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Korzhova, Valentina.
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Motion analysis of fluid flow in a spinning disk reactor
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by Valentina Korzhova.
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[Tampa, Fla] :
b University of South Florida,
2010.
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Dissertation (PHD)University of South Florida, 2010.
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Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
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ABSTRACT: The flow of a liquid film over a rapidly rotating horizontal disk has numerous industrial applications including pharmaceuticals, chemical engineering, bioengineering, etc. The analysis and control of complex fluid flows over a rapidly rotating horizontal disk is an important issue in the experimental fluid mechanics. This work focuses on developing a novel approach for fluid flow tracking and analysis. Specifically, the proposed algorithm is able to detect the moving waves, compute the wave parameters and determine controlling film flow parameters for the fluid flowing over a rotating disk. The input to the algorithm is an easily acquired video data. It is shown that under single light illumination it is possible to track the specular portion of the reflected light on the moving wave. Hence, the fluid wave motion can be tracked and fluid flow parameters can be computed. The fluid flow parameters include wave velocities, wave inclination angles, and distances between consecutive waves. Once the parameters are computed, their accuracy is analyzed and compared with the solutions of the mathematical fluid dynamics models based on the NavierStokes equations. The fluid model predicts wave characteristics based on directly measured controlling parameters, such as disk rotation speed and fluid flow rate. It is shown that the calculated parameter values coincide with the predicted ones. The average computed parameters are within 5 − 10% of the predicted values. In addition, given recovered fluid characteristics and fluid flow controlling parameters, full 3D wave description is obtained. That includes 3D wave location, speed, and distance between waves, as well as approximate wave thickness. Next, the developed approach is generalized to modelbased recovery of fluid flow controlling parameters: the speed of the spinning disk and the fluidflow rate. The search in space for model parameters is performed to minimize the error between the predicted flow characteristics predicted by the fluid dynamics model (e.g. distance between waves, wave inclination angles) and parameters recovered from video data. Results demonstrate that the speed of a disk and the flow rate, when compared to the ground truth available from direct observation, are recovered with the error less than 10%.
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Advisor: Dmitry Goldgof, Ph.D.
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Image processing
Fluidflow tracking
Wave detection
Wave velocity
Wave inclination
Mathematical modeling
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Dissertations, Academic
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x Computer Science & Engineering
Masters.
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.4556
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Motion Analysis of Fluid Flo w in a Spinning Disk Reactor b y V alen tina N. Korzho v a A dissertation submitted in partial fulllmen t of the requiremen ts for the degree of Do ctor of Philosoph y Departmen t of Computer Science and Engineering College of Engineering Univ ersit y of South Florida Ma jor Professor: Dmitry B. Goldgof, Ph.D. Sudeep Sark ar, Ph.D. Grigori Siso ev, Ph.D. Aydin Sunol, Ph.D. Date of Appro v al: Septem b er 18, 2009 Keyw ords: image pro cessing, ruidro w trac king, w a v e detection, w a v e v elo cit y w a v e inclination, mathematical mo deling c r Cop yrigh t 2010,V alen tina N. Korzho v a
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DEDICA TION This thesis is de dic ate d, esp e cial ly, to my p ar ents and, also, to my husb and, son, and daughter for their love, supp ort, guidanc e, and understanding.
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A CKNO WLEDGEMENTS I w ould lik e to thank Dr. Dmitry Goldgof for giving me the opp ortunit y to w ork with him and the in v aluable academic guidance and hours that he dedicated in the leading of this dissertation. I also thank Dr. Sudeep Sark ar, Dr. Grigory Siso ev and Dr. Aydin Sunol, for the time they to ok to review this dissertation and their helpful commen ts. I w ould lik e to thank m y family and all of m y friends who ga v e me inspiration, supp ort, and comfort throughout this tough road.
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T ABLE OF CONTENTS LIST OF T ABLES iii LIST OF FIGURES v ABSTRA CT vii CHAPTER 1 INTR ODUCTION 1 1.1 Ov erview of the Relativ e W orks 1 1.2 Con tributions of This Dissertation 8 1.3 La y out of the Dissertation 9 CHAPTER 2 THEORETICAL BA CK GR OUND 12 2.1 Theory of Fluid Flo w 12 2.1.1 General Case 12 2.1.2 Fluid Flo w o v er Rotating Disk The Karman's Problem 13 2.2 Mathematical Mo deling 14 2.2.1 Ev olution Equations 14 2.2.2 The Exp erimen tal Mo del 18 2.3 Camera Calibration Accuracy Analysis 20 2.3.1 Ov erview of Camera Calibration 20 2.3.2 Camera Calibration Accuracy 21 CHAPTER 3 ESTIMA TION OF FLUID FLO W P ARAMETERS 24 3.1 T rac king Sp ecular Rerection P atterns 25 3.2 Algorithm for Detecting and T rac king Sp ecular P oin ts on W a v es 30 3.3 Estimation of P arameters of Spiral W a v es 32 3.3.1 Asymptotically Optimal Numerical Metho d of Differen tiation 32 3.3.2 Radial V elo cit y Comp onen t Computation 34 3.3.3 W a v e Thic kness and Azim uthal V elo cit y Comp onen t Computations 35 3.3.4 Inclination Angle Computation 35 3.3.5 Distance Bet w een Consecutiv e W a v es Computation 36 CHAPTER 4 D A T A A CQUISITION 37 4.1 Exp erimen tal Setup 37 4.2 Data Collection 38 i
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CHAPTER 5 RESUL TS AND COMP ARISON TO FLUID FLO W MODEL 42 5.1 Regimes of Fluid Flo w 42 5.2 Exp erimen tal FluidFlo w P arameter Estimates 42 5.2.1 Estimation of Radial V elo cit y Comp onen t 42 5.2.2 Estimation of W a v e Thic kness and Azim uthal V elo cit y Comp onen t 44 5.2.3 Estimation of Inclination Angles 45 5.2.4 Estimation of Distances Bet w een Consecutiv e W a v es 46 5.3 Comparison to Fluid Flo w Mo del 47 5.3.1 Comparison of Radial V elo cities 48 5.3.2 Comparison of Inclination Angles 50 5.3.3 Comparison of Distances 51 5.4 Statistical Analysis of Estimated Fluid Flo w P arameters 53 CHAPTER 6 MODELBASED RECO VER Y OF CONTR OLLING P ARAMETERS 55 6.1 Algorithm of Reco v ery Con trolling P arameters 56 6.2 Estimation of Absolute and Relativ e Errors 57 CHAPTER 7 SUMMAR Y 65 7.1 Conclusion 65 7.2 F uture Researc h 66 REFERENCES 68 APPENDICES 73 App endix A T rac king the P eaks of W a v es Using the Sp ecular Rerection P atterns 74 App endix B A Sto c hastic Case 77 App endix C A Regression Mo del 79 ABOUT THE A UTHOR End P age ii
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LIST OF T ABLES T able 4.1 Collected video data. 40 T able 4.2 Usage of video data. 41 T able 5.1 Exp erimen tal w a v e v elo cities ( mm=s ). Video data are tak en using camcorder Optura 20. 43 T able 5.2 Exp erimen tal w a v e v elo cities ( mm=s ). Video data are tak en using high denition camera (JV C GYHD 100). 43 T able 5.3 Exp erimen tal w a v e inclinations (in radian). Video data are tak en using camcorder Optura 20. 45 T able 5.4 Exp erimen tal w a v e inclinations (in radian). Video data are tak en using high denition camera (JV C GYHD 100). 46 T able 5.5 Exp erimen tal distances b et w een consecutiv e w a v es ( mm ). Video data are tak en using camcorder Optura 20. 46 T able 5.6 Exp erimen tal distances b et w een consecutiv e w a v es ( mm ). Video data are tak en using high denition camera (JV C GYHD 100). 47 T able 5.7 Radii of the rst and second w a v es R 1 and R 2 resp ectiv ely 47 T able 5.8 Input data and mo del co ecien ts. 48 T able 5.9 Exp erimen tal a v erage radial v elo cities and predicted radial v elo cities ( mm=s ). 49 T able 5.10 Calculated a v erage w a v e inclinations and predicted w a v e inclinations (in radian). 50 T able 5.11 Calculated a v erage distances o v er fteen videos and theoretically calculated distances ( mm ). 51 T able 6.1 The absolute errors of ruid ro w parameters 58 T able 6.2 The absolute errors of ruid ro w parameters with the sp eed of 300 r pm and the ruidro w rate of 0 : 2 l pm 62 iii
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T able 6.3 The relativ e errors of ruid ro w parameters for the v arious rotation disk sp eeds at the radius of 40 mm 62 T able 6.4 The relativ e errors of ruid ro w parameters for the v arious rotation disk sp eeds at the radius of 60 mm 63 T able 6.5 The relativ e errors of reco v ered disk sp eeds (RDS) at the com bined minimal relativ e errors (CMRE) for video segmen ts 1 through 8. 64 T able 6.6 The relativ e errors of reco v ered disk sp eeds (RDS) at the com bined minimal relativ e errors (CMRE) for video segmen ts 9 through 15. 64 iv
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LIST OF FIGURES Figure 1.1 Dep endence of the heat transfer co ecien t on the radius and the sp eed of the disk [63 ]. 2 Figure 1.2 Dissertation organization ro w c hart. 11 Figure 2.1 Denition of and 18 Figure 3.1 (a) Blo c ksc heme of algorithms; (b) Sc hematic view. 25 Figure 3.2 The optical paths of rerected b eams. 26 Figure 3.3 The inciden t angle and the emittance angle 27 Figure 3.4 The graph z = sin( 2 x ) A 2 x 2 and z = A ( x ) 2 2 x 27 Figure 3.5 A camera and ligh t are in the same lo cation, x c z c 28 Figure 3.6 (a) Detected p oin ts of w a v es. (b) A detected w a v e. 32 Figure 3.7 Sample of images with detected p oin ts on w a v es. 36 Figure 4.1 Exp erimen tal setup. 37 Figure 4.2 Calibration pattern. 38 Figure 4.3 Sc hematic setup. 39 Figure 4.4 Rotating disk closeup. 39 Figure 5.1 The regimes of the disk rotation at (a) 300 r pm and (b) 500 r pm 42 Figure 5.2 (a) Phase v elo cities. (b) Amplication factors. 48 Figure 5.3 Comparison of the exp erimen tal estimation of w a v e v elo cities and the theoretical v elo cities of ruid ro w. 49 Figure 5.4 Comparison of the exp erimen tal and theoretical w a v e inclinations. 50 Figure 5.5 Comparison of the exp erimen tal and theoretical distances b et w een consecutiv e w a v es. 51 v
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Figure 6.1 A blo c ksc heme of determining the disk sp eed and ruidro w rate. 56 Figure 6.2 (a) The relation b et w een absolute errors of inclination angles, rotating disk sp eeds, and ruidro w rates; (b) the relation b et w een absolute errors of distances, rotating disk sp eeds, and ruidro w rates. 59 Figure 6.3 The relation b et w een absolute errors of (a) inclination angles and sp eeds of rotating disk; (b) distances and sp eeds of rotating disk. 60 Figure 6.4 The relation b et w een absolute errors of (a) inclination angles and ruid ro w rates; (b) distances and ruid ro w rates. 61 Figure 6.5 Relations of ruidro w parameter relativ e errors and rotation sp eeds of disk at the radius of 40 mm 63 Figure 6.6 Relations of ruidro w parameter relativ e errors and rotation sp eeds of disk at the radius of 60 mm 64 Figure C.1 (a) LogLik eliho o d; (b) Residuals vs Fitted v alues; (c) Standardized Residuals; (d) Square ro ot of Standardized Residuals vs Fitted v alues. 80 Figure C.2 Standardized Residuals vs Lev erage. 81 Figure C.3 (a) LogLik eliho o d; (b) Residuals vs Fitted v alues; (c) Standardized Residuals; (d) Square Ro ot of Standardized Residuals vs Fitted v alues. 82 Figure C.4 Standardized Residuals vs Lev erage. 83 vi
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MOTION ANAL YSIS OF FLUID FLO W IN A SPINNING DISK REA CTOR V alen tina N. Korzho v a ABSTRA CT The ro w of a liquid lm o v er a rapidly rotating horizon tal disk has n umerous industrial applications including pharmaceuticals, c hemical engineering, bio engineering, etc. The analysis and con trol of complex ruid ro ws o v er a rapidly rotating horizon tal disk is an imp ortan t issue in the exp erimen tal ruid mec hanics. The spinning disk reactor exploits the b enets of cen trifugal force, whic h pro duces thin highly sheared lms due to radial acceleration. The h ydro dynamics of the lm results in excellen t ruid mixing and high heat or mass transfer rates. This w ork fo cuses on dev eloping a no v el approac h for ruid ro w trac king and analysis. Sp ecically the dev elop ed algorithm is able to detect the mo ving w a v es and compute con trolling lm ro w parameters for the ruid ro wing o v er a rotating disk. The input to this algorithm is an easily acquired nonin v asiv e video data. It is sho wn that under single ligh t illumination it is p ossible to trac k sp ecular p ortion of the rerected ligh t on the mo ving w a v e. Hence, the ruid w a v e motion can b e trac k ed and ruid ro w parameters can b e computed. The ruid ro w parameters include w a v e v elo cities, w a v e inclination angles, and distances b et w een consecutiv e w a v es. Once the parameters are computed, their accuracy is analyzed and compared with the solutions of the mathematical ruid dynamics mo dels based on the Na vierStok es equations for the case of a thin lm. The ruid mo del predicts w a v e c haracteristics based on directly measured con trolling parameters, suc h as disk rotation sp eed and ruid ro w rate. It is sho wn that vii
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the calculated parameter v alues appro ximately coincide with the predicted ones. The a v erage computed parameters w ere within 5 10% of the predicted v alues. In addition, giv en reco v ered ruid c haracteristics and ruid ro w con trolling parameters, full 3 D w a v e description is obtained. That includes 3 D w a v e lo cation, sp eed, and distance b et w een w a v es, as w ell as appro ximate w a v e thic kness. Next, the dev elop ed approac h is generalized to mo delbased reco v ery of ruid ro w con trolling parameters: the sp eed of the spinning disk and the initial ruidro w rate. The searc h in space for mo del parameters is p erformed as to minimize the error b et w een the ro w c haracteristics predicted b y the ruid dynamics mo del (e.g. distance b et w een w a v es, w a v e inclination angles) and parameters reco v ered from video data. Results demonstrate that the sp eed of a disk and the ro w rate are reco v ered with high accuracy When compared to the ground truth a v ailable from direct observ ation, w e noted that the con trolling parameters w ere estimated with less than 10% error. viii
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CHAPTER 1 INTR ODUCTION 1.1 Ov erview of the Relativ e W orks The analysis of ruid ro w is an imp ortan t issue in the exp erimen tal ruid mec hanics, including calculating forces and momen ts on aircraft, determining the mass ro w rate of p etroleum through pip elines, and understanding nebulae in in terstellar space. Understanding atmospheric dynamics, o ceanographic streams, and cloud motion is of great imp ortance for w eather, climate prediction, etc. Rotating ro ws ha v e b een used to study a v ariet y of ph ysical pro cesses including geostrophic turbulence, baro clinic instabilit y con v ection, and c haos. The ro w of a liquid lm o v er a rapidly rotating horizon tal disk has n umerous industrial applications (pharmaceutical, c hemical engineering, bio engineering, etc.), ranging from spincoating of silicon w afers to the atomization of liquids. One of the most imp ortan t applications in presen t time is the transfer gases, for instance of carb on dio xide, in to liquids. Under certain conditions, this ro w is accompanied b y the formation of nonlinear w a v es leading to remark ably large increases in the rates of heat and mass transfer. Therefore, the analysis and con trol of complex ruid ro ws o v er a rapidly rotating horizon tal disk is a ma jor scien tic issue. Here, w e prop ose a no v el videobased tec hnique for extraction of ruid ro w c haracteristics and estimation of their accuracy The study [2, 39 63] has in v estigated the heat and mass transfer c haracteristics of the lm considered as a steady state, giving rise to lo cal heat and mass transfer co ecien ts as a function of radius, ro w rate, and rotation sp eed. It has b een sho wn that the presence of surface w a v es leads to a signican t enhancemen t in transfer pro cesses 1
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at the lm surface, and as suc h are desirable features of the ro w. Dep endence of the heat transfer co ecien t on the radius of the disk and the sp eed of the disk is demonstrated in Figure 1.1. T o in tensify these pro cesses, con trol mec hanism needs Figure 1.1 Dep endence of the heat transfer co ecien t on the radius and the sp eed of the disk [63 ]. to adjust h ydro dynamic parameters of ro w with relev an t transfer of pro cesses. So, the con trolled lm ro w is requested. Since v arious regimes of lm ro w in a spinning disk reactor strongly inruence these pro cesses, it is imp ortan t to con trol the formation of the regimes to increase pro cess pro ductivit y [33 ]. Video observ ation can pro vide a costeectiv e w a y to observ e the lm ro w and to determine the actual ro w parameters. Recen t review [33 ] summarizes exp erimen tal and theoretical studies of lm ro w o v er a rotating disk. Most exp erimen tal in v estigations of ro w o v er a spinning disk attempt to measure the lo cal maxim um or the mean of a lm thic kness in order to obtain information ab out the surface w a v es [2 ]. V arious mec hanical [6 ], electrical [3 38], and optical [30 63 ] tec hniques w ere emplo y ed. The most promising w as the optical tec hnique that w as used to collect information ab out the w a v es observ ed. In the considered exp erimen ts, a camera w as placed b elo w the disk and connected to a computer that pro vided video imaging hardw are and soft w are. T o measure the lm thic kness o v er a disk domain, calibration of mec hanical or optical to ols and estimation of absorption 2
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co ecien ts w ere p erformed. All three tec hniques (mec hanical, electrical, and optical) ga v e insucien t information to classify w a v e regimes and select most ecien t regime for sp ecic tec hnological applications. Early exp erimen tal in v estigations [6 ] pro vided some qualitativ e and quan titativ e understanding of the eect of ro w rate and rotational sp eed on the ro w c haracteristics for a giv en set of ph ysical parameters. Exp erimen tal observ ations [2 3 30 55 63] ha v e demonstrated that at a small ro wrate, a smo oth lm is formed, and at a mo derately higher ro wrate, circumferen tial w a v es mo ving from the disk cen ter to the disk p eriphery are formed. F urther increase in ro w rate leads to the app earance of spiral w a v es un winding in the direction of the rotation [6]. It is sho wn in [63 ] that the initially uniform lm breaks do wn in to w elldened spiral w a v es, whic h then break do wn further in to a confused assem bly of w a v elets. Circumferen tial and spiral w a v es w ere found to deca y at large disk radii. Comparison of those observ ed w a v es and their asso ciated parameters with the w a v es observ ed in falling lms sho ws their similarit y The w a v es in the falling lms w ere rst studied exp erimen tally in the seminal w ork [22 ]. The monograph [4 ] had collected most imp ortan t results concerning falling lm. In [31 ] the authors presen t measuremen ts of the instabilities of thin liquid lm ro wing do wn an incline. The results are in go o d agreemen t with linear stabilit y prediction. The gro wth rates and w a v e v elo cities ha v e b een measured as a function of w a v en um b er. They also estimated the lm thic kness in real time with high accuracy using ruorescence imaging. Theoretical explanation of exp erimen tal results has receiv ed increasing atten tion in recen tly published researc h. There are three main directions for theoretical in v estigation of a lm ro w o v er a rotating disk: calculation of w a v eless ro w, analysis of its linear stabilit y and nonlinear sim ulations of niteamplitude w a v es. The w a v eless solutions, asymptotic and n umerical, w ere in v estigated in [13 40, 52 ]. These studies ha v e b een successful in determining selfsimilar and asymptotic solutions in the limit of large Ec kman n um b ers [40 45 ] as w ell as n umerical solutions for nite Ec kman n um b ers [13 52]. 3
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The linear stabilit y analysis w as examined and p erformed using asymptotic metho ds [6 ] and the full Na vier Stok es equation for nite Ec kman n um b ers [50 51 ]. In recen t pap ers [32 47, 48 49 ] an ev olution system of equations w as deriv ed and analyzed to mo del axisymmetric niteamplitude w a v es; this mo del ma y b e extended for nonaxisymmetric ro ws to explain the exp erimen tal results. Nev ertheless, there are problems that could b e treated b y parallel application of exp erimen tal and theoretical approac hes: w a v e regime sensitivit y to ro w conditions and threedimensional structures observ ed in exp erimen ts. In the last decade, there has b een signican t w ork in image pro cessing related to the motion analysis of nonrigid ob jects [8 11 10 21 36 69 ]. Most of the w orks ha v e concen trated on articulated and elastic motion [1]. In [21 ], the authors demonstrate that the Finite Elemen t Mo deling allo ws to realize the motion analysis of biological ob jects. The sim ulation of deformable ob jects is essen tial for man y applications. Computer Aided Design uses deformable mo dels to sim ulate the deformation of industrial materials and tissues. In image analysis, deformable mo dels w ere used for tting curv ed surfaces, b oundary smo othing, registration, and image segmen tation. Later, deformable mo dels w ere deplo y ed in c haracter animation and computer graphics for the realistic sim ulation of skin, facial m uscles, clothing, and h uman or animal c haracters. The mo deling of deformable soft tissue is of great in terest for a wide range of medical imaging applications. A comprehensiv e review of deformable mo dels for medical image analysis and clothing mo deling applications can b e found in [34 46 ] and [19 ], resp ectiv ely More recen tly suc h mo deling tec hniques ha v e b een used for tasks suc h as age estimation [29 ] and p erson iden tication [65 ]. Eorts ha v e b een made to assist face animation and recognition using a highly accurate mo del that tak es in to accoun t anatomical details of a face suc h as b ones, m usculature, and skin tissues [66 ]. This is based on the premise that the n uances recognizable b y h umans can b e syn thesized (and fully explained) only b y an elab orate 4
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biomec hanical mo del. The early studies in this direction dev elop ed mo dels with a hierarc hical biomec hanical structure that w ere capable of sim ulating linear and sphincter facial mo v emen ts [37]. F or example, Zhang et al. [36, 38] studied a mo del that incorp orates a more detailed 3la y er skin mo dule to c haracterize the b eha vior and in teraction among the epidermis, dermis, h yp o dermis, and m uscular units. Analysis of ruidlik e motion w as also attempted [11 36 69 ]. Recen tly w ork w as b egun in an eort to com bine precise exp erimen tal setup, theoretical deriv ation, and basic image analysis tec hniques [56 55 57, 63 ]. The equations constitute a set of ph ysical constrain ts that are dieren t [21 ] from those commonly used in the study of solid motion (e.g., the rigidit y constrain t). F or the fast ruidlik e motion in the air, ha ving w a vy or turbulen t c haracter, detecting in terface b et w een ruid and air is imp ortan t. A sp ecial socalled particle image v elo cimetry (PIV) tec hnique w as dev elop ed [44 ] to measure the kinematics of turbulen t ruid ro w in con trolled lab oratory exp erimen ts. Giv en a t ypical ensem ble of PIV images, the aim is to calculate the instan taneous in terface. This, ho w ev er, requires sp ecialized imaging devices. A no v el approac h dedicated to measuring v elo cit y in ruid exp erimen tal ro ws through image sequences w as dev elop ed in [10 ]. The prop osed tec hnique is the extension of opticalro w sc hemes that includes a sp ecic enhancemen t for ruid mec hanics applications. In pap er [8 ], the authors dened a complete framew ork for pro cessing large image sequences for a global monitoring of short range o ceanographic and atmospheric pro cesses. They used a no v el regularization tec hnique in optical ro w computation that preserv es ro w discon tin uities. The study [16 ] conrms that the optical ro wbased dense v ector elds sho w motion basically consisten t with traditional atmospheric motion v ector calculation metho ds. The motion information demonstrates that some areas are not co v ered b y 'traditional' v ectors. Dierences are observ ed in areas of strong winds (suc h as jetstreams) where the opticalro w metho d tends to underestimate the strength of winds, including instan taneous v elo cit y on the surface of ruid and air con tact. Using 5
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the no v el regularization tec hniques, the strength of winds is estimated ecien tly and with a reasonable degree of accuracy Algorithms used are t ypically based on lterlik e motion. In the pap er [5] a no v el approac h for estimating ruid motion elds is presen ted. First, a lo cal ro w probabilit y distribution function at eac h pixel w as estimated using the ST AR mo del and the data from a spatiotemp oral neigh b orho o d. Then, the set of distribution functions w as fed in to a global optimization framew ork. Exp erimen ts with real ruid sequences sho w that this metho d can successfully estimate their motion elds. Analysis of ruidlik e motion w as also attempted [11 36 69 ]. In [69 ], authors prop osed a new metho d for reco v ering nonrigid motion and structure of clouds under ane constrain ts using timev arying cloud images obtained from meteorological satellites. This problem is c hallenging not only due to the corresp ondence problem but also due to the lac k of depth cues in the 2D cloud images (scaled orthographic pro jection). In this pap er, ane motion w as c hosen as a suitable mo del for small lo cal cloud motion. In [11 ], authors addressed the problem of estimating and analyzing the motion of ruids in image sequences. They in v estigated a dedicated minimizationbased motion estimator and demonstrated the p erformance of the resulting ruid ro w estimator on meteorological satellite images. The pap er [36 ] presen ts a ph ysicsbased metho d to compute the optical ro w of a ruid. Authors prop osed a metho d in whic h ph ysical equations describing the ob ject are used as supplemen tary constrain ts. The ph ysical mo del emplo y ed is a com bination of the con tin uit y equation and the Na vierStok es' equations. The authors demonstrated the eectiv eness of the prop osed metho d b y presen ting exp erimen tal results of sim ulated and real Karman ro ws. In the pap er [41 ], a ruid ro w estimation metho d for o cean/riv er w a v es, clouds, and smok e based on the ph ysical prop erties of w a v es suc h as the v elo cit yfrequency relationship and a w a v e statistical prop ert y w as dev elop ed assuming that man y ruidlik e motion c hanges are due to w a v e phenomena that lead to a brigh tness c hange. The author sho ws that the results of the exp erimen ts with syn thetic and real images are 6
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impro v ed compared to the w orks [11 35 62]. In the pap er [37 ], authors in v estigate estimation of v elo cit y w ater elev ation and con taminan t concen tration in a riv er curren t using the Kalman lter nite elemen t metho d (KEFEM). Close agreemen t b et w een the observ ed and the computed results w as obtained. The describ ed w ork in this dissertation fo cuses on dev eloping imaging tec hniques and image analysis algorithms to detect the tra v eling w a v es, determine the w a v e regimes, and compute ph ysical and con trolling lm ro w parameters. The input to these algorithms is an easily acquired nonin v asiv e video data. The pro duction of thin lms o v er a spinning disk and the formation of w a v es in realistic conditions are of in terest here. In this w ork, a t w o part algorithm is prop osed. The rst part includes image analysis, detecting, trac king, and reconstructing of measuring w a v e shap e and w a v e propagating sp eed. Based on the image in tensities and geometrical constrain ts of disk and surface w a v es, the algorithm is prop osed. The ruid ro w parameters and c haracteristics are calculated and compared with the solutions of the relev an t mathematical mo dels. Calculations of radial v elo cit y and inclination angle are obtained with the asymptotically optimal steps. New results for step selection are deriv ed. Initial v ersion and further v ersion of algorithms and analysis of spiral w a v es in a spinning disk reactor are presen ted in [25 26 27, 28 ]. The second part of the algorithm concen trates on mo delbased reco v ery of ruid ro w con trolling parameters. Here the searc h in the space of mo del parameters is p erformed to minimize the dierence b et w een the predicted ro w c haracteristics (e.g. distances b et w een w a v es, w a v e inclination angles) and the ones measured from video data. The o v erall purp ose of this w ork is to dev elop a system of visual scanning, recording, and trac king of the lm ro w o v er a spinning disk with the in ten tion of detecting regimes of the ruid ro w with regard to dieren t conditions using a single camera system, of calculating ruid ro w parameters and c haracteristics, and of comparing them with the solutions of the relev an t mathematical mo dels. 7
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1.2 Con tributions of This Dissertation This dissertation has the follo wing con tributions: 1. A no v el approac h and analysis w ere dev elop ed for ruid w a v e detecting and trac king o v er a spinning disk. The no v elt y consists of w a v e studying in real conditions, with regard to certain disk surface frictions and air resistances. 2. It w as sho wn that under single ligh t illumination it is p ossible to trac k sp ecular p ortion of the rerected ligh t on the mo ving w a v e. 3. A no v el approac h for computing ruid ro w parameters (w a v e v elo cit y comp onen ts, w a v e inclinations, thic kness of lm, and distances b et w een consecutiv e w a v es) from observ ed w a v e patterns w as dev elop ed. The no v elt y consists of dev eloping new mo del and video based algorithms and their accuracy analysis. 4. F or practical realization, the optimal metho ds (asymptotically optimal and quasioptimal) are used for estimating the v elo cities and the inclination angles. 5. An arbitrary step along the azhim utal angle is used for the exp erimen tal estimation of w a v e inclinations. 6. The dev elop ed approac h is generalized to a mo delbased reco v ery of ruid ro w con trolling parameters: the sp eed of spinning disk and the ruidro w rate. The searc h in space for mo del parameters is p erformed to minimize the error b et w een the predicted ro w c haracteristics predicted b y the ruid dynamics mo del (e.g. distance b et w een w a v es, w a v e inclination angles) and the parameters reco v ered from video data. P ortions of the w ork in this dissertation ha v e b een presen ted on the Seminar of Image Pro cessing, June 2005 and Graduate Studen t Researc h Comp etition, No v em b er 2005 at USF, Departmen t of Computer Science and Engineering. Also, parts are published 8
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in the ICPR2006, Ukrobraz 06 Conferences, 59th Ann ual Meeting of the Division of ruid Dynamics, and Optical 3D Measuremen t T ec hniques Conference 2009 [25 26 27 28 ]. The detailed pap er is submitted in the journal "In ternational Journal P attern Recognition and Articial In telligence" [24 ]. 1.3 La y out of the Dissertation The organization of this pap er is depicted in Figure 1.2. Chapter 2.1 oers description of the general theory of ruid ro w based on the equations of Na vierStok es, including the imp ortan t particular case of the thin lm ro w, whic h are later compared with the resp ectiv e exp erimen tal data and uses for the prop er estimations of w a v e parameters suc h as w a v e v elo cit y comp onen ts, w a v e inclinations, and distances b et w een consecutiv e w a v es. Also, this c hapter presen ts the deriv ations of the camera calibration accuracy whic h is used in Section 5 under estimation of errors of exp erimen tal v elo cities of w a v es and inclination angles. In Chapter 3, a video based algorithm for detecting and trac king of w a v es in a spinning disk reactor is presen ted. The input to this algorithm is an easily acquired nonin v asiv e video data. Details of the video input information are analyzed. It is sho wn that under a single ligh t illumination assumption it is p ossible to trac k w a v e motion b y observing sp ecular p ortion of the rerected ligh t on the mo ving w a v e. Hence, the ruid w a v e motion can b e trac k ed and ruid ro w parameters can b e computed. The ruid ro w parameters include w a v e v elo cities, w a v e inclination angles, and distances b et w een consecutiv e w a v es. Determination of v elo cit y comp onen ts and inclination angles is an illp osed problem. So, the visionbased asymptotically optimal b y accuracy metho d w as applied. Chapter 4 is dev oted to data acquisition for the exp erimen tal disk reactor used in this study 9
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Chapter 5 presen ts exp erimen tal results for estimating parameters of w a v e spirals suc h as their v elo cit y comp onen ts, inclination angles, and distances b et w een consecutiv e w a v es and their estimated accuracy Those results are compared with the related results of theoretical mo dels in tro duced in Section 2.1. It w as sho wn that no statistically signican t dierence exists b et w een theoretical and calculated v alues of w a v e parameters at the signican t lev el = 0 : 05. V ariations of the rotational sp eed and the ro w rate lead to the mo dication of the shap e, amplitude, and v elo cit y of the observ ed w a v es. So, the computerized system for estimation of sp eed of a spinning disk and ro w rate using the video data and the relev an t system of ev aluation equations are dev elop ed and describ ed in Chapter 6. Conclusion c hapter follo ws. It should b e noted that the presen ted w ork fo cuses on computerized pro cessing of the input video data in order to extract the ruid ro w c haracteristics and their analysis, including estimation of errors of n umerical results. Qualitativ e and quan titativ e comparison sho ws go o d coincidence (within 510%) of exp erimen tal and theoretical results. Also, sp ecics of video input information are analyzed. It is sho wn that under a single ligh t illumination assumption it is p ossible to trac k w a v e motion b y observing sp ecular p ortion of the rerected ligh t on the mo ving w a v e. 10
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Figure 1.2 Dissertation organization ro w c hart. 11
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CHAPTER 2 THEORETICAL BA CK GR OUND 2.1 Theory of Fluid Flo w 2.1.1 General Case Considering an in viscid ruid, describ ed b y the follo wing v ariables: the ruid densit y ( X ; t ) ; the v elo cit y v ector eld u ( X ; t ), and the pressure p ( X ; t ); X 2 R 3 d is the spatial co ordinate. In a Cartesian system co ordinate, an incompressible viscous ruid can b e describ ed with the Na vierStok es equations that are four coupled nonlinear partial dieren tial equations for four unkno wn functions (the three comp onen ts of u and the pressure p ): @ u 1 @ t + u 1 @ u 1 @ x + u 2 @ u 1 @ y + u 3 @ u 1 @ z + 1 @ p @ x = @ u 1 @ x ; @ u 1 @ y ; @ u 1 @ z ; @ u 2 @ t + u 1 @ u 2 @ x + u 2 @ u 2 @ y + u 3 @ u 2 @ z + 1 @ p @ y = @ u 2 @ y ; @ u 2 @ y ; @ u 2 @ z ; @ u 3 @ t + u 1 @ u 3 @ x + u 2 @ u 3 @ y + u 3 @ u 3 @ z + 1 @ p @ z = @ u 3 @ x ; @ u 3 @ y ; @ u 3 @ z ; @ u 1 @ x + @ u 2 @ y + @ u 3 @ z = 0 ; where is the kinematics' viscosit y Also, there is the matter of b oundary conditions [12 ]. Using the cylindrical system co ordinates ( r ; ; z ), the steady Na vierStok es equation for a radial v elo cit y u is: u @ u @ r + @ u @ z v 2 r = 1 @ p @ r + ( @ 2 u @ z 2 + @ 2 u @ r 2 + 1 r @ u @ r u 2 r ) ; (2.1) 12
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where u; v ; w are the comp onen ts of v elo cit y in the r ; ; z direction. The authors [40 ] sho w that for a thin lm, h=r 1, where h is the lm thic kness, the radial ro w can b e obtained from (2.1) to lo w est order with resp ect to h=r b y solving the equations n 2 r = @ 2 u @ z 2 ; where n is the angular v elo cit y of the disk. Later, the last t w o form ulas ab o v e are used to estimate the w a v e amplitude in the case of thin lm ro w. 2.1.2 Fluid Flo w o v er Rotating Disk The Karman's Problem Considering an innite plane disk, rotating at constan t angular v elo cit y n with an un b ounded ruid and taking cylindrical p olar co ordinates ( r ; ; z ), the steady axisymmetric solution for v elo cit y comp onen ts ( u; v ; w ) and pressure ( p ) can b e found in the form [64 ]: u = n r f ( ) ; v = n r g ( ) ; w = ( n) 1 = 2 h ( ) ; p = n P ( ) ; = z n 1 = 2 (2.2) under the follo wing b oundary conditions: u = 0 ; v = n r ; z = 0; u; v 0 ; z 1 : (2.3) Substituting (2.2) and (2.3) in the Na vierStok es equations written in the cylindrical p olar co ordinates, the functions f ( z ), g ( z ), h ( z ), and P ( z ) can b e found [64 ] as solution of a set of ordinary dieren tial equations: 13
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2 F + dH d = 0 ; F 2 G 2 + H dF d = d 2 F d 2 ; 2 F G + H dG d = d 2 G d 2 ; H dH d = dQ d + dH d ; (2.4) with b oundary conditions F = H = 0, G = 1, when = 0, F 0, G 0, as 1 F or small v alues of a solution of (2.4) can b e written in p o w ers of F or large v alues of a solution can b e written in the exp onen tial form (see [64 ]). 2.2 Mathematical Mo deling This section describ es the mathematical mo dels of the lm ro w o v er a disk rotating with angular v elo cit y n. 2.2.1 Ev olution Equations A mo del deriv ation giv en b elo w follo ws to [47 ] with accoun ting nonaxisymmetric terms. The authors [48] consider the ro w of a thin, Newtonian, incompressible liquid lm of the densit y kinematic viscosit y surface tension and the ro w is describ ed b y the v elo cit y comp onen ts u v w and the pressure p dep ending on the the cylindrical co ordinates r and z and time t The liquid lm is b ounded from ab o v e b y an essen tially in viscid gas; the gas liquid in terface is lo cated at z = h while the underlying solid disc is situated at z = 0. The full Na vierStok es system accompanied b y the b oundary conditions (noslip and no{p enetration at the disc surface, the kinematic b oundary condition, shear and normal stress balances at the lm surface) on the disk and the free surface is form ulated [47 ]. Analysis of exp erimen ts, carried out in [47 ], rev ealed that a relation 2 = 2 1 is satised in all data a v ailable when capillary w a v es 14
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are observ ed, where and are determined b y = H c n 2 R 4 c 1 3 ; = H c ~ r ; (2.5) where H c and R c are scales of a thic kness and a radius. After omitting terms of O ( 2 = 2 ) in the problem statemen t, the pressure ma y b e eliminated and the appro ximate mo del follo ws in the form: @ u @ x + @ w @ z + 2 u + @ v @ # = 0 ; @ u @ t + u @ u @ x + w @ u @ z + v @ u @ # + u 2 ( v + E) 2 = 1 45 e 2 x @ @ x e 2 x @ 2 h @ x 2 + 2 @ 2 h @ # 2 + @ 2 u @ z 2 ; @ v @ t + u @ v @ x + w @ v @ z + v @ v @ # + 2 u ( v + E) = 1 45 e 4 x @ 3 h @ x 2 @ # + 2 @ 3 h @ # 3 + @ 2 v @ z 2 ; z = 0 : u = 0 ; v = 0 ; w = 0 ; z = h ( x ; #; t ) : @ h @ t + u @ h @ x + v @ h @ # = w ; @ u @ z = 0 ; @ v @ z = 0 ; (2.6) where # = E t is the azim uthal angle related to the spinning disc; and the similarit y parameter and the Ec kman n um b er E = 45 E 2 1 = 1 45 2 n 8 R 4 c H 11 c 1 3 ; (2.7) E = = n H 2 c (2.8) ha v e b een in tro duced. Here x = x ; t = t ; w = w ; (2.9) 15
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These equations are rendered dimensionless via the follo wing scaling: t E t n ; r R c e x ; z H c z ; u r n r u E ; u n r 1 + v E ; u z n H c w E ; p n 2 r 2 p; h H c h; (2.10) where H c is c hosen so that the dimensionless radial ro w rate is equal to unit y for a giv en v alue of R c under steady conditions: H c = Q c 2 n 2 R 2 c 1 3 : (2.11) The observ ed w a v es ha v e a c haracteristic length scale, whic h is m uc h smaller than R c In exp erimen ts is a small co ecien t. Th us, the problem (2.6) ma y b e considered as dep ending on t w o parameters: the lm parameter that also app ears in a falling lm problem [4 ] and the Ec kman n um b er E; then = 45 E 2 1 Using the parab olic v elo cit y prole, an appro ximate system of ev olution equations for the lm thic kness h and t w o v alues, q ( u ) and q ( v ) c haracterizing ro w rates in the radial and azim uthal directions, is deriv ed: @ h @ t + @ q ( u ) @ x + 2 q ( u ) + @ q ( v ) @ = 0 ; q ( u ) = Z h 0 u r dz ; q ( v ) = Z h 0 u dz ; @ q ( u ) @ t + a 11 @ @ x q ( u ) 2 h + a 12 @ @ q ( u ) q ( v ) h + a 13 q ( u ) 2 h + a 14 q ( v ) 2 h = E 2 e 2 x h @ @ x 2 e 2 x @ 2 h @ x 2 + @ 2 h @ 2 b 1 q ( u ) h 2 + h + 2 E q ( v ) ; @ q ( v ) @ t + a 21 @ @ x q ( u ) q ( v ) h + a 22 @ @ # r q ( v ) 2 h + a 23 q ( u ) q ( v ) h = E 2 3 he 4 x @ 3 h @ x 2 @ + @ 3 h @ 3 b 2 q ( v ) h 2 2 E q ( u ) : (2.12) 16
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The system (2.12) includes the constan t co ecien ts a 11 = 6 5 ; a 12 = 17 14 ; a 13 = 18 5 ; a 14 = 155 126 ; b 1 = 3 ; a 21 = 17 14 ; a 22 = 155 126 ; a 23 = 34 7 ; b 2 = 5 2 : The axisymmetric v ersion of (2.12) w as deriv ed in [47 ]. System (2.12) has a steady axisymmetric solution of spiralt yp e slo wly v arying along the radius; stabilit y of this solution is in v estigated to small p erturbations ^ h; ^ q ( u ) ; ^ q ( v ) = h; q ( u ) ; q ( v ) exp i ( x + n t ) ; where and n are giv en real w a v e n um b ers, and is an unkno wn complex frequency determining stabilit y or instabilit y of the basic ro w. F urther, distances and v elo cities of most unstable p erturbations whic h p osses largest amplication factors are obtained. These p erturbations are compared with the exp erimen tal data. The corresp onding instan t lo cal inclination of the linear w a v e spiral is: tan = 1 r dr d = n ; (2.13) where is the angle b et w een direction of the spiral and tangen t, and n are w a v e n um b ers along radial and azim uthal directions. These n um b ers are compared with exp erimen tal measuremen ts. The denitions of and are sho wn in Figure 2.1. 17
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Figure 2.1 Denition of and 2.2.2 The Exp erimen tal Mo del Let ( x; y ) b e the Cartesian system co ordinates in the plane of an observ ed disk with the origin at the cen ter of that disk; and let ( r ; ) b e the resp ectiv e p olar system co ordinates. Using the theory [43 ], the follo wing spiral equations w ere utilized with regard to the ruid friction and air resistance [42 58 60]: a r = r 00 ( t ) = n 2 r 8 c f r 0 ( t ) c r es 2 sin r 0 2 ( t ) ; a s = v 0 s ( t ) = 8 f (n r v s ( t )) c 2 cos v 2 s ( t ) ; tan = r 0 ( t ) v s ( t ) = ( y 0 x + y x )(1 + y 0 x y x ) = 1 r dr d ; 0 ( t ) = r 0 ( t ) r ( t ) 1 sin ; dr d = r ( t ) r 0 ( t ) v s ( t ) ; d 2 r d 2 = r 0 2 + r ( t ) r 00 ( t ) v s ( t ) v 0 s ( t ) r 0 ( t ) r ( t ) v 2 s ( t ) ) 1 0 ( t ) ; x ( t ) = r ( t ) cos ; y ( t ) = r ( t ) sin ; y 0 x = r 0 sin + r cos 0 r 0 cos + r sin 0 ; v ( t ) = ( r 0 2 ( t ) + v 2 s ( t )) 1 2 r (0) = 0 ; r 0 (0) = r 0 0 ; r ( T ) = 100 ; v s (0) = 0 ; (2.14) 18
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where a r and a s are accelerations of ruid resp ectiv ely along radius and p erp endicular to radius, c f is a co ecien t of ruid friction, and c r es is resistance of air to ruid. In the case of w ater w e ha v e 8 c f = 0 : 4( mm 1 ) ; c r ef = 2( mm 1 ) ; The form ula (2.14) is not v alid in the vicinit y of r (0) = 0. Therefore, w e need to use another mo del for ( t ) on the segmen t (0 ; b ) for small b Let ( t ) = a t on this segmen t. Then 0 (0) = 0 ; a = 0 ( b ) = r 0 ( b ) r ( b ) tan ( b ) : It is clear that is the angle b et w een direction of the spiral and tangen t of the circle. T o nd ( t ), the angle b et w een the radius at the momen t t and the axes x the righ t triangle AB C where legs AB = dr = r 0 ( t ) dt B C = r d = r 0 ( t ) (angle is opp osite to r d ) (see Figure 2.1) is considered. Note that the second and the third form ulas for tan ab o v e can b e used b y image algorithms, since they do not dep end on the time but only on geometric prop ert y of the resp ectiv e solution. The appro ximate solution for this nonlinear system is found using Euler's n umerical metho d with decreasing steps of computation un til appro ximate solutions are stabilized. In our exp erimen ts, in the case of w ater, n is obtained from the resp ectiv e graph for the reactor used, r 0 (0) is obtained from the exp erimen t and equals: r 0 0 = 3 : 8 10 6 285 2 2 : 5 3 280( mm s ) ; 19
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where 3 : 8 10 6 mm 3 is v olume of a gallon of w ater, 285 s is the time for the w ater to run out of the resp ectiv e capacit y 2 : 5 mm is the radius of the tub e, and 3 mm is the gap b et w een the end of the tub e and the disk surface. 2.3 Camera Calibration Accuracy Analysis 2.3.1 Ov erview of Camera Calibration Camera calibration is a necessary step in 2D and 3D computer vision in order to extract metric information from video images; and it is imp ortan t for accuracy in 2D and 3D reconstruction. Muc h w ork has b een done, starting in the photogrammetric comm unit y [59 ], and more recen tly in computer vision [7 14 18, 23, 54 67 68]. Camera calibration is the pro cess of relating the ideal mo del of the camera to the actual ph ysical device and determining the p osition and orien tation of the camera with resp ect to a w orld reference system. Dep ending on the mo del used, there are dieren t parameters to b e determined. F or the pinhole camera mo del the parameters to b e calibrated are classied in to t w o groups: 1. In ternal (or in trinsic) parameters. In ternal geometric and optical c haracteristics of the lenses and the imaging device. 2. External (or extrinsic) parameters. P osition and orien tation of the camera in a w orld reference system. The relationship b et w een a p oin t ( X ; Y ) and its image pro jection ( x; y ) is giv en [68] b y 2 6 6 6 6 4 x y 1 3 7 7 7 7 5 = A [ r 1 ; r 2 ; t ] [ X ; Y ; 1] T ; A = 2 6 6 6 6 4 s r c x 0 s c y 0 0 1 3 7 7 7 7 5 (2.15) 20
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where [ r 1 ; r 2 ; t ] are the extrinsic parameters (the rotations and translation) that relate the w orld co ordinate system to the camera co ordinate system; A is the camera in trinsic matrix, in whic h ( c x ; c y ) is the principal p oin t, s = f =s x = f =s y f is the fo cal length, s x = s y is the eectiv e size of the pixel, s is the scale factor (accuracy of whic h to b e accoun ted for an y uncertain t y due to imp erfections in the viewing camera), and r is the sk ewness of the image axes. 2.3.2 Camera Calibration Accuracy The relations b et w een observ ed ( x d ; y d ) co ordinates and the ideal (distortionfree) pixel co ordinates ( x; y ) are: x d = x + k ( x c x ) ; y d = y + k ( y c y ) ; k = k 1 r 2 d + k 2 r 4 d ; r 2 d = x 2 d + y 2 d ; where k 1 and k 2 are the co ecien ts of the radial distortion. These relations allo w us to calculate x and y after whic h v alues X and Y are calculated using the follo wing form ulae (2.16). X ; Y ; 1] T = r 1 ; r 2 ; t ] 1 A 1 [ x; y ; 1] T ; x = x d + c x k 1 + k ; y = y d + c y k 1 + k : (2.16) Due to the fact that the main error of the calibration metho d [14 ] used in this w ork is determined b y v alues of distortion, the relativ e deviations for x y and X Y are estimated, assuming for simplicit y that c x = c y = 0. In this regard, the result in [68 ], page 13, is used. The relativ e deviations for the estimates of k 1 and k 2 do not exceed 21
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34%. Since in the case considered x = x d 1 + k ; y = y d 1 + k ; with regard to the main terms (assuming k is small, k 0 : 03, and k 1 = k 2 ), the absolute v alues of v ariations j x j j x d j + x d j k j j y j j y d j + y d j k j ; k = k 1 ( r 2 d + r 4 d ) ; k k = k 1 k 1 0 : 03 ; (2.17) from where, j x j x j x d j x d + k j k j k ; j y j y j y d j y d + k j k j k ; r d = r x j x j + y j y j ( x 2 + y 2 ) 1 = 2 : Using the fact that j x d j x d ; j y d j y d < 0 : 3 720 where 720 is maximal v alue of the n um b er of pixels and 0 : 3 is the upp er b ound for x d and y d it is easy to estimate that x x < 0 : 3 720 + 0 : 03 0 : 03 0 : 0013 ; y y 0 : 0013 : In the case when r = p x 2 + y 2 w e ha v e r r x x + y y 0 : 0026 : Similarly in the case when R = p X 2 + Y 2 with regard to R = 1 s ( x 2 + y 2 ) 1 2 ; s s < 0 : 001 ; 22
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w e ha v e = R R x x + y y + s s 1 s R 0 : 0026 + 0 : 000001 < 0 : 003 : (2.18) The estimates of the relativ e errors of the initial data are calculated and used for the exp erimen tal estimation of the radial v elo cit y comp onen ts and the spiral w a v e inclination angles of the lm ro w (see Subsections 3.3.3 and 3.3.4). 23
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CHAPTER 3 ESTIMA TION OF FLUID FLO W P ARAMETERS In this section videobased algorithms for detecting and trac king of w a v es in a spinning disk reactor are presen ted. Using exp erimen tal video data and the dev elop ed mo dels and algorithms, the c haracteristics of w a v e regimes are estimated. Therefore, in order to calculate the ruid ro w parameters the follo wing steps need to b e p erformed: 1. Detection of p oin ts on the p eaks of w a v es along the same radius, 2. Estimation of radial v elo cities, 3. Estimation of w a v e inclination angles, and 4. Estimation of distances b et w een the consecutiv e p eaks of w a v es or the resp ectiv e w a v e spirals. The algorithms use the p olar system co ordinates and Cartesian system co ordinates with the origin in the cen ter of the disk. The blo c ksc heme and the resp ectiv e sc hematic view are giv en in Figure 3.1. Prior to the pro cess of detecting pro jected p oin ts with maximal in tensit y on w a v es (with the xed camera p osition relativ ely to the disk), the prepro cessing of images is p erformed. This is done using lo cal con tract enhancemen t, whic h impro v es the visual app earance of the w a v efron t for h uman observ ation and normalizes the in tensit y v alues. That follo ws b y lo cal thresholding op eration whic h detects p oin ts at maximal in tensit y along the radial (in resp ect to the spinning disk) direction. In the exp erimen tal setup, videos w ere tak en from a side view b ecause of the b etter visibilit y of the w a v es. Moreo v er, in man y industrial applications the imaging m ust b e from a side view suc h as in the syn thesis of aero gels. In suc h applications the rotating disk is inside the closed cylinder with a windo w on the side of the cylinder. 24
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(a) (b) Figure 3.1 (a) Blo c ksc heme of algorithms; (b) Sc hematic view. 3.1 T rac king Sp ecular Rerection P atterns The total rerection is an additiv e of sev eral comp onen ts: diuse rerection ( L d ), sp ecular rerection ( L c ) from the fron t surface [23 ], scattering ( L h ) from the v olume of 25
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liquid of thic kness ( h ), and rerection ( L b ) from the b ottom [61 ], L = L d + L s + L h + L b : (3.1) The optical path of the rerected ligh t is illustrated in Figure 3.2. Figure 3.2 The optical paths of rerected b eams. W e sho w that at eac h time instan t ( t 0 ) there is only one p oin t on crosssection on the w a v e with the sp ecular rerection comp onen t L s whic h dominates when nonclear ruid is imaged. Consequen tly this sp ecular p oin t on the w a v e has the highest in tensit y in the image. A t this p oin t the inciden t angle (the angle b et w een the view er direction and the surface normal orien tation at the giv en p oin t) and the emittance angle (the angle b et w een the illumination direction and the surface normal orien tation at the giv en p oin t) are equal due to geometrical optics (rerection la w) [15 17] (see Figure 3.3). Since the appro ximation of a ruid w a v e has a sinelik e shap e [35 ], the pro jection of w a v e on the plane xoz p erp endicular to w a v e motion of spatial structure of ruid w a v es in a t w odimensional space is appro ximated b y the parab olic equation. Within the in terv al [ ; ] the pro jected sine function and parab olic function are similar as sho wn in Figure 3.4. Th us, the appro ximation of A sin( ( 2 x )) A with the parab ola on the in terv al [ ; ] has the form 26
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Figure 3.3 The inciden t angle and the emittance angle Figure 3.4 The graph z = sin( 2 x ) A 2 x 2 and z = A ( x ) 2 2 x A 2 2 2 A sin( ( 2 )) + A = A 2 2 2 + A (1 cos ( ( 2 )) = A 2 2 2 2 A sin 2 ( 2 ) = A 2 2 2 2 A sin 2 ( 2 ) = A 2 2 2 2 A 2 2 3 + ::: 2 = A 2 2 2 2 A ( ) 2 4 ( ) 4 48 + ( ) 6 24 2 ::: A ( ) 4 24 ; (3.2) Using parab olic appro ximation of w a v e prole, w e consider, rst, a sp ecial case when t w o co ordinates of a camera and ligh t are the same, x l = x c and z l = z c (see Figure 27
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3.5), ha ving z l >> R where R is a radius of a disk, w e deriv e the follo wing equations: Figure 3.5 A camera and ligh t are in the same lo cation, x c z c z l z 0 x l x 0 = 1 dz 0 dx 0 ; z 0 = A 2 2 x 2 0 ; dz 0 dx 0 = A! 2 x 0 ; x 3 0 + 2 A! 2 z l + 1 A 2 4 x 0 2 x l A 2 4 = 0 ; (3.3) where ( x 0 ; z 0 ) is the sp ecular p oin t at the time instan t ( t 0 ) on the w a v e. This equation has one real ro ot, since the deriv ativ e 3 x 2 0 + 2 A! 2 z l +1 A 2 4 do es not c hange the sign. The solution of (3.3)is sho wn b elo w. x 0 = (27 A! 2 x l + 3 p c 0 ) 1 = 3 3 A! 2 2( A! 2 z l + 1) A! 2 (27 A! 2 x l + 3 p c 0 ) 1 = 3 ; c 0 = 24 A 3 6 z 3 l + 72 A 2 4 z 2 l + 72 A! 2 z l + 24 + 81 A 2 4 x 2 l ; (3.4) where x 0 is the distance from the top of the w a v e to the sp ecular p oin t on the w a v e. If the w a v e is mo ving b y in the xdirection, the equation b ecomes z 1 = A 2 2 ( x 1 ) 2 ; dz 1 dx 1 = A! 2 ( x 1 ) ; ( x 1 ) 3 + 2 A! 2 z l + 1 A 2 4 ( x 1 ) 2 A 2 4 ( x l ) = 0 ; (3.5) 28
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where ( x 1 ; z 1 ) is the sp ecular p oin t at the time instan t ( t 1 ) on the mo v ed w a v e. Solving the equation (3.5), w e ha v e x 1 = (27 A! 2 x l 27 A! 2 + 3 p c 0 ) 1 = 3 3 A! 2 2( A! 2 z l + 1) A! 2 (27 A! 2 x l 27 A! 2 + 3 p c 0 ) 1 = 3 ; c 0 = c 0 + 81 A 2 4 2 162 A 2 4 : (3.6) Estimation of dierence j x 0 ( x 1 ) j appro ximately equals zero when x l ; z l x l z l and z l >> R where R is the radius of the disk. Since the distance x 1 x 0 it is reasonable to use the sp ecular p oin ts for trac king w a v es. Hence, trac king the sp ecular p oin ts allo ws us to trac k the w a v e of ruid ro wing o v er a rotating disk. In a general case, when co ordinates of a camera ( x c ; z c ) and co ordinates of a ligh t ( x l ; z l ) are not the same, w e deriv e the follo wing equations for the sp ecular p oin t p ositions: z c z 0 x c x 0 m 0 1 + z c z 0 x c x 0 m 0 = z l z 0 x l x 0 m 0 1 + z l z 0 x l x 0 m 0 ; m 0 = 1 A! 2 x 0 ; x 5 0 ( x l + x c ) x 4 0 + 2 A 2 4 x 3 0 ( 3 A 2 4 x l + 2 A! 2 x c z l + 3 A 2 4 x c + 2 A! 2 x l z c ) x 2 0 ( 4 A 2 4 z c z l + 2 z c z l A 3 6 4 A 2 4 x l x c ) x 0 + 2 x c z l + x l z c A 3 6 = 0 : (3.7) The equation (3.7) has only one solution whic h is negativ e if 2 x c z l + 2 x l z c is p ositiv e and p ositiv e if 2 x c z l + 2 x l z c is negativ e. Mo ving the w a v e b y the distances j x 0 j j x 1 j when z c z l x l z l and A! 2 = const Th us, under general camera and ligh t p ositions, the sp ecular p oin ts do not coincide with the p eaks of the w a v es. Ho w ev er, as sho wn ab o v e, the distances along the radial 29
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direction b et w een sp ecular p oin ts and the p eaks are appro ximately constan t for the mo ving w a v e. Hence, trac king the sp ecular p oin ts allo ws us to trac k the w a v e of ruid ro wing o v er a rotating disk. Moreo v er, the conditions under whic h the sp ecular p oin ts practically coincide with the p eaks of w a v es are found (see App endix A). This coincidence is more precise when the ratio, r z c is smaller, where z c is a z co ordinate of a camera and r is a radius of the rotating disk. 3.2 Algorithm for Detecting and T rac king Sp ecular P oin ts on W a v es In the pro cess of detecting p oin ts along the radiusv ector the maxim ums of in tensities are found corresp onding to the sp ecular p oin ts of the w a v es. W e consider the spiral b elo w as p erio dic functions due to their stationary prop ert y [30 ] with resp ect to the rotating disk. Let b e the p erio d of the spiral equations in the p olar system of co ordinates r = r j ( ) : r j ( + ) = r j ( ) ; where is the anglestep of the calibration in the p olar system co ordinates, N = > 1 is an in teger, and = i = 0 + i ; i = 1 ; 2 ; :::; N ; r j ( 0 ) = r 0 = min r j ( ) ; r 0 < r i 1 < r i 2 < ::: < r iS 100 ; r ij = r j ( i ) ; i = 1 ; 2 ; :::; N ; j = 1 ; :::; S ; (3.8) where S is the n um b er of spirals for eac h i r ij are exp erimen tal data for = i ; the p oin ts ( i ; r ij ) are on the resp ectiv e spirals. T o determine the n um b er of spirals S the training frame for eac h video is used. The n um b er of spirals S is equal to the n um b er of in tensit y increases in the radial direction from the cen ter of a disk. Then 1st spiral: ( 1 ; r 11 ), ( 2 ; r 21 ), ( 3 ; r 31 ), ..., ( N ; r N 1 ); ( N +1 ; r 12 ), ( N +2 ; r 22 ), ..., ( 2 N ; r N 2 ); ...; ( ( S 1) N +1 ; r 1 S ), ( ( S 1) N +2 ; r 2 S ), ..., ( S N ; r N S ); 30
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2d spiral: ( N +1 ; r 11 ), ( N +2 ; r 21 ), ( N +3 ; r 31 ), ..., ( 2 N ; r N 1 ); ( 2 N +1 ; r 12 ), ( 2 N +2 ; r 22 ), ..., ( 3 N ; r N 2 ); ...; ( S N +1 ; r 1 S ), ( S N +2 ; r 2 S ), ..., ( S 2 N ; r N S ); ...; (n+1)th spiral: ( nN +1 ; r 11 ), ( nN +2 ; r 21 ), ( nN +3 ; r 31 ), ..., ( ( n +1) N ; r N 1 ); ( ( n +1) N +1 ; r 12 ), ( ( n +1) N +2 ; r 22 ), ..., ( ( n +2) N ; r N 2 ); ...; ( ( n + S 1) N +1 ; r 1 S ), ( ( n + S 1) N +2 ; r 2 S ), ..., ( ( n + S ) N ; r N S ). Let R ij ; i = 1 ; :::; N ; j = 1 ; :::; S ; b e the giv en data on the con tracted disk in the form of the standard ellipse with the parameters a = R and 0 < b < R Then r ij = r j ( i ) = R ij R [( a cos i ) 2 + ( b sin i ) 2 ] 1 = 2 ; where 0 = 0 ; and i = 1 ; :::; N ; j = 1 ; :::; S : Let r b e the radiusstep of calibration in the p olar system co ordinates, and ( i ; k r ; I ik ) ; i = 1 ; 2 ; ::: ; k = 1 ; 2 ; :::; b e the calibration net on the con tracted disk, where I ik are the in tensities of the p oin ts ( i ; k r ) : Then R ij = 1 2 M j + M k = j M k r ; I ik 6 = 0 ; k = j M ; j M + 1 ; :::; j + M ; i = 1 ; 2 ; :::; N ; j = 1 ; 2 ; :::; S ; 31
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where 2 M means maximal n um b er of pixels in the vicinit y of the p oin t ( i ; R ij ) along the radius with the angle i and with the cen ter in that p oin t. This pro cessing is con tin ued for all frames and rep eated for eac h video sequence. Samples of resulting images are sho wn in Figure 3.6. (a) (b) Figure 3.6 (a) Detected p oin ts of w a v es. (b) A detected w a v e. Note: With the purp ose of presen ting clarit y pixel detected p oin ts are enhanced in Figure 3.6. 3.3 Estimation of P arameters of Spiral W a v es 3.3.1 Asymptotically Optimal Numerical Metho d of Dieren tiation Estimation of the rst deriv ativ es for a giv en function is the illp osed problem [20 ], i.e., for suc h problems, arbitrary small errors of the initial data can giv e, in general, arbitrary large errors of the resp ectiv e results. In the deterministic case, when ~ X i is the v ector at the time t X i is its exp erimen tal v alue, h is the step of dieren tiation, and j X i ~ X i j < for an y t Then, with regard to the main terms 32
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= ~ X 0 i X i +1 X i 1 2 h = ~ X 0 i ~ X i +1 ~ X i 1 2 h + ( ~ X i +1 ~ X i 1 ) ( X i +1 X i 1 ) 2 h = ~ X 000 i 6 h 2 + h = ( h ) ; 0 ( h ) = ~ X 000 i 3 h h 2 = 0 ; h opt = 3 ~ X 000 i 1 = 3 ; 4 opt = 2 0 @ ~ X 000 i 3 1 A 1 = 3 2 = 3 : (3.15) Instead of unkno wn ~ X 000 i w e apply X 000 i = X g ( i +2) 2 X g ( i +1) + 2 X g ( i 1) X g ( i 2) 2 g 3 6 = 0 ; g = 1 = 5 ; h = 3 X 000 i 1 = 3 ; 4 = 2 j X 000 i j 3 1 = 3 2 = 3 : (3.16) Theorem 1. T he estimate 4 in (3.16) is asy mptotical l y optimal Pr o of. W e ha v e d = ~ X 000 i X 000 i = ~ X 000 i X g ( i +2) 2 X g ( i +1) + 2 X g ( i 1) X g ( i 2) 2 g 3 j ( ~ X g ( i +2) 2 ~ X g ( i +1) + 2 ~ X g ( i 1) ~ X g ( i 2) ) 2 g 3 ( X g ( i +2) 2 X g ( i +1) + 2 X g ( i 1) X g ( i 2) ) 2 g 3 j cg 2 + 3 g 3 = ( g ) ; (3.17) where c is a certain existing constan t (fth deriv ativ e). Using (3.17), w e nd g = 1 = 5 ; d = ( c + 3) 2 = 5 ; 33
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that is under g order 1 = 5 the v alue of X 000 i ~ X 000 i has the order 2 = 5 Then opt 3 = ~ X 000 i j X 000 i j ; 1 ( c + 3) 2 = 5 j X 000 i j j X 000 i j X 000 i ~ X 000 i j X 000 i j ~ X 000 i j X 000 i j ~ X 000 i + X 000 i ~ X 000 i j X 000 i j 1 + ( c + 3) 2 = 5 j X 000 i j ; that is, opt 1, 0, and hence is asymptotically optimal. The sto c hastic case, describ ed in App endix B, is applicable to the estimation of the w a v e v elo cit y and w a v e inclination. It is based on the normal distribution of errors and estimation of a v ariance. 3.3.2 Radial V elo cit y Comp onen t Computation T o determine the v elo cit y of the w a v es, the sequences of the images of lm ro ws are used with the step of dieren tiation t > 0. Cho osing the system of co ordinates at the cen ter of the rotating disk, the estimate of the radial v elo cit y comp onen t is giv en b y r 0 exp = r t = r t i +1 r t i t ; t = j t i +1 t i j : where r t and r t +1 are the v alues of the radii from the cen ter to the p oin ts on the w a v e at the momen ts t i and t i +1 The problem of estimating radial v elo cit y is illp osed, i.e., for suc h problems, arbitrary small errors of the initial data can giv e, in general, arbitrary large errors of the resp ectiv e results [20 ]. So, the asymptotically optimal metho d for minimizing error of estimate under a kno wn error of initial data is used (see Equation (3.15)). 34
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3.3.3 W a v e Thic kness and Azim uthal V elo cit y Comp onen t Computations Due to small thic kness of the ro w, estimation of the third co ordinate z with high accuracy is rather dicult using traditional metho ds. So, w e use z b elo w from its mo del v alues. According to [30 ], co ordinate z = 3 Q 2 n 2 r 2 1 = 3 ; (3.18) where Q is the constan t ro w rate, n is the angular v elo cit y of a disk, and is viscosit y of ruid. This form ula is applied in [30 ] for the condition that the con v ectiv e terms are small compared to the cen trifugal terms. The azim uthal v elo cit y comp onen t of w a v e is determined b y v a;theor = r 0 ( t ) tan (3.19) (see [30 ]). The v elo cit y r 0 ( t ) and the inclination angle are exp erimen tally estimated. 3.3.4 Inclination Angle Computation Figure 2.1 sho ws the denition of the inclination angle Again, the asymptotically optimal metho d for minimizing the error of estimate under a kno wn error of initial data is used (see Equation (3.15)). The follo wing estimates are used: tan = 1 ~ r d ~ r d 1 r r ( + ) r ( ) 2 ; j r ~ r j ; = 3 r 000 1 = 3 ; d ~ r d r ( + ) r ( ) 2 r 000 6 2 + = ( ) ; r 000 = r i +2 2 r i +1 + 2 r i 1 r i 2 2 g 3 ; g = 1 = 5 : (3.20) 35
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Note that form ulae (3.20) do not dep end on time but only on geometric prop ert y of the spirals and on resolution of the video. 3.3.5 Distance Bet w een Consecutiv e W a v es Computation In order to calculate the distance l the dierence b et w een t w o neigh b oring sp ecular p oin ts of w a v es, in radial direction, w as calculated: l = r 2 r 1 : (3.21) Note that v alues r 1 and r 2 ha v e to b e used in the pro cess of a v eraging. The part of the sequence with detected p oin ts on the fron t of the w a v es that w as used for calculating the a v erage distance is sho wn in Figure 3.7. (1) (2) (3) (4) (5) (6) (7) (8) (9) Figure 3.7 Sample of images with detected p oin ts on w a v es. 36
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CHAPTER 4 D A T A A CQUISITION 4.1 Exp erimen tal Setup The exp erimen tal setup consists of: a motor, alumin um rat = round sto c k, reserv oir, tubing, brass adapters, bunged cords, ro wmeter, copp er tubing, alumin um con trol b o x, switc hes, return pump, miscellaneous hardw are (Figure 4.1). Figure 4.1 Exp erimen tal setup. The main c haracteristic of the motor is giv en b y the turnable calibration. Measuremen ts w ere p erformed in the follo wing w a y W ater con tained in a plastic con tainer with an adjustmen t v alv e for the ro w w as drained through copp er tubing at a constan t starting ro w rate Q whic h can b e c hanged in the range of 0.20.8 l pm (liter p er min ute). Liquid emerged from the nozzle as a free jet p ouring out on to the cen ter of a constan tly rotating alumin um disk with a diameter of 200 mm The rotational frequency of the disk w as monitored b y a motor con trol. W ater lea ving the rotating disk w as collected at the b ottom reserv oir and recycled to the top reserv oir b y a pump. 37
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The pattern of 8 x 10 squares (see Figure 4.2) w as used to calibrate the camera. Three h undred t w en t y p oin ts (corners of squares on the exp erimen tal pattern) w ere Figure 4.2 Calibration pattern. c hosen in the Cartesian system co ordinate with the origin in the cen ter of the disk. The tec hnique describ ed in [68 ] w as used for nding the in trinsic and extrinsic parameters of the camera. 4.2 Data Collection The sc hematic setup for data acquisition is sho wn in Figure 4.3. Video data w as collected for the range of ro w rates from 2.0 l pm to 8.0 l pm and for the range of angular sp eed of the disk from 200 r pm to 800 r pm Videos of the ro w pro cess w ere tak en at dieren t settings with con trolling parameters (ro w rate and sp eed of the disk), dieren t arrangemen ts of ligh t illuminations, and dieren t settings of the camera using the p ortable camcorder Canon Optura 20 capable of capturing images at 30 f ps (frames p er second)(resolution (720x480) and high denition camera(JV C GYHD 100), capable of capturing images at 30 f ps (resolution 1280x720, color) (see T able 4.1). Duration of 38
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Figure 4.3 Sc hematic setup. videos in time is from 1 sec to 5 sec. Note: In the pro cess of detecting and trac king of w a v es, the video data with clear visibilit y of w a v es are used. The collected videos w ere used as is sho wn in T able 4.2 Figure 4.4 sho ws the sample image of the liquid lm that ro ws o v er a disk, rotating with the angular v elo cit y (n) of 500 r pm (rev erses p er min ute) and the ro w rate ( Q ) of 0.8 l pm It can b e seen (Figure 4.4) that the lm surface is co v ered b y spiral w a v es. Figure 4.4 Rotating disk closeup. 39
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T able 4.1 Collected video data. C amer a D isk r otating F l uid f l ow N umber of speed ( r pm ) r ate ( l pm ) v ideo cl ips S tandar d 500 0 : 8 10 S tandar d 300 0 : 2 10 H ig h 800 0 : 8 5 D ef inition H ig h 700 0 : 8 5 D ef inition H ig h 600 0 : 6 5 D ef inition H ig h 500 0 : 8 5 D ef inition H ig h 500 0 : 6 5 D ef inition H ig h 500 0 : 4 5 D ef inition H ig h 400 0 : 4 5 D ef inition H ig h 300 0 : 2 5 D ef inition H ig h 200 0 : 2 5 D ef inition 40
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T able 4.2 Usage of video data. D isk r otating F l uid f l ow T r aining sets T esting sets Gr ound tr uth speed ( r pm ) r ate ( l pm ) of v ideos of v ideos sets of v ideos & total # & total # & total # of f r ames of f r ames of f r ames 800 0.8 5 5 5 150 700 0.8 5 5 5 150 600 0.6 5 5 5 150 500 0.8 15 15 15 15 450 150 500 0.6 5 5 5 150 500 0.4 5 5 5 150 400 0.4 5 5 5 150 300 0.2 15 15 15 15 450 150 200 0.2 5 5 5 150 41
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CHAPTER 5 RESUL TS AND COMP ARISON TO FLUID FLO W MODEL 5.1 Regimes of Fluid Flo w The ruid ro w o v er a spinning disk of 300 r pm and 500 r pm are illustrated in Figure 5.1. (a) (b) Figure 5.1 The regimes of the disk rotation at (a) 300 r pm and (b) 500 r pm 5.2 Exp erimen tal FluidFlo w P arameter Estimates 5.2.1 Estimation of Radial V elo cit y Comp onen t The estimate of the step of dieren tiation (Equation (3.16)) is t = 3 r 000 ( t ) 1 = 3 1 = 3 1 30 ; (5.1) 42
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where 0 : 0026 + 0 : 000001 < 0 : 003 (Equation (2.18)) is an error in the initial data. The video data are used for estimation of v elo cit y comp onen ts. The a v erage exp erimen tal v elo cities are estimated using the deriv ation step equal to 1 30 The radii and the a v erage exp erimen tal v elo cities of w a v es for ten videos (tak en with camera Optura) with the disk rotation of 520 r pm and the ro w rate of 0 : 8 l pm are giv en in T able 5.1. T able 5.1 Exp erimen tal w a v e v elo cities ( mm=s ). Video data are tak en using camcorder Optura 20. h h h h h h h h h h h h h h h h h h V ideo number s R adii ( mm ) 40 50 60 70 80 90 100 1 805 750 700 630 560 540 520 2 820 755 690 650 570 536 530 3 801 760 685 635 570 542 535 4 803 740 685 650 585 540 515 5 770 750 675 625 580 535 525 6 810 735 685 625 585 545 520 7 795 740 687 630 565 550 530 8 819 720 690 645 570 540 525 9 804 730 675 640 580 535 530 10 816 752 680 630 560 560 540 Av er ag ed 804 : 3 743 : 2 685 : 2 636 572 : 0 542 : 3 527 S tandar d dev iation 14.6 12.4 7.48 9.66 9.50 7.76 7.53 Using the videos tak en with High Denition camera, with the disk rotating at 520 r pm and the ro w rate of 0 : 8 l pm the a v eraged v elo cities of w a v es for v e videos are giv en in T able 5.2. T able 5.2 Exp erimen tal w a v e v elo cities ( mm=s ). Video data are tak en using high denition camera (JV C GYHD 100). h h h h h h h h h h h h h h h h h h V ideo number s R adii ( mm ) 40 50 60 70 80 90 100 1 795 745 680 640 570 525 515 2 785 740 685 635 565 530 525 3 800 735 680 625 570 524 515 4 805 746 690 630 575 530 510 5 790 730 670 625 570 525 510 Av er ag ed 795 739 : 2 681 631 570 526 : 8 515 S tandar d dev iation 7.91 6.76 7.42 6.52 3.54 2.95 6.12 43
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5.2.2 Estimation of W a v e Thic kness and Azim uthal V elo cit y Comp onen t The form ula (3.18) is applied (see [30 ]) for the condition that the con v ectiv e terms are small compared to the cen trifugal terms. Exp erimen tal setup in [30 ] is similar to the considered one in this w ork, but dieren t b y t w o p oin ts: the surface of a disk w as made practically without an y friction and a rotating disk w as placed in a v acuum c ham b er. That means that w e ha v e to consider in additional t w o forces: forces due to friction and resistance of air. It is sho wn b elo w that those forces will b e negligible compared to the cen trifugal force for considered v alues of Q n, r The cen trifugal force and the forces due to friction and air resistance [42 ], resp ectiv ely are: f 1 = mv 2 r = m n 2 r ; f 2 = 8 pif mv ; f 3 = 0 : 5 c a Av 2 ; where m is the mass of w ater, A is the crosssection area p erp endicular to w a v e v elo cit y v ector, and a air densit y Under the radial v elo cit y v in the considered range of [280 ; 1000], the radius r in the range of [0 ; 100], and the densit y of w ater w e ha v e f 2 f 1 = 8 f r 8 0 : 005 100 280 4 : 5 100 < 1 20 ; f 3 f 1 0 : 5 a c r 0 : 5 1 100 1009 < 1 20 ; i.e., the forces due to friction and air resistance are negligible compared to the cen trifugal force. Using the form ula (3.19), r 0 ( t ) w as estimated ab o v e and the inclination angle will b e estimated b elo w. Th us, using those t w o estimates, an azim uthal v elo cit y comp onen t can b e found. F or instance, if the radial v elo cit y for the radius of 60 mm is tak en from T able 5.1 and the v alue for the inclination angle from T able 5.10, then the azim uthal 44
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v elo cit y v a;theor r 0 tan 684 tan 0 : 53 1167 mm=s Also, the lm thic kness h can b e calculated using the follo wing form ula h = q 2 r 0 R Q 2 ; where is kinematic viscosit y 5.2.3 Estimation of Inclination Angles The asymptotically optimal step = 3 r 000 1 = 3 1 = 3 1 25 ( r adian ) (see (3.20)). The obtained asymptotically optimal step is consisten t with the size of the pixel for the camera used. An error is estimated as 4 = 2 r 000 3 1 = 3 2 = 3 6%, where r 000 11 and = 0 : 003 (see (2.18)). The video sequences of ruid ro w o v er a rotating disk of 500 r pm and the ro w rate of 0.8 l pm are used to calculate a v eraged w a v e inclinations. The results for ten videos (tak en with camera Optura) are illustrated in T able 5.3. The results for v e videos T able 5.3 Exp erimen tal w a v e inclinations (in radian). Video data are tak en using camcorder Optura 20. h h h h h h h h h h h h h h h h h h V ideo number s R adii ( mm ) 40 50 60 70 80 90 100 1 1 : 01 0 : 73 0 : 54 0 : 47 0 : 36 0 : 34 0 : 30 2 0 : 97 0 : 73 0 : 53 0 : 46 0 : 36 0 : 33 0 : 3 3 0 : 96 0 : 7 0 : 5 0 : 45 0 : 35 0 : 32 0 : 28 4 1 : 01 0 : 73 0 : 54 0 : 47 0 : 34 0 : 32 0 : 29 5 0 : 96 0 : 71 0 : 5 0 : 43 0 : 33 0 : 31 0 : 28 6 1 0 : 72 0 : 54 0 : 44 0 : 35 0 : 34 0 : 3 7 0 : 96 0 : 71 0 : 5 0 : 45 0 : 33 0 : 33 0 : 29 8 0 : 97 0 : 73 0 : 51 0 : 44 0 : 35 0 : 31 0 : 29 9 0 : 99 0 : 69 0 : 5 0 : 43 0 : 33 0 : 31 0 : 28 10 0 : 96 0 : 71 0 : 52 0 : 47 0 : 35 0 : 34 0 : 3 Av er ag e 0 : 979 0 : 716 0 : 518 0 : 451 0 : 346 0 : 325 0 : 291 S tandar d dev iation 0.021 0.014 0.018 0.016 0.011 0.013 0.009 (tak en with High Denition camera) are illustrated in T able 5.4. The video sequences of ruid ro w o v er a rotating disk of 500 r pm and the ro w rate of 0.8 l pm are used to calculate a v eraged w a v e inclinations. 45
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T able 5.4 Exp erimen tal w a v e inclinations (in radian). Video data are tak en using high denition camera (JV C GYHD 100). h h h h h h h h h h h h h h h h h h V ideo number s R adii ( mm ) 40 50 60 70 80 90 100 1 0 : 97 0 : 72 0 : 53 0 : 47 0 : 36 0 : 33 0 : 28 2 1 : 01 0 : 72 0 : 52 0 : 43 0 : 34 0 : 32 0 : 3 3 0 : 98 0 : 71 0 : 5 0 : 45 0 : 36 0 : 32 0 : 28 4 1 : 0 0 : 70 0 : 52 0 : 44 0 : 35 0 : 33 0 : 29 5 0 : 94 0 : 72 0 : 49 0 : 45 0 : 36 0 : 31 0 : 30 Av er ag e 0 : 98 0 : 714 0 : 512 0 : 448 0 : 354 0 : 322 0 : 29 S tandar d dev iation 0.027 0.009 0.016 0.015 0.009 0.008 0.01 5.2.4 Estimation of Distances Bet w een Consecutiv e W a v es The ten videos (captured with camera Optura) with the sequence of frames for the ro w rate of 0.8 l pm and the rotation of disk of 500 r pm w ere used to determine the distance b et w een consecutiv e w a v es (see T able 5.5). T able 5.5 Exp erimen tal distances b et w een consecutiv e w a v es ( mm ). Video data are tak en using camcorder Optura 20. h h h h h h h h h h h h h h h h h h V ideo number s R adii ( mm ) 40 50 60 70 80 90 100 1 4 : 9 3 : 9 3 : 0 2 : 5 2 : 2 2 : 1 1 : 9 2 4 : 6 3 : 8 3 : 0 2 : 4 2 : 1 1 : 9 2 3 4 : 8 3 : 9 3 : 1 2 : 5 2 : 2 2 : 1 2 : 1 4 4 : 9 3 : 7 2 : 9 2 : 3 2 : 2 2 : 1 1 : 9 5 4 : 7 3 : 9 2 : 9 2 : 3 2 : 3 1 : 9 1 : 9 6 4 : 6 3 : 9 3 : 1 2 : 5 2 : 2 2 : 1 2 7 4 : 8 3 : 7 3 : 0 2 : 4 2 : 2 2 2 8 4 : 6 3 : 9 2 : 9 2 : 5 2 : 1 2 : 1 1 : 9 9 4 : 9 3 : 6 3 : 1 2 : 3 2 : 1 2 : 0 1 : 9 10 4 : 7 3 : 8 3 : 1 2 : 4 2 : 4 2 : 1 2 Av er ag e 4 : 75 3 : 81 3 : 01 2 : 41 2 : 18 2 : 04 1 : 96 S tandar d dev iation 0.127 0.110 0.088 0.088 0.063 0.084 0.070 The sequence of frames for the ro w rate of 0.8 l pm and the disk rotation of 500 r pm w ere pro cessed to determine the distance b et w een consecutiv e w a v es, using the High Denition camera (see T able 5.6). 46
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T able 5.6 Exp erimen tal distances b et w een consecutiv e w a v es ( mm ). Video data are tak en using high denition camera (JV C GYHD 100). h h h h h h h h h h h h h h h h h h V ideo number s R adii ( mm ) 40 50 60 70 80 90 100 1 4 : 8 3 : 9 3 : 0 2 : 5 2 : 2 2 : 0 2 : 1 2 4 : 8 3 : 8 3 : 1 2 : 5 2 : 3 2 : 1 2 : 0 3 4 : 6 3 : 9 3 : 0 2 : 3 2 : 2 2 : 1 1 : 9 4 4 : 9 3 : 7 3 : 1 2 : 5 2 : 2 2 : 0 2 : 1 5 4 : 6 3 : 6 3 : 1 2 : 4 2 : 1 2 : 1 1 : 9 Av er ag e 4 : 74 3 : 78 3 : 06 2 : 44 2 : 20 2 : 06 2 : 00 S tandar d dev iation 0.134 0.130 0.0548 0.089 0.071 0.055 0.1 5.3 Comparison to Fluid Flo w Mo del The system (2.12) ma y b e applied to describ e spiral w a v e regimes if t w o parameters 2 and ( "= ) are small; these conditions allo w one to use the b oundary la y er appro ximation. The conditions ( 2 and ( "= ) are small) w ere examined for the observ ed spiral w a v es describ ed ab o v e. The co ordinates of p oin ts on the fron t of the curv es in the radial direction w ere tak en at a constan t incremen t of azim uthal angle = = 6. The estimated radii, R 1 and R 2 are presen ted in T able 5.7. The parameters relev an t to T able 5.7 Radii of the rst and second w a v es R 1 and R 2 resp ectiv ely ( r adian ) R 1 ( mm ) R 2 ( mm ) = 6 39.94 43.04 = 3 48.99 49.28 1 = 2 70.03 72.29 2 = 6 89.97 91.97 5 = 6 98.00 99.91 T able 5.7 are sho wn in T able 5.8. It can b e seen from T able 5.8 that the principal conditions ab out small v alues of 2 and ( "= ) are fullled for the spiral w a v es as w ell as for the axisymmetric w a v es. Th us, the b oundary la y er appro ximation extended b y the terms accoun ting for dep endence on the azim uthal angle ma y b e form ulated. T o estimate the inclination angle of nonaxisymmetric w a v es, the eigen v alues w ere calculated for dieren t v alues of radius under exp erimen tal conditions with n = 520 47
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T able 5.8 Input data and mo del co ecien ts. R 1 R 2 2 10 3 2 2 10 Re mm mm 39.94 43.04 3.284 0.6086 69.64 48.99 49.28 0.8604 0.5225 138.4 70.03 72.29 0.8290 0.5211 140.0 89.97 91.97 0.1765 0.4666 230.2 98.00 99.91 0.1543 0.4622 240.4 r pm and the rate of initial ruid ro w Q c equal to 0 : 8 l pm Examples of calculations are giv en in Figure 5.2. It is seen that the nonaxisymmetric p erturbations are more Figure 5.2 (a) Phase v elo cities. (b) Amplication factors. unstable than the axisymmetric ones. Then for a few radius v alues, w a v e n um b er and inclination parameter n w ere found in the case of maxim um amplication factors. 5.3.1 Comparison of Radial V elo cities The a v eraged exp erimen tal v elo cit y of w a v es o v er fteen videos w as calculated. The calculated a v erage and the predicted results are giv en in T able 5.9. The last ro w of this table con tains the standard deviations of calculated results. The nal v ariation of the results o v er dieren t videos is estimated as the maxim um of absolute dierences b et w een the a v eraged results o v er fteen videos. Exp erimen tally 48
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T able 5.9 Exp erimen tal a v erage radial v elo cities and predicted radial v elo cities ( mm=s ). ` ` ` ` ` ` ` ` ` ` ` ` ` ` V el ocity R adii ( mm ) 40 50 60 70 80 90 100 E xper imental ( av er ag ed ) 801 742 684 634 572 537 523 P r edicted 721 670 630 599 573 551 542 V ar iation ( max min ) 50 40 30 25 25 36 30 E r r or (%) 6 5 4 4 4 7 6 S tandar d dev iation 13.2 10.8 7.8 8.8 7.9 9.9 9.0 calculated a v erage with v ariations and theoretically predicted results of radial v elo cities of ruid ro w are illustrated in Figure 5.3. The qualitativ e b eha vior of the v elo cit y of Figure 5.3 Comparison of the exp erimen tal estimation of w a v e v elo cities and the theoretical v elo cities of ruid ro w. w a v es, according to Figure 5.3, is in go o d agreemen t with the asymptotic theory of w a v e v elo cit y [30 47 ]. The error of quan titativ e exp erimen tal and predicted v alues is within 10%. 49
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5.3.2 Comparison of Inclination Angles The sequences of ten videos of ruid ro w o v er a rotating disk of 500 r pm and the ro w rate of 0.8 l pm w ere pro cessed to nd the a v eraged inclination angles for the radii in the range of 40100 mm The calculated a v erage inclination angles and predicted inclination angles are sho wn in T able 5.10. T able 5.10 Calculated a v erage w a v e inclinations and predicted w a v e inclinations (in radian). P P P P P P P P P R adii 40 50 60 70 80 90 100 E xper imental ( av er ag e ) 0.98 0.71 0.52 0.45 0.35 0.31 0.29 P r edicted 1.02 0.69 0.52 0.42 0.36 0.32 0.31 V ar iation 0.07 0.04 0.05 0.04 0.03 0.07 0.06 Error (%) 7 6 9 9 8 9 7 S tandar d dev iation 0.021 0.012 0.017 0.015 0.01 0.011 0.009 The calculated a v erage and the theoretically predicted results for w a v e inclinations are sho wn in Figure 5.4. Figure 5.4 Comparison of the exp erimen tal and theoretical w a v e inclinations. In accordance with exp erimen tal observ ations and theoretical predictions, the inclination angle decreases as the radius increases. 50
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5.3.3 Comparison of Distances The ten videos with the sequence of frames for the ro w rate of 0.8 l pm and the rotation of disk at 500 r pm w ere used to determine the distance b et w een consecutiv e w a v es (see T able 5.11). The fteen videos with the sequence of frames for the ro w rate of 0.8 l pm and the disk rotation of 500 r pm w ere used to determine the distance b et w een consecutiv e w a v es (see T able 5.11). T able 5.11 Calculated a v erage distances o v er fteen videos and theoretically calculated distances ( mm ). ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` D istances R adii ( mm ) 40 50 60 70 80 90 100 E xper imental ( av er ag ed ) 4 : 7 3 : 8 3 : 1 2 : 4 2 : 1 2 : 0 2 : 0 P r edicted 4 : 5 3 : 9 3 : 0 2 : 6 2 : 3 2 : 2 2 : 0 V ar iation 0.3 0.3 0.2 0.2 0.2 0.2 0.2 Error (%) 6 8 7 8 9 9 10 S tandar d dev iation 0.12 0.11 0.077 0.083 0.062 0.072 0.077 The calculated a v erage and the theoretically predicted results for distances are sho wn in Figure 5.5. Figure 5.5 Comparison of the exp erimen tal and theoretical distances b et w een consecutiv e w a v es. 51
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It can b e seen in Figure 5.5 that there is a corresp ondence b et w een theoretical prediction of distances and exp erimen tal data as for other v ariables. In addition, as predicted, the distance decreases as the radius increases. 52
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5.4 Statistical Analysis of Estimated Fluid Flo w P arameters F or in v estigating the dierence b et w een calculated and predicted mean v alues of ruidro w parameters, the metho d of paired comparisons, t test, or nonparametric Wilco xon signed rank test [9] can b e used. Those metho ds are designed to test if the second random v ariable in the pair has the same mean as the rst. Since t w o samples (exp erimen tal and predicted) are indep enden t, it is appropriate to use those metho ds. Using the Wilco xon signed rank test, the n ull h yp othesis is E D = 0 and the alternativ e h yp othesis is E D 6 = 0, where E D is the mean of dierences b et w een predicted and exp erimen tal parameters, D i = Y i X i The absolute dierences j D i j = j Y i X i j i = 1 ; :::; n; are computed for eac h of the n pairs. Ranks from 1 to n are assigned to these n pairs. Then the signed rank R i is dened for eac h pair as follo ws: R i = sig n ( D i ) R j D i j ; i = 1 ; :::; n; (5.2) where n is the n um b er of pairs. Then the test statistic is the sum of the p ositiv e signed ranks W + = X ( R i ; w her e D i is positiv e ) : (5.3) Or if n > 30, the normal appro ximation can b e used and the test statistic is: W = P i n R i p P i n R 2 i : (5.4) The n ull h yp othesis is rejected at the lev el if W + (or W ) is less than 2 quan tile or greater than its 1 2 quan tile from the T able 'Quan tiles of the Wilco xon Signed Ranks T est Statistic' for W + and for W from T able 'Normal Distribution' [9 ]. 53
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The t w otailed p v alue is t wice smaller than the one tailed p v alues, appro ximated from the normal distributions as l ow er tail ed p v al ue = P Z P i n R i + 1 p P i n R 2 i (5.5) or upper tail ed p v al ue = P Z P i n R i 1 p P i n R 2 i : (5.6) F or in v estigation of the dierence b et w een the calculated and the predicted parameter v alues of ruidro w parameters, the nonparametric Wilco xon signed rank test [9 ] w as used. The test statistics for w a v e v elo cities (see T able 5.9) are W = 1 : 183 > 1 : 96 and t = 0 : 7418 So, the n ull h yp othesis E D = 0,where E D is the mean of dierences b et w een predicted and exp erimen tal parameters, is accepted at the signican t lev el = 0 : 05. Th us, there is not a statistically signican t dierence b et w een calculated and predicted radial v elo cities. Using T able 5.10, the test statistic for inclination angle W equals to 0 : 043033, whic h is higher then 1 : 96 at a signicance lev el of = 0 : 05. So, w e fail to reject the n ull h yp othesis at that lev el. Also, using the ttest, the test statistic is t = 0 : 0421 and w e accept the n ull h yp othesis of no dierence b et w een the calculated and the predicted mean v alues at a signicance lev el of = 0 : 05. Th us, there is not a statistically signican t dierence b et w een the calculated and the predicted inclination angles. By in v estigating the dierence b et w een calculated and predicted mean v alues of distances b et w een consecutiv e w a v es (see T able 5.11), w e compute the test statistics W = 0 : 378 > 1 : 96 and t = 0 : 0271. So w e accept the n ull h yp othesis of no dierence b et w een the calculated and the predicted distance mean v alues at a signicance lev el of = 0 : 05. 54
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CHAPTER 6 MODELBASED RECO VER Y OF CONTR OLLING P ARAMETERS The eect of the con trolling parameters suc h as the ro w rate and rotational sp eed on the ro w c haracteristics for a giv en set of ph ysical parameters of ruid w as studied in n umerous exp erimen tal and theoretical in v estigations (see, for example, [2 3, 30, 33 48 63]). In the case of small ro w rates, the lm w as observ ed to break up in to rivulets. The classication of dieren t ro w regimes along the radius at mo derate ro w rates is giv en in [3 ]. A t higher lo w rates, a smo oth uniform lm w as formed. Up on further increase in ruid ro w rate, circumferen tial and spiral w a v es mo ving from the disk cen ter to the disk p eriphery w ere observ ed. A t the highest ro w rates, a com bination of helical and circumferen tial w a v es w as observ ed. V ariations of the rotational sp eed, on the other hand, led to the mo dication of the shap e, amplitude, and the v elo cit y of the observ ed w a v es. Photographic evidence for w a v e formation is presen ted in [63 ]. Therefore, the pro cess of reco v ery of con trolling parameters consists of t w o parts: determination of a sp ecic ro w regime (circular w a v es, spiral w a v es, complex c haotic w a v es, etc.) and reco v ery of sp ecic v alues of con trolling parameters for a sp ecic w a v e regime. This is done based on the observ ed w a v e parameters. In this w ork w e concen trate on the second part and assume man ual ro w regime iden tication. This is reasonable as the application of this w ork is in visual con trol of the pro cesses on the spinning disk. As suc h, recommended v alues of the con trolling parameters are kno wn. The spiral w a v e regime is observ ed in our exp erimen ts, and an appropriate mathematical mo del of the lm ro w pro cess is selected. F or this regime w e en umerate 55
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observ ed w a v e parameters (w a v e inclination angle, distance b et w een w a v es,v elo cit y of the w a v e) computed from video. Giv en all that, algorithms for the mo delbased reco v ery of con trolling parameters are prop osed. 6.1 Algorithm of Reco v ery Con trolling P arameters The main idea is to v ary con trolling parameters un til predicted w a v e parameters are within predened error tolerance of the actual observ ed w a v e parameters. The blo c ksc heme of suc h metho ds is giv en in Figure 6.1. Figure 6.1 A blo c ksc heme of determining the disk sp eed and ruidro w rate. The steep est descen t algorithm [53 ] is used to nd the con trolling parameters. This metho d starts its searc h in the direction of the steep est slop e (an tigradien t) un til an 56
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impro v emen t in this direction is found. After eac h new step a new steep est slop e is determined and the pro cess con tin ues un til the algorithm arriv es at some p oin t where it cannot see an y impro v emen t an ymore, at this p oin t the algorithm terminates. This algorithm is simple and fast but could get stuc k in a lo cal minim um. T o o v ercome this problem to some exten t, the W eigh ted Latin Hyp ercub e sampling (WLHS) is used to nd the starting p oin t for the gradien tbased searc hing algorithm. Practically w e do not need to use WHLS b ecause, ha ving images giv en, w e can assume a kno wn certain range of the sp eed of the disk and the ro w rate. Th us, the problem is reduced to the case of more exact determination of the con trol parameters in the frame of a priory giv en. Kno wing of starting con trol parameters close to the desired ones can b e imp ortan t for the con v ergence of the resp ectiv e algorithms of searc h. If the starting p oin t is unkno wn, the regression mo del describ ed in (C) can b e used to determine the con trolling parameters close to the realistic ones. The NewtonRaphson metho d is not used since it is based on the second order deriv ativ es, whic h are less reliable in noisy images. 6.2 Estimation of Absolute and Relativ e Errors Ob viously to emplo y an y optimization metho ds the error criterion needs to b e dened. The leastsquare error criterion is used. E a = 1 N N X i =1 j x ci x pi j 2 1 = 2 ; E r k = N X i =1 j x ci x pi j 2 1 = 2 N X i =1 j x ci j 2 1 = 2 ; (6.1) where E a is an absolute error, E r k is a relativ e error, x ci and x pi are the calculated and predicted parameters resp ectiv ely and N is a data size. Sp ecically to this problem, 57
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x :i are ruid ro w parameters ( l )(see Equations (3.20), (3.21)) and E r k is a relativ e error corresp onding to the inclination angle or the distance. Predicted parameters are estimated using the ev aluation mo del describ ed in Section 2.2. The sample absolute errors b et w een the exp erimen tal and predicted w a v e inclinations and distances b et w een consecutiv e w a v es for dieren t rotation disk sp eeds (n r pm ) and v arious ruid ro w rates ( Q l pm ) are sho wn in T able 6.1. In this sample, the a v eraged ground truth exp erimen tal v alues (o v er 15 video clips with sequences of 10 frames eac h) of w a v e inclinations and distances are tak en under the follo wing conditions: the sp eed of disk equal to 500 r pm and the initial ruidro w rate equal to 0 : 8 l pm at the radius of 40 mm The relations b et w een the absolute error of inclination angles T able 6.1 The absolute errors of ruid ro w parameters Absol ute er r or Absol ute er r or of w av e incl inations of distances ( r adian ) ( in mm ) H H H H H H n Q 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 200 0 : 86 0 : 83 0 : 79 0 : 76 0 : 81 3 : 96 3 : 88 3 : 78 3 : 74 3 : 78 300 0 : 74 0 : 68 0 : 64 0 : 59 0 : 63 2 : 41 2 : 36 2 : 22 2 : 15 2 : 20 400 0 : 65 0 : 62 0 : 59 0 : 57 0 : 59 1 : 29 1 : 24 1 : 1 0 : 97 0 : 99 500 0 : 17 0 : 11 0 : 09 0 : 03 0 : 08 0 : 27 0 : 21 0 : 18 0 : 06 0 : 19 600 0 : 52 0 : 49 0 : 47 0 : 45 0 : 47 1 : 19 1 : 10 0 : 99 0 : 91 0 : 94 700 0 : 77 0 : 73 0 : 71 0 : 71 0 : 76 1 : 72 1 : 67 1 : 62 1 : 57 1 : 61 800 0 : 80 0 : 79 0 : 77 0 : 74 0 : 77 2 : 52 2 : 47 2 : 41 2 : 37 2 : 44 and distances and sp eeds of the rotating disk and rates of ruidro w are illustrated in Figure 6.2. The relation b et w een absolute errors of the inclination angles and rotation disk sp eeds and the xed ruid ro w rate of 0 : 8 l pm is sho wn in Figure 6.3 ( a ); and the relation b et w een absolute errors of the distances and rotation disk sp eeds and the xed ruid ro w rate of 0 : 8 l pm is sho wn in Figure 6.3 ( b ). The relation b et w een absolute errors of inclination angles and ruid ro w rates and the xed rotating disk sp eed of 500 58
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0 0.5 1 1.5 2 2.5 200 400 600 800 0.2 0.4 0.6 0.8 1 1.2 Fluidflow rate (lpm) Absolute error of wave inclinations Rotating disk speed (rpm) Absolute error (mm) (a) 0 0.5 1 1.5 2 2.5 200 400 600 800 0 1 2 3 4 5 6 Fluidflow rate (lpm) Absolute error of distances Rotating disk speed (rpm) Absolute error (mm) (b) Figure 6.2 (a) The relation b et w een absolute errors of inclination angles, rotating disk sp eeds, and ruidro w rates; (b) the relation b et w een absolute errors of distances, rotating disk sp eeds, and ruidro w rates. r pm is sho wn Figure 6.4 ( a ); and the relation b et w een absolute errors of distances and ruid ro w rates and the xed rotating disk sp eed of 500 r pm is sho wn Figure 6.4 ( b ). Another example with the a v eraged ground truth exp erimen tal v alues (o v er 15 video clips with sequences of 10 frames eac h) of w a v e inclinations and distances, tak en with the disk sp eed of 300 r pm and the initial ruidro w rate of 0 : 2 l pm at the radius of 40 mm is giv en in T able 6.2. 59
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(a) (b) Figure 6.3 The relation b et w een absolute errors of (a) inclination angles and sp eeds of rotating disk; (b) distances and sp eeds of rotating disk. Estimations of inclination angles ha v e more uncertain t y than measuremen ts of the distances. So, relativ e errors (6.1) are calculated and dieren t measuremen ts of the com bined relativ e errors are computed using the follo wing form ula, E c = E r + (1 ) E r l ; where E r is a relativ e error of inclination angles, E r l is a relativ e error of distances, and is a constan t. One of the results is sho wn in T able 6.3 and Figure 6.5. In this example, the a v erage ground truth ruidro w parameters w ere tak en from video data 60
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(a) (b) Figure 6.4 The relation b et w een absolute errors of (a) inclination angles and ruid ro w rates; (b) distances and ruid ro w rates. with the disk sp eed of 500 r pm and the ratero w of 0 : 8 l pm at the radius of 40 mm Another example is sho wn in T able 6.4 and Figure 6.6. The ruidro w parameters w ere tak e from the same videos at the radius of 60 mm The a v erage results of ground truth w a v e parameters for fteen videos are compared with the predicted w a v e parameters and, then, the disk sp eed is c hosen at a minim um com bined relativ e error (see T ables 6.5 and 6.6). As w e can see from T ables 6.5 and 6.6, the con trolling parameters can b e estimated with the relativ e error less than 10%. 61
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T able 6.2 The absolute errors of ruid ro w parameters with the sp eed of 300 r pm and the ruidro w rate of 0 : 2 l pm Absol ute er r or Absol ute er r or of w av e incl inations of distances ( r adian ) ( mm ) H H H H H H n Q 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 200 0 : 48 0 : 53 0 : 61 0 : 68 0 : 71 2 : 08 2 : 15 2 : 55 2 : 67 2 : 98 300 0 : 04 0 : 05 0 : 09 0 : 15 0 : 23 0 : 09 0 : 46 0 : 54 0 : 72 1 : 05 400 0 : 65 0 : 72 0 : 79 0 : 81 0 : 89 2 : 29 2 : 44 2 : 51 2 : 93 2 : 91 500 0 : 77 0 : 86 0 : 94 1 : 05 1 : 17 2 : 27 2 : 56 2 : 68 2 : 96 3 : 19 600 0 : 82 0 : 89 0 : 97 1 : 15 1 : 27 2 : 39 2 : 46 2 : 66 2 : 71 2 : 94 700 0 : 98 1 : 01 1 : 07 1 : 21 1 : 34 2 : 42 2 : 67 2 : 72 2 : 77 2 : 95 800 1 : 02 1 : 09 1 : 17 1 : 34 1 : 47 2 : 52 2 : 77 2 : 81 2 : 99 3 : 14 T able 6.3 The relativ e errors of ruid ro w parameters for the v arious rotation disk sp eeds at the radius of 40 mm S peed of disk r otation R el ativ e er r or of R el ativ e er r or of C ombined ( r pm ) w av e incl inations distances er r or 200 0 : 78 0 : 79 0 : 785 250 0 : 69 0 : 46 0 : 575 300 0 : 56 0 : 28 0 : 42 350 0 : 42 0 : 20 0 : 31 400 0 : 26 0 : 12 0 : 19 450 0 : 15 0 : 09 0 : 12 500 0 : 05 0 : 08 0 : 065 550 0 : 028 0 : 11 0 : 069 600 0 : 18 0 : 18 0 : 18 700 0 : 45 0 : 32 0 : 385 800 0 : 64 0 : 48 0 : 56 62
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Figure 6.5 Relations of ruidro w parameter relativ e errors and rotation sp eeds of disk at the radius of 40 mm T able 6.4 The relativ e errors of ruid ro w parameters for the v arious rotation disk sp eeds at the radius of 60 mm S peed of disk r otation R el ativ e er r or of R el ativ e er r or of C ombined ( r pm ) w av e incl inations distances er r or 200 0 : 69 0 : 81 0 : 75 250 0 : 58 0 : 51 0 : 545 300 0 : 44 0 : 32 0 : 38 350 0 : 32 0 : 21 0 : 265 400 0 : 2 0 : 14 0 : 17 450 0 : 11 0 : 08 0 : 095 500 0 : 026 0 : 06 0 : 043 550 0 : 1 0 : 13 0 : 115 600 0 : 27 0 : 19 0 : 23 700 0 : 43 0 : 33 0 : 38 800 0 : 61 0 : 52 0 : 565 63
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Figure 6.6 Relations of ruidro w parameter relativ e errors and rotation sp eeds of disk at the radius of 60 mm T able 6.5 The relativ e errors of reco v ered disk sp eeds (RDS) at the com bined minimal relativ e errors (CMRE) for video segmen ts 1 through 8. V ideo # 1 2 3 4 5 6 7 8 C M R E 0 : 026 0 : 037 0 : 063 0 : 046 0 : 053 0 : 021 0 : 063 0 : 037 R D S ( r pm ) 500 510 540 520 475 500 535 505 R el ativ e er r or s (%) 0 2 8 4 5 0 7 1 T able 6.6 The relativ e errors of reco v ered disk sp eeds (RDS) at the com bined minimal relativ e errors (CMRE) for video segmen ts 9 through 15. V ideo # 9 10 11 12 13 14 15 C M R E 0 : 056 0 : 052 0 : 057 0 : 036 0 : 087 0 : 051 0 : 0534 R D S ( r pm ) 470 470 525 505 545 480 485 R el ativ e er r or s (%) 6 6 5 1 9 4 3 64
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CHAPTER 7 SUMMAR Y 7.1 Conclusion This dissertation presen ts a no v el videobased algorithm to detect mo ving w a v es, to determine w a v e regimes, and to compute con trolling lm ro w parameters. The input to this algorithm is an easily acquired nonin v asiv e video data. The rst part of the algorithm includes image analysis, trac king, and reconstruction algorithms to measure the w a v e shap e and the w a v e propagating sp eed. It is sho wn that it is p ossible to trac k w a v e motion b y observing sp ecular p ortion of the rerected ligh t on the mo ving w a v e under a single ligh t illumination assumption. The ruid ro w parameters and c haracteristics (v elo cities, thic kness of lm, inclination angles, and distances b et w een consecutiv e w a v es) are calculated. V elo cities and inclination angles are estimated using the socalled quasioptimal metho d, whic h minimizes error of dieren tiation estimate under kno wn error of initial data. The ruid ro w parameters are compared with the solutions of the relev an t computation ruid dynamics mo dels based on the Na vierStok es equations. The ruid mo dels predict w a v e c haracteristics based on directly measured con trolling parameters (suc h as disk rotation sp eed and ruid ro w rate). F rom the results, w e observ e that the a v erage computed parameters are within 510% of the predicted v alues. The second part of algorithm concen trates on mo delbased reco v ery of ruid ro w con trolling parameters. The searc h in space of mo del parameters is p erformed so that the predicted ro w c haracteristics (e.g. distance b et w een w a v es, w a v e inclination angles) are close to those measured from video data. The aim of suc h algorithms is visual con trol 65
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of the pro cesses in spinning disk reactors. The metho d of steep est descen t is used to nd the con trolling parameters using b oth the exp erimen tal videobased parameters and the theoretical mo del of ruid ro w. Exp erimen tal results demonstrate that the sp eed of a disk and the ro w rate are reco v ered with high accuracy whic h supp orts the v alidit y of the approac h. When compared to the ground truth a v ailable from direct observ ation, the con trolling parameters are estimated with less than 10% error. Results presen ted in this w ork substan tiate that using the dev elop ed algorithms, it is p ossible to accomplish the ab o v e tasks (estimation of ruid ro w parameters and con trolling parameters) with reasonable accuracy W e b eliev e that the demonstrated approac h will b e v aluable in exp erimen tal studies of w a v e patterns as w ell as suitable for practical applications of visual qualit y con trol of c hemical pro cesses. 7.2 F uture Researc h Some issues that need to b e addressed in future in v estigations are: 1. Exp erimen ts with dieren t sizes of the rotating disk; 2. Exp erimen ts with v arious ph ysical parameters, including viscosit y of liquid; 3. 3D surface reco v ery using m ultiple ligh t illuminations; 4. Analysis of those exp erimen ts based on nonlinear solutions of the ev olution system; 5. F urther comparing the results of the exp erimen ts with the resp ectiv e mo dels; 6. The applicabilit y of the theory of computer vision for the ev olution systems; 7. P erforming the dev elop ed system for a broader set up, for example, for the synthesis of aero gels, when a rotating disk is inside of a closed cylinder; 66
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8. Using the dev elop ed system for ruid w a v es in natural condition problems including the astroph ysical and geoph ysical problems, lik e global scale ro w in the atmosphere, the o ceans; 9. Applying this w ork to the medical problems, for example, for analysis of cerebral spinal ruid ro w w a v eforms. 67
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[30] G. Lenew eit, K. Ro esner, and R. Ko ehler. Surface instabilities of thin liquid lm ro w on a rotating disk. Exp eriments in Fluids 26(12):75{85, 1999. [31] J. Liu, J. P aul, and J. Gollub. Measuremen ts of the primary instabilit y of lm ro ws. Journal of Fluid Me chanics 250:69{101, 1993. [32] O. Matar, G. Siso ev, and C. La wrence. Ev olution scales for w a v e regimes in liquid lm ro w o v er a spinning disk. Phys. Fluids 16:1532{1545, 2004. [33] O. Matar, G. Siso ev, and C. La wrence. `the ro w of thin liquid lms o v er spinning discs. The Canadian Journal of Chemic al Engine ering 84(6):625{642, 2006. [34] T. McInerney and D. T erzop oulos. Deformable mo dels in medical images analysis: A surv ey Me dic al Image A nalysis 1(2):91{108, 1996. [35] H. Murase. Surface shap e reconstruction of nonrigid transparen t ob ject using refraction and motion. IEEE T r ans. P AMI 14(10):1045{1052, 1992. [36] Y. Nak a jima, H. Inomata, H. Noga w a, Y. Sato, S. T am ura, K. Ok azaki, and S. T orii. Ph ysicsbased ro w estimation of ruids. Pattern R e c o gnition 36:1203{ 1212, 2003. [37] Y. Ojima and M. Ka w ahara. Estimation of riv er curren t using reduced k alman lter nite elemen t metho d. Computer Metho ds in Applie d Me chanics and Engine ering 198(912):904{911, 2009. [38] O. P o v aro v, E. V asilc henk o, and P P etro v. W a v e ro ws of thin liquid la y ers in a cen trifugal force eld. A c ademy of Scienc es of USSR, Power Engine ering and T r ansp ort 16(1):172{176, 1978. [39] M. Rahman. T ransp ort to a c hemically activ e thin liquid lm o v er a spinning disk. ASME Journal of Ener gy R esour c es T e chnolo gy 120(4):293{298, 1998. [40] J. Rausc her, R. Kelly and J. Cole. An asymptotic solution for the laminar ro w of thin lms on a rotating disk. Appl. Me chanics 40:43{47, 1973. [41] H. Sak aino. Fluid motion estimation metho d based on ph ysical prop erties of w a v es. CVPR 2008 pages 1{8, 2008. [42] J. Saleh. Fluid Flow. Handb o ok McGra wHill, 2002. [43] A. A. Sa v elo v. Plane curves. Systematics, applic ations Nauk a, 1960. [44] S. Sh ubhra K. Misra, M. Thomas, K. C., J. Kirb y F. V eron, and M. Bro cc hini. Estimation of complex airw ater in terfaces from particle image v elo cimetry images. Exp eriments in Fluids 2006. [45] A. F. Sh v ets, L. P P ortno v, G. G. Filipp o v, and A. I. Gorbuno v. Flo w of an axisymmetric lm of viscous liquid on the surface of a spinning disc. The or. F ound. Chem. Eng. 26(895), 1992. 70
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[46] A. Singh, D. Goldgof, and D. T erzop oulos. Deformable Mo dels in Me dic al Image A nalysis IEEE Computer So ciet y Press, 1998. [47] G. Siso ev, O. Matar, and C. La wrence. Axisymmetric w a v e regimes in viscous liquid lm ro w o v er a spinning disk. Fluid Me chanics 495:385{411, 2003. [48] G. Siso ev, O. Matar, and C. La wrence. Mo delling of lm ro w o v er a spinning disk. J. Chem. T e chnol. Biote chnol. 78(23):151{155, 2003. [49] G. Siso ev, O. Matar, and C. La wrence. Stabilizing eect of the Coriolis forces on a viscous liquid lm ro wing o v er a spinning disc. Comptes R endus Me c anique 332:203{207, 2004. [50] G. Siso ev and V. Shk ado v. Flo w stabilit y of a lm of viscous liquid on the surface of a rotating disc. Journal of Engine ering Physics 52(6):671{674, 1987. [51] G. Siso ev and V. Shk ado v. Helical w a v es in a liquid lm on a rotating disc. Journal of Engine ering Physics 58(4):573{577, 1990. [52] G. Siso ev, A. T al'drik, and V. Shk ado v. Flo w of a viscous liquid lm on the surface of a rotating disc. Journal of Engine ering Physics 51(4):1171{1174, 1986. [53] J. Sn yman. Pr actic al Mathematic al Optimization: A n Intr o duction to Basic Optimization The ory and Classic al and New Gr adientBase d A lgorithms Springer, 2005. [54] P Sturm and S. Ma ybank. On planebased camera calibration: a general algorithm, singularities, applications. In CVPR'99 v olume 6, pages 1432{1437, 1999. [55] S. Thomas, A. F aghri, and W. Hank ey Exp erimen tal analysis and ro w visualization of a thin liquid lm on a stationary and rotating disk. Fluid Engine ering 113(1):73{80, 1991. [56] S. Thomas, W. Hank ey and A. F aghri. Onedimensional analysis of the h ydro dynamic and thermal c haracteristics of thin lm ro ws including the h ydraulic jump and rotation. ASME J. He at T r ansfer 112(3):728{735, 1990. [57] S. T ogashi, T. Oh ta, and H. Azuma. Fingering ro w pattern of a liquid lm on a rotating substrate. Chem. Engng. Jap an 34(11):1402{1406, 2001. [58] D. T ritton. Physic al Fluid Dynamics Clarendon Press, 1988. [59] R. Y. Tsai. An ecien t and accurate camera calibration tec hnique for 3D mac hine vision. In Pr o c e e dings of CVPR'86 1986. [60] J. W alk er. Physics Pren tice Hall, 2002. [61] W. W elford. Optics Oxford Univ ersit y Press, 1976. 71
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[62] R. Wildes, M. Amabile, A. Lanzillotto, and T. S. Leu. Reco v ering estimates of ruid ro w from image sequence data. CVIU 80:246{266, 2000. [63] W. W o o ds. The h ydro dynamics of thin liquid lms ro wing o v er a rotating disc. Ph.D. thesis, University of Newc astle up on T yne, UK 1995. [64] R. Zeytounian. The ory and Applic ations of Visc ous Fluid Flows Springer, 2004. [65] Y. Zhang, S. Kundu, D. Goldgof, S. Sark ar, and L. Tsap. Elastic face, an anatom ybased biometrics b ey ond visible cue. Pr o c e e dings of International Confer enc e on Pattern R e c o gnition 2:19{22, 2004. [66] Y. Zhang, E. Prak ash, and E. Sung. Ecien t mo deling of an anatom ybased face and fast 3d facial expression syn thesis. Computer Gr aphics F orum 22(2):159{170, 2003. [67] Z. Zhang. Flexible camera calibration b y viewing a plane from unkno wn orien tations. In Pr o c e e dings of ICCV'99 1999. [68] Z. Zhang. A rexible new tec hnique for camera calibration. IEEE T r ansactions on Pattern A nalysis and Machine Intel ligenc e 22:1330{1334, 2000. [69] L. Zhou, C. Kam bhamettu, D. Goldgof, K. P alaniappan, and A. Hasler. T rac king nonrigid motion and structure from 2d satellite cloud images without corresp ondences. IEEE T r ansactions on Pattern A nalysis and Machine Intel ligenc e 23:1330{1336, 2001. 72
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APPENDICES 73
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App endix A T rac king the P eaks of W a v es Using the Sp ecular Rerection P atterns Assuming [35 ] that tops of w a v es o v er a disc can b e represen ted with enough accuracy b y collections of piecewise sin usoids, let one of those sin usoids ha v e a form Z = Asin 2 X A on the segmen t [ ; ] and let ( x; z ) b e a p oin t on that sin usoid, x ((0 ; 0) is the co ordinate of the resp ectiv e top of w a v e pro jection on the plane 0 X Z ), where A and are parameters of w a v e appro ximation. Then the condition that the normal to the sin usoid in the sp ecular p oin t [15 ] ( x; z ) crosses the p oin t ( x c ; z c ) has the form x c x z c z = A! sin ( x ) ; (A.1) from where under small sin ( x ), small A and large A! 2 z c the solution is: x x c 1 + a! 2 ( z c A x c 1 + A! 2 z c x c A! 2 z c : (A.2) It follo ws from A.2 that x can b e close to zero under sucien tly small x c z c Lets consider a condition when t w o straigh t lines passing through p oin ts ( x; z ), ( x c ; Z c ), and ( x l ; z l ), ha v e equal angles with the normal to the same sin usoid [15 ] in the p oin t ( x; z ). Using the kno wn form ula for an angle b et w een t w o lines, tan = k 2 k 1 1 + k 1 k 2 ; where slop es of those lines, the conditions of equal angles are: 1 k 1 k 2 1 k 2 + k 1 = k 3 k 2 1 1 k 2 + k 3 ; (A.3) 74
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App endix A (Con tin ued) where k 3 is a slop e of the second line. In our case, k 1 = z c z x c x ; 1 k 2 = A! x; k 3 = z l z x l x ; (A.4) where w e can put x c = and, for the simplicit y x c = x l = x with the v alue for con v enien t disp osition of a camera and a ligh t. Then the ob vious solution is x = 0 ; z = 0), the normal to the sin usoid is axis 0 Z x l = k 1 = z and the angle = arctan z Ho w ev er, if the co ordinate of ( x c ; z ) and ( x l ; z ) are xed and j x c x l j 6 = 2 then in the case that the segmen t of the size 2 under the top of the w a v e is in the cen ter of the segmen t [ x c ; x l ], then x = x c + x l 2 z = 0 the normal is parallel to 0 Z and the angle = arctan( 2 z x l x c ). If in addition the segmen t of the size 2 under the top of the w a v e pro jection is arbitrary disp osed on a disk, then under condition of equalit y of angles ab o v e the normal in the desired p oin t ( x; z ) will not b e parallel to 0 Z and w e ha v e to nd the solution when ( x; z ) will b e the nearest to the middle of the resp ectiv e segmen t. Let the left side of the sin usoid ha v e the co ordinate ( x 0 ; z 0 ), z 0 = A cos ( x 0 ) A and let it ha v e maximal p ossible distance from x l the righ t end of the segmen t [ x l ; x c ]. This distance do es not exceed 2 r where r is the radius of a disk. The normal in the p oin t ( x 0 ; z 0 ) to the sin usoid has the slop e k 2 1 A! 2 x 0 so that w e ha v e to nd the condition on z suc h that the lines passing through the p oin ts ( x c ; z ? ), ( x 0 ; z 0 ), and the p oin ts ( x l ; z ? ), ( x 0 ; z 0 ) ha v e equal angles with the normal to the sin usoid in the p oin t ( x 0 ; z 0 ). In our case, k 1 = z ? z 0 x c x 0 ; k 3 = z ? z 0 x l x 0 ; k 2 = 1 A! 2 x 0 : (A.5) 75
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App endix A (Con tin ued) Therefore, instead of (A.3) w e ha v e ( z ? z 0 ) 2 1 ( x c x 0 ) 2 1 ( x l x 0 ) 2 A! 2 x 0 ( z ? z 0 ) 1 x c x 0 1 x l x 0 A! 2 x 0 2 1 + 2 A! 2 x 0 = 0 ; (A.6) from (A.6) z ? z 0 = ( A! 2 x 0 ) 2 1 1 x c x 0 + 1 x l x 0 1 2 A! 2 x 0 0 @ ( A! 2 x 0 ) 2 1 1 x c x 0 + 1 x l x 0 2 1 2 A! 2 x 0 2 2 1 ( x c x 0 ) 2 1 ( x l x 0 ) 2 1 A 1 = 2 (A.7) In the case, when << r and x 0 = r x c = r x l = r w e ha v e z ? z 0 ( A! 2 r ) 2 1 1 2 A! 2 r + 0 @ ( A! 2 r ) 2 1 1 2 A! 2 r 2 + 2 2 1 A 1 = 2 = A! 2 r 1 A! 2 r 1 2 + 1 A! 2 r 1 A! 2 r 2 1 4 + 2 1 = 2 ; (A.8) i.e., z will b e the order 1 Th us, arbitrary xed disp osition of x c and x l requires rather high disp osition of z and certain restriction on parameters of w a v es. F or the b est accuracy not the high disp osition of z and without the restriction on parameters, w e ha v e to mo v e x c and x l along radii in suc h a w a y that the measured tops turn out to b e appro ximately in the middle of those co ordinates. 76
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App endix B A Sto c hastic Case Under the condition f ( x ) 2 N ( f ( x ) ; 2 ) ; P [ f ( x 1 x 2 ] = P [ f ( x 1 )] P [ f ( x 2 )] ; (B.1) w e nd estimate of 2 ? 2 = 1 M M X =1 [ f ( x ) f ( x )] 2 ; j ? j c 1 p M ; (B.2) and also E f 0 ( x ) f ( x + h ) f ( x h ) 2 h = f 0 ( x ) f ( x + h ) f ( x h ) 2 h c 6 h 2 ; E f 0 ( x ) f ( x + h ) f ( x h ) 2 h f 0 ( x ) + f ( x + h ) f ( x h ) 2 h 2 = E f ( x + h ) + f ( x + h ) + f ( x h ) f ( x h ) 2 h 2 = 1 4 h 2 E [ f ( x + h ) f ( x + h )] 2 + E [ f ( x h ) f ( x h )] 2 = 1 2 h 2 ; = f 0 ( x ) f ( x + h ) f ( x h ) 2 h c 6 h 2 + h 2 ; P 0 : 95 ; (B.3) from where c 3 h 2 2 h 3 = 0 ; h 4 = 6 c 2 ; h 2 = 6 c 1 = 2 ; c 6 1 = 2 + c 6 1 = 2 = 2 c 6 1 = 2 2 c 6 1 = 2 ? + 2 c 6 1 = 2 c 1 p M : (B.4) 77
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App endix B (Con tin ued) W e can nd c 1 using that ( M 1) 2 2 has 2 M 1 distribution. Ho w ev er, P 2 2 c 1 M 1 = P ( M 1) 2 2 ( M 1) c 1 = P 2 M 1 ( M 1) c 1 = 0 : 95 : (B.5) 78
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App endix C A Regression Mo del The video data are used to build a regression mo del. The a v erage results of exp erimen tal w a v e parameters (inclination angles, distances, and v elo cities) for v arious con trolling parameters (sp eed of rotating disk and initial ruid ro w rate) are calculated. The relation b et w een inclination angle, disk sp eed, and ro w rate is: 1 = 1 + 2 x disk + 3 x r ate + 4 x disk x r ate ; (C.1) where is the inclination angle and i ; i = 1 ; 2 ; 3 ; 4 ; are co ecien ts, 1 = 9 : 071 e 02, 2 = 1 : 718 e 03, 3 = 2 : 596 e 02, and 4 = 3 : 598 e 04. The residual standard error is 0 : 01489 on 12 degrees of freedom, the m ultiple Rsquared is 0 : 9986, and the adjusted Rsquared is 0 : 9983. T o c hec k if the mo del is correct and if the assumptions are satised, w e plot the residuals v ersus the the tted v alues. This plot should not rev eal an y ob vious pattern. Figure C.1 (b) plots the residuals v ersus the tted v alues for the exp erimen tal inclination angles. No un usual structure is apparen t. An extremely useful pro cedure is to construct a normal probabilit y plot of the residuals. If the underlying error distribution is normal, this plot will resem ble a straigh t line (see Figure C.1 (c)). Scaled residuals suc h as the standardized and studen tized are useful in lo oking for outliers. Most of standardized residuals should lie in in terv al [3,3], and an y observ ation with a standardized residual outside of this in terv al is p oten tially un usual with resp ect to its observ ed resp onse. A residual v ersus a lev erage plot is a scatterplot of residuals against hat v alues. The ScaleLo cation plot, also called SpreadLo cation or SL plot, tak es the square ro ot of the absolute residuals in order to diminish sk ewness (square ro ot of the absolute residuals is m uc h less sk ew ed than the absolute residuals for Gaussian zeromean residuals). The ResidualLev erage plot sho ws con tours of equal Co ok's 79
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App endix C (Con tin ued) (a) (b) (c) (d) Figure C.1 (a) LogLik eliho o d; (b) Residuals vs Fitted v alues; (c) Standardized Residuals; (d) Square ro ot of Standardized Residuals vs Fitted v alues. distance, for v alues of Co ok's lev els (b y default 0.5 and 1) and omits cases with lev erage one (see Figure C.2). 80
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App endix C (Con tin ued) Figure C.2 Standardized Residuals vs Lev erage. The relation of distances of sp eed disk and ro w rate is: 1 dist = r 1 + r 2 x disk + r 3 (1 =x r ate ) + r 4 x disk (1 =x r ate ) ; (C.2) where dist distances and co ecien ts: r 1 = 3 : 162 e 02, r 2 = 5 : 430 e 04, r 3 = 1 : 198 e 03, and r 4 = 2 : 469 e 05. The residual standard error is 0 : 006687 on 12 degrees of freedom, the m ultiple Rsquared is 0 : 9982, and the adjusted Rsquared is 0 : 9978. T o c hec k if the mo del is correct and if the assumptions are satised, w e plot the residuals v ersus the the tted v alues. This plot should not rev eal an y ob vious pattern. Figure C.3 (b) plots the residuals v ersus the tted v alues for the exp erimen tal distances. No un usual structure is apparen t. A normal probabilit y plot of the residuals resem bles the straigh t line (see Figure C.3 (c)). Ho w ev er, the underlying error distribution is normal. The ResidualLev erage plot is sho wn in Figure C.4. 81
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App endix C (Con tin ued) (a) (b) (c) (d) Figure C.3 (a) LogLik eliho o d; (b) Residuals vs Fitted v alues; (c) Standardized Residuals; (d) Square Ro ot of Standardized Residuals vs Fitted v alues. 82
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App endix C (Con tin ued) Figure C.4 Standardized Residuals vs Lev erage. 83
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ABOUT THE A UTHOR V alen tina N. Korzho v a receiv ed the MS degree in computer science from the Univ ersit y of South Florida (USF) in 2006, where she is curren tly a PhD candidate. She w orks as a graduate researc h assistan t at USF, sp ecializing in image pro cessing and pattern recognition. Her additional in terests include algorithm optimization and mathematical mo deling with applications in medicine. She has published more than 28 scien tic w orks in sev eral conference pro ceedings and journals. She is a studen t mem b er of the IEEE.
