3-D least squares matching for volumetric velocimetry data processing

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Jeremy Rodgers
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1 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul -D let que mth fo volumet velomet dt oe Ptk Wetfeld H-Ged Olve Put Je Kthofe Chtoh Büke : Ittute of Photommet d Remote Se Tehhe Uvetät Dede Gem {tk.wetfeldh-ed.m}@tu-dede.de : Dte Dm A/S Skovlude Demk olve.ut@dtedm.om : Itute of eh d Flud Dm Tehhe Uvetät Bekdeme Fee Gem {kthofeueke}@mfd.tu-fee.de Att The e eet thee-dmeol let que mth oh led to tme-eolved volumet tle me velomet dt to deteme -D velot feld. It wll todue the futol d toht model of the tk lothm whh lude the eomet oet the meteto of the tfomto the devto of fomto o eo detemlt d ellt of the tfomto mete well the teto of ddtol ott. The mlemetto of -D let que mth h ee vldted wth oth multed d el dt. The u otetl well the ellt wll e doumeted omtvel fo ll dt et. The meteto fo dffeet flow d the eete of ft mete eted tjetoe wll e led d omo etwee -D o oelto d -D let que mth ve.. Itoduto Photommet -D moto l well-etlhed feld of loe-e hotommet d llow the etto of eomet fomto fom me wth hh eo d ellt. I th otet let que tehque uh let que mth det oweful flele d wdel ued tool fo the omutto of velot veto fom -D o -D me equee. Volumet PIV eete tomoh eotuto of tle otellto fom lmted ume of hoed me vew l fele eotuto lothm lke ART Hem & Let 976 o ART et l. 9. The dvte of PIV the etvt to hh eed dete. Th kd of tomoh eotuto fltte the mlemetto of volume-ed tle tk tehque the th dete PTV tle tk velomet oh whh m e oe to mute t hh eed dete Ptoou & Do 989. F. Volumet Dt Poe Wlleff. Volume-ed tk tehque delve dee flow velot feld fomto dvd tme- - -

2 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul eolved volumet velomet dt to uod of edefed e d tk thee uod thouh the eotuted equee of voel e. Hee -D o oelto -D CC Kee & Ad 99 the thtfowd tehque to deteme -D dlemet veto etwee uod of two oeutve eoh wth u-voel eo lult -D o oelto oeffet feld d ftt Gu futo to t to ot u-voel u. A mlemetto of th oh qute mle ut lmted to the detemto of the hft eh deto ol. Thu uod wth ft defomto wll ot e tked ell. So tll move th tetve me defomto method ut t muh le omuttol effot. Hee -D let que mth -D LS offe the dvte of e dtve to uod defomto otto d he whh mke t the teet ltetve to -D CC.. Ple -D let que mth -D LS volumet mth tehque et l I lo to -D let que mth Akem 984; Föte 984; Gü 985 -D LS utle tetve eomet d domet tfomto etwee two o moe oeutve uod w tht the um of the que of the vlue dffeee etwee the voel of the uod ehe mmum. The eomet tfomto -D ffe tfomto wth mete the otol domet djutmet m ue -mete le oeto tem F.. F. -D ffe tfomto of uod. Fom left to ht: tlto le otto he. The eetl dvte of thee ddtol o-tltol uod tfomto mete the dtvt to he le otto o he. If ot odeed the m hme the detemto of velot veto o temtll deteote the qult o eult. Comed to the -D o oelto -D CC method uod tk fomulted let que olem h the follow mlto: The eomet tfomto lude ddtol le he d otto mete. Thee mete llow dt to le defomto of uod. The mthemtl model e eteded ddtol fomto o ott. Whe led to lqud flow dt omelt ott h to e todued to foe the volume of uod to em ott du the tetve tfomto. Aum oet mthemtl model d uffet ott wth the uod -D LS heve ve hh u-voel eo dow to / of voel. -D LS ovee few teto d m edue the omuttol effot. - -

3 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul The oevto equto e o-le. The leed equto eque tl vlue uull oted e-kowlede o hehll l the tehque o eoluto md. The djutmet h hh edud llow fo eel ete deo d the deteto d elmto of outle. Au etmte of the ukow mete e delveed.. themtl odel. Futol odel Temlte uod d eh uod tke fom oeutve volume dt et d ovde vlue oevto fo the djutmet t eh oto e.. The eomet d domet elto etwee thoe uod e fomulted e whee d model he hte d ott d e ode mll oe fto. The eomet -D ffe tfomto model ve wth e hft le d otto d he. Wth Gu-kov odel G the mete e etmted mm the um of the que of the vlue dffeee etwee d. Suoe omte vlue e vlle fo ll todued mete of the uod t t tl oto Eq. e leed d d d e wth { } j d d d d d d j j j 4 Wth the vlue det thee oodte deto 5 th led to leed oevto equto - -

4 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul 6 d d d d d d d d d d d d e Φ Lke ll o-le let que ohe the etmto of the ukow tke le tetve w. The e udted fte eh teto utl eet teto te h ee ehed. I eel the edul mout e o-tee vlue d the vlue the eh uod wll eed to e detemed teolto. Fo the el e of tk tle lqud uod whle h t he due to defomto wll hve ott volume. Coequetll volume ott w mlemeted to ode the omelt of the lqud. The omelt ott fomulted 7 d eult oe ddtol leed equto fo the G: Ψ Ψ Ψ Ψ d d d 8. Stoht odel The toht model dee the ve d o-ve of the oevto. I the eet e homoeeou dt e vlle d ll vlue oevto e todued wth me - o u efto tem of equl ve e. equl toht weht. I P O 9 whee the -o tdd devto of ut weht d the -o tdd devto of the oevto k k{...}. Due to deedet d homoeeou oevto the weht mt of the oevto P wll ofte e dett mt I. The omelt ott Eq. 7 todued wth hhet ot ett t -o tdd devto to 'eo' Ψ P Ψ.. Gu-kov odel Eh voel uod odue oe oevto equto Φ. Fo eh oeutve of uod ddtol equto h to e et u whh ode the omelt ott Ψ. The omlete equto tem olved eteded Gu-kov odel odtol let - 4 -

5 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul que djutmet to etmte u ukow tfomto mete vlue oevto e uod: v A ˆ l u u w Β ˆ u u Aum oml dtuted eo the follow odto fo the edul v e. the otee w eult Bˆ w m T v Pv k whh led to the eteded oml tem of equto olved fo the veto of ukow: T ˆ A PA k B B T P Ψ T A Pl w Hee the Jo mt A ot of the devtve of Φ Eq. 6 wth eet to the ukow d dee the futol elto etwee the mete. I loou w to the oevto equto B ot of the devtve of Ψ Eq. 8. A u B u Φ Φ Φ Ψ Ψ Ψ [ ] Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ The veto l deoted the veto of the edued oevto: l The veto of L multle k e ot omll of teet. The veto of the ukow ot the etmted mete: ˆ u [ ] T 4-5 -

6 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul 4. Itel Qult eue A let que djutmet method -D let que mth -D LS delve fomto o the eo detemlt d ellt of the ffe tfomto mete. Th lude the tdd devto of eh of the mete well the oelto etwee mete. The -oteo tdd devto of ut weht ve T v Pvld v ˆ 5 wth edud u. The -oteo tdd devto of etmted mete ve ˆ ˆ 6 q Q A PA whee q the th T dol elemet of the ofto mt. Futhemoe the level of fe e lulted fo eh mete eh uod. To dede whethe tfomto mete ft o ot the oe-dmeol Studet tet futo e lulted fo eh todued mete. The mete e ft f t ˆ > q 7 fulflled. The qutle q e oted the vee of the Studet' t vee umultve dtuto futo u the deee of feedom d ve olt P tht the mete e ft. No-ft mete m e eluded fom the etmto oe d et to eo ode to move the teth of the oluto. Aum t let oe o-ft mete the todued et of mete the let que djutmet eeted utl ll ued mete e ft. 5. Some Reult 5. Smulted Dt Set A multed dt et w eeted omo two oeutve voel e e. wth volume of voel. Thee volumet eotuto ot the flow of vote. The eod tte of the vote w heved defed tlto d otto of the ft oe. Futhe ome dom oe w dded to the vlue of oth voel e. The velot feld fomto wee detemed omtvel -D o oelto -D CC d -D let que mth -D LS. A eul d of 7 voel uod w defed to. Fo eh d oto the oeod uod wee detemed ethe lulto the -D o oelto oeffet o lult the -D LS ffe tfomto mete. All - 6 -

5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul tk eult wee eted. Nethe thehold fo the o oelto oeffet o -D LS outle te wee led. Fo -D CC the temlte uod w hfted thouh the eh volume.

7 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul tk eult wee eted. Nethe thehold fo the o oelto oeffet o -D LS outle te wee led. Fo -D CC the temlte uod w hfted thouh the eh volume. At eh oto the omled o oelto oeffet etwee the vlue of the temlte d the oeod eh uod wee lulted. A Gu futo w ftted to the omuted -D o oelto oeffet feld to ot u-voel u. The oto wth the hhet oeffet eeet the loto of. Fo -D LS the mete of the -D ffe tfomto wee omuted deed Se.. Pmete wee eluded fom the tfomto f the tued out ft the fe tet Se. 4. Veto leth Noml/tul vto -D CC -D LS F. Co eto of olo-oded veto leth left d oml/tul vto ht of multed vote heved -D CC to d -D LS ottom. The eult e ee F. d T.. At ft the o eto of the eult velot veto feld look ml fo -D CC d -D LS. Al the vto etwee the oml d tul tle oto how the et dvte of -D LS. I ott to -D CC the tjetoe wee detemed moe eel d ellt. l the ete of the vote moved eo fto -6 e heved whh ove the eleve of detem ot ol tlto mete uod tk. σ σ σ -D CC.8e-.566e-.656e- -D LS.77e-.6948e- 4.5e- T. Stdd devto of tul/oml vto d voel

5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul 5.

A ott mo eete llel lht heet le wth thke of mm.

Neutll uot eed tle e jeted to the ete of vote eeto. See Kthofe et l.

Voel e of the volumet eotuto t oe eoh Wetfeld &. F. 6. Vote med wth fou me t oe eoh et l. 9.

The oe e o multle ojetve tfomto of eh me vew to eh deth le of the voel eeetto of the ojet e d

The velot feld fomto the e oted l the eeted -D LS to the tmeeolved voel e eeetto.

The tl vlue fo ll mete wee omuted utomtll l -D LS uevel o -level md of the ojet e.

8 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul 5. Rel Dt Set The el dt et ote fom the follow eemetl etu: A vote wte tk llumted -D le em deve. A ott mo eete llel lht heet le wth thke of mm. Th volume of out m eoded tem of fou hoed hh eed me 4 4 el f equed wth teleet lee F. 4 d 6. Neutll uot eed tle e jeted to the ete of vote eeto. See Kthofe et l. 9 fo detled efto of th eemetl etu. F. 4. Eemetl etu wth teleet lee Kthofe et l. 9. X Y Z F. 5. Voel e of the volumet eotuto t oe eoh Wetfeld &. F. 6. Vote med wth fou me t oe eoh et l. 9. The volumet eotuto F. 5 eult fom moved eotuto tehque eeted et l. 9 d Wetfeld &. The oe e o multle ojetve tfomto of eh me vew to eh deth le of the voel eeetto of the ojet e d ART mtoe le eotuto tehque to fll-u the voel e wth vlue fomto. The velot feld fomto the e oted l the eeted -D LS to the tmeeolved voel e eeetto. A eul d of 5 uod w defed to the volumet eotuto of voel d the ffe mete wee detemed. The tl vlue fo ll mete wee omuted utomtll l -D LS uevel o -level md of the ojet e. If dequte tl vlue e vlle -D LS ovee few teto F. 7 d 8. No-ft mete wee eluded. Tjetoe whh do ot fulfll the te of outle deteto heme wee emoved: Affe tfomto mete tdd devto: The eult of uod wth tdd devto eeed eet thehold wee deleted. Coveee ehvo: Cuod wth dve o ollt oluto wee ejeted

5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul Veto leth: Tlto veto eeed eet thehold wee elmted. Nehohood oelto: The dffeee of the tlto veto omoet etwee eho uod wee led.

F. how the deto of the velot veto fo oe hlf of the vote fotl vew. F. 9. Co eto of olo-oded velot voel e et l. 9. F.. Co eto of olo-oded velot voel e et l. 9. The tdd devto of ut weht w.

9 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul Veto leth: Tlto veto eeed eet thehold wee elmted. Nehohood oelto: The dffeee of the tlto veto omoet etwee eho uod wee led. Veto wth devto fom the ehohood eeed eet lmt wee elmted. F. 7. Temlte uod d eh uod thouh eve teto. F. 8. Coveee ehvo of ll ffe tfomto mete. The follow fue vule the -D LS eult. F. 9 how the o eto of olo-oded velot voel e. F. how the deto of the velot veto fo oe hlf of the vote fotl vew. F. 9. Co eto of olo-oded velot voel e et l. 9. F.. Co eto of olo-oded velot voel e et l. 9. The tdd devto of ut weht w.5 vlue ved ove ll eted uod. T. how the vee tdd devto of the ffe tfomto mete. A oe ee the tel eo of the uod tlto mete the ode of / of voel. Howeve oe h to ode tht thee tel eo fue e ol elt f the umed futol d toht model oet -D ffe tfomto d let que djutmet um Gu eo dtuto. Futhemoe T. ve ovevew o the eete of ft -D ffe tfomto mete ove ll eted uod. A the uod tlto mete wee ot eluded ule the fe tet the ll hve % hee. The le mete oted the omelt odto wee - 9 -

10 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul ol ft eltvel few uod whle the otto d he mete wee ft eell the ete of the vote F.. I totl out % of the uod howed t let oe ft o-tlto mete ov the dequtee of the -D LS oh. Futhe the ed -D LS o-tlto mete e ued to etmte the -D defomto teo well the -D ottol teo detl Kthofe et l.. σ [v]..5.9 S [%] σ [v].4e-.8e-.6e- S [%] σ [v].4e-.e-.7e-.8e-.6e-.6e- S. [%] T. Avee tdd devto of tfomto mete d eete of ft mete eted tjetoe. F.. Velot veto dl wth veto elo to uod wth t let oe ft otlto -D ffe tfomto mete oded ee et l Outlook The e eeted volume ed tk tehque to omute Eule -D velot feld fomto fom tme-eolved voel e eeetto. l due to t dtvt to uod defomto d otto -D let que mth -D LS fom the teet - -

11 5th It Sm o Alto of Le Tehque to Flud eh Lo Potul 5-8 Jul ltetve to ovetol -D o oelto. It ould e how tht ee eo d ellt e heved f le otto d he fomto wee odeed the tk oe. oeove thee mete ele the detemto of he teo fo eh teoto ue. Futue wok wll oette o eteo of the le tfomto model -D LS todu hhe ode oloml. The eoluto of the velot feld m lo e moved detf dvdul tle voel e d tk thoe tle u the eult of the volume-ed tk ood omto. Refeee Akem F Hh eo dtl me oelto. I: Poeed of the 9th Photommet Week Shfteehe de Uvetät Stuttt Vol Föte W Qult emet of ojet loto d ot tfe u dtl me oelto tehque. I: Itetol Ahve of Photommet Vol. 5-III A Gü A Adtve let que oelto - oweful me mth tehque. I: South Af Joul of Photommet Remote Se d Ctoh Vol Hem G. T. & Let A Itetve eotuto lothm. Comute Bolo d ede 6:7-94. Kee R. D. & Ad R. J. 99. Theo of o-oelto l of v me. Aled Setf Reeh Kthofe J. Büke Ch. & Put O. 9. Tomo PTV u D S Illumto d Teleet Im. Poeed of the 8th Itetol Smoum o Ptle Ime Velomet 5-8 Auut eloue Vto Autl. Kthofe J. Wetfeld P. Put O. H.-G. & Büke Ch.. Det Etmto of D Defomto d Rotto Rte Teo fom G Vlue Wehted Voel Se v Let Sque th. Aeted fo ulto the Poeed of the 5th Itetol Smoum o Alto of Le Tehque to Flud eh 5 8 Jul Lo. H.-G. Stefd A. & Gü A Fom el to voel - tk volume elemet equee of -d dtl me. I Itetol Ahve of Photommet d Remote Se Volume /. H.-G. Wetfeld P. Pute T. Bøtkjæ N. Kthofe J. Büke C. 9. Photommet tehque mult-me tomoh PIV. I: So J. Atko C. ed. Poeed of the 8th Itetol Smoum o Ptle Ime Velomet 5-8 Auut eloue Vto Autl Ptoou D. & Do T Al -d tuulet moto oe hel flow ue of teeoo d tle tk. I: H.-H. Hehol d H. E. Fedle ed Adve Tuulee Vol. Se Hedele. So F.. Itetve me defomto method PIV. euemet See d Teholo :R-R9 IOP Pulh Ltd. Wetfeld P. & H.-G.. -D Let Sque Tk Tme-eolved Tomoh Reotuto of Dee Flow ke Feld. Aeted fo ulto the Poeed of the Commo V WG V/ Smoum 4 Jue Newtle. Wlleff. J.. A Sto-Temol th Alothm fo D Ptle Tk Velomet. D. Tehhe Wehfte ETH Zuh N. 576 Zuh Swteld. - -

2. Elementary Linear Algebra Problems

2. Elementary Linear Algebra Problems . Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N 3 4 5 3 6 4 7 8 Full y tee Depth te tep log()n Veto updte the f- poe wth N : ) ( ) ( ) ( )