o cobie oe h oe ige ke b he se videoce bu fo diffee views (fe oio give b Eule s gles ( ) ( ) ( ) k k k γ β α d/o slio give b ( ) ( ) ( ) k k k ). I such cse he iisic ees ei he se fo ll he iges d he clibio sk es couig he followig ee veco : ( ) ( ) ( ) ( ) ( ) ( ) T v u ) ( 3 6 9 γ β α γ β α K Φ I hs bee oved h esuees wih subiel ccuc (/ iel) c be obied wih secil es like cosses o dos. I h cse he jo souce of clibio eos elies o clibio es. High quli clibio es e difficul o chieve. The hve o be echicll sble i ie (coed wih he eeue chge) o o be oved ve ccuel o isue euclidi efeece fe wihou bis. Tke io ccou his oi of view i hs bee deosed h i is ossible iside uli-ige clibio och o esie he clibio oi coodies ogehe wih he iisic d eisic clibio ees. The ew ee veco o be deeied if he coodies i i i Z Y X of he clibio ois hve o be esied wih he diiol clibio ees kes he followig fo: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T Z Y X Z Y X v u ) ( Φ 3 3 6 9 K K γ β α γ β α (.7)

396 Sessio 7 whee eeses he ol ube of ge ois d he ol ube of iges ke duig he clibio sge. Fig. shows he clssicl se of views used duig clibio. The videoce is oed oud he ge o isue coe of obsevio views d ioduce geoeicl cosis i he 3D couio of clibio ois. Noe h oig he ce oud he objec is ideicl o oe he objec oud fied ce. Fig.. Clssicl ce geoe duig self-clibio sequece.. Solvig he oble The eos ε d u ε v beig o-lie fucios of Φ he iiiio of (.6) is o-lie oiiio oble. Oe w of solvig he oble is o lieie (.6) wih soe iiil vlue Φ d solve fo Φ. The soluio is he obied b ddig Φ o Φ s he ew iiil vlue d eeig he ocess uil cei covegece is sisfied. Give 3D ois d hei coesodig D ige ois i views we c wie he lieied esuee o eo equios i i fo: V L A Φ (.8)

Seeoscoic PIV 397 wih L ε ( ) d A ε ( ) Φ Φ. Le he weigh i of he esuees be W he les sques soluio o (.8) is iiiio oble of ( V T WV) i 9 6 3 Φ R The soluio o (.9) c be obied s: T T ( A WA) ( A WL) (.9) Φ (.) Fo he les sque esie of (.8) d (.9) we c coue he esie of he esidul veco V s T T Vˆ ( I A( A WA) A W)L (.) he esie of he so clled sdd eo of ui weigh which is he oseioi esie of he sdd deviio σ of he oise o he ige coodies if he odel is coec d hee e o sse eos: T V WV (.) σˆ N d he esie of he covice i of he ees Φ : C Φ T ( A ) WA (.3) Fo ech idividul eeφ i we c he coue he esie of is ecisio o he sdd deviio: ˆ σ (.4) i ˆ σ c ii I ode o ssue lgoih covegece he clibio veco se s iiil vlue hs o be o oo f fo he soluio. I he fil eeiel esuls we will show h his iiil vlue is o ell difficul o obi d h good covegece c be chieved if he clibio views of he e e ke fo diffee oieio obsevios. This is equivle o isuig ve cosied igulio gles i sce fo he 3D oi ecosucio. As udelied i (Bow 97) we hve oiced h ulile views ke wih ± 45 degee oio oud he objec leds o bee esie (bee covegece d bee ccuc) of he clibio ees. Sice he 3D coodies of he clibio ois e siuleousl esied wih he clibio ees he eisic geoe of he visio sse is deeied u o scle fco. Acull i hese codiios he ecosucio of bigge clibio e obseved fo fhe disce will ovide idei-

398 Sessio 7 cl ige. This loss of eic diesios does o hve ge cosequeces o sigle video-ce clibio. Ke ifoio fo fuhe licio sks e ol coied i he iisic ees. Le us ecll h eisic ees give he 3D locio of he clibio e eessed i he video-ce fe d i ccodce wih give view of he ige se. I ode o clibe wih wo (o oe) video-ces i will be ecess o ioduce eic diesio o fi he eisic geoe of he video-ce cofiguio. Such sk c esil be efoed wih he ccue kowledge of he Euclidi disce bewee ol ois og o he bsolue legh of slio bewee wo views. 3 Scheiflug odel I gul SPIV ech seso is iled ccodig he Scheiflug codiio i ode o focus o he lse le wih low eue ube (Hisch e l. 993 Psd d Jese 995). The Scheiflug codiio is obied s he les suoed b he seso esecivel he les iesec wih he lse le i coo lie. I is deiced i Fig. 3 wih he "ie coveio". As sesos e o loge ohogol o he oicl is oicl disoios e o oe isooic wih esec o he ige cee. The geel eessio fo he ige coodies c be obied b ojecig he disoed ige coodies ( δ ( ) δ( ) ) i he fo ige le oo he iled ige le : ~ δ s S 33 ~ RSch δ s S 33 (3.) s whee he Scheiflug oio R Sch defies he oio fo he videoce coodie sse o he iled ige le coodie sse. Whe he disoios d esuee eos due o he sub-iel e deeco e of he se ode of giude he Scheiflug oio is coesed wih he eisic oio h defies he objec le osiio i he videoce fe. Accodig o esecive oi of view his is o el oble. Ideed clibio defies he seso d he objec i sce d i h cse he clssicl ojecio schee hs o be choose. Reebe h o ccuc is equied boh i he fl clibio ge defiiio d i is dislcee becuse ll hese ees will be esied duig he o lie oiiio. I secio 5 we will use l clibio objec ied b lse ie d ull oved o cee he se of iges.

Seeoscoic PIV 399 Fig. 3. Scheiflug codiio ccodig o he "ie coveio" (he "ige" le is lced fo of he ojecio cee b se of he oicl ige le wih esec o he ojecio cee). 4 Lse le equio Afe clibio of he seeo sse he igid sfoio ( R S S ) fo he secod video-ce coodie sse o he fis oe fo he i h view is give b : ( ) ( ) R i RS i S R i i (4.) Le us ssue h he oicl disoios e coeced. As he ojecios e esecive oes hee eiss ue hoogh sfoio H bewee ( ) ( ) ois (of iel hoogeeous coodies u u v ) d ( ) T ( u v ) T ( ) (of iel hoogeeous coodies u ( ) ( ) ) (Hle d Zisse ) : ( ) ( ) s u H u (4.)

Sessio 7 4 whee H is 3 b 3 i defied wih ol 3 ukow ees d s scl ioduced i ode o olie he hid cooe of ( ) u. I fc he ige ( ) u is he ige of he bck-ojecio of u o he lse le. Fig. 4. Hoogh sfoio bewee iges whe viewig he lse le. The lse le beig defied b he disce d fo he fis ojecio cee d b he ol veco ( ) T he hoogh H c be eessed s : ( ) ( ) ( ) ( ) R H v u d d d v u S S (4.3)

Seeoscoic PIV 4 T The uose is o fid he hee ukows ( b c) which leds o d he lse le equio. The soluio ( ~ ~ b c~ ) T c be obied b iiiig T he su i εi εi whee ε i is he deviio veco bewee he esued osiio u ( ) i (coesodig o he i h esued osiio u i ) d he cue vlue H b cui. s 5 Eeiel esuls The ehod oosed fo SPIV self-clibio hs bee lied o i of 4 768 /" CCD ces oued o he Scheiflug devices desiged b he Lboo of Mechics of Lille (Fce) i he fe of he Euoiv ojec. The CCD ces wee equied wih f5 leses focused o he esuee le wih disce of 5 d viewig i 45. Fo self-clibio we hve ol ube of 963 ukow ees (icluded iisic d eisic ees d lso clibio oi coodies fo (.7)) fo equios. B usig 5 clibio dos o fl suo obseved fo 8 views i.e. 4 esuees fo 3 ukow ees he sse o be solved is clel edud. The views e ke b hd-osiioig he fl clibio ge ccodig o ve diffee oieios (± 45 ) i ode o bee esie he clibio ees. The Scheiflug video-ces hve bee self-clibed ccodig o clssicl ojecio schee (secio 3) s disoios eed egligible i dislcee esuees cied ou i his eeie. The coodies of he ige cee e (4983) iel i fe d (6857) iel i fe. The vlues of d (iied equl s iels e sque) e 86 iel fo videoce d 8 iel fo video-ce. The coesods o objec disce of 47. The igid sfoio of he seeo device ( R S S ) (which llows o ss fo he secod video-ce coodie sse o he fis oe) is give o ble. The clibio ee veco hs bee obied wih bsolue esiduls less h.4 iel. Tble. Rigid sfoio of he seeo device. α β γ 68.4 86.8 69.7-499.56-6.55 467.74

4 Sessio 7 Tble. Tslio esuees i slio oio. α ( ) β ( ) γ ( ) Agle ( ) Mgiude..6 -..999.45 -.45.9.9994.9.3...43 -.58..4..7 -..9993.5 -.63.7.9995 -...3.9987.44 -.44..999.3.4.3.9975.4 -.3.9.9977.6.8..9989.44 -.43.4.999.5.55.6.9974.4 -.48.7.9976.5.6 -.3.9996.44 -.4.7.9998.5..5.4.4 -.4.4.6 -.3.7 -..6.44 -.58.7.8..7 -..999.5 -.56.3.999..3..9996.4 -.43.3.9998...3.998.4 -.3.6.998.7..5.9984.4 -.37.3.9986.5.4..998.4 -.45.5.9984.4..4.9997.4 -.4..9999.6...8.4 -.3.3..3. -.6..45 -.39.7.3 Me.9996 Deviio.3 I ode o ssess he ecisio of his clibio he fl clibio ge ws oued o eoe cool sges. The ge ws fis sled log he hoiol X-is i he objec le wih se d secodl oed oud he Y-veicl is b -.5 ses. Fo ech locio he osiio of ech do ws esued b he self-clibed seeo device usig he sub-iel e deeco he he ge l loclised. The oio-slio sfoio bewee he k h d (k) h osiios of he ge eessed i he k h ge coodie sse is eoed i ble fo he slio oio ( sles) d i ble 3 fo he oio oio (8 sles). The oios hve bee ecoveed wihi µ fo he slio d wihi.3 fo he oio which gees wih he esoluio of he eoe cool sges. The ehod oosed i secio 4 fo ecoveig he lse le equio hs bee esed o dol-ied e sicked o CD-ROM b w of suo d osiioed i fo of he clibed seeo device. The ehod lied o he whole iges of he seeo i gve he equio of he suo le i 3 ieios. Fig. 5 shows he bili of he ehod which hs deeced sll bu he locio of he cel hole of he CD-ROM. 5 ois defied i fe wee ched wih hei coesodig ois i fe b coelio. Afewds he coesodig 3D ois wee ecosuced b igulio (i he coodie sse of video-ce ). The e disce of he 3D ois fo he ecoveed le is less h 7 µ d he sdd deviio equl o 93 µ.

Seeoscoic PIV 43 Tble 3. Dislcee esuees i oio oio. α ( ) β ( ) γ ( ) Agle ( ) Mgiude.8 -.497.48 -.84.3 -.566.497.358.8 -.53 -.59 -.83.6 -.57.54.356.88 -.56 -.7 -.83.7 -.578.58.363.8 -.4983 -.77 -.85.5 -.548.4984.3.48 -.55 -.5 -.85.7 -.64.55.395.75 -.5 -.58 -.838.5 -.587.5.373.7 -.538 -.67 -.843.6 -.593.539.38 Me.58 Deviio. Fig. 5. Vecos fo he ecoveed le o he esued ois ove-ied o he do e (equio of he e le ecoveed i he coodie sse of video-ce :.75 X -.4 Y -.734 Z 347.9 ). 6 Coclusio We hve deosed h he clibio of gul Seeo PIV sse i Scheiflug codiio c be efoed b hd-osiioig e-ied ge d ecodig se of views. Afe deecig he ge ks i he views oil i hole odel is seched fo s he se ie h he locliio of he views d he coodies of he ks o he ge b o lie les e sque. We showed h he islige oble c be solved b sechig fo

44 Sessio 7 he hoogh bewee he wo ige les whe viewig he esuee le. Such oocol hs bee evlued i i o sheic objecs. Afe self-clibig he eeiel se-u dislcees wee esued wihi ecisio close o he esoluio of he eoe cool devices used i uli-le clibio. The ehod oosed fo he ecove of he esuee le equio llowed o deec bu of few of icoees. The edudc fo self-clibig c be icesed b elcig he locliio of he secod video-ce b he igid sfoio of he seeo device. The oosed oocol hs ow o be vlided i esece of iefces. The och will be woh seeig i eleceic cofiguio which hs bee sudied i he fe of he Euoiv ojec (Fouel e l. 3). Refeeces Aeic Socie fo Phooge (984) Mul of Phooge. Fouh ediio Bjokquis D.C. (998) Desig d clibio of seeoscoic PIV sse. Nih Ieiol Sosiu o Alicios of Lse Techiques o Fluid Mechics Lisbo Pougl Jul 3-6 Bow D.C. (97) Close-ge ce clibio. Phoogeic Egieeig 37:8:855-866 Coude S. Scho J.P. Bck ojecio lgoih wih islige coecios fo D3C seeoscoic PIV. Mes. Sci. d Techol. :37-38 De Agio L. H E. Reid I. Self-clibio of oig d oo ig ces. I. Joul of Coue Visio 45: Fouel T. Coude S. Fouie C. Ducoe C. (3) Seeoscoic icle ige velocie usig eleceic leses. Mes. Sci. d Techol. 4:494-499 Hle R. Zisse A. Mulile view geoe i coue visio. Cbidge Uivesi Pess Hisch K.D. Hiichs H. Rosho A. Deese F. (993) Hologhic d seos coic dvces i 3-D PIV. Hologhic Picle Ige Velocie. Poc. of Fluids Egieeig Divisio Aeic Socie of Mechicl Egiees ed. E.P. Rood 48:33-36 Lves J.M. Vil M. Dhoe M. (998) Do we ell eed ccue clibio e o chieve elible ce clibio. Poc. of ECCV98 Feibug Ge 58-74 Li M. Lves J.M. (996) Soe secs of oo les clibio. IEEE PAMI 8: Psd A.K. Jese K. (995) Scheiflug seeoce fo icle ige velocie i liquid flows. Alied Oics 34:79-799 Nqwi A. Disoio coesio fo PIV sses. Teh Ieiol Sosiu o Alicios of Lse Techiques o Fluid Mechics Lisbo Pougl Quéo G. Rbe A. Lusse F. Goug P. Sile d ccue PIV

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