3D Particle Position Measurement via the Defocusing Concept

Transcription

1 Internatonal Journal of Advanced Scence and Technology Vol. 24, November,, 2 3D Partcle Poston Measurement va the Defocusng Concept Xaol Bao, Muguo L State Key Laboratory of Coastal and Offshore Engneerng, Dalan Unversty of Technology, Dalan 624, Chna E-mal: echo_allen@hotmal.com Abstract Ths paper presents a 3D partcle poston measurement method based on the defocusng concept ntroduced by A.Pentland wth a sngle camera. Dfferent from the exstng approaches, the blur crcle s used to dentfy the blur extent for varous defocused partcles, and the depth-blur relaton s formulated by the modfed A.Pentland mathematcal formula. And, an eght-step algorthm based on Crcular Hough Transform and Quadrant Radus Hstogram was gven to calculate the 3D partcle poston. Experments on real defocused partcle mages have demonstrated the feasblty of the proposed method for 3D partcle poston recovery, and the mnmum Mean Absolute Error of depth s about.8 mm. Keywords: partcle poston measurement; Crcular Hough Transform; Quadrant Radus Hstogram.. Introducton Partcle Image Velocmetry (PIV) has developed to the stage of measurement of 3D velocty components n 3D flow felds. As a result, lots of 3D PIV technques have been nvestgated. Many of them are base on seedng partcles, whch vsualze the flow and can be recorded by an magng system. 3D partcle poston measurement of tracer partcle becomes one of the essental problems n 3D PIV research. Althougharous approaches are proposed for determnng partcle 2D poston n object plane, partcle depth measurement s not well establshed yet. The stereoscopc 3D partcle poston measurement approach requres at least two, but rather three or four synchronzed cameras [][2]. The confguraton of a mult-camera system may pose fnancal and techncal challenges. Holographc approach has receved much attenton [3][4], prased for ther hgh potental but reluctantly mplemented due to ther delcate optcal set-up and senstvty to envronment perturbatons. The defocusng approach, unlke stereoscopc approaches, has one unque optcal axs and s based on pattern matchng rather than on stereoscopc matchng of partcle mages [4]. Orgnally, a sngle camera wth a modfed three-hole aperture was used [5]. Due to the pnhole mask, there exsts the problem of a sgnfcant lack of llumnaton n a conventonal lghtng setup. Lately, three ndvdual cameras are used nstead of pnhole mask [6]. Some smulatons usng synthetc mages show that partcle densty should be lower than. partcles per pxel. Partcle mage shape s used by Han and Kähler [7]. Anastgmatc magng lens produces ellpsodal partcle mages whose elongaton and orentaton depend on the separaton of the partcle from the plane of best focus. The mplementaton of these two technques was dscussed n PIV [8]. It s well known that defocus blur of a pont object depends on aperture sze f the focal length and depth are fxed [9]. When the lens aperture was set at f /4, one expermental

2 Internatonal Journal of Advanced Scence and Technology Vol. 24,, November, 2 defocused partcle mage n the llumnated volume (about 3 cm thck) was acqured and shown n Fg.. Fgure Expermental defocused partcle mage One nterestng dscernble pont whch can be notced n Fg. s that the defocused partcles are hollow blur crcles and have sharp edges. So, 2D Gaussan functon s not a good model of pont-spread functon for defocusng [][]. It s possble to use a blur crcle, nstead of usng a 2D Gaussan functon, to descrbe the defocused partcle. In ths paper, we present a 3D partcle poston measurement method based on depth from partcle defocus [2]. 3D partcle poston can be calculated drectly usng the mathematcal formula wthout correspondence problem and expensve calculaton. The remander of the paper comprses four sectons. Secton 2 revews the measurement prncples based on partcle defocus. Secton 3 represents the basc theores used n ths study. Secton 4 explans the procedure for calculatng the 3D partcle poston. Secton 5 presents the expermental results. Secton 6 concludes the paper. 2. Measurement prncples A typcal magng system s represented n Fg. 2. In Fg. 2 s the dstance between the mage plane and the lens, u s the dstance between the lens and the locaton of focus, r s the radus of aperture and σ s the spatal constant of pont spread functon. B` 2σ Image plane Aperture r Lens Object plane A A` B v v u u Fgure 2 Geometry of defocused magng Fg. 2 shows the stuaton n whch a lens system havng a small f-number s used to project a pont at a dstance u onto an mage plane at dstance v. The pont A appears focused on the mage plane. The pont B appears blurred on the mage plane. Gven the confguraton shown n Fg. 2, A.Pentland derved the formula [2]: Fv D = () v F +σkf 2

3 Internatonal Journal of Advanced Scence and Technology Vol. 24, November,, 2 where D s the depth, f s the f-number of the camera lens, F s the focal length of the camera lens and k s a constant of proportonalty. The basc lmtaton of ths formula s that these parameters are based on the deal thn lens model and n fact they can never be measured precsely on any camera. For ths reason, equaton () was modfed as follow: Fv D== q (2) v F +σkf where q s the compensaton factor. After partcle depth measurement, the X-coordnate and Y-coordnate of partcle poston can be computed usng equaton (3) and equaton (4), respectvely. D u s x= u+ (3) f D v s y = v+ (4) f where s s the pxel sze of camera. In ths case, s s set equal to 7.6 µm. 3. Basc Theores Crcular Hough Transform (CHT) s the method commonly used to detect crcular objects and locate ther centers. In [3], D.loannou appled the radus hstogram to verfy the exstence of crcular object and determne ts radus. In ths study, they are used to calculate the blur crcle center and radus, respectvely. In the real stuaton, the blur crcle center and radus cannot be determned by CHT and radus hstogram exactly due to dstorton, nose and pxels mssng from edge. For ths ssue, Quadrant Radus Hstogram (QRH) was proposed, and t wll be descrbed n ths secton. 3.. Crcular Hough Transform The Crcular Hough Transform can be a very effcent approach for detectng crcles and crcular arcs, and locatng the centers of crcular objects [4-6]. CHT need just one parameter plane to store all the nformaton for locatng crcles of varous raduses, whch accumulates not just one pont per edge pxel but whole lne of ponts along the edge normal at each edge pont locaton [6]. In fact, the lne need not be extended ndefntely n ether drecton but only over the restrcted range of raduses on the blur crcle. The peaks, whch are corresponds to the blur crcle centers, are located by drect searchng n CHT parameter space Radus Hstogram In ths secton, we use the output of CHT as the blur crcle center. The constructon of radus hstogram conssts of fve parts[3][6-8]: () choose one detected partcle; (2) extract the square sub-regon of partcle edge mage wth the center placed at the blur crcle center and the sde length set equal to the maxmum blur crcle radus, as shown n Fg. 3; (3) calculate the dstance between the sub-regon center and the pxel over the sub-regon; (4) compute the hstogram of dstance ( radus hstogram ), as shown n Fg. 4; (5) the hghest peak n the radus hstogram, whose poston corresponds to the blur crcle radus, s detected. 3

4 Internatonal Journal of Advanced Scence and Technology Vol. 24,, November, 2 Fgure 3 The extracted square sub-regon of partcle edge mage 3.3. Quadrant Radus Hstogram Fgure 4 The radus hstogram of blur crcle Gven the sub-regon extracted n secton 3.2, we compute the Quadrant Radus Hstogram for t as follows [9][2]. We use the output of radus hstogram n secton 3.2 as the blur crcle radus. Frstly, the sub-regon s splt nto four quadrants, depcted n Fg. 5. Secondly, the radus hstogram for each quadrant ( quadrant radus hstogram ) s computed, as shown n Fg. 6. Fnally, the four hghest peaks, whose postons correspond to four blur crcle raduses ( quadrant radus ), are detected n four quadrant radus hstograms. Second Quadrant Frst Quadrant r2 r r3 r4 Thrd Quadrant Fourth Quadrant Fgure 5 Four-quadrant splt of extracted square sub-regon 4

5 Internatonal Journal of Advanced Scence and Technology Vol. 24, November,, 2 (a) Frst quadrant (b) Secton quadrant (c) Thrd quadrant (d) Fourth quadrant Fgure 6 Four quadrant radus hstograms 4. Partcle 3D poston measurement 4.. Partcle center and radus update scheme In the deal case, the partcle edge s the contnuous crcle. The center can be determned exactly by searchng the hghest peak n the CHT parameter space. In fact, because of dgtzaton error, dstorton of partcle edge and pxels mssng from the partcle edge, the partcle center cannot be determned accurately by drect searchng. In ths secton, we use the result of analyss of QRH to update the partcle center coordnate and radus, and then modfy more accurately. The update scheme for the partcle center coordnate and radus s stated as follows. Intalzaton Intal partcle center coordnate ( u ) ( =,..., N, N s the number of detected partcles) and radus r are obtaned by usng CHT and radus hstogram Four quadrant raduses r, r 2, r 3 and r 4 are obtaned by usng QRH r r2 If. 5 then r + r2 r = max( r, r3 ), r 3 = max( r, r3 ) End {f} r2 r4 If. 5 then r2 + r4 r 2 = max( r2, r4 ), r 4 = max( r2, r4 ) 5

6 Internatonal Journal of Advanced Scence and Technology Vol. 24,, November, 2 End {f} r = mean( r + r2 + r3 + r4 ) u u + r r3 v = v + r2 r4 =, u = u = v D partcle poston measurement procedure In summary, the 3D partcle poston measurement s conssted of the followng eght steps: Step : Preprocess the partcle mage, and then detect the partcle edge. In ths approach, Sobel edge detecton method s used to detect the partcle edge, because t s able to estmate edge orentaton to o and very smple to apply. Step 2: Detect the blur crcle and calculate the blur crcle radus center coordnate p p ( u ) usng CHT. Step 3: Extract the square sub-regon of the partcle edge mage wth the center of regon p p placed at the blur crcle center ( u ) and the sde length set equal to the maxmum blur crcle radus r, and then determne the blur crcle radus r p usng radus hstogram. Step 4: Choose one of the detected partcles, and set: ( u )= ( u ) (5) p p r = r (6) Step 5: Extract the square sub-regon of the partcle edge mage wth the center of regon placed at the blur crcle center ( u ) and the sde length set equal to the blur crcle radus r, and then construct the QRH. Step 6: Analyze the QRH, and update the partcle center coordnate ( u ) and radus r wth the partcle center and radus update scheme. Step 7: Judge whether the teratve condton s met or not. If η s less than %, the algorthm s contnued wth next step. If not, ( u ) and radus r are set as the ntal blur crcle center and radus, respectvely, and return to step 5. r r3 + r2 r4 η = % (7) 2 where r, r 2, r 3 and r 4 are the four quadrant raduses of the chosen partcle. Step 8: Calculate the partcle depth usng equaton (2) wth blur crcle radus, and then calculate the 2D partcle poston usng equaton (3) and (4). Step 9: Label the chosen partcle, and return to step 4 untl all detected partcles are labeled. 5. Implementaton and results 5.. Expermental setup and defocused partcle mage acquston We have employed an magng system wth a sngle CCD camera to capture dfferent focused and defocused partcle mages. Fg. 6 llustrates the magng system used n ths study. In Fg. 6, L s the dstance between the camera lens and ts focal pont, and R s the dstance between the camera lens and the glass plate. They can be obtaned by readng the scale on the optcal ral. The tracer partcles whose sze s about.4 mm are placed on the glass plate at known coordnate postons. The glass plate s llumnated from the sde wth p 6

7 Internatonal Journal of Advanced Scence and Technology Vol. 24, November,, 2 laser lght. Partcle mages are captured wth a CCD camera (IMPERX IPX-2M3-L). The focal length of the camera lens s set equal to 5 mm, and L s set equal to 45 mm. The captured mage s dgtzed nto 6(H) 2(V) pxel wth 8 bt gray level resoluton. After the camera calbraton, partcle mages are captured at nne glass plate postons rangng from 45 mm to 4 mm at nterval of -5 mm, as shown n Fg. 7. (In order to show clearly, we cut the mage sze 6 2 to ). From Fg. 7, t s obvous that the partcles are defocused on the mage plane as the glass plate moves away from focal pont. Fgure 6 Expermental setup Fgure 7 Expermental defocused partcle mage (R=4 mm) 5.2. Parameter estmaton The rght-hand sde of equaton (2) contans two unknown parameters, k and q. If two dfferent values of varable D and σ are known, we can solve the equatons for k and q. In ths study, the values of varable D and σ are measured by the defocus calbraton procedure. Frstly, the two defocused partcle mages are captured wth glass plate at two or more known postons. Secondly, the blur crcle radus s calculated by usng blur crcle radus measurement algorthm mentoned above. Fnally, the equatons are solved for k and q The results and dscusson The partcle radus measurement: The above computatonal method for computng the blur crcle radus was mplemented, and the mean and the standard devaton of blur crcle radus were computed, respectvely, as shown n Fg. 8 and Fg. 9. 7

8 Internatonal Journal of Advanced Scence and Technology Vol. 24,, November, 2 8 Mean of partcle radus (rmean/pxel) Depth (d/mm) Fgure 8 The mean of partcle radus.22.2 STD of partcle radus (rstd/pxel) Depth (d/mm) Fgure 9 The standard devaton of partcle radus It can be seen that the partcle radus ncreases approxmately lnearly wth decreasng the depth n Fg. 8. From Fg. 9, f the depth s less than 43 mm the stand devaton of partcle radus ncreases wth ncreasng the depth, or, ncreases wth decreasng the depth The partcle 3D poston measurement: We defne e x as the Mean Absolute Error (MAE) of x-coordnate, e y as the MAE of y-coordnate and e d as the MAE of depth. They are gven as between: n ex = x x (8) n = n ey = y y (9) n = n ed = d d () n = where x, y and d are the X-coordnate, Y-coordnate and Z-coordnate of partcle poston whch are known on the glass plate, n s the number of partcles n the mage plane. They are shown n Fg. and Fg.. 8

9 Internatonal Journal of Advanced Scence and Technology Vol. 24, November,, MAE of x MAE of y Fgure The MAE of X-coordnate and Y-coordnate.3.2. MAE of depth (ed/mm) Depth (d/mm) Fgure The MAE of depth The MAE of X-coordnate and Y-coordnate are much less than the MAE of depth. The mnmum MAE of depth s.8 mm, and the maxmum MAE of depth s.27 mm. Acknowledgement Ths work was fnancally supported by Natonal Natural Scence Foundaton of Chna (Grant No ). References [] Keft, R.N., Schreel, K.R.A.M.an der Plas, G.A.J., and Rndt, C.C.M.: The applcaton of a 3D PTV algorthm to a mxed convecton flow, Experments n Fluds, 22, 33, (4), pp [2] Wllneff, J.: 3D partcle trackng velocmetry based on mage and object space nformaton. ISPRS Commsson V Symposum, Internatonal Archves of Photogrammetry, Remote Sensng and Informaton Scences, Corfu, Greece, Sep 22, pp. -6 [3] Hnsch, K.D.: Holographc partcle mage velocmetry. 4th Internatonal Symposum on Partcle Image Velocmetry, Gottngen, Germany, Sep 2, pp [4] Wllert, C.E., and Gharb, M.: Three-dmensonal partcle magne wth a sngle camera, Experments n Fluds, 992, 2, (6), pp [5] Perera, F., and Gharb, M.: Defocusng dgtal partcle mage velocmetry and the three dmensonal characterzaton of two-phase flows, Measurement Scence and Technology, 22, 3, (5), pp [6] Grohte, R.L., and Dabr, D.: An mproved three-dmensonal characterzaton of defocusng dgtal partcle mage velocmetry based on a new magng volume defnton, Measurement Scence and Technology, 28, 9, (6), pp. -4 [7] Han, R., and Kähler, C.J.: 3D3C tme-resolved measurement wth a sngle camera usng optcal aberratons. 3th Int Symp on Applcaton of Laser Technques to Flud Mechancs, Lsbon, Portugal, June 26, pp. -9 9

10 Internatonal Journal of Advanced Scence and Technology Vol. 24,, November, 2 [8] Cerpka, C., Segura, R., Han, R., and Kähler, C.J.: A smple sngle camera 3C3D velocty measurement technque wthout errors due to depth of correlaton and spatal averagng for mcro-fludcs, Measurement Scence and Technology, 2, 2, (4), pp. -4 [9] Pham, D.T., and Aslantas, V.: Depth from defocusng usng a neural network, Pattern Recognton, 999, 32, (5), pp [] Murata, S., and Kawamura, M.: Partcle depth measurement based on depth-from-defocus, Optcs and Laser Technology, 999, 3, (), pp [] Ln, H.Y., and Gu, K.D.: Depth recovery usng defocus blur at nfnty. 9th nternatonal conference on pattern recognton, Tampa, USA, Dec 28, pp [2] Pentland, A.P.: A new sense for depth of feld, IEEE Transactons on Pattern Analyss and Machne Intellgence, 987, 9, (4), pp [3]Marcn, S., and Ignacy, D.: Crcular object detecton usng a modfed Hough Transform, Internatonal Journal of Appled Mathematcs and Computer Scence, 28, 8, (), pp [4] Welang, D., and James, Y.: A robust Hough Transform algorthm for determnng the radaton centers of crcular and rectangular felds, Physcs n Medcne and Bology, 29, 54, pp [5] Julen, C., Valére, F., and Dder, V.: Optmzaton of an Hough Transform algorthm for the search of a center, Pattern Recognton, 28, 4, (2), pp [6] Daves, E.R., Theory, Algorthms, Practcaltes (Elsever Press, 25, 3rd edn. ), pp [7] Heung, S.K., and Jong, H.K.: A two step crcle detecton algorthm from the nterestng chords, Pattern Recognton Letters, 2, 22, (6-7), pp [8] Yuen, H.K., Prncen, J., and Kttler, J.: Comparatve study of Hough Transform methods for crcle fndng, Image and vson computng, 99, 8, (), pp [9] Danyu, L., Yu, C.,Wallapak, T., Johnny, W., Jung, H.O., and Pet, C.D.G.: Quadrant Coverage Hstogram: A new method for measurng qualty of colonoscopc procedures. Proceedngs of the 29th Annual Internatonal Conference of the IEEE EMBS Cté Internatonale, Lyon, France, August 27, pp [2] Moyer, A.C., Evans, J.L., and Powell, M.: Comparson of observed gale radus statstcs, Meteorology and Atmospherc physcs, 27, 97, (-4), pp Authors Xaol Bao receved the M.S. degree n communcaton and nformaton system from Shenyang Lgong Unversty, Shenyang, Chna, n 29. He s currently workng toward the Ph.D degree n Dalan Unversty of Technology. Hs research nterests nclude computer vson, mage processng and pattern recognton. Muguo L receved the B.S. degree n automatc engneerng from Dalan Unversty of Technology, Dalan, Chna, n 978. He s currently a Professor wth the State Key Laboratory of Coastal and Offshore Engneerng, Dalan Unversty of Technology. Hs research nterests nclude computer vson, mage processng, automatc control.