3-D LEAST SQUARES TRACKING IN TIME-RESOLVED TOMOGRAPHIC RECONSTRUCTION OF DENSE FLOW MARKER FIELDS

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Commsson V Symposum, Newcastle upon Tyne, UK.

Unverstät Dresden, Helmholtzstr. 1, D-162 Dresden, Germany patrck.westfeld@tu-dresden.

de/pf/photo/ Commsson V, WG V/1 KEY WORDS: PIV Processng, Tomographc Reconstructon, Trackng, Flow Measurement Technques ABSTRACT: Flow measurement technques determne velocty vector felds n lqud or

The tomo-piv (tomographc partcle mage velocmetry) technque presented n ths paper generates tme-resolved volumetrc reconstructons of a partcle constellaton from a lmted number of synchronzed camera

Whle 3-D PTV comes wth the necessty to detect, dentfy and match ndvdual partcles for establshng mult-mage and mult-temporal correspondences, tomo-piv facltates volume-based trackng schemes appled to

1 Commsson V Symposum, Newcastle upon Tyne, UK D LEAST SQUARES TRACKING IN TIME-RESOLVED TOMOGRAPHIC RECONSTRUCTION OF DENSE FLOW MARKER FIELDS Patrck Westfeld and Hans-Gerd Maas Insttute of Photogrammetry and Remote Sensng Technsche Unverstät Dresden, Helmholtzstr. 1, D-162 Dresden, Germany patrck.westfeld@tu-dresden.de, Commsson V, WG V/1 KEY WORDS: PIV Processng, Tomographc Reconstructon, Trackng, Flow Measurement Technques ABSTRACT: Flow measurement technques determne velocty vector felds n lqud or gas flows. In flud mechancs, many methods are based on seedng partcles to vsualze the flow maged by an adequate camera system. The tomo-piv (tomographc partcle mage velocmetry) technque presented n ths paper generates tme-resolved volumetrc reconstructons of a partcle constellaton from a lmted number of synchronzed camera vews. The advantage of tomo-piv n contrast to the establshed 3-D PTV (partcle trackng velocmetry) technque s ts nsenstvty to hgh seedng denstes. Whle 3-D PTV comes wth the necessty to detect, dentfy and match ndvdual partcles for establshng mult-mage and mult-temporal correspondences, tomo-piv facltates volume-based trackng schemes appled to voxel cubods flled wth partcles. The paper presents mproved photogrammetrc technques for the determnaton of 3-D flow velocty felds. Ths ncludes a mult-camera system confguraton and calbraton, approaches for a full tomographc reconstructon n gas and lqud and 3-D least squares trackng for volume-based trackng n object space. 1. INTRODUCTION Elsnga et al. (25) have proposed an approach to 3-D PIV, whch s based on a tomographc reconstructon of the observaton volume and subsequent 3-D cross correlaton n tme-resolved voxel data. Tomographc PIV generates a tomographc reconstructon of a partcle constellaton from a lmted number of camera vews, for nstance by applyng Herman and Lent s (1976) MART algorthm (multplcatve algebrac reconstructon technque). 3-D velocty feld nformaton can be obtaned from tme-resolved voxel data by dvdng the data nto cubods of a pre-defned sze and trackng these cubods. Heren, 3-D cross correlaton s a straghtforward enhancement when advancng from 2-D PIV to 3-D PIV. Dsadvantages of both, MART and 3-D cross correlaton can be seen n the computatonal effort causng rather long processng tmes. Putze & Maas (28) and Maas et al. (29) have already ntroduced more effcent approaches on volumetrc reconstructon and partcle trackng. Ths artcle summarzes the efforts of our workng group and presents results whch prove the sutablty of photogrammetrc technques n voxel data sequences analyss. buoyant seedng partcles are njected nto the center of a vortex generator. See (Ktzhofer et al., 29) for detaled specfcatons of ths expermental setup. Fgure 1. Vortex rng at one epoch. 2. SENSOR AND DATA The tomographc reconstructon methods and the cubod trackng have been mplemented and tested n two dfferent expermental setups. Frst, a vortex rng n a water tank s llumnated by a 3-D laser beam devce. A rotatng mrror generates parallel lght sheet planes wth a thckness of 1 mm. Ths volume of about (1 1 1) cm 3 s recorded by a system of four synchronzed hgh speed cameras ( pxel, 1 fps) equpped wth telecentrc lenses (Fg. 1 and 2). Neutrally Fgure 2. Expermental setup wth telecentrc lenses (Ktzhofer et al., 29). Usng telecentrc lenses results n a parallel projecton for the observed volume. A consderaton of mult-meda geometres s 597

2 Commsson V Symposum, Newcastle upon Tyne, UK. 21 not necessary for ths specal case, but the collnearty equatons have to be adapted to a parallel projecton n space: x' pp x K( r11( X X ) r21( Y Y ) r31( Z Z )) x' (1) y' pp K( r ( X X ) r ( Y Y ) r ( Z Z )) y' y where cc: Focal length x,y : Image pont ppx,ppy: Prncpal pont Δx, Δy : Correcton functons X,Y,Z: X,Y,Z: rr,c: 32 Projecton center Object pont Elements of a rotaton matrx R The four-camera system was calbrated by takng mage sequences of a target, whch was moved through the observaton volume by a 3-D translaton stage. From these reference postons and ther respectve mage coordnates, the orentaton parameters of each camera were determned n a parallel projecton telecentrc optcs camera model (Eq. 1). The transton of the optcal paths from the camera through the plan glass nterface nto the water could be neglected due to the fact that the cameras were equpped wth telecentrc lenses warrantng a parallel projecton rather than central perspectve projecton. In a second expermental confguraton, dfferent partcle constellatons n a water basn are llumnated by a fber optc lght source and are captured by a four-camera-recordngsystem (1 1 pxel; Fg. 3). Heren, the cameras are equpped wth central perspectve lenses and are able to capture mages wth an nterval of a few mcroseconds between two mages n a trggered double exposure mode. geometry, two knds of ray tracng approaches are mplemented; a forward (FRT) and a backward ray tracng (BRT). For a spatal ntersecton, the complete mage ray path has to be reconstructed (Fg. 4): () Intersect the mage ray wth the frst nterface Σ 1 and calculate the percng pont P 1. () Calculate the angle of ncdence α 1 resp. the angle of refracton α 2 n accordance to Snell s law usng the refracton ndces of the meda ar and glass. () Calculate the refracted drecton vector usng the ncomng drecton vector, the surface normal vector of P 1 n Σ 1 and the relatve refractve ndex. (v) Repeat - for the second refracton at the followng nterface Σ 2. (v) Fnally, the coordnates of an object pont X can be calculated by ntersectng the reconstructed ray wth a desred depth layer resp. wth two (or more) other ray paths. For a spatal resecton (BRT) the coordnates of the percng ponts P 1 resp. P 2 have to be calculated sequentally n an teratve manner by solvng a non-lnear system of equatons of condtons. These constrants are: () The percng pont P 1,2 s located on the plane Σ 1,2. () Snell s law has to be fulflled. () The normal vector of P 1,2 n Σ 1,2 as well as the ncdent and reflected ray are coplanar. A detaled mathematc descrpton for FRT and BRT ncludng all necessary equatons s gven n (Mulsow, 21). X Σ 1 * x α 1 P 1 ar α 2 glass Σ 2 P 2 lqud α 3 Fgure 4. Multmeda geometry. Mulsow s (21) flexble multmeda bundle approach s used to determne the nteror and exteror camera parameters as well as the meda and nterface parameters. The latter, namely the refractve ndces and the normal vectors of the planes, are necessary to reconstruct the mage ray paths through the dfferent meda gas, glass and lqud as descrbed above. X 3. VOLUMETRIC RECONSTRUCTION Fgure 3: Expermental setup wth central perspectve lenses. If used n one medum only, the well-known collnearty equatons descrbe the mage ray paths: x' pp y' pp x y r11( X X ) r cc r ( X X ) r r12 ( X X ) r cc r ( X X ) r ( Y Y ) r ( Y Y ) r ( Y Y ) r ( Y Y ) r ( Z Z ) x' ( Z Z ) ( Z Z ( Z Z ) y' ) Ths collnearty of the mage pont, the projecton center and the object pont s not gven f partcles are observed n lquds by central perspectve lenses. To consder the multmeda (2) 3.1 Prncple Tomographc partcle mage velocmetry (Tomo-PIV, Elsnga et al., 25) generates a tomographc reconstructon of a 3-D partcle constellaton from typcally four camera vews. and has the potental to solve the spatal resoluton lmt n 3-D PTV (partcle trackng velocmetry), whch s set by ambgutes occurrng at hgh seedng denstes. A reconstructon can be performed by a MART (multplcatve algebrac reconstructon technque; Herman & Lent, 1976): 1. Project every voxel nto the frst mage space, usng the model equatons 1 resp. 2. The voxel gets the gray value GV obtaned by nterpolaton the gray value gv 1 from the correspondng pxel. (Fg. 5, left) 598

Commsson V Symposum, Newcastle upon Tyne, UK. 21 2. Project every voxel nto the second mage space. Multply the exstng voxel gray value GV wth the gray value gv 2 of the correspondng pxel. (Fg.

3 Commsson V Symposum, Newcastle upon Tyne, UK Project every voxel nto the second mage space. Multply the exstng voxel gray value GV wth the gray value gv 2 of the correspondng pxel. (Fg. 5, rght) 3. Proceed wth all other camera vews j. Fnally, the voxel space wll contan multplcatvely accumulated mage ntensty nformaton of the nstantaneous partcle constellaton. It s obvous, that only voxels at vald partcle postons wll show hgh values. Repeatng the MART for each epoch results n a tme-resolved 3-D voxel space representaton of the object space. Dsadvantages of a pxelwse reconstructon and the fnal realzaton by the MART can be seen n the computatonal effort. In the second case, the projectve transformaton model can be solved drectly by takng the four corner voxel of each depth layer. All other mage coordnates can be obtaned by nsertng the correspondng set of parameters nto Eq. 3. A layer-wse reconstructon can be performed wth orthographc projectons, too. In comparson to the use of a projectve transformaton, the determnaton of correspondences between voxel and pxel for each camera vew s even more straghtforward. Agan, only the corner voxel of a depth layer have to be transformed n a dscrete way by usng the accordng model equaton 1. Due to orthographc projectons, all other mage coordnates can be obtaned by a blnear nterpolaton. Volumetrc Reconstructon n Lqud Fgure 5: Prncple of a tomographc reconstructon (Putze & Maas, 28) 3.2 Improvements Improvements can be seen n the radometrc as well as geometrc reconstructon process and wll be presented and dscussed n the followng sub-sectons. Algebrac Reconstructon Technque To fll-up the voxel space wth gray value nformaton, Herman and Lent s MART can be replaced by a MnART (mnstore algebrac reconstructon technque; Maas et al., 29) applyng a mnmum operator rather than a multplcaton to the gray values of all camera pxels n each voxel. A 3-D partcle constellaton can then easly be obtaned by a thresholdng n voxel space. Volumetrc Reconstructon n Gas For a volumetrc reconstructon n gas, the process bases on a multple projectve transformaton of each camera vew nto each depth layer of a voxel representaton of the object space. Compared to pxel-wse lne-of-sght based mplementatons, ths approach saves plenty of computaton tme. After the ntalzaton of the voxel space wth an adequate resoluton, the calculaton of the correspondng mage coordnates of each voxel of each depth layer for each vew j s performed by ether usng homogeneous coordnates as suggested by Putze & Maas (28) or even more straghtforward by utlzng the projectve transformaton drectly: x' a a X a Y 1 c X c Y y' (3) 1 2 b b X b Y 1 c X c Y In homogeneous coordnates, t s suffcent to go through the transformaton for the corner voxels of a depth layer only. All other mage coordnates can be obtaned by a blnear nterpolaton. See (Putze & Maas, 28) for a detaled mathematc descrpton. 1 2 For a volumetrc reconstructon n lqud, the layer-wse rectfcaton approaches are not feasble anymore. To avod a throwback to a voxel-wse reconstructon from object to mage space, the mage rays orgnatng n each pxel of each camera are ntersected wth each layer of the voxel space, takng nto account the fact that each ray s twce broken at the ar-glass and glass-water nterfaces (Sec. 2). Though ths soluton s performed pxel-wse, t s qute fast. A huge mprovement of the computaton tme can be acheved by thresholdng the mages before the voxel space transformaton, performng the ray tracng only for pxels above the threshold. Obvously, the use of MnART s not feasble here because not every pxel wll be projected nto the voxel space. As an alternatve, the MART can be used. Further, a volume-wse transformaton of each mage content nto the voxel space can be performed by the determnaton of the parameters of a polynomal wth three varates n X, Y and Z (Eq. 4). Consequently, the use of the mproved MnART s feasble agan to map the mage content nto the actual depth layer. x ' a X Y Z,,, (4) To solve the transformaton model of Eq. 4, the mage coordnates of a suffcent amount of control ponts are calculated drectly usng the collnearty equatons 2 plus multmeda correcton. The dstrbuton of those control ponts can be seen n analogy to the dstrbuton of ground control ponts of a set of aeral photographs for a block adjustment; namely a dense pattern along the edges of the voxel space and some sngle ponts n the center of each depth layer. The correspondng mage coordnates of all other voxels can then be effcently calculated by solvng Eq. 4 usng the set of parameters determned pror. 3.3 Reconstructon Results The followng Fg. 6 shows the reconstructon results at one epoch of a vortex rng n a water tank, llumnated by one of the 1 thckened laser lght sheets (Sec. 2). The volumetrc representaton s voxel. Each voxel corresponds to (9 µm)

Commsson V Symposum, Newcastle upon Tyne, UK. 21 Fgure 6. Voxel space of the volumetrc reconstructon of a vortex rng at one epoch.

Convergence behavor: Cubods wth a dvergng or oscllatng soluton were rejected. Vector length: Translaton vectors exceedng a preset threshold were elmnated.

The 3-D LST steerng parameters were set on the bass of a- pror knowledge on the flow and emprcally on the bass of a seres of program runs.

The numerc results of the dfferent geometrc reconstructon methods do not dffer, whch means that the accuracy of the mage ray path reconstructon only depends on the accuracy of the calbraton routne. 4.

4 Commsson V Symposum, Newcastle upon Tyne, UK. 21 Fgure 6. Voxel space of the volumetrc reconstructon of a vortex rng at one epoch. All voxels wth gray values less than 15 were elmnated for a better vsual representaton. Further, the rght of Fg. 6 s shows a reduced resoluton of 1:1. The secton n the left s at full resoluton 1:1. Convergence behavor: Cubods wth a dvergng or oscllatng soluton were rejected. Vector length: Translaton vectors exceedng a preset threshold were elmnated. Neghborhood correlaton: The dfferences of the translaton vector components between neghborng cubods were analyzed. Vectors wth devatons from ther neghborhood exceedng a preset lmt were elmnated. The 3-D LST steerng parameters were set on the bass of a- pror knowledge on the flow and emprcally on the bass of a seres of program runs. The parameters controllng the outler elmnaton process were set automatcally followng 3-sgma rules. Optonally, gaps n the vector feld can be closed by neghborhood based nterpolaton. The numerc results of the dfferent geometrc reconstructon methods do not dffer, whch means that the accuracy of the mage ray path reconstructon only depends on the accuracy of the calbraton routne. 4. VOXEL SPACE TRACKING D Least Squares Trackng Euleran 3-D velocty feld nformaton can be obtaned by volume-based trackng technques appled to tme-resolved voxel space representatons. Here, 3-D least squares trackng (3- D LST) forms a rather nterestng alternatve to conventonal 3- D cross correlaton. 3-D LST s a volumetrc trackng technque, whch s adaptve to cubod deformaton and rotaton. It mnmzes the sum of the squares of voxel value dfferences by determnng the coeffcents of a 3-D affne transformaton between cubods at consecutve tme steps. In addton to the three dsplacement vector components, the 12 parameters of the 3-D affne transformaton n 3-D LST contan scale, rotaton and shear nformaton. Ths allows for a hgher precson and relablty n case of velocty gradents n the nterrogaton volume. Moreover, these parameters enable the determnaton of a shear tensor for each nterrogaton cube. When appled to lqud flow data, an ncompressblty constrant s ntroduced to force the volume of a cubod to reman constant durng the teratve transformaton. The result of 3-D LST appled to sequences of tomographcally reconstructed voxel structures s a dense 3-D velocty vector feld wth addtonal shear tensor nformaton. A more detaled descrpton of the functonal model of the 3-D LST can be found n (Maas et al., 1994). Fgure 8. Cross sectons of color-coded velocty n voxel space. Fgure 8 shows a color-coded vsualzaton of selected layers n the 3-D LST results. Fgure 9 shows the translaton vector lengths of one half of the vortex rng n a frontal vew. As one can see, some velocty vectors n the center of the vortex were elmnated as potental outlers. Ths has to be attrbuted to the fnte cubod sze and the fact that the 3-D affne transformaton parameters can only recover lnear cubod deformatons. The results mght be mproved by some parameter fne tunng or by a hgher seedng densty allowng for smaller cubods. 4.2 Trackng Results A regular grd of 25 3 voxel cubods was defned nto the volumetrc reconstructon ganed form Sec. 3 to apply the 3-D LST. For each cubod, the 12 parameters of the 3-D affne transformaton were determned. Parameters, whch turned out nsgnfcant n the sgnfcance test, were excluded from the transformaton. A volume constrant was appled to consder the ncompressblty of the lqud. Outlers n the results were removed n an outler detecton procedure based on the followng crtera: Affne transformaton parameter standard devaton: The results of cubods wth standard devatons exceedng a preset threshold were deleted. Fgure 9. Color-coded vector lengths of one half of the vortex rng (frontal vew, X=const.=13.14 mm). In the experment descrbed here, the standard devaton of unt weght produced by the least squared adjustment process, averaged over all accepted cubods, was 2.5 gray values. Tab. 1 6

5 Commsson V Symposum, Newcastle upon Tyne, UK. 21 shows the average standard devatons of the 12 affne transformaton parameters. As one can see, the nternal precson of the cubod translaton parameters s n the order of 1/1 of a voxel. However, one has to consder that these nternal precson fgures are only realstc f the assumed functonal and stochastc model s correct (3-D affne transformaton and least squares adjustment assumng Gaussan error dstrbuton). Further verfcaton tests have to be performed to get a better estmate of the real accuracy potental of the method. a b c σ = [vx] Sg = [%] a 1 b 2 c 3 σ = [vx] 2.4e-3 1.8e-3 1.6e-3 Sg = [%] a 2 a 3 b 1 b 3 c 1 c 2 σ = [vx] 2.4e-3 2.3e-3 1.7e-3 1.8e-3 1.6e-3 1.6e-3 Sg. = [%] Table 1 Average standard devatons of transformaton parameters and percentage of sgnfcant parameters n accepted trajectores. Furthermore, Tab. 1 gves an overvew on the percentage of sgnfcant 3-D affne transformaton parameters over all accepted cubods. As the cubod translaton parameters (a,b,c) were not excluded as a rule n the sgnfcance tests, they all have 1% here. The scale parameters (a1,b2,c3), constraned by the ncompressblty condton, were only sgnfcant n relatvely few cubods, whle the rotaton and shear parameters (a2,a3,b1,b3,c1,c2) were sgnfcant especally n the center of the vortex (Fg. 1). In total, about 2% of the cubods showed at least one sgnfcant non-translaton parameter, provng the adequateness of the 3-D LST approach. Further, the ganed 3-D LST non-translaton parameters can be used to estmate the 3-D deformaton tensor as well as the 3-D rotatonal tensor drectly (Ktzhofer et al., 21). 5. SUMARY AND OUTLOOK The suggested approaches for volumetrc reconstructons under dfferent expermental setups and 3-D least squares trackng turned out to be effcent and accurate volumetrc PIV technques. The sequental projectve transformaton for volumetrc reconstructons n gas, resp. n lquds f telecentrc lenses are used, has the advantage of beng fast and graphcs card mplementaton frendly. For the more common case of a tomographc voxel space reconstructon n lqud usng mages captured wth central perspectve lenses, a fast pxel-wse technque and a polynomal approach were proposed. 3-D LST as a cubod trackng technque has the great advantage of nherently determnng 12 affne transformaton parameters of each cubod. These 12 parameters allow to adapt to lnear deformatons of cubods, thus mprovng the precson and relablty of cubod translaton parameters. Moreover, they form a bass for the determnaton of a shear tensor for each tracked cubod. The fact that about 2% of tracked cubods n a vortex rng experment showed at least one sgnfcant nontranslaton parameter proves the relevance of determnng not only transformaton parameters n cubod trackng. Future work wll concentrate on a relable mplementaton of the polynomal approach to reconstruct the whole voxel space usng one set of hgher order polynomal parameters only. Beyond, the lnear transformaton model n 3-D LST can be extended by ntroducng hgher order polynomals. The resoluton of the velocty feld may also be mproved by dentfyng ndvdual partcles n voxel space and trackng those partcles, usng the results of the volume-based trackng as good approxmaton. ACKNOWLEDGMENTS The authors would lke to thank ther cooperaton partners Prof. Dr.-Ing. habl Chrstoph Brücker and Dpl.-Ing. Jens Ktzhofer (Insttute of Mechancs and Flud Dynamcs, TU Freberg, Germany) and Dr.-Ing. Olver Pust and Thomas I. Nonn (Dantec Dynamcs A/S, Skovlunde, Denmark) for provdng the vortex rng data set as well as for helpful, supportve and enjoyable suggestons and dscussons. REFERENCES Elsnga, G. E., Scarano, F., Weneke, B. & van Oudheusden, B. W., 25. Tomographc partcle mage velocmetry. In 6th Internatonal Symposum on Partcle Image Velocmetry (PIV5). Herman, G. T. & Lent, A., Iteratve reconstructon algorthms. Computers n Bology and Medcne, 6: Fgure 1. Velocty vector dsplay wth vectors belongng to cubods wth at least one sgnfcant non-translaton 3-D affne transformaton parameter coded n green. Ktzhofer J., Brücker Ch. & Pust O., 29. Tomo PTV usng 3D Scannng Illumnaton and Telecentrc Imagng. Proceedngs of the 8th Internatonal Symposum on Partcle Image Velocmetry, August, Melbourne, Vctora, Australa. Ktzhofer, J., Westfeld, P., Pust, O., Maas, H.-G. & Brücker, Ch., 21. Drect Estmaton of 3D Deformaton and Rotaton Rate Tensor from Gray Value Weghted Voxel Space va Least 61

6 Commsson V Symposum, Newcastle upon Tyne, UK. 21 Squares Matchng. Accepted for publcaton n the Proceedngs of the 15th Internatonal Symposum on Applcatons of Laser Technques to Flud Mechancs, 5 8 July 21, Lsbon. Maas, H.-G., Stefands, A. & Grün, A., From pxels to voxels - trackng volume elements n sequences of 3-d dgtal mages. In Internatonal Archves of Photogrammetry and Remote Sensng, Volume 3, 3/2. Maas, H.-G., Westfeld, P., Putze, T., Bøtkjær, N., Ktzhofer, J., Brücker, C., 29. Photogrammetrc technques n mult-camera tomographc PIV. In: Sora, J., Atknson, C. (eds.), Proceedngs of the 8th Internatonal Symposum on Partcle Image Velocmetry, August, Melbourne, Vctora, Australa, pp Mulsow, C., 21. A Flexble Mult-meda Bundle Approach. Accepted for publcaton n the Proceedngs of the Commsson V, WG V/1 Symposum 21, June 21, Newcastle. Putze, T. & Maas, H.-G, 28. 3D determnaton of very dense partcle velocty felds by tomographc reconstructon from four camera vews and voxel space trackng. In: Jang J. Chen J. and H.-G. Maas, edtors, Internatonal Archves of Photogrammetry, Remote Sensng and Spatal Informaton Scences, Volume XXXVII, Part B5, pages ISPRS. 62

REFRACTION. a. To study the refraction of light from plane surfaces. b. To determine the index of refraction for Acrylic and Water.

$REFRACTION. a. To study the refraction of light from plane surfaces. b. To determine the index of refraction for Acrylic and Water.$ Purpose Theory REFRACTION a. To study the refracton of lght from plane surfaces. b. To determne the ndex of refracton for Acrylc and Water. When a ray of lght passes from one medum nto another one of dfferent