Cellular Neural Network Based PTV
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1 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 Cellula Neual Netwok Based PT Kazuo Ohmi, Achyut Sapkota : Depatment of Infomation Systems Engineeing, Osaka Sangyo Univesity, Daito Shi, Osaka, Japan ohmi@ise.osaka-sandai.ac.jp : Depatment of Infomation Systems Engineeing, Osaka Sangyo Univesity, Daito Shi, Osaka, Japan sapkota@wind.ise.osaka-sandai.ac.jp Abstact In the ecent yeas, atificial neual netwoks and genetic algoithms have been attactive in the field of visualization. Among them neual netwoks seem paticulaly attactive whee the use of the leaning pinciple facilitates the development of fully automated knowledge based system of measuement. Hopfield neual netwok based appoaches had been caied out by some of the authos fo two-dimensional Paticle Tacking elocimety (PT) but thei application is inconvenient due to inheent demeit of high computation time with the incease in numbe of paticles. So fa as thee-dimensional measuement technique, whee two diffeent stages of paticle matching namely: spatio-diffeential and time-diffeential paiing ae caied out, is concened, the two dimensional algoithms with slight modification can be used fo the second stage while fo the fist stage the basic methodology is to check the poximity of a paticle image and its elated epipola line in eithe of the two o moe camea fames. In the pesent study, netwok model based on cellula neutal netwok, a massive paallel computing paadigm defined in discete N-dimensional space, has been implemented fo time-diffeential as well as spatio-diffeential paiing of PT. Unle the Hopfield neual netwok, connections ae limited to units in local neighbohood of individual units in cellula neual netwok. Thus, the point of this study lies on the connection scheme of the neuons, accoding to which the netwok is educed fom fully connected to patially connected scheme. The netwok has been successful to yield bette esults to both of the cases with significantly low computation time and adequate accuacy.. Intoduction In the ecent yeas, atificial neual netwoks and genetic algoithms have been attactive in the field of flow measuement and thus have been implemented, with geat enthusiasm, in the paticle paiing pocess of -D o 3-D paticle tacking velocimety (Knaak et. al., 997; Ohmi et al., 00; Ohmi & Yoshida, 00; Ohmi, 003; Ohmi & Sapkota, 004; Ohmi, 004). Among them neual netwoks seem paticulaly attactive whee the use of the leaning pinciple facilitates the development of fully automated knowledge based system of measuement. In PT, to explain summaily, two-dimensional and thee-dimensional measuement techniques ae in pactice. Obviously, paticle tacking opeations ae caied out in diffeent manne fo twodimensional and thee-dimensional paticle images. In two dimensional (-D) PT, the paticle images taken at cetain time inteval ae pocessed. In othe wods, all the paticles in an image (at t = t n, say) ae compaed to the paticles in the anothe image (at t = t n+, say) to detemine physically coespondent paticle pais. In this context, Knaak et al. (997) caied out the implementation of neual netwok techniques to paticle paiing pocess of -D PT using Hopfield neual netwok (Hopfield, 98). But this algoithm has the inheent demeit of long computation time with the incease in numbe of paticles (Ohmi and Sapkota, 004). Similaly, 3-D PT is basically composed of two successive steps of paticle matching. The fist one is the spatial paticle paiing, in which the paticles viewed by two (o moe than two) steeoscopic cameas with a paallax (diffeent viewing angles) have to be coectly paied at evey synchonized time step. This is quite impotant because the 3-D coodinates of individual paticles cannot be computed without being assued of identification of two paticle images. The basic conventional methodology of spatial paticle paiing is to check the poximity of a paticle image - -
2 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 and its elated epipola lines in eithe of the two camea fames (Nishino et. al., 989). Conventional methods, accompanied by cetain concepts, yielded the bette paiing esults but the computation time and CPU load ae lagely inceased with numbe of paticles as epoted by Ohmi (003). So fa as the second step i.e. time-diffeential paticle paiing is concened, the algoithms explained in the above case of -D PT can be used with slight modification. In the pesent study, the concept of cellula neual netwok has been implemented fo timediffeential as well as spatio-diffeential matching of the paticles. In case of time-diffential matching the concept fomulated by Knaak et.al. has been impoved in the sense of connection scheme of neuon whee netwok has been visualized as patially connected scheme athe than fully connected case of Hopfield while fo the case of spatio-diffential paiing new object function has been defined.. Cellula Neual Netwok Cellula neual netwok (CNN) poposed by Chua and Yang (988) is a massive paallel computing paadigm defined in discete N-dimensional spaces. Unle Hopfield neual netwok, the connections ae limited to units in local neighbohood of individual units in this case. In othe wods, any cell is connected only to its neighbo units, i.e. adjacent units inteact diectly with each othe. Units not in the immediate neighbohood have indiect effect because of the popagation effects of the dynamics in the netwok. The basic achitectue of cellula neual netwok is shown in Fig. Fig.. Cellula neual netwok achitectue If the unit located in position (j, l) of a two dimensional M*N aay is denoted by C, and its neighbohood N is defined by: N { C max { j i, l k } ; j M, l N} = () Whee,, a positive numbe, is a size of neighbohood. The basic pinciple now is that the Lyapunov enegy function of the netwok system should be minimized as fomulated below, - -
3 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 E = T,.. k ) ( j, l ) k ) I () Whee is the neuon state at the cossing of ow i and column k (so is in le manne), T, is the connection weight of the two neuon units and, and I the enegy theshold of unit. N and M defined in () epesents the numbe of paticles in the fist and second fames. The neuon units undego seies of state updates in ode to minimize the function, and the initial state and the updated state of neuon units ae not computed in a simple theshold scheme (0 o ) but accoding to the following nomalized function (new) Whee, u = + tanh 0.5 ( new) u = T j C The mapping of the paticles fo paticle paiing algoithm is as shown in the Fig.. N, + I (3) (4) Paticle ID in the nd fame Paticle ID numbe in the st fame Fig.. Neuon units aangement fo PT paticle paiing poblem The physical state of neuon units is expessed as active o inactive as shown in Figue. The active neuons (designated by lage dots) stand fo coect paiing of paticle ID numbes and the inactive ones (designated by small dots) fo incoect paiing. As a geneal ule of this type of paticle paiing poblem, the paticles in the fist and second fames have to meet with a unique oneto-one coespondence. So, evey ow o column of Fig. ought to have at least but no moe than one active neuon... Application fo time-diffeential paticle matching of PT The impotant thing in the implementation of this type of netwok in PT is the definition of a physically appopiate object function in the fom of constaint conditions. Knaak et al. (997) intoduced fou diffeent physical constaint conditions fo the object function in the case of two dimensional paticle images, which ae e-fomulated as below in accodance with the poposed new scheme
4 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 φ 0 = C0 f k ) R φ = C f X + C fy k ) k ) (5) (6) φ = C 3 k ) ( j, l) g( i, j) g( k,l) (7) φ = C 3 4 k ) j j i jk + l l k il (8) The fist condition (5) equies the minimization of the total sum of the distances between all the paied paticles, whee f R (i, stands fo the Euclid distance between the paticles i and k in diffeent fames. The second condition (6) asks fo the minimization of the sum of local fluctuations of the paticle displacement, whee f X (i, stands fo the x-component of the displacement vecto fom paticle i to paticle k and f Y (i, the y-component. The thid condition (7) equies the minimization of the sum of local fluctuations of the distances to evey neighbo paticle, whee g(i,j) stands fo the Euclid distance between the two neighboing paticles i and j in the same fame. The fouth condition (8) equies the uniqueness of solution i.e. one paticle in one fame will have only one unique patne in the othe fame. The five coefficients C 0, C, C, C 3, C 4 ae weight constants by which the weight balance of the fou diffeent constaint conditions can be contolled. Using this new combination of constaint conditions, the final object function is given by φ = φ φ + φ φ 3 (9) Compaing the second-ode tem of (9) with that of the Lyapunov function in (), the connection weight T, is mathematically detemined as below T, = C C 3 4 ( δ ) ( δ ij kl C4 δ ij ( δ kl ) ) g( i, j) g( k, l) δ kl ( δ ) ij (0) And, by compaing the fist-ode tems, the theshold I is detemined. I = f R C C 0 f X C fy () In (), δ ij stands fo the Konekke delta and is equal to if i=j, o 0 if othewise. By substituting the values of T, and I to (5) fom (0) and (), the unit output u is calculated as - 4 -
5 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 follows: u C3 = C4 ( j, l) M g il l j) g ( k,l) g ( k,l) g j) + N j jk l C0 f R il j + C f C f X jk Y () Fom thus obtained unit output u, the updated unit value (new) in (3) is calculated. This seies of computation is iteated until the object function φ tends to a minimum and the unit values come to thei espective optima, which ae basically active o inactive. Moe pecisely, since the unit values in (3) ae not given in binay digits, the maximal unit value fo each ow (o column) of Fig. is egaded as active and all the othes inactive. The teminal condition of the iteative calculation is that all the active neuons emain unvaied ove ten successive iteations. The values of the weight constants C 0 to C 4 ae detemined on a ty-and-eo basis. The initial values of and ae given andom decimal numbes anging between 0.0 and.0. In ode to facilitate the detemination of the weight constants, the distance paametes in the object functions φ 0 toφ 3 ae also nomalized in the ange between 0.0 and.0. Fo the time-diffeential matching of 3-D PT, the elation (6) is changed to accommodate the thid component (i.e. z component) as follows. φ = C + C 5 z f X f + C f Y Whee the additional component i.e. f z (i, is the z-component of the displacement vecto fom paticle i to paticle k. C 5 is an additional weight constant. All the othe constaint conditions fom φ 0 toφ 3 emain unchanged. The pocess simila to above is followed aftewads. (3).. Implementation fo spatio-diffeential paticle matching of PT Spatio-diffeential paticle matching is the fist and quite impotant stage of 3-D PT because the 3-D coodinates of individual paticles cannot be computed without being assued of identification of two paticle images. In this stage paticles viewed by two (o moe than two) steeoscopic cameas with a paallax (diffeent viewing angles) have to be coectly paied at evey synchonized time step. If the paallactic angle between the two cameas is small (say 0 degees o less), some of the cuent tempoal paticle paiing algoithms can be applied to the spatial paticle paiing. But, in this case, the esultant measues of paticle coodinates ae not so much esolved in depth diection as in the two plana diections. Fo moe depth esolution, the paallactic angle has to be lage. The basic methodology of the lage-paallax steeoscopic paticle paiing is to check the poximity of a paticle image and its elated epipola lines in eithe of the two camea fames. The mathematical fom of this epipola line can be fomulated fom the following mapping equations epesenting a pespective tansfom: - 5 -
6 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 c c d d x + c x + c x + d x + d y + c 3 y + c y + d 3 3 y + d z + c 3 4 z + c 4 z + d 4 z + d - c c 3 - d xx - c 3 - d 3 xx 3 xy - c 3 - d 3 xy - d yx - c 3 yy - c yx zx - d 33 yy - d 33 = X zy = Y zx zy = X = Y (4) whee x, y and z ae the physical-space 3-D coodinates of a paticle centoid, X and Y ae the -D paticle coodinates on the left-camea pojection sceen, and X and Y on the ight-camea pojection sceen. The two sets of matix coefficients c yy and d yy ae the camea paametes fo left and ight cameas, which ae detemined by means of calibation using a given numbe of calibated taget points viewed by the same two cameas. In these equations, if eithe set of (X, Y ) o (X, Y ) is given, the othe set of X and Y come into a linea elation, poviding an aithmetic equation of the elevant epipola line. The nomal distance between given paticle image and a given epipola line is calculated fom a basic geometic fomula. Having explained so, a new physically appopiate object function has been defined, fo spatial paticle tacking, in the fom of constaint conditions which ae fomulated as follows, φ 0 s = C0s ( f n + f n ( k, i)) φ s = C s j j i jk + l l k il (5) (6) The fist condition (5) equies the minimization of the sum of nomal distance of epipola line of given paticle fom its coect pai, whee f n (i, stands fo the nomal distance of epipola line in ight fame (of paticle i in left fame) fom paticle k in the ight fame. The second condition (6) asks fo paticles in the left and ight fames to meet unique one to one coespondence. The two coefficients C 0s and C s ae weight constants by which the weight balance of the two diffeent constaint conditions can be contolled. The pocess goes simila to the computation pocedue in time-diffeential matching explained above. 3. Results The paticle matching algoithm is tested by using the 3-D PI Standad Images (Okamoto et. al., 997) available fom the isualization Society of Japan. These images ae composed of vaious sets of synthetic time-seies paticle images geneated fom DNS (Diect Numeical Simulation) esults of a 3-D impinging jet in a squae cavity and 6 sets of them ae steeoscopic images fo with diffeent potions of the flow filed and/o diffeent viewing angles of the cameas. One of the advantages of these standad images is that the data sets come with text files of the oiginal paticle coodinates, so that one can compae the analysis esults of paticle matching with the coect data. In this wok, the matching esults ae pesented fo image data sets #35 and #35. These two images (#35 and #35) sets epesent the same potion of the flow field with same viewing angles of the cameas. The only diffeence is the numbe of paticles distibuted in the view field. The camea viewing angle is inclined to left o ight by 30 degees with espect to the vetical axis, - 6 -
7 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 which is pependicula to the majo obsevation plane of the flow field. Fig. 3 shows sample fames of above mentioned images seies (a) Left fame #35 (b) Right fame #35 (c) Left fame #35 (d) Right fame #35 Fig 3. Sample fames fom the PI Standad Image 3. Spatio-diffeential matching esults Table and show the spatio-diffeential paticle paing esults fo #35 and #35 image seies. As shown in the table, paticle paiing pefomance is geatly impoved with the new algoithm in compaison to conventional method of epipola line poximity analysis. Even when tested with lossof-pai paticle images also, the esults wee satisfactoy, though conceptually, the technique poposed hee deals the pefectly paied data sets. The total computation time fo 509 pefect pais of #35 image seies was only 85 seconds with Celeon.5 GHz CPU which takes aound half an hou fo conventional poximity analysis, this is indeed a geat impovement in computation time
8 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 Table. Test esults fo #35 image seies Numbe Paiing esults Fame # of Conventional paticles method With CNN Pefectly paied % 98.9% paticle images % 98.9% Loss-of pai % 9.4% Paticle images % 89.53% Table. Test esults fo #35 image seies Numbe Paiing esults Fame # of Conventional paticles method With CNN Pefectly paied % 93.63% paticle images % 94.30% Loss-of pai % 73.69% paticle images % 73.4% 3. Time-diffeential paticle matching esults Table 3 and 4 shows the time-diffeential paticle paing esults afte the computation of 3-D coodinates though the successful matching of the steeoscopic images as shown in Tables and. The esults have been geatly impoved even in this case which is moe evident in the case of #35 image seies whee the accuacy has been impoved by moe than thee times than that of conventional method. But, the algoithm has slightly lost the accuacy with the incease in numbe of paticles especially in the case of loss-of-pai paticle images. Sample vectos maps ae shown in the Fig.4. Pefectly paied paticle images Loss-of-pai paticles images Table 3. Test esults fo #35 image seies Numbe Paiing esults Fame # of Conventional paticles method With CNN 000/ 77/ % 97.83% 000/ 37/ % 79.4% - 8 -
9 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 Pefectly paied paticle images Loss-of pai paticles images Table 4. Test esults fo #35 image seies Numbe Paiing esults Fame # of Conventional paticles method With CNN 000/ 509/ % 88.53% 000/ 88/ % 53.54% (a) Pefectly paied image pais (b) Loss-of-pai paticle images Fig. 4. ectos maps obtained fom successful matching of the econstucted 3-D coodinates (Image seies #35) Similay, when tested with the Standad Image 3-D coodinates (the coodinate data which wee used to make the standad images) the eoneous paiing is actually zeo afte optimization of the weight constants C 0 to C 5, with significantly low computation time as shown in Table 5. It shows that the algoithms pefomance is compaatively excellent in the case of time-diffeential matching than in the case of spatio-diffeential matching. The vecto map is shown in Fig 5. Table 5. Time diffeential paticle paiing using Standad Image 3-D coodinates No of paticles C0 C C C3 C4 C5 CPU Time Iteations ' ' ' ' ' ''' 4-9 -
10 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, Conclusions Fig.5 ectos maps obtained by matching the standad 3-D coodinates (000 Paticles of #35 Image seies) The concept of Cellula neual netwok has been successfully implemented fo paticle paiing algoithm of Paticle tacking velocimety with significantly low computation time and adequate accuacy. The significant impovement in computation time has cited a comfotable way fo the use of such netwoks, which was peviously seemed tedious due to long computation time. The limitation fo the time being lies on dependency of the pefomance of algoithm on the value of weight constants. The definition of new object function fo spatial paticle paiing went well with high degee of accuacy in the pefectly paied data sets but accuacy seems to be degaded with the inclusion of loss-of-pai paticle images. This is especially effective fo highe numbe of paticle images, while in case of lowe numbe of paticle images the degee of accuacy is still moe than satisfactoy. It seems that special teatment is equied fo loss-of-pai paticle images with highe numbe of paticles pe fame. In eithe of the cases, pefectly paied images and loss-of-pai images, the pefomance of the algoithm is excellent both in tems of accuacy and computation time in compaison to conventional method of epipola line poximity analysis. Centeing on this fact, the algoithm is the bette altenative fo conventional methods of spatial paticle paiing. Moeove, since the algoithm has been tested with synthetic time-seies paticle images geneated fom DNS (Diect Numeical Simulation), thee ae still diffeent avenues, to go though, fo the algoithm in the seies of pefomance test whee pactically captued image data ae used. As a matte of fact, paticle tacking by itself is typically not capable of successfully tacking paticles at the vey high seed paticle densities, and its altenative in flow visualization i.e. coelation based PI ae not efficient solution fo low seed paticle density egimes. A hybid system combining both the techniques is the best solution fo the development of high-esolution PI algoithm. The algoithm poposed fo PT hee can be the majo component of such system
11 3th Int Symp on Applications of Lase Techniques to Fluid Mechanics Lisbon, Potugal, 6-9 June, 006 Refeences [] Chua L.O, Yang L, 988, Cellula Neual Netwoks: Theoy and Applications, IEEE Tans. on Cicuits and Systems, (CAS), ol.35, pp [] Hopfield J.J., 98, Neual netwoks and physical systems with emegent collective computational abilities, Poc. Nat l Acad. Sci. USA, ol.97, pp [3] Knaak M., Rothlübbes C., Oglmeiste R., 997, A Hopfield neual netwok fo flow field computation based on Paticle Image elocimety/ Paticle Tacking elocimety image sequences, IEEE Intenational Confeence on Neual Netwoks, PP [4] Nishino K., Kasagi N., Hiata M., 989, Thee-dimensional paticle tacking velocimety based on automated digital image pocessing, Tans. ASME, J. Fluids Eng., ol., pp [5] Ohmi K., Yoshida N., Huynh Thai Hoang, 00, "Genetic algoithm PI", Poc. 4th Intenational Symposium on Paticle Image elocimety, P05. [6] Ohmi K., Yoshida N., 00, "3-D Paticle Tacking elocimety Using a Genetic Algoithm", Poc. 0th Intenational Symposium on Flow isualization, F033. [7] Ohmi K., 003, "3-D Paticle Tacking elocimety with an impoved genetic algoithm", Poc. 7th Intenational Symposium on Fluid Contol,Measuement and isualization, #7. [8] Ohmi K., Sapkota A., 004, "Paticle Tacking elocimety using a Hopfield Neual Netwok", Poc. 5th Intenational Symposium on Tanspot Phenomena, #0. [9] Ohmi K., Sapkota A., 004 Impovement in Hopfield Neual Netwok PT, Poc. Intenational confeence on Advanced Optical Diagnostics in Fluids, Solids and Combustion, Tokyo, Japan,#000 [0] Okamoto K., Nishio S., Kobayashi, T., SagaT., 997, Standad images fo paticle imaging velocimety, Poc. PI-Fukui, pp
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