DESIGN AND SIMULATION OF PHOTOGRAMMETRIC NETWORKS USING GENETIC ALGORITHMS INTRODUCTION

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1 DESIGN AND SIMULATION OF PHOTOGRAMMETRIC NETWORKS USING GENETIC ALGORITHMS Gustavo Olague Departamento de Cencas de la Computacón Dvsón de Físca Aplcada Centro de Investgacón Centífca y de Educacón Superor de Ensenada BC Km 07 carretera Tuana-Ensenada 860 Ensenada BC Méxco golague@ccesemx ABSTRACT Ths wor descrbes the use of genetc algorthms n the desgn of photogrammetrc networs A system made for ths purpose EPOCA (Evolvng POstons of CAmeras) s presented Wth the system a photogrammetrc networ can be desgned usng three-dmensonal CAD-models of the obect The system provdes the atttude of each camera n the networ tang nto account the magng geometry of the system (contrbuton to ntersecton angles) as well as several maor constrants le the ncdence angle constrant worspace constrant and the problem of vsblty When plannng a photogrammetrc networ the cameras should be placed so that each pont can be seen at least from two cameras When the obect s three-dmensonal a combnatoral problem s presented Genetc algorthms are stochastc optmzaton technques whch have proved useful at solvng computatonally dffcult problems In order to solve the problem we pose t n terms of a global optmzaton process The system s capable of mnmzng the error whle tang nto account the computatonal burden (the mathematcal error plays a role as t leads to the error to be optmzed) Our system reproduces confguratons reported n the lterature of well-nown nternatonal experts le Fraser and Mason Moreover the system can desgn networs for several adonng planes and complex obects openng new nterestng research ways The results obtaned confrm the effectveness and effcency of the soluton INTRODUCTION Networ desgn s referred to the process of placng a set of cameras n order to acheve photogrammetrc tass However photogrammetrsts consder networ desgn as a non-easy tas Close range photogrammetry has the goal of measurng an obect through a spatally dstrbuted set of cameras also nown as confguraton We obtan hghly accurate measurements through one of these networs The accuracy s related to the camera confguraton used at the moment of carryng out the photogrammetrc proect Indeed the spatal dstrbuton s drectly related to the accuracy and relablty of the system We wll call camera networ one of these confguratons and the subect of ths study wll be the optmzaton of such a networ Selectng a well-desgned photogrammetrc networ presents however a problem of fundamental mportance Although ts defnton seams smple t reaches a hgh complexty manly due to the numerous constrants and desgn decsons that need to be taen Camera networ desgn s hard to obtan due to the unnown number of confguratons havng all a very smlar accuracy but wth a very dfferent magng geometry Automaton of camera networ desgn s an mportant step towards autonomous nspecton systems Approaches to Networ Desgn Prevous approaches to camera networ desgn have attempted to dentfy the man stages n the process Followng the wdely accepted classfcaton scheme by Grafarend (974) networ desgn has been dvded nto four desgn stages from whch only the frst three are used n close range photogrammetry Zero Order Desgn (ZOD): Ths stage attempts to defne an optmal datum n order to obtan accurate obect pont coordnates and exteror orentaton parameters Frst Order Desgn (FOD): Ths stage nvolves defnng an optmal magng geometry whch n turn determnes the accuracy of the system Second Order Desgn (SOD): Ths stage s concerned wth adoptng a sutable measurement precson for the mage coordnates It conssts usually n tang multple mages from each camera staton

2 Thrd Order Desgn (TOD): Ths stage deals wth the mprovement of a networ through the ncluson of addtonal ponts n a wea regon Photogrammetrc measurement operatons attempt to satsfy n an optmal manner several obectves as precson relablty and economy The ZOD and SOD are greatly smplfed n comparson to geodetc networs for whch the four stages were orgnally developed Indeed FOD the desgn of networ confguraton or the sensor placement tas needs to be comprehensvely addressed for photogrammetrc proects Ths desgn stage must provde an optmal magng geometry and convergence angle for each set of ponts placed over a complex obect (Fraser 996) Photogrammetrsts have acnowledged the degree of expertse needed to carry out a photogrammetrc proect In ths way Mason et al (995) developed a wor called CONSENS that follows the expert system approach and uses multple cameras n combnaton wth optcal trangulaton It outlnes a method of overcomng the set of constrants and obectves presented n camera staton placement The method s based on the theory of generc networs whch consttutes compled expertse descrbng an deal confguraton of four camera statons that can be employed to provde a strong magng geometry for the class of planar networ desgn problems Complex obects are dvded nto planes; each one s evaluated through one of these networs and then connected wth some extra cameras wth the purpose to establsh ust one common datum However the expert system approach has shown unlely that full automaton of the networ desgn process wll be acheved due n large part to human expert s extensve use of commonsense reasonng (Fraser 996) On the other hand the classfcaton ust presented serves the photogrammetrc user to dentfy what set of tass need to be carred out n desgnng a networ Our Approach Photogrammetrc experts regard smulaton as a vable strategy to the problem of networ desgn (Fraser 996) Ths strategy s based on the stages ZOD FOD and SOD Gven the crtera related to requred trangulaton precson the ntal step s to adopt a sutable observaton and measurng system Ths entals the selecton of an approprate camera format focal length and mage measurement system as well as a frst approxmaton to a sutable networ geometry Once fnshed ths desgn stage the networ s evaluated aganst the specfed crtera If the networ fals to acheve the crtera a new stage to dagnose and dentfy the problem s carred out FOD or SOD wll be appled to the current soluton If both correctons are nsuffcent a new networ wll be proposed untl a soluton to the problem s acheved In ths way networ desgn s teratve n nature However the man queston: how to obtan an ntal confguraton wth an optmal magng geometry s unsolved and left as responsblty to the desgner In ths paper we propose a method for solvng the most fundamental stage n networ desgn We set the problem n terms of a global optmzaton desgn whch s capable of managng the problem usng an adaptve strategy It explores the soluton space usng both non-contnuous optmzaton and combnatoral search Our approach then s to mnmze the uncertanty of the three-dmensonal measurements usng as a crteron the maxmum standard devaton of the 3D-measurements consderng that the optmzaton satsfes a number of basc constrants Emphass n ths paper s gven to the optmzaton process from an algorthm pont of vew and how the numerous desgn decsons are consdered to overcome the computatonal burden Paper Organzaton Ths paper s organzed as follows: Frst we present the mathematcal model we have used as well as the procedure we have followed n order to obtan an expresson of the three dmensonal error Then we present a bref summary of the constrants on networ desgn We descrbe the problem of networ desgn n terms of a stochastc global optmzaton and how we have mplemented a system usng genetc algorthm strategy We study dfferent aspects (vsblty occluson) related to the complexty of the search space Ths study allows us to dentfy the way of analyzng the search space Thus we present our mult-cellular genetc algorthm whch optmzes the crteron Fnally we present some examples followed by a concluson MATHEMATICAL MODEL The problem nvolves a set of cameras vewng P ponts placed over S surfaces We want to now the best camera networ n order to reduce the uncertanty of the reconstructed ponts The way of solvng the system here s smlar to that proposed by several photogrammetrsts (Brown 980 Fraser 987) nown as Lmtng Error Propagaton It maes the assumpton that proectve parameters are error free and that the varances n the obect pont coordnates

3 arse solely from the propagaton of random errors n the mage coordnates measurements Ths method has proven to be effectve n the case of strong networs Three Dmensonal Reconstructon Let u and v denote the photo coordnates of the mage of pont n photograph For each par of mage coordnates (u v ) observed on each mage the followng relatonshp exsts: m X + m Y + m 3Z + m 4 u = () m 3X + m 3Y + m 33Z + m 34 v m = m 3 X X + m + m 3 Y Y + m + m 3 33 Z Z + m + m 4 34 Ths system of equatons assumes that lght rays travel n straght lnes that all rays enterng a camera lens system pass through a sngle pont and that the lens system s dstorton less or as usually n hgh accurate measurement that dstorton has been cancelled out after havng been estmated The magng process ust descrbed s nown as the pnhole camera model Ths model s based on the fundamental assumpton that the exposure center the ground pont and ts correspondng mage pont all le on a straght lne In ths way a pont n the scene P = n of homogeneous coordnates (X Y Z ) T s proected nto p ponts of mage coordnates (u v ) through a proecton matrx M = m of sze 3 4 correspondng to the th mage Therefore three-dmensonal measurements can be obtaned from several mages Each matrx M represents a mappng composed of a transformaton W C from the world coordnates W to the camera coordnates C gven by x X y R wc Twc Y = z 0 3 Z where the rotaton matrx R wc whch s a functon of three rotaton parameters ( β γ ) and the translaton vector T wc also of three degrees of freedom characterzes the camera s orentaton and poston wth respect to the world coordnate frame Under perspectve proecton the transformaton from the 3D-world coordnate system to the Dmage coordnate s X su R wc Twc Y where the matrx sv = K 0 3 Z s u f K = 0 0 represents the ntrnsc parameters of the camera f s the focal length of the camera ( u v ) are the horzontal and vertcal pxel szes on the mage plane and ( u v 0 0 ) s the proecton of the camera s center (prncpal pont) on the mage plane In ths way a camera can be consdered as a system that performs a nown lnear proectve transformaton from the proectve space P 3 nto the proectve plane P Consderng the proectve matrx of each camera and mage ponts as nown we can develop an error propagaton study usng Equaton () that can be rewrtten as follows: X ( )( )( ) um 3 m um 3 m um 33 m 3 m 4 u m 34 ( )( )( ) Y = v m v m 34 m m vm m vm m or n matrx notaton consderng one obect pont A p M ) P = b( p M ) ( 0 v 0 f u v Z where A s a 3 matrx and b s a vector Snce the nverse of squares soluton A T A can be computed we can fnd the least

4 T T P = ( A A) A b ( ) AP b whch mnmzes Ths equaton expresses the lnear relatonshp between the mage ponts P through the proectve matrces M p and the three-dmensonal ponts 3D Error Estmaton Untl now we have studed the functon to transform a pont n space nto an mage pont P = f ( p) gven by Equaton () whch s gong to be useful to develop an analyss of error propagaton (Faugeras 993) The ey to manpulatng geometrc uncertanty s to be able to transform the nformaton or probablty densty functon on a feature avalable n one form (mage pont) nto another form of nterest (pont n space) Ths transformaton of nformaton can be grouped nto a famly of transformatons that we approxmate to the exact transformaton by a frst-order relaton usng a Taylor seres Successve moments can be found by equatng hgher order terms; however usng hgher order terms s nether vable nor desrable as any computatonal smplcty would be lost In ths way a lnear approxmaton s to be used n whch we assume a Gaussan dstrbuton Then the mean E[P] and covarance ΛP are suffcent nformaton to completely defne the feature densty functon All ths s gven by the followng proposton (Faugeras 993 Chap 5): p m R of Gaussan dstrbuton mean [P] Proposton Gven a random varable n and P R the random vector gven by P = f ( p) where f s a functon of class approxmated to a frst-order Taylor expanson by f ( E[ p]) and ts covarance by: T f ( E[ p]) f ( E[ p]) ΛP = Λp p p Therefore ΛP s a symmetrc postve defnte matrx whch descrbes the bounds on P = f ( p) E [ P] = f ( E[ p]) gven those of p n the vcnty of E [ p] E and covarance P Λ C the mean of P can be n the vcnty of Ths proposton lets us compute the uncertanty of the three-dmensonal pont nowng the uncertanty n the mage ponts Another model s needed to gve an nterpretaton of matrx Λ p whch descrbes bounds on the possble values of the coordnates of p Ths mples the need for an mage error model Image Error Estmaton In order to complete the uncertanty analyss a study concernng the mage ponts must be done In ths way experments can be carred out to detect the crcular targets of the calbraton grd see Fgure Ths grd s composed of 60 retro-reflectve crcular targets wthn a volume of cm 3 Images have been acqured wth a PULNIX TM-6EX camera and a KINOPTIK lens of focal length 5-mm The frame grabber s an "Imagng Technology 50"

5 Fg The calbraton grd s composed of retro-reflectve targets Fg Ths graph shows the best-ft curve whch corresponds better to the ncdence angle constrant The X-axs corresponds to the convergence angle (n degrees) and the Y-axs to the mage error As a means to estmate the covarance matrx Λp of our D measured ponts we wll use Proposton Therefore we must have a functon to relate mage pont errors taen from several photographs over dfferent angles The relaton wll be establshed usng the cross-rato: ( c a)( d b) f ( ) = ( a b c d) = ( 3) ( d a)( c b) whch s a proectve nvarant It lets us compute the uncertanty of the cross-rato as a functon of the uncertanty of the mage ponts In the case of homographc confguratons of 4 ponts the uncertanty vares n a way nversely proportonal to the dstances between the ponts In ths way consderng the uncertanty of the mage ponts as dentcal we can select a confguraton of 4 ponts f ( ) = [03] = 4 usng the retro-reflectve targets of the calbraton grd n order to produce stable confguratons The cross-rato can be lnearzed locally wth respect to confguraton f ( ) = [03 ] The relatonshp between the standard devaton of the cross-rato and the uncertanty of the mage ponts can be obtaned by a frst-order Taylor expanson of Equaton (3): T Λ = f ( ) Λp f ( ) ( 4) where Λ of sze s equal to the cross-rato varance Ths value s computed consderng all the crossratos f ) of 4 algned and equdstant targets usng the statstcal method ( N σ = ( f ( ) f ( )) N = f () s the Jacoban of f () ( b d)( c d) ( a c)( d c) = ( a d) ( b c) ( a d)( b c) σ where the partal dervatves wth respect to each pont are computed as follows: ( b d)( b a) ( a c)( b a) ( a d)( b c) ( a d) ( b c) J Consderng that under a homographc transformaton J ( λ a λ b λ c λ d) = J ( a b c d) λ we obtan (03) = 80 J p Λ s the covarance matrx of the 4 algned ponts used to compute the crossrato f () Snce all targets are dentfed wthout systematc errors Λp = σ p I4 4 where σ p represents the standard devaton of the target extracton and I represents the unt matrx In ths way Equaton (4) can be rewrtten as

6 λ σ = σ p J σ p = σ ( 5) J where σ represents the standard devaton of the cross-rato λ s the average dstance between the targets and J s the Jacoban of f () We assume that the mage errors are Gaussan and dentcal (smlar targets) along ther algnment drecton and constant durng the experment (smlar dstance of observaton and vewng angle) We use a subpxel target detector on the calbraton grd of Fgure smlar to one presented by (Gruen 985) From hundred of retro-measures of four approxmately equdstant ponts along each drecton we compute the uncertanty n the mage pontsσ p from the measured uncertanty of the cross-ratoσ usng Equaton (5) σ s a functon of the observaton angle along ther algnment drecton As the targets are crcular and vewed as an ellpse t can be assumed that the covarance of ts two-dmensonal locaton can be consdered as a dagonal matrx The graphs n Fgure show fnal results of our experments along the x-drecton (smlar results are obtaned along the y-drecton) Consequently the reference frame of the ellpse represented by two axes has each one-assocate varance correspondng to the vewng angle n the two correspondng drectons These results can be approxmated to a model usng the Levenberg-Marquardt method (Press et al 988) and the mert functon (Note that these observatons are better approxmated to a hyperbolc functon): υ 90+ y = ρ ( e 90 x + e x ) + ϑ υ yeldng the best ft parameters:υ =7974 ρ =3 0-3 and ϑ =8 0-3 for both experments Ths model corresponds to the ncdence angle constrant n camera networ desgn Notce that for an angle larger than 80 degrees we have no more measurements and approachng ths value leads to an nfnte uncertanty The Λ p model s useful to compute the covarance matrx of the 3D ponts ΛP CONSTRAINTS ON NETWORK DESIGN Camera networ desgn must deal wth a seres of constrants n order to propose an optmal camera dstrbuton The accuracy of the system s related to the magng geometry as well as the convergence angle that each camera does wth respect to each obect surface In order to answer the most basc queston of a favorable magng geometry (FOD or the confguraton problem) we must dstngush among the several constrants lmtng the search space Mason (994) has proposed a set of constrants and obectves that we prefer to separate nto two parts: Prmary Constrants Consderng the constrants lmtng the search space we dentfy the followng four due to the characterstcs of the FOD problem: Contrbuton to ntersecton angles or the magng geometry Wthn a camera placement system the man obectve s to now the contrbuton of each camera wth respect to the others Two fundamental questons need to be answered e decdng on how many cameras wll be needed and where should be placed However before to answer the frst queston we need to answer the second one Once we now where to place a gven number of cameras s a trval matter decde on the number Convergence angle The relablty of mage measurements from drectons close to coplanar are dffcult and even mpossble to obtan The mnmum allowable ncdence angle s dependent on the type of feature ts geometry and materal The accuracy of the measurement wth respect to the convergence angle s a functon of the vewng drecton and the surface normal at the feature A hyperbolc model has already been computed (Olague 998) usng retro-reflectve targets In the case of crcular targets the mnmum convergence angle s about 0 degrees and about 30 degrees for the nd of targets we have used Worng space constrant The worspace n whch the photogrammetrc survey s conducted can mpose restrctons on the selecton of an deal magng geometry Ths constrant ncludes the walls of the room any obstructons n the worng envronment and even the robot s worspace n whch a camera could be mounted Vsblty Ths constrant s related to the problem of obstructons n the envronment Vewponts affected by occlusons caused by other obects n the worspace or the obect tself should be avoded f possble

7 Ray tracng technque has been used n order to obtan vsblty nformaton of an obect from dfferent vewponts We create a database that s used then nto our optmzaton process Secondary Constrants Camera networ desgn s a functon of the magng geometry as well as the convergence angle Consequently the other constrants (optcal constrants) wll not be taen nto account when estmatng a favorable magng geometry These constrants lac sgnfcant mportance once the camera observes the entre obect In ths way an optmal dstance of the camera to the obect can be defned a pror n order to measure the dfferent obect ponts Thus we can consder that all features appear wthn the feld-of-vew n focus at a gven resoluton and depth of feld On the other hand photogrammetrsts affrm the total number of ponts as rrelevant once a suffcent number s used to compute the exteror orentaton parameters NETWORK DESIGN AS AN OPTIMIZATION PROBLEM The problem of camera networ desgn presents dscontnutes manly due to the occluson of targets leadng to a combnatoral optmzaton process whch we have approached usng a mult-cellular genetc algorthm Genetc algorthms are probablstc parallel search technques based on the mechansm of natural selecton and natural genetcs Snce ther development n the late 960s (Holland 99) genetc algorthms have been proven effectve n searchng large non-lnear and poorly understood search spaces where expert nowledge s scarce or dffcult to encode and where tradtonal optmzaton technques fal Automaton of camera networ desgn wth the goal of achevng hgh accurate measurements s not drect Many decsons need to be taen wth the purpose of proposng one optmal confguraton The complexty of the problem becomes evdent when we study complex obects usng multple cameras Determnstc methods do not adapt very well Ths s manly due to the hgh number of desgn decsons n the form of thresholds The system must tae nto account all the constrants n order to solve the numercal and combnatoral problem In ths way genetc algorthms provde a general framewor useful n solvng ths tas Moreover t s necessary to notce that networ desgn s based on very precse rules as magng geometry and ncdence angle Ths allows us to dfferentate a good confguraton from a bad one Genetc algorthm strategy s based only n the drect comparson of solutons The set of solutons correspondng to the local mnma does not have a sgnfcant dfference wth respect to the global mnmum as photogrammetrsts affrm (Mason 995) Ths set of solutons called here ``alternatve solutons'' provdes smlar characterstcs about the homogenety of the ellpsod of uncertanty Consequently we can conclude that they are of the same nature However all these confguratons can be very dfferent wth respect to the magng geometry Stochastc methods do not have these problems Ths s the man reason we have selected them In ths way a process of combnatoral search le genetc algorthms must fnd the dfferent topologes that a set of cameras presents wth respect to an obect tang nto account the constrants that lmt the search space The nowledge of these confguratons s an mportant step towards camera networ desgn Genetc Algorthms for Networ Desgn The camera networ desgn can be acheved followng genetc algorthm methodology Ths methodology s composed of fve maor components: The defnton of a structure A whch represents a tentatve soluton to the problem We represent t here as a set of varables whch are grouped nto ust one common structure The envronment E ε whch lmts the structure s represented here as the set of geometrcal and optcal constrants 3 A measure µ E of performance e the ftness of the structures for the envronment whch s represented here as the value µ (P) 4 The adaptve plan Γ τ whereby the system's structure s modfed to effect mprovements Ths s the genetc algorthm detaled later on 5 The operator's set Ω ω whch are used by the adaptve plan Ths s represented here by the crossover and mutaton operatons These fve man ponts are fundamental to establsh a strategy for the camera networ desgn problem Structure defnton s mportant from the photogrammetrc pont of vew as well as the genetc algorthm Indeed followng these fve man ponts a genetc algorthm usng bnary representaton wll be developed Bnary representaton s not the only alternatve In fact ths representaton gves us the possblty of carryng out larger mutatons

8 Moreover the codng sze has not ncdence over the qualty of the soluton Evaluaton of a networ s drectly related to the angle separaton between each camera composng the networ In ths way once we have found an optmal soluton small dsplacements do not mprove the accuracy of the system On the other hand the algorthm must tae nto account the spatal constrants as well as the occluson problems n order to solve the numercal and combnatoral problem n the case of a mult-dmensonal search space Complexty of the Search Space Descrpton of the search space ncludes the defnton of forbdden zones for whch the geometrc constrants and vsblty prohbt the camera placement An example s the convergence angle whch lmts the number of cameras observng a set of surfaces creatng n ths way sub confguratons C s for each surface Decodng of each structure produces several error computatons wth respect to the number of surfaces conformng the obect In ths way after all these computatons we select as the evaluaton of the structure the maxmum error value In summary nterpretaton of structure s realzed dynamcally contrary to tradtonal genetc algorthms Thus smlar to genetc programmng decodng s carred out over a varable structure Consequently sub confguratons C s C are produced wth respect to the surfaces manly due to target s orentaton n the followng way: true f θ < θ max where θ s the ncdence angle Cam C s = false otherwse In ths way a camera belongs to several confguratons due to target s orentaton Camera networ needs to optmze the dstrbuton of each camera wth respect to the others wth the purpose to mprove the convergence of the networ Therefore the system needs to tae nto account the ncdence that each camera does wth respect to the surfaces and at the same tme to measure all targets wth at least two cameras for each surface Under these crcumstances our system must confront a combnatoral problem Contrary to expert photogrammetrsts or the strategy developed by Mason (994) the soluton to the problem can be obtaned wthout reducng the complexty of the problem In other words we do not need to dvde the obect nto several parts wth the purpose to solve each surface ndvdually; nether do we need to nsert new cameras wth the goal to mprove the system The soluton s obtaned gven a certan number of cameras as we wll see n the examples Search Space for Convex Obects Wthn a stochastc optmzaton process such as the genetc algorthm we have used the search space C must be taen nto account n order to generate sutable camera postons We need to consder then a genetc algorthm worng under constrants Strateges need to be mplemented n order to avod forbdden zones Consequently a penalzaton process has been developed n order to access the vald space

9 C y C x = ( 0********000 ******) evaluaton error calculaton error calculaton error calculaton Fg 3 Ths Fgure shows how we dvde the search space contanng the cube wth respect to the (X Y Z) axes The ntersecton of two schemas defnes precsely wth respect to axes (Y X) a zone of the space whch lets observe both surfaces at the same tme Smlar consderatons are consdered for the other zone of the space Camera networ needs to optmze the mage dstrbuton of each camera wth respect to the others tang nto account the ncdence angle It produces sub confguratons for each measured surface that we need to consder n our algorthm The search space n the case of a convex obect s represented n Fgure 3 Ths Fgure shows how the search space s dvded usng the vewng sphere model where the cameras move on a sphere loong nwards towards a central pont In ths example f we want to place a set of cameras n order to measure three contguous surfaces several sub confguratons are defned as s represented n the Venn dagram of Fgure 3 In ths manner the search space s dvded n S = 3 zones wth respect to the number of nterest surfaces composng the obect The set of regons Esp _ sol U S = s = C s called the vald soluton space whch s constraned n our search Wthn ths space a zone correspondng to the ntersecton of all the sub spaces desgnate the poston space for whch we can observe at the same tme all nterest surfaces of the cube Ths space zone was the obectve searched by the systems devoted to the placement of only one camera However for a system conceved for three-dmensonal measurements a spatal dstrbuton of the camera networ must be acheved over all the vald space n order to obtan an optmal confguraton from the convergence vewpont Cameras must be separated wthn the space n order to mprove the convergence of the system A compromse s mpled over the space zones desgnated by the dsuncton of two neghborng surfaces:

10 C y + Cz : Cz + Cx : Cx + Cy where U + V = { x U or x V : x U V} Thus these space regons do not allow each camera to observe all the surfaces of the obect at the same tme A camera poston can be defned n the followng way: true f θ x y < θmax for each S Cam ( β) ( Cx + Cy ) = Cam ( β) ( Cx Cy ) false otherwse Moreover we can observe that wthn ths space exsts a prvlege space: ( C + C ) C x y z whch represents the sub spaces where a camera can observe each surface S x or S y at the same tme than surface S z Ths mples a space zone where a better contrbuton to the trangulaton of the system could be obtaned wth respect to the cameras placed wthn the regon Cy Cx Ths nd of consderatons can be extended to complex obects presentng convexes and concaves regons A combnatoral exploson generated by these constrants ncrease as more cameras are added Occluson Problems Another dffcult aspect wthn the optmzaton process s related to the vsblty of the targets Ths mples new prohbted zones whch must be taen nto account wthn the combnatoral search process Thus soluton space s redefned as the vsblty space We need to avod the camera placement where a camera cannot measure the targets Ths constrant lmts the space zones where we can observe all the nterest surfaces Consequently the set of regons called the vsblty space s defned as follows: Esp _ Vs = Esp _ Sol = U S = Cs at least one part of the obect s vsble Dvson of the vsblty space could be done for each surface studed n order to optmze the test of vsblty In ths way we approxmate the sze of each vewng drecton n the sphere n order to obtan homogeneous regons However the vsblty space ust obtaned ncreases the combnatoral problem Whch sub set of cameras needs to measure the surface S x? Ths problem s solved partally by a prevous computaton of the vsblty space (to solve t we use ray tracng technque as many researchers have done for smlar problems) As a result of the dvson of the sphere the test of vsblty s carred out over each secton of the sphere n order to now the vsble targets for a gven vewpont In ths way the global optmzaton process can use ths nformaton prevously n order to search wthn the combnatoral space savng a great amount of computatonal tme The Mult-Cellular Genetc Algorthm The mult-cellular genetc algorthm then proceeds as follows: An ntal random populaton of N convergent networs that satsfy the envronment constrants s chosen and t s represented by ( β ) coded nto a bnary strng representaton Next we evaluate each networ and store the correspondng maxmum value of the dagonal of ΛPn for each tree structure Ths corresponds to the ftness value whch says how good the networ s compared wth other solutons n the populaton P (t) 3 Then we select a populaton of good networs by tournament selecton: two networs are selected from P (t) and are compared selectng the best ndvdual accordng to ts ftness yeldng the populaton P ( t +) 4 From ths populaton we recombne the bnary strngs ( n β n ) for each camera usng the followng operatons: a Crossover wth a probablty Pc = 07 Ths operaton was mplemented usng one-cut pont Actually due to the classfcaton of the MGA ths operaton wors le a multple-cut-pont Let the two parents be: xy

11 = ] x [ x x x3 x4 x5 x6 x7 x8 x9 [ y y y3 y 4 y5 y6 y7 y8 y9 = ] y If they are crossed after the random th poston = 4 the resultng offsprng are: = [ ] ' x x x x3 x4 y5 y6 y7 y8 y9 ' y [ y y y3 y4 x5 x6 x7 x8 x9 = ] b Mutaton wth a probablty Pm = 0005 Ths operaton alters one or more genes Assume that the = y5 gene of the chromosome ' x s selected for a mutaton Snce the gene s t would be flpped nto 0 These operatons yeld a new populaton whch we copy nto P (t) 5 Steps 3 and 4 are repeated untl the optmzaton crteron stablzes Fnally ths algorthm mnmzes the maxmum value n the dagonal of ftness ( ΛP ) = mn = K N max = K3 Λ P : Thereby the camera placement M relatve to the world coordnate frame s optmzed Geometrcally each represents a hyper ellpsod whch changes ts orentaton and sze as each sensor placement optmal placement soluton s proposed where the combned uncertanty of all ponts s mnmal EXAMPLES Λ P M does Thus an We have run a seres of experments to test the valdty of our approach We present some results n Fgures 4a-d whch show several confguratons desgned by EPOCA The cameras are loong to a set of targets represented by ther error ellpsods algned n one or two planes as well as over a complex obect These confguratons are a product of our evolutonary system In fact wthn a stochastc optmzaton process we cannot mae conclusons from ust one tral Actually each confguraton presented s the product of about 50 ndependent trals Fgure 4a llustrates a soluton wth four cameras loong at a planar surface Ths soluton s not the standard one used by the experts: a photogrammetrst usually puts the four cameras at four-corners of a cube whose center contans the targets to be measured In fact Fraser (98) has already dscussed our confguraton; he notced that ths confguraton s not atypcal Our experments confrm Fraser's statement hence the equvalence between both confguratons (Mason 994) Fgure 4b presents an nterestng result Cameras are placed over the same places of Fraser's confguraton Ths result corresponds to the second order desgn (SOD) used by photogrammetrsts Ths operaton conssts n the acquston of multple exposures from each camera composng the networ SOD operaton s used normally after havng selected a basc confguraton The multple exposures thus yelded a good mprovement n the mean standard error value over that obtaned n the basc networ In the case of Fgure 4d ust 0 trals were necessary Ths result shows how the obect shape constrants the search space where cameras could be placed These results correspond well to the generc networ theory Ths demonstrates how our system s able to propose solutons used by experts Moreover our system can contrbute to fnd solutons n the case of complex obects

12 a) 4 cameras over a plane b) 6 cameras over a plane c) Confguraton smlar to one proposed by d) 6 cameras over a complex obect Mason (995) Fg 4 Among the several desgns proposed by EPOCA we have dentfed desgn a) as one used by Fraser (98) whch s not atypcal of an magng geometry Other examples of confguratons reported n the lterature by well nown nternatonal experts were reproduced by EPOCA see Fgures b) and c) Moreover t can be used n the case of complex obects as can be apprecated from Fgure 4d CONCLUSIONS In ths paper we have presented a soluton to the problem of optmal camera placement wth the goal of achevng hghly accurate 3D measurements n terms of an optmzaton desgn The problem has been dvded nto two man parts The frst was devoted to an analytcal error model from whch a crteron was derved The crteron we have chosen s the maxmum element n the dagonal extracted from the covarance matrx whch s based on the error propagaton phenomenon Ths crteron was chosen n order to speed up the MGA process The second part was devoted to a global optmzaton method whch has mnmzed the crteron Due to the occluson of ponts caused by the dfferent constrants the problem presents dscontnutes whch leads to combnatoral aspects n the optmzaton process These constrants are naturally ncorporated nto the genetc algorthm methodology The system s a dynamc process where the global state emerges by co evoluton through the behavor and nteracton of the cells whch are related by co adaptaton It s necessary to menton that the behavor of these cells s nfluenced by the global state An extrnsc parallelsm together wth the ntrnsc

13 parallelsm s executed A phenomenon of nche s present n each camera representaton These nches evolve wth respect to the others manly due to the rules of the system Our wor confrms results obtaned by other researchers: Holland (975) genetc algorthms are able to solve effcently such nd of optmzaton problems wth hgh combnatoral aspects Our EPOCA system successfully produces two and three camera networs desgns smlar to those used by photogrammetrsts In the case of four cameras a non-standard desgn was proposed whch should gve smlar results to those obtaned from the more classcal networs Moreover the system can desgn networs for several adonng planes and complex obects All the confguratons are good n terms of the camera dstrbuton and ray nclnaton It s necessary to notce that all results agree well wth emprcal results obtaned by photogrammetrsts Ths wor has to be consdered as an addtonal step towards automated camera networ desgn The strength of our approach s ts generalty Consderaton of alternatve optmzaton processes worng on a non-unform crteron could easly extend ths wor However several smplfyng assumptons were made here that have to be explored n the future We assume that the camera external and nternal parameters are perfectly nown In fact the user has good estmates for them but f hgh accuracy were needed the bundle adustment would have to refne ther estmates together wth the 3D-measurement estmaton Ths complcates the computaton of the crteron at each evaluaton step n the optmzaton Ths aspect wll ncrease the optmzaton tme as well as the non-lnearty and the open questons that arse are: Wll such system converge? How long wll t tae? If ths mght not be a bg theoretcal ssue t s a practcal one as a huge scale factor could multply the computaton tme These aspects are both related to the computatonal complexty of the optmzaton; to explore them s our goal for future research Acnowledgements Author s grateful for research foundng from CONACYT Méxco (repatraton proect) Prof Roger Mohr contrbuted to the development of EPOCA wth many useful deas crtcsms and suggestons Fgures 3 and 4 were generated wth software wrtten at the Geometry Center Unversty of Mnnesota I am also grateful to Dr Scott Mason and Dr Marc Schoenauer for hs helpful comments and nterest n ths research REFERENCES Faugeras O (993) Three-Dmensonal Computer Vson A Geometrc Vewpont Artfcal Intellgence The MIT Press Cambrdge MA USA Fraser CS (98) Optmzaton of Precson n Close Range Photogrammetry Photogrammetrc Engneerng and Remote Sensng 53(5): May Fraser CS (987) Lmtng Error Propagaton n Networ Desgn Photogrammetrc Engneerng and Remote Sensng 48(4): March Fraser CS (996) Networ Desgn In KB Atnson edtor Close Range Photogrammetry and Machne Vson chapter 9 pages 56-8 Whttles Publshng Roselegh House Latheronwheel Cathness KW5 6DW Scotland UK 996 Gruen AW (985) Adaptve least squares correlaton: a powerful mage matchng technque S Afr Journal of Photogrammetry Remote Sensng and Cartography 4(3): Holland JH (99) Adaptaton n Natural and Artfcal Systems: An Introductory Analyss wth Applcatons to Bology Control and Artfcal Intellgence The MIT Press Cambrdge MA USA Frst appear n 975 Mason SO and Gruen A (995) Automatc Sensor Placement for Accurate Dmensonal Inspecton Computer Vson and Image Understandng 3(6): Mason SO (994) Expert System-Based Desgn of Photogrammetrc Networs PhD thess Insttut für Geodáse und Photogrammetre Zürch May 994 Olague G and Mohr R (998) Optmal Camera Placement to Obtan Accurate 3D Pont Postons In Proceedngs of the 4 th Internatonal Conference on Pattern Recognton Brsbane Australa Vol pages 8-0 August IAPR-APRS Olague G and Mohr R (000) Optmal Camera Placement for Accurate Reconstructon Submtted to Machne Vson and Applcatons Prevous verson appears as INRIA Research Report RR-3338 January 998 Olague G (998) Planfcaton du placement de caméras pour des measures 3D de précson PhD thess Insttut Natonal Polytechnque de Grenoble Octobre 998 ftp://ftpmagfr/pub/medathequeimag/theses/98- OlagueGustavo/notce-francashtml