UNIVERSIDADE DE COIMBRA

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1 UNIVERSIDADE DE COIMBRA DEPARTAMENTO DE ENGENHARIA ELECTROTÉCNICA E DE COMPUTADORES INSTITUTO DE SISTEMAS E ROBÓTICA COIMBRA, PORTUGAL ARTICLE: Pose Estmaton for Non-Central Cameras Usng Planes Author: Pedro MIRALDO mraldo@sr.uc.pt Co-Author: Helder ARAUJO helder@sr.uc.pt CONTENTS I Introducton I-A Proposed Approach I-B Notaton II Relatonshp of Incdence usng the Homography 2 III Proposed Algorthm 2 III-A Estmaton of the Homography Matrx III-B Ambgutes III-C Recovery of the Pose Parameters IV Refnement of the Parameters 4 V Experments 4 V-A Synthetc Experments V-B Experments wth real data VI Conclusons 5 VI-A Analyss of the Experments VI-B Closure References 6 Preprnt: submtted to the IEEE Int l Conf. Autonomous Robot Systems & Compettons ROBÓTICA 4 February, 4.

2 Pose Estmaton for Non-Central Cameras Usng Planes* Pedro Mraldo and Helder Araujo Insttute of Systems and Robotcs, Department of Electrcal and Computer Engneerng Unversty of Combra, COIMBRA, PORTUGAL Emal: {mraldo, Abstract In ths paper we study pose estmaton for noncentral cameras, usng planes. The method proposed uses nonmnmal data. Usng the homography matrx to represent the transformaton between the world and camera coordnate systems, we descrbe a non-teratve algorthm for pose estmaton. In addton, we propose a parameter optmzaton to refne the pose estmate. We evaluate the proposed solutons aganst the state-of-the-art method n terms of both robustness to nose and computaton tme. From the experments, we conclude that the proposed method s more accurate aganst nose. We also conclude that the numercal results obtaned wth ths method mprove wth ncreasng number of data ponts. In terms of processng speed both versons of the algorthm presented are faster than the state-of-the-art algorthm. I. INTRODUCTION Consderng only geometrc enttes, an magng system s a mappng between the 3D world and a 2D mage []. Ths mappng can be represented by an ndvdual assocaton between world 3D lnes and pxels n the mage plane [2], [3]. Camera calbraton conssts n the estmaton of the correspondences between mage pxels and the correspondng projectng 3D straght lnes. Usually mage space does not change and, as a result, we can defne a 3D coordnate system for the mage coordnates. On the other hand, a camera s a moble devce and as a consequence we can not defne a fxed global coordnate system to represent the lnes mapped nto the mage ponts. Therefore we defne a 3D reference coordnate system assocated wth the camera to represent the 3D lnes mapped nto the mage pxels []. As a consequence, to estmate the coordnates of 3D enttes represented n a dfferent coordnate system, we need to estmate a rgd transformaton mappng the camera coordnate system nto the world coordnate system. For central camera models whch can be modeled by a perspectve projecton Fgure a, several approaches to absolute pose estmaton, such as [4], [5], have been proposed for both mnmal confguratons sutable for hypotheszeand-test archtectures lke RANSAC [6] and non mnmal cases [7], [8], [9], [0]. Some work has also been developed extendng pose estmaton methods so that ponts and lnes can be used, such as e.g. [], [2]. Most of the algorthms for the estmaton of pose are based on arbtrary 3D target pont confguratons. In many problems such as moble robotcs and augmented realty, t s pratcal to use planar patterns to compute the absolute pose. For the case of central cameras several approaches usng 3D coplanar ponts were developed, such as [3], [4]. In the last few years, cameras whose projecton rays do not ntersect at a sngle effectve vew pont magng devces that can not be modeled by a central projecton, see Fgure b Ths work was supported by the Portuguese Foundaton for Scence and Technology Ref: PTDC/EIA-EIA/22454/0 and the Portuguese program Mas-Centro Ref: CENTRO-07-ST24-FEDER started to be used, due essentally to the large felds of vew that can be obtaned. Examples of such confguratons occur when the magng rays are subject to reflecton and/or refracton. For these magng devces new methods and algorthms have to be consdered for the estmaton of the absolute pose. For non-central camera models, there are algorthms for the mnmal case, [5], [6]. For the non-mnmal case, Schweghofer and Pnz at [7] proposed an teratve soluton for global pose estmaton. Despte the fact that the method proposed by Schweghofer and Pnz can be appled to both planar and non-planar cases, the method s teratve. In ths artcle, we address the problem non-mnmal absolute pose estmaton for general non-central cameras, when consderng the case where the world ponts belong to a plane. We present a non-teratve algorthm to estmate the pose. In addton, we also propose a refnement of the estmaton of the pose parameters by means of an optmzaton usng the Quas-Newton algorthm. A. Proposed Approach For the estmaton of the 3D pose, the calbraton of the magng devce s assumed to be known. We use the Generalzed Camera Model [2], whch can represent any type of magng devce central or non-central. Ths model assumes that an mage pxel s mapped nto an arbtrary ray n 3D world. Snce we assume that the camera has been prevously calbrated, for all mage pxels we know the correspondng 3D straght lne coordnates n the camera coordnate system. Pose s gven by the estmates of the rotaton and translaton parameters that defne the transformaton between the camera and the world coordnate systems. In ths artcle we use the homography map to represent ths transformaton. Usng the homography matrx and based on the relatonshp of ncdence between ponts and lnes n 3D space, we defne an algebrac relatonshp for the pose. However, the homography matrx s a functon of both transformaton and 3D plane parameters [8]. As a result, we dvded the estmaton of the homography nto two steps: we defne a space of solutons wth three degrees of freedom for the homography matrx, based on the algebrac relatonshp between the 3D ponts and 3D lnes; usng the nformaton of the 3D plane, three constrants that the space of solutons must satsfy are defned. The homography matrx s estmated usng these constrants. B. Notaton In general, bold captal letters e.g. A R n m, n rows and m columns, bold small letters e.g. a R n, n elements and small letters e.g. a represent matrces, vectors and one dmensonal elements respectvely. The matrx represented as â lnearzes the exteror product such that a b = âb. Let us consder: known matrces U R n m, V R k l and C; and an unknown matrx X R n l. Usng Kronecker

3 Equaton 3. Thus, from Equaton 4 and snce m C m C = 0, we derve the followng relaton d C Hp W m C = m C dc Hp W = 0. 5 Image Plane a b Fg.. Depcton of the pose estmaton problem, usng planar patterns. Fgure a shows pose estmaton usng central cameras. Fgure b shows the pose estmaton confguraton n the case of a general non-central camera. product we can defne the followng relaton UXV T = C V U vec X = vec C where represent the Kronecker product wth V U R nk nl and vec. s a vector formed by the stackng of the columns of the respectve matrx. II. RELATIONSHIP OF INCIDENCE USING THE HOMOGRAPHY Pose estmaton requres the estmaton of a rotaton matrx R SO 3 and a translaton vector t R 3 that defne the rgd transformaton between the world and camera coordnate systems. Snce we consder that the magng devce s calbrated, pose s specfed by the rgd transformaton that satsfes the relatonshp of ncdence between ponts n the world coordnate system and 3D straght lnes represented n the camera coordnate system, Fgure. To dstngush between features represented n the world coordnate system and enttes n the camera coordnate system, we use the superscrpts W and C respectvely. The rgd transformaton between a pont n world coordnates p W and the same pont n camera coordnates p C s gven by p C = Rp W + t. 2 Snce we use the assumpton that all the ponts belong to a plane Π W, from the homography map [], [8], we can rewrte Equaton 2 as p C = R + ζ tπt p W 3 }{{} H where Π W. = ζ, π R 4, H R 3 3 s called the homography matrx, ζ and π are the dstance from the plane to the orgn and the unt normal vector to the plane Π W respectvely. For pose estmaton, we assume that the non-central camera s calbrated. Therefore for each pxel the correspondng 3D lne s known n the camera coordnates system. Let us consder that lnes are defned usng Plucker coordnates l C R =. d C, m C, where d C and m C are the drecton and moment of the lne respectvely, constraned to d C, m C = 0. From the 3D ncdent relaton between a lne and a pont [9], we have d C p C = m C d C p C m C = 0. 4 Snce our goal s to estmate the pose usng coplanar ponts p W Π W, we can use the homography map to transform ponts from camera coordnates nto the world coordnates Usng the Kronecker product, we solate the unknown matrx H, such that p W T m C dc vec H = 0. 6 From the propertes of the Kronecker product [], the dmenson of the column-space of p W T m C d C s equal to the product of the dmenson of the column-space of p W T and m C d C. Snce d C, m C = 0, t can be shown that the dmenson of the column-space of both p W T and m C d C are equal to one. As a result, the dmenson of the column-space of p W T m C d C s one. Snce the dmenson of the column space s equal to the number of lnearly ndependent columns/rows, we conclude that Equaton 6 only has one lnearly ndependent rows. III. PROPOSED ALGORITHM In the prevous secton we descrbe an algebrac relatonshp between the coordnates of ponts represented n the world coordnate system and the coordnates of lnes represented n the camera coordnate system, for an unknown homography matrx. However, we note that t does not contan all the known nformaton. Snce the coordnates of the ponts are known, the coordnates of the plane Π W. = ζ, π are also known whch, accordng to the defnton of the homography matrx Equaton 3, must be taken nto account on the estmaton of the homography map. However, and f the data are not corrupted wth nose, these constrants need not to be taken nto account n the estmaton of the homography matrx. If, on the other hand, data are affected by nose the estmated homography map wll be an approxmaton. If the constrants assocated to the plane coordnates Π W are not mposed, the error on the estmaton of the parameters wll affect the elements of ζ, π, whch wll decrease the accuracy of the method. In the rest of ths secton we derve an approach whch takes nto account the plane parameters n the computaton of the homography matrx. Wthout loss of generalty, we consder a rgd transformaton { R, t} of the coordnates of the world ponts, such that the coordnates of the 3D ponts and of the plane are p W = Π W. = ζ, π = Rp W + t, 7 t T Rπ + ζ, Rπ 8 such that π s proportonal to the z axs and ζ =. A. Estmaton of the Homography Matrx Usng the representaton of the world ponts descrbed n the prevous secton and the algebrac constrants defned n Equaton 6, for a set of ponts n the world and respectve 3D lnes, we am to estmate a homography matrx H, such that p W T C C m d. p W T C C N m N d N }{{} M vec H = 0. 9

4 Mean Fve Smallest Sngular Values Medan Fve Smallest Sngular Values Fg. 2. In ths fgure we show the mean and medan of the varaton of the fve smallest sngular values of the matrx M as a functon of the nose we use the varable mentoned n the secton descrbng the experments. In theory, and wthout nose, for N 8, we wll have one sngular value of M equal to zero. Ths means that the space of solutons for the homography matrx H s one-dmensonal. In that case the soluton s gven by the rght sngular vector that corresponds to the zero sngular value. Snce matrx H must have the second smallest sngular value equal to one, ths condton can be used to determne the correct soluton from the one-dmensonal space of solutons. However, wth nosy data, and n general, no sngular value s equal to zero. The varaton of the fve smallest sngular values as a functon of the nose level s shown n the Fgure 2. As we can see from ths fgure, when the nose standard devaton ncreases, the three smallest sngular values take smlar values. { As a result, we select three rght sngular vectors e, e 2, e 3} that correspond to the three smallest sngular values of the matrx M. Usng ths set of rght sngular vectors we defne the space of solutons for vec H as vec H { }. = α e + α 2 e 2 + α 3 e 3 : α R,. 0 Unstackng the vectors e to matrces F, we defne the matrx H as a functon of the unknowns α, such that H = α F + α 2 F 2 + α 3 F 3. However and from the fact that ζ =, π must be parallel to the z axs and from Equaton 3, the homography matrx must verfy p C = R + [ ] 0 0 t p W 2 } {{ } H where R and t are respectvely the unknown rotaton and translaton that defne pose. Snce R SO 3, h = r and h 2 = r 2 h and r are the th columns of matrces H and R respectvely, t s possble to defne the followng constrants that apply to the frst and second column of the estmated homography matrx h T h =, h T 2 h 2 = and h T h 2 = 0. 3 From the space of solutons for the homography matrx defned at Equaton, we can defne the columns h and h 2 as h = α f + α 2 f 2 + α 3 f 3 4 h 2 = α f 2 + α 2 f α 3 f where f j s the j th column of the matrx F. Wthout loss of generalty, we can defne h = h /α, whch means h = f + bf 2 + cf 3 and h 2 = f 2 + bf cf and b = α 2 /α and c = α 3 /α. Usng ths formulaton we rewrte the constrants of Equaton 3 as h T h 2 = 0 and h T h h T 2 h 2 = 0. 7 Replacng the columns of the homography matrx n these constrants usng Equaton 6, we defne two constrants that apply to the space of the unknowns b and c. These constrants can be expressed by two functons g b, c = 0, for =, 2, of the form g b, c = κ b2 + κ 2 bc + κ 3 c2 + κ 4 b + κ 5 c + κ 6. 8 Thus, the soluton for the proposed problem s the set of unknowns b and c such that g b, c = g b, c = 0. 9 The formulaton of the Equaton 9 represents the estmaton of the ntersecton ponts between two quadratc lnes. From the Bézout s theorem [2], the theoretcal maxmum number of soluton for ths problem s four. In the remanng of ths secton we descrbe a method to solve ths problem. Let us consder the constrant g b, c = 0. Solvng ths equaton for the unknown b we get two solutons b = p [c] 2κ ± v[c]/2 2κ where p [c] and v[c] are two polynomal equatons wth unknown c and degrees one and two respectvely. Substtutng the unknown b on g 2 b, c = 0 usng Equaton, we get the constrant p 2 [c] ± p 3 [c]v[c] /2 = 0 p 2 [c] = p 3 [c]v[c] /2 2 where the degree of the polynomal equatons p 2 [c] and p 3 [c] are respectvely two and one. Squarng both sdes of Equaton 2 we get p 2 [c] 2 = p 3 [c] 2 v[c] p 4 [c] = p 2 [c] 2 p 3 [c] 2 v[c] = 0 22 where the polynomal equaton p 4 [c] has degree four. Thus, to fnd c that solves the problem defned by Equaton 9 we just need to fnd the roots of the four degree polynomal equaton p 4 [c], whch can be solved n closedform e.g. usng the Ferrar s technque for solvng the general quartc roots. For each real soluton of c we get the unknown b selectng the correct soluton on Equaton. To conclude the algorthm, we recover the soluton for α usng α = ±, α 2 = bα and α 3 = cα. 23 h Note that f we have a soluton array α, α 2, α 3, from Equatons 0 and 9, the solutons array α, α 2, α 3 wll also verfy the same contrants, and that s why we attrbute both sgns to α.

5 B. Ambgutes From the prevous secton, we see that we can have multple solutons for the set of unknowns α, α 2, α 3. For the computaton of the soluton descrbed at Secton III-A, t s only requred that N 6. However for N 8 the dmenson of the null space of M wll be equal or smaller than one and, as a result, for the dfferent set of possble arrays α, α 2, α 3 and as a result vec H we wll get non zero solutons for the algebrac relaton of Equaton 9. Only when the data s noseless, we can get a sngle zero soluton. As a result and from the set of possble solutons, we can choose the one that mnmzes the Equaton 9. Note that the solutons are obtaned n pars α, α 2, α 3 and α, α 2, α 3. Thus, two solutons wll be selected from the prevous paragraph. However these two solutons wll generate two homography matrces. Moreover, these two solutons wll be dffer only wth respect to the sgn, ± H. From Equaton 4, the estmated solutons for the homography matrx must verfy the followng condton ± C d H p W = m C,. 24 As a result, we choose the sgn of the estmated homography matrx that mnmzes ths equaton, for all the mappngs between 3D ponts and lnes. C. Recovery of the Pose Parameters To recover the pose parameters, R, t, we frst have to decompose the matrx H nto R and t. Snce h = r and h 2 = r 2 and from Equaton 3, usng H we can defne R = h h 2 h h 2, 25 t = h 3 h h Note that the constrants defned n Equaton 4 are verfed, whch means that R SO 3. To conclude, the estmaton of the absolute pose R and t, takng nto account the rgd transformaton defned by R and t, are gven by R = R R and t = R t + t. 27 IV. REFINEMENT OF THE PARAMETERS In addton to the non-teratve algorthm descrbed n Secton III-A, we propose an teratve refnement of the rotaton and translaton parameters that defne the pose. Usng the geometrc dstance between a 3D lne and a world pont, d C p C m C d l C, p C. = d C, 28 and snce we are consderng coplanar ponts such that π s parallel to the z axs and ζ = and usng Equaton 3, we defne the geometrc dstance between a world pont and a 3D lne as d l C, p W = d C R + [ 0 0 t ] p W m C d C. 29 We am to mnmze the sum of the squares of the geometrc dstance defned n the prevous equaton 2 argmn d l C, p W 30 R,t for all the mappngs between world ponts and 3D lnes. We consder the rotaton parametrzaton usng quaternons [8]. To fnd the soluton for Equaton 30, we consder the nonteratve soluton at Secton III-A and use the Quas-Newton optmzaton technque teraton method [] to refne the parameters of both rotaton and translaton. V. EXPERIMENTS We evaluate the proposed algorthm by comparng t to the method proposed by Schweghofer and Pnz at [7], usng both synthetc and real data. A. Synthetc Experments In ths secton we evaluate both the non-teratve algorthm and the non-teratve algorthm plus the refnement of the pose parameters, usng synthetc data sets, aganst the state-of-theart method. For that purpose consder a cube wth 800 unts of sde length. The data was generated by randomly mappng 3D lnes and ponts { } l C p C, for =,..., N. A random rgd transformaton was randomly generated R and t, where the translaton parameter s defned n the same cube wth 800 unts of sde length and appled to the set of ponts such that p C p W. The dataset for the pose problem s { } l C p W. Let us consder that the estmated pose s gven by { R, t}. We consder both rotaton and translaton metrcs for the computaton of the error such that: Rotaton error: d rotaton = R R ; frob 2 Translaton error: d translaton = t t. To generate the 3D lnes l C we consder the followng procedure. For each p C, an addtonal world pont q C s computed and thus, the lne n Plücker coordnates l C s computed as shown n [9]. A varable labeled as Devaton From Perspectve Camera s also defned: the value of ths varable represents the length of the sdes of the cube to whch the set of ponts { } q C must belong. Note that when ths value tends to zero, the camera model tends to central and that s the reason why the varable s named Devaton From Perspectve Camera. In addton, we also defne a varable to represent nose. Instead of consderng the set { } p C, q C to compute the lne, we consder { } p C + r, q C, where vector r has random drecton and whose the norm s dstrbuted accordng to a normal dstrbuton whose standard devaton s varable. Ths varable s named n the experments. We compare the state-of-the-art algorthm proposed by Schweghofer and Pnz at [7] wth both the non-teratve method and ts varaton wth the teratve parameter refnement method. The comparson s performed takng nto account the accuracy and the processng tme. The accuracy was evaluated as a functon of the number of ponts used to compute the pose, Fgure 3a; the Nose Level, Fgure 3b; and the Devaton From Perspectve Camera, Fgure 3c.

6 Mean Rotaton Error Our+LM Our SP 0 0 Medan Rotaton Error 0 2 Mean Translaton Error 0 2 Medan Translaton Error a Rotaton and translaton errors means and medan as a functon of the number os ponts. We use a of 7.5 unts and a Devaton from Perspectve Camera value of 50 unts. The y-scale of the graphcs s represented n logarthmc bass Mean Rotaton Error Our+LM: 50 Our: 50 SP: 50 Our+LM: 50 Our: 50 SP: 50 Our+LM: 400 Our: 400 SP: Medan Rotaton Error Mean Translaton Error Medan Translaton Error b Rotaton and translaton errors mean and medan as a functon of the. We use three dfferent number of ponts represented at three dfferent colors: black, blue and orange for 50, 50 and 400 number of ponts respectvely. Mean Rotaton Error Our+LM: 50 Our: 50 SP: 50 Our+LM: 50 Our: 50 SP: 50 Our+LM: 400 Our: 400 SP: Devaton From Prespectve Camera Medan Rotaton Error Devaton From Prespectve Camera Mean Translaton Error Devaton From Prespectve Camera Medan Translaton Error Devaton From Prespectve Camera c Rotaton and translaton errors mean and medan as a functon of the dstrbuton of the 3D lnes. We use a of 7.5 unts and three dfferent values for the number of ponts as the Fgure b. Fg. 3. In ths fgure we show the accuracy results of the applcaton of the proposed non-teratve wthout and wth parameters refnement algorthms compared to the state-of-the-art methods of Schweghofer and Pnz we dentfy our non-teratve soluton as Our, our non-teratve soluton plus a parameter refnement by Our + LM and the Schweghofer and Pnz algorthm as SP. We evaluate both algorthms n terms of: number of ponts used, Fgure a; n terms of the standard devaton of the nose, Fgure b; and n terms of the dstrbuton of the 3D lnes, Fgure c. In all the cases, the accuracy of the pose was measured for both rotaton and translaton mean and medan. To conclude the experments wth synthetc data we show a comparson between the processng tmes for the proposed algorthm and for the Schweghofer and Pnz algorthm, n Fgure 4. We note that our algorthm was fully mplemented n MATLAB whle the algorthm of Schweghofer and Pnz uses the SEDUMI optmzaton toolbox [22], whch s mplemented n C/C++. B. Experments wth real data In addton to the experments wth synthetc data, we evaluate our algorthm the state-of-the-art algorthms usng real data. We consder calbrated non-central sphercal catadoptrc camera as shown n Fgure 5. Usng a chess-board plane wth 60 ponts, we get a set of mages, movng the plane to dfferent postons and wth dfferent orentatons. The metrcs for the errors used n ths secton were the same as those used n the prevous secton. The results were: for our non-lnear method, we obtaned a mean error of for the rotaton matrx and a mean of [mm] for error on the translaton vector; for our nonlnear method plus wth teratve refnement, we obtaned a mean error of for estmate of the rotaton matrx and 5.59[mm] for error on the translaton vector; for the algorthm proposed by Schweghofer and Pnz, we obtaned a mean of for the rotaton error and 6.5[mm] for the translaton error. VI. CONCLUSIONS A. Analyss of the Experments From the experments wth synthetc data we notce that the results obtaned wth non-teratve algorthms are, n general, worse than both non-teratve plus parameters refnement proposed also n ths artcle and the teratve soluton proposed by Schweghofer and Pnz. As we can see from Fg. 3a, the results for the non-teratve case are nferor to both teratve cases when the number of ponts s smaller than 300. In general, as t can be seen from Fgure 3b, the nonteratve soluton yleds worse results when compared wth both teratve methods. In some cases, t can be seen that the non-teratve method proposed n ths paper yelds better results than the teratve state-of-the-art method proposed by Schweghofer and Pnz. However and from the same fgure, we can see that the non-teratve approach plus the teratve

7 Processng Tme Tme n Seconds Our+LM Our SP Fg. 4. In ths fgure we dsplay the processng tmes correspondng to the method proposed by Schweghofer and Pnz and to our algorthm, as a functon of the number of ponts. These results correspond to the experment descrbed n Fgure 3a. We note that whle our method s fully mplemented n MATLAB, the optmzaton of the Schweghofer and Pnz algorthm s mplemented n C/C++ we label our non-teratve soluton as Our, our non-teratve soluton plus a parameter refnement by Our + QN and the Schweghofer and Pnz algorthm as SP. refnement, suggested n Sec. IV, gves better results than the non-teratve method proposed by Schweghofer and Pnz. We have to take also nto account that, as shown n Fg. 4, both our methods are sgnfcantly faster than Schweghofer and Pnz approach, whch by tself s an advantage. The method non-teratve plus a parameter refnement gves better results then the non-teratve and Schweghofer and Pnz for almost all cases n both Fgs 3a and 3b. To conclude the analyss of the errors, we note that from Fg. 3c, the applcaton of the method proposed by Schweghofer and Pnz deterorates when the Devaton from Perspectve Camera decreases. On the other hand, and for the non-teratve method proposed n ths paper, such effect s not notceable. In terms of computaton tme Fg. 4, the non-teratve method s clearly faster. The processng tme for the non-lnear plus parameter refnement tends to grow more rapdly than the processng tmes for the Schweghofer and Pnz algorthm. B. Closure In ths paper we addressed the planar pose problem for noncentral camera models. To the best of our knowledge, ths s the frst planar-based algorthm for general non-central cameras. We propose two methods: a fast non-teratve soluton; and ths soluton plus a parameter refnement. From the expermental results, we can conclude that both our approaches are sgnfcantly faster than the state-of-the-art method, specally the non-teratve soluton. 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