Accurate Feature Extraction and Control Point Correction for Camera Calibration with a Mono-Plane Target

Transcription

1 Accurate Feature Etracton and Control Pont Correcton for Camera Calbraton wth a Mono-Plane Target Y. Xao Unversty of Ednburgh yao@nf.ed.ac.uk R.B. Fsher Unversty of Ednburgh rbf@nf.ed.ac.uk Abstract The paper addresses two problems related to 3D camera calbraton usng a sngle mono-plane calbraton target wth crcular control marks. The frst problem s how to compute accurately the locatons of the features (ellpses) n mages of the target. Snce the structure of the control marks s known beforehand, we propose to use a shape-specfc searchng technque to fnd the optmal locatons of the features. Our eperments have shown ths technque generates more accurate feature locatons than the state-of-the-art ellpse etracton methods. The second problem s how to refne the control mark locatons wth unknown manufacturng errors. We demonstrate n a case study, where the control marks are laser prnted on a A4 paper, that the manufacturng errors of the control marks can be compensated to a good etent so that the remanng calbraton errors are reduced sgnfcantly. 1. Introducton Recent years have seen the popularty of camera calbraton usng a mono-plane target (on whch the control ponts are all coplanar). A mono-plane target can be economcally made by ust smply attachng a paper prnted wth dentfable marks (control marks) to a flat surface [1], offerng great convenence for camera calbraton n vson applcatons such as DVS (Desktop Vson System) [2]. However, the easy-to-use property of the mono-plane target should not compromse the accuracy of calbraton. In ths paper, we dscuss ssues related to how to mprove calbraton accuracy wth a mono-plane target. Specfcally we nvestgate two problems: 1) how to etract accurate features (proecton of control marks) n mages of the target and 2) how to obtan accurate locatons of the actual control marks on the target, gven unknown manufacturng errors. It s clear that a better calbraton can be acheved f we have more accurate features and control marks. The control marks on the calbraton target used n ths study are a 2D array of crcles (Fg.1). Crcular marks are commonly adopted for calbraton targets [3]. Compared wth checkerboard patterns, another popular category of control marks for calbraton, crcular marks offer at least the followng advantages: 1) crcular marks can be effcently manufactured wth good precson; 2) accuracy of ellpse (proecton of crcle) detecton n mages s arguably hgher ( for nstance, [4] reports a 1/100 pel level of ellpse detecton accuracy, whle corner detecton from checkerboard mages can acheve only about 1/10 pel level of localzaton accuracy [17]); 3) the symmetry n shape of crcles and ellpses can be used to mprove sgnfcantly the accuracy of localzaton of the crcles and ellpses, makng them very blur-resstant (see Secton 2.1); 4) the smoothness of crcular/ellptcal shapes allows the applcaton of effectve optmzaton technques n searchng for optmal locatons of the crcles/ellpses. The proecton of a crcle s often an ellpse. In prevous work, varous ellpse detecton technques were used for calbraton [4-8]. However, a maorty of these methods etract ellpses ndvdually n mages wthout reference to the structure of the crcles on the calbraton target. We argue that such methodology may not be optmal for accurate feature etracton, snce detecton of a sngle ellpse can be vulnerable to mage defcences such as nose, non-unform llumnaton, etc. There are methods that map the entre structure of the control marks to the mage planes and match them wth real calbraton mages [9]. Whle seemngly more accurate, they suffer from modelng unknown factors n the magng process such as llumnaton, surface reflectance, pont-spread functon of the lens, etc., degradng ther performance n real world applcatons. In ths paper, we propose a method that employs the structure of the control crcles on the calbraton target to gude the etracton of the correspondng ellpses n mages. It stll etracts each ellpse ndvdually, however, the shape of the ellpse s constraned by the locaton and orentaton of ts correspondng crcle on the calbraton target, therefore potentally more accurate and relable ellpse locatons can be acheved. The other problem studed n ths paper s related to the locatons of the control crcles on the calbraton target. Ideally the locatons should be known beforehand. However, for the calbraton target we used on whch the

2 control crcles are prnted by a hgh-precson laser prnter, we have found the mnor offsets of the prnted crcle locatons relatve to ther deal locatons contrbute consderably to the overall resdual calbraton errors. We propose a method to compensate for the offsets based on the assumpton that the offsets are caused by non-unform paper loadng speed by the prnter. Our eperments support ths hypothess, and demonstrate that calbraton errors are sgnfcantly reduced after the compensaton. Whle the paper s largely about 2D ssues, camera ntrnsc, etrnsc and lens dstorton parameter estmaton s fundamental to and s prmarly seen n 3D contets, where stereo cameras or lght strpe trangulaton systems depend on accurate camera calbraton. The remander of the paper s organzed as follows. Secton 2 dscusses the problem of ellpse etracton and Secton 3 addresses the problem of crcle locaton correcton. 2. Feature etracton 2.1. Technques The control marks used for calbraton n ths study are crcles. When a camera observes the calbraton target, the crcles on the target surface are proected to the mage plane as ellpses (Fg.1). In order to calculate parameters of the camera, the locatons of the ellpses n the mages of the target are needed, whch s usually done through ellpse etracton technques. Frst, the mult-step approach (mage-edge-ellpse) employed n prevous work does not have an overall measurement of the qualty of etracted ellpses. Instead, each step ntroduces ts own errors, allowng errors to propagate. Second, t s well known that edge detecton and gradent calculaton are senstve to mage defcences, whch may render many etracton results nvald. Fnally, the specfc nformaton about the calbraton scene s not used. We argue that better ellpse etracton results can be acheved by eplotng the scene nformaton and not separatng ellpse etracton nto multple steps. Look at the calbraton scene n Fg.1. The crcles are arranged n a 2D array. The rad of the crcles and the dstances between the crcles are known. The background s a whte paper and the foreground obects are sold black crcles. In such a specfc crcumstance, t s qute possble that the scene nformaton, e.g., the structure of the control crcles and the background/foreground propertes, can be used to assst ellpse etracton to acheve better accuracy and relablty. To ths end, we devsed a method to fnd optmal ellpse parameters from the target mages drectly, avodng the problems wth the prevous mult-step approaches. The method mamzes the dfference of ntensty between the nsde and outsde of the ellpse, whch provdes a smple and yet effectve measurement of the qualty of ellpse etracton ustfed by the background/foreground property of the mages. The metrc structure of the control crcles s used to ntalze the optmzaton for the ellpse to avod local mnma. Fgure 1: The calbraton target at dfferent vews. Ellpse etracton has a long hstory of study n computer vson wth applcatons rangng from obect recognton [5] to 3D reconstructon [6]. Early methods often take three steps: etract edge ponts, then use the Hough transform [10] to fnd ponts belongng to ellpses, and fnally use ellpse fttng technques [11] to obtan ellpse equatons. Latest methods acheve subpel accuracy n ellpse etracton by employng a template to compute subpel edge locatons [7] and usng mage gradents to derve addtonal curvature nformaton [8]. Whle these methods are for general purpose, they are not necessarly optmal n the specfc case of ellpse etracton for camera calbraton. Fgure 2: 1D profle of an ellpse from Fg. 1 and half ponts. The graylevel propertes of target mages are llustrated n Fg. 2, where the 1D graylevel profle of an ellpse n Fg.1 s plotted. Ideally the profle should be a bnary sgnal wth a flat bottom and a flat top representng the foreground and background respectvely, and a step edge between them representng the boundary of the ellpse. However, the real profle has a nosy top, nosy bottom and a few pels of wdth n transton, due to varous factors ncludng llumnaton, magng nose, mage blur, etc. It s nontrval to recover the true locaton of the edge (ellpse boundary) from a contamnated sgnal lke Fg.2, especally when the models of llumnaton, mage nose, and mage blur are not

3 known. In ths paper, we propose to use the half-ponts of the sgnal to capture the edge locatons. A half pont of a sgnal s the pont where the sgnal has a graylevel value of the mddle between the top and the bottom of the deal bnary sgnal. Snce the deal sgnal s usually unknown, we use the followng formula to estmate a half pont: argma( ω + ω g ( u) du u) du ) (1) where u) denotes a graylevel sgnal, ω s a coeffcent to defne an ntegraton wndow. The wndow should be set around an edge of the sgnal and not etend to the other edges. For the sgnal n Fg. 2, ω=5 works suffcently. It s not dffcult to prove that formula (1) gves the eact half pont f the sgnal s lnearly nterpolated between the top and the bottom. If t s not, our hypothess s that formula (1) generates consstent systematc shfts on the edge locatons and the shfts can be compensated when an ellpse s etracted as a whole wth ts boundary all satsfyng formula (1) snce shfts n opposte drectons of the ellpse have opposte sgns. The ellpse can then be estmated usng the followng formula: 1 arg ma( A V C 1 u, v) dudv A Ω + Ω + u, v) dudv ) In formula (2), Ω+ denotes the regon between the ellpse boundary C and ts offset curve that s parallel to C wth dstance ω, and the area of Ω+ s A +. Smlarly, Ω s the regon between the ellpse boundary and ts offset curve wth dstance -ω, and the area of Ω s A. V C denotes the parameters of the ellpse C. In our mplementaton of the method proposed n ths paper, the ellpse parameters are chosen as sem aes rad, ellpse orentaton and ellpse center. The ellpse orentaton s epressed by the angle between the maor as and the -as n the mage coordnate frame. By defnton, formula (2) s equvalent to mamzng the dfference of mean graylevel values between the nsde and the outsde of the ellpse. In prncple, t s smlar to the SUSAN edge detector [12]. The numercal appromaton of formula (2) s as follows: argma VC 0 + n ) sω = sω s = 0 + n ) s In formula (3), the ntegrals n formula (2) are turned to sums of graylevel values on grds n Ω + and Ω. The lattudes of the grds are parallel to the ellpse boundary C whch s sampled at ponts, and the longtudes are (2) (3) decded by n whch are unt normal vectors of the ellpse boundary C at. s s used to adust the samplng nterval along a longtude. The graylevel values of the sampled ponts on the grds can be calculated from the orgnal mage usng nterpolaton technques [13]. Equaton (3) can be optmzed usng the Smple technque [14]. To ensure the optmzaton fnds the correct ellpse boundary locatons, t s mportant to ntalze the ellpse parameters near the optmal soluton. The ntalzaton s done as follows. Frst a blob detecton technque called BCoM (see detals n the Secton 2.2) s used to roughly estmate the ellpse centers, and then the camera s calbrated usng the feature ponts, fnally the parameters of the ellpses are obtaned by proectng the crcles on the calbraton target to the mage plane. It has been shown n our eperments that the BCoM s able to acheve subpel accuracy n ellpse center etracton (some quanttatve evaluaton n Secton 2.2), whch serves well to provde ntal estmate of the ellpse parameters Results We carred out eperments to evaluate the ellpse etracton technque above. The frst eperment s wth synthetc data that was generated as follows. Fve parameters: sem aes rad a and b, orentaton φ, ellpse center u 0, v 0 were used to specfy the geometry of an ellpse. The parameters were randomly chosen n the ranges a [7, 13], b [3, 7], φ [-π, π], and u 0, v 0 [20,21]. The graylevel value nsde the ellpses s 150 and outsde s 50. The mages were frst produced n hgher resoluton (410 by 410) and then smoothed usng Gaussan flter wth σ = 10, fnally rescaled to resoluton 41 by 41 and contamnated by addtve Gaussan nose wth σ = test mages were generated. The condton of syntheszng the ellpse mages s the same as [4], n whch two subpel ellpse etracton methods: the moment preservng (MP) ellpse detector [7] and the moment and curvature preservng (MCP) detector [4] were tested and compared. The smulated mages were used to test our mamzng outsde-nsde dfference (MOID) ellpse etracton method, therefore we can make a far comparson wth the prevous MP and MCP methods. The parameters of MOID n ths test were chosen as ω=3 and s= 0.1. Snce the blur kernel n synthetc data s Gaussan wth σ = 1 (mplemented n Gaussan wth σ = 10 n the 10X enlarged mages), ω=3 proved a good balance between algorthm effcency and accuracy n our eperments. Larger ω only produced neglgble accuracy mprovement wth a ncreased computatonal cost lnearly proportonal to ω. Smlarly, s= 0.1 was chosen emprcally from a few trals of the MOID performance. Wth ths setup

4 of parameters, the MOID detected all 100 ellpses n a tme of seconds. Our mplementaton of the algorthm was wrtten n Matlab scrpt, and the computer s a wndows PC wth 2GB RAM and Intel Xeon 2.99 GHz CPU. The comparson results are summarzed n Table 1, where the RMS (root mean square) errors of the etracted ellpse centers are lsted. It can be seen that our MOID method acheves hgher precson n ellpse center etracton from the synthetc mages than the MP and MCP methods, wth about 20% and 10% mprovement each. The accuracy of the MOID method has also been tested n real calbraton tasks. Table 3 reports the calbraton results from ellpse centers usng the MOID method for three stereovson systems. The reported measurements are descrbed below. Each stereovson system has a par of cameras of dfferent type, as summarzed n Table 2. Bnary Center of Mass (BCoM) and Graylevel Center of Mass (GCoM) are also computed from the mages to make comparson. The BCoM method computes the centrod of a blob (representng an ellpse) segmented from the orgnal mage (the technque s equvalent to computng the center of mass of the bnary mage of the blob). The segmentaton s done by usng Ostu s method [15]. The graylevel center of mass [10] s calculated on an mage wndow whch s epanded 10 pels from the boundng bo of the segmented blob. The BCoMs and GCoMs can be used as estmate to the centers of ther underlyng ellpses. Table 1 RMS of detected ellpse centers method MP MCP MOID y Table 2 Specfcatons of the stereo systems evaluated System Camera Make Mkrotron Puln Canon Camera Type Vdeo B/W Vdeo B/W Statc Colour Resoluton Lenses 75mm 35mm 58mm Lghtng LED Panel Dffusve Flash Baselne Horzontal Vertcal Horzontal Table 3 Calbraton errors usng dfferent feature etracton methods System 1 System 2 System 3 Reproecton Rectfc aton MOID BCoM GCoM MOID BCoM GCoM The ellpse centers etracted by the three methods were used for calbraton of the stereovson systems respectvely. Once a calbraton was done, two types of calbraton errors were calculated to evaluate the qualty of the calbraton. The frst calbraton error s RMS of the calbraton resduals (also known as reproecton errors). The other calbraton error s the RMS of the vertcal offsets of correspondng ellpse centers on the left and rght mages after stereo rectfcaton. Ideally a par of correspondng ponts on the left and rght mages can be rectfed n such a way that only horzontal dspartes est. The rectfcaton algorthms n [16] was chosen n ths study snce t mnmzes mage dstorton after rectfcaton. The vertcal offsets between rectfed correspondng ponts n real-world data are caused by the errors n the estmated eppolar geometry between the stereo cameras. The reproecton and rectfcaton errors are two ndcators to the accuracy of calbraton of stereovson systems. It can be seen that the MOID method consstently produces smaller calbraton errors than the other two methods. GCoM produces nosy results especally n the LED lghtng system (system 1) where the llumnaton s non-unform, suggestng some senstvty to lghtng condtons. BCoM performs consstently as well n all the tests, but t generates larger calbraton errors than MOID n most of the tests, especally rectfcaton errors. The result suggests the MOID method s more accurate and precse than the other two methods. 3. Control pont correcton When calbratng, the metrc structure of the control marks (crcles n ths study) on the calbraton target s often supposed to be accurate so that t can be used as known nformaton to derve other parameters of the camera system. However, the precson of control marks comes wth manufacturng cost, whch may lmt the achevement of hgh accuracy for a camera system. In ths research, we proposed a method that can compensate for a far amount of errors n the structure of control marks. Snce the shfts between the real and deal 3D coordnates of the control marks are fed for a specfc calbraton target, our hypothess s that they can be estmated and corrected usng a suffcent number of observatons. Let us assume the locaton errors of control ponts are fed and the locaton errors of feature ponts are..d. Random. Then ther relaton can be epressed as below: = f ( X + ) + δ, = 1, K, n, = 1, L m (4) where, f s the proecton functon of camera n the -vew, X are the deal coordnates of the -th control pont, are the mage coordnates of the feature pont, are a

5 constant 3-vectors representng the shft of the control pont, and δ s..d 2D zero-mean whte nose. Gven ths model, camera parameters and shfts of control ponts can be estmated by mnmzng the followng least squares error: arg mn( f ( X + ) V, Formula (5) gves an unbased estmate of the camera parameters V and the shft of the control pont f the number of observatons m s suffcent. However, f δ s not..d random and m s small fnte number, whch s qute often the case n a real calbraton task, systematc errors may occur n the calbraton, and the errors wll be partly reflected n the estmated. Ths s the so-called overfttng problem. Snce δ may behave dfferently due to dfferent condtons n the calbraton tests such as llumnaton, mage nose, etc., estmated usng Formula (5) may not be consstent n all tests, whch nevertheless wll render the nvald. Whle realstcally t s hard to devse a general method for estmatng gven unknown propertes of δ, etra constrants must be used to make the problem tractable. In the specal case of ths study, the calbraton crcles were prnted on an A4 paper by a laseret prnter and then attached to a flat surface. We hypotheszed the shfts of crcles are caused by the non-unform loadng speed of the paper, therefore they only occur along the paper feedng drecton (vertcally on the A4 paper). The control ponts (centers of the control crcles) form a 2D array where deally a row of control ponts have the same vertcal coordnate and a column of control ponts have the same horzontal coordnate. We assume the horzontal coordnates of the control ponts are unchanged (zero horzontal shft) and a row of control ponts are affected dentcally by the paper rollng speed and have the same shft n ther vertcal coordnate (dentcal vertcal shft). Wth these assumptons, we only need to assocate a varable representng a vertcal shft to a row of control ponts rather than a translaton vector to each ndvdual control pont. Then Formula (5) can be re-wrtten as: 2 ) (5) arg mn( f ( Xk + ) k ) (6) V, k The ndces,,k n Formula (6) are used to denote -vew of the calbraton target, -row and k-column of the control ponts. It s clear that problem (6) has less degree of freedom (number of parameters) than problem (5), and therefore s potentally able to generate more relable results. Even 2 though the overfttng problem may stll est, ts nfluence has been allevated. (a) (c) (d) Fgure 3: Calbraton resduals for control ponts of the target n Fg 1. Before control pont correcton: (a) y-resduals when target was placed n normal orentaton as n Fg. 1; (b) -resduals when target was placed n orentaton 90 o to Fg. 1. After control pont correcton: (c) y-resduals for the calbraton data n (a); y-resduals for the calbraton data n (b). Our eperments show the zero-horzontal-shft and dentcal-vertcal-shft assumptons are vald. Frst, we calbrated a camera usng the calbraton target placed n normal orentaton as n Fg. 1, and we obtaned a calbraton resdual map as n Fg.3(a). Horzontal strpe patterns appeared n the resduals n y-coordnate, whch ndcates that a row of control ponts have smlar vertcal shfts on the target plane. When we rotated the target by 90 o and used t to calbrate the same camera agan, we obtaned vertcal strpe patterns n -resduals of the calbraton. Ths can be eplaned as the vertcal shfts of control ponts have been turned to horzontal due to the 90 o rotaton of the target. Second, we have estmated the vertcal offsets of the control pont rows from dfferent sets of calbraton mages of the same target from 3 dfferent cameras. Fg. 4 depcts the vertcal shfts of 10 rows of control ponts for the calbraton target n Fg. 1 estmated from 5 frames of mages acqured from the left and rght cameras of stereo system 1 (Cam 1F and Cam 1R) and the top and bottom cameras of stereo system 2 (Cam 2T and Cam 2B) mentoned n Table 2. It can be seen that the vertcal shfts are consstent across the cameras and stereo systems, whch supports the valdty of the zero-horzontal-shft and the dentcal-vertcal-shft assumptons about the control crcles. In the meantme, the results also demonstrate that (b)

6 the degree of overfttng for each estmate of the vertcal shfts s not severe. We hypothesze that the unknown errors of feature pont locatons have behaved relatvely randomly for each row of control ponts and therefore been compensated well n solvng problem (6). Snce the varaton of the vertcal shfts for each row of control ponts s not large, we calculate ther mean value and use t to correct the coordnates of the control ponts of that row. After the control pont correcton, we can use the new control pont coordnates for camera calbraton. Fg.3(c,d) llustrate the calbraton resdual maps usng the corrected control pont postons. The mage feature data are the same as those n Fg.3(a,b) respectvely. It can be seen that Fg.3(c,d) do not have the strpe patterns n Fg.3(a,b), and also the magntude of resduals n Fg.3(c,d) s sgnfcantly reduced, ndcatng that the systematc drft of the control ponts have been compensated effectvely. Note that the resdual maps n Fg.3(c,d) are stll not completely spatal-random, whch suggests that there may be stll some other sources of systematc error n the calbraton process, e.g., unknown llumnaton. was chosen to acqure feature ponts snce the MOID method has been proved accurate. It can be seen that the correcton of the control pont coordnates sgnfcantly reduced the calbraton errors (especally the re-proecton errors) despte the dfference n camera types, lghtng condtons, and stereo confguratons. It also shows that a small amount of locaton compensaton (0.05 mm mamum n Fg. 4) can sgnfcantly reduce calbraton errors and mprove calbraton accuracy. 4. Concluson Ths paper has reported our study on two ssues related to camera calbraton, namely, how to acheve accurate features and how to obtan accurate control ponts. It has been shown that more accurate feature etracton can be done by employng the nformaton of the metrc structure of the control marks and ntegratng the multple steps of feature etracton to a sngle optmzaton framework. The obectve functon of the optmzaton, namely, the dfference between the nsde and outsde of the features has been proved vald. For 3D locatons of control ponts, t s demonstrated that the offsets between deal locatons and real locatons can be estmated by hypotheszng correctly the cause of the offsets. In ths partcular case of study, t has been shown that the locaton errors occurred manly on vertcal drecton. A correcton of the control pont locatons s done to effectvely mprove calbraton performance. The prncples n the feature etracton and control pont correcton n ths paper may be easly etended to other types of control marks and calbraton targets. Fgure 4: Estmated vertcal shfts of control ponts for the calbraton target n Fg. 1. Table 4 Calbraton errors before and after control pont correcton (n pels) System 1 System 2 System 3 Reproecton before after Rectfc aton before after We verfed the corrected control pont coordnates n new camera calbraton applcatons. The calbraton errors lsted n Table 3 are the results of calbraton usng the corrected control pont coordnates for all 3 algorthms. The results usng uncorrected control pont coordnates are compared n Table 4. The MOID ellpse etracton method References [1] B. Trggs, Autocalbraton from Planar Scenes, Proc. European Conf. Computer Vson, pp , Freburg, Germany, June [2] Z. Zhang, "A Fleble New Technque for Camera Calbraton", IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 22, no. 11, pp , 2000, do: / [3] J. Hekkla, Geometrc Camera Calbraton Usng Crcular Control Ponts, IEEE Trans. Pattern Analyss and Machne Intellgence, vol. 22, no. 10, pp , 2000, do: / [4] J. Hekkla, Moment and Curvature Preservng Technque for Accurate Ellpse Boundary Detecton, Proc. Int. Conf. Pattern Recognton,(ICPR 98), pp , vol. 1, [5] E.R. Daves, Fndng Ellpses Usng Generalsed Hough transform, Pattern Recognton Letters, vol. 9, no. 2, pp , 1989f. [6] L. Quan, Conc Reconstructon and Correspondence from Two Vews, IEEE Trans. Pattern Analyss and Machne

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