Research Article A Decoupled Calibration Method for...

Research ArticleA Decoupled Calibration Method for Camera IntrinsicParameters and Distortion Coefficients

Kun Yan12 Hong Tian1 Enhai Liu1 Rujin Zhao1 Yuzhen Hong12 and Dan Zuo12

1 Institute of Optics and Electronics Chinese Academy of Sciences Photoelectric Sensor Laboratory Photoelectric Avenue ChengduSichuan 610209 China2University of Chinese Academy of Sciences Beijing 100149 China

Correspondence should be addressed to Kun Yan yankunioe163com

Received 13 July 2016 Revised 17 November 2016 Accepted 27 November 2016

Academic Editor Salvatore Strano

Copyright copy 2016 Kun Yan et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Camera calibration is a necessary process in the field of vision measurement In this paper we propose a flexible and high-accuracymethod to calibrate a camera Firstly we compute the center of radial distortion which is important to obtain optimal resultsThenbased on the radial distortion of the divisionmodel the camera intrinsic parameters and distortion coefficients are solved in a linearway independently Finally the intrinsic parameters of the camera are optimized via the Levenberg-Marquardt algorithm In theproposed method the distortion coefficients and intrinsic parameters are successfully decoupled calibration accuracy is furtherimproved through the subsequent optimization process Moreover whether it is for relatively small image distortion or distortionlarger image utilizing our method can get a good result Both simulation and real data experiment demonstrate the robustness andaccuracy of the proposed method Experimental results show that the proposed method can be obtaining a higher accuracy thanthe classical methods

1 Introduction

Camera calibration is an important part of the applicationof photogrammetry that aims to compute the camera modelparameters from two-dimensional images [1 2] Cameramodel parameters include intrinsic and extrinsic parametersThe intrinsic parameters describe the geometry of imagingprocess while the extrinsic parameters indicate cameraposition and attitude in the world coordinate system Cameracalibration precision directly affects the measuring accuracyof vision measurement system Therefore the study of aflexible and high-precision camera calibration method hasvery important significance

Nowadays the techniques of camera calibration can bedivided into two categories the traditional camera calibrationand the self-calibration of the cameraThe traditional cameracalibration methods use scene information including thepoints or lines with precise coordinates to solve camera

parameters while self-calibration methods only use therelationship between the sequences of images to solve thecamera parameters In general to satisfy the high-accuracyrequirement we always make use of the traditional cameracalibration method Besides the traditional method consistsof three types the linear method nonlinear optimizationmethod and the two-step method Hall et al [3] introducedthe first linear method by computing the 3 times 4 transforma-tion matrix based on the pinhole model Later the nonlinearcalibration method is developed by introducing a varietyof lens distortion [4ndash10] Zhang [11] proposed a flexiblecalibration technique for desktop vision system (DVS) byusing a printed planar calibration pattern which is a typicalrepresentative of the two-step method

Recently more research is dedicated to improving theperformance of the camera calibration Wang et al [12]proposed a new calibration model of camera lens distortionwhich is according to a transform from ideal image plane

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1392832 12 pageshttpdxdoiorg10115520161392832

2 Mathematical Problems in Engineering

to real sensor array plane Ahmed and Farag [13] proposeda robust approach to distortion calibration by using a least-median-of-squares estimator based on the analysis of dis-torted straight lines in the images Later Ricolfe-Viala andSanchez-Salmeron [14] presented a robust metric calibrationmethod with nonlinear camera lens distortion which com-putes the camera lens distortion isolated from the cameracalibration process under stable conditions independentlyof the computed lens distortion model or the number ofparameters Many other researchers [15ndash19] also proposedthe nonlinear objective function in three-dimensional spaceor distortion free space to minimize the calibration error toimprove the accuracy of the calibration

However in the above literatures most of the calibrationmethods employ the same nonlinear optimization methodsAnd almost completely the lens distortion coefficients andother intrinsic and extrinsic camera parameters are estimatedin an optimization framework at the same time Accordingto the report of Hartley [20] such nonlinear iteration can betroublesome and may converge to the local minima withoutselecting a good initial value Moreover due to the distortionoften coupling parameters in the camerarsquos internal parametersand external parameters methods which extend the calibra-tion of the pinhole model to obtain the camera distortionparameters lead to high internal parameters miscalculation[7] In addition the coupling between the different param-eters can make the estimation results quite unreliable [713] Therefore it is important to use a different method toestimate the camera distortion coefficients apart from thepinhole model In [14] Ricolfe-Viala and Sanchez-Salmeroncomputed lens distortion model or the number of distortionparameters independently But they solved the distortion cen-ter by a nonlinear minimization method Although Ahmedand Farag [13] successfully decoupled intrinsic parametersand distortion coefficient they relied on the fact that straightlines in the scenemust always perspectively project to straightlines in the imageMoreover they assumed that the distortioncenter is known In [21] Fitzgibbon proposed a noniterativemethod to estimate radial distortion by introducing thedivision model He also successfully decoupled intrinsicparameters and distortion coefficient However the radialdistortion includes only one parameter in his work

In this paper a high-accuracy calibration method isproposed Different from [13 14 21] this method computesthe center of radial distortion firstly which is important toobtain optimal results Afterwards based on the divisionmodel the interior parameter and distortion coefficients ofthe camera are estimated by linear method which providea good initial value for the subsequent optimization More-over our method can calculate camera intrinsic parametersand any number of distortion coefficients In addition theintrinsic parameters of the camera are optimized via theLevenberg-Marquardt algorithm and then we obtain moreaccurate results Experimental results show that the proposedmethod can be obtaining a higher accuracy than the classicalmethods

This paper is organized as follows Section 2 gives a briefdescription of the camera model and the lens distortionmodel Section 3 describes the detail procedure of the

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Figure 1 The model of camera

proposed camera calibration method Section 4 verifies theproposed calibration method by simulation experiments andreal data experiments At last this paper ends with severalimportant conclusions in Section 5

2 Camera Model

The camera model is shown in Figure 1 Given one homoge-neous coordinates point P119908 = (119883119908 119884119908 119885119908 1)119879 in 3D spaceP119906 = (119883119906 119884119906 1)119879 is the perspective projection of P119908 in theimage plane based on pinhole model without lens distortionP119889 = (119883119889 119884119889 1)119879 is the real projection of P119908 in the imageplane of camera considering the lens distortion

Without consideration of lens distortion the mappingbetween the 3D point P119908 and 2D image point P119906 is given by

119904P119906 = K [R t]P119908K = [[[

119891119906 119904119896 11990600 119891V V00 0 1 ]]] (1)

where 119904 represents the nonzero scale factors K is the cameraintrinsic matrix with (1199060 V0) being the coordinates of theprincipal point 119891119906 and 119891V being the effective focal lengthin pixels and 119904119896 being the parameter describing the skewof the two image axes [R t] represent the rotation andtranslation vector from the world coordinate system to thecamera coordinate system respectively

Mathematical Problems in Engineering 3

Because of the plane calibration model used in ourcalibration procedure we assume that the model plane 119885119908 =0 with no loss of generality From (1) we have

119904(1198831199061198841199061 ) = K ( r1 r2 r3 t)(11988311990811988411990801 )= K (r1 r2 t)(1198831199081198841199081 )

(2)

where r119894 is the 119894th columnof the rotationmatrixR By abuse ofnotation we still denote the point on themodel plane byP119908 =(119883119908 119884119908 1)119879 Therefore 3D point P119908 and its image point P119906are related by a homographyH119904P119906 = HP119908

H = K (r1 r2 t) (3)

With the influence of lens distortion as shown in Figure 1the actual image point P119889 is not the point P119906 which is theimage plane intersection of the connection of the 3D pointP119908 and the optical center O119888 but it has some deviationTheoretical calculations will be carried out through theactual image coordinates after correction for the ideal imagecoordinates the calibration process can be achieved throughdistortion compensation model Camera lens distortion wasfirst introduced by Conrady in 1919 with the decentringlens distortion Afterwards Brown [22] proposed the radialdecentring and prism distortion model which has beenwidely used In general the radial distortion is sufficientfor a high-accuracy measurement Brown proposed thePolynomial Model (PM) which is the most popular modelto describe radial distortion

P119906 minus e = (P119889 minus e) sdot 119871 (119903119889 119896) with 119871 (119903119889 119896) = (1 + 11989611199032119889 + 11989621199034119889 + sdot sdot sdot) (4)

where 1198961 1198962 are the distortion coefficients e = (1198891199060119889V0 1)119879 is the homogeneous coordinate of the center ofdistortion (COD) and 119903119889 is the pixel radius to e

PM works best for lens with small distortions Forthe wide-angle lens or fisheye lens there is considerabledistortion and they often need too many terms compared tothe actual So Fitzgibbon [21] suggested the division model(DM) to describe the radial distortion model

P119906 minus e = (P119889 minus e)119871 (119903119889 119896) (5)

where 119871(119903119889 119896) is the same as in (4) Combining (4) and (5) wecan get a more generic Rational Model (RM)

P119906 minus e = (P119889 minus e) sdot 1198711 (119903119889 1198961)1198712 (119903119889 1198962) (6)

However the calculation speed of the imaging modelbased on the RM is relatively slow and the convergenceresults are more dependent on the accuracy of the initialvalue On the other hand the most remarkable advantage oftheDMover the PM is that it is able to express high distortionat much lower order So our work for camera calibration isbased on the DM

3 Camera Calibration Procedure

In this section the process of the proposed calibrationmethod is described in detail The whole process is dividedinto four steps Firstly in order to make the center ofdistortion (COD) to the original the COD is estimatedaccurately and the calibration image is corrected Thenthe corresponding points between the model plane and itsimage are used to compute the homography and distortioncoefficients simultaneously based on the radial distortiondivision model (DM) [21] In the third step all the cameraintrinsic parameters are analytically calculated from thehomographies of calibration images Finally the intrinsicparameters of the camera are optimized via the Levenberg-Marquardt algorithm

31 Find theCOD In the literatures [9 14 23] the researchersassume that the COD is the center of the image or theprincipal point of the camera However we all know that thisis not a good assumption The COD can be displaced by anumber of factors from the center of the image such as anoffset from the center of theCCD lens a tilt with respect to theplane of the lens sensor or the installation of a combinationof lenses and image cropping In general consumer levelcameras (such as several hundred dollar costs) it shouldnot be assumed that the photocenters of the camera areaccurate because these effects are not much different fromthe subjective image quality [24]

Experiments have been done by Hartley and Zisserman[25] to show that the general assumption that the COD isat the center of the image is not exactly right Besides theyhave also proved that the point may be remarkably deviatedfrom the center of the image or the principal point of thecamera So the accurate estimation of COD is important toobtain optimal results in camera calibration Here we adoptthe novel method proposed in [24] to accurately estimatethe position of COD in this paper This method is simplebut it produces perfect results We briefly introduce thisapproach here For more details the reader can refer to theliterature [24] The known point P119908 on the calibration modelplane and its corresponding image point P119889 are related by aradial fundamental relationship and it can be represented asfollows [P119889]119879 ([e]timesH)P119908 = 0 (7)

where H is the same as in (3) and [e]times (the skew symmetricmatrix) represents the cross-product Writing F119903 = [e]timesH (in[24] the matrix F119903 is called as radial fundamental matrix)then (7) can be rewritten as[P119889]119879 F119903P119908 = 0 (8)


Note that F119903 can be calculated in a general way [25](such as the eight-point algorithm) from some correspondingpoints Then the COD can be estimated from its left epipole

e119879F119903 = e119879 [e]timesH = 0 (9)

Of course in the absence of radial deformation the basicmatrix of the above calculation is not stable and the estimatedvalue of e is essentially arbitrary and meaningless Thereforeif there is no radial distortion then it is not much of a senseto talk about the distorted center [24]

32 Compute the Distortion Coefficients and HomographyWe will compute the distortion coefficients and homographyin this section Based on the reports in the literatures [622 26] it is likely that the distortion function is completelydominated by the radial components in particular the dom-inant first term In addition any more detailed modelingnot only does not help (ignore when compared with sensorquantization) but also leads to numerical instability [6 26]

After getting the COD e we can compute the H byfactoring the matrix F119903 (F119903 = [e]timesH) Although [e]times issingular this decomposition is not unique Here a novelmethod is presented to solve this problem In doing this wechange the coordinates of the image so that the COD is theoriginal As is shown in Figure 1 the image plane coordinatesystem O119909119910119909119910 is transformed to O1198891006704119883119889 1006704119884119889 Where O119889 is thereal position of the COD we represent it as 1006704e = [0 0 1]119879in homogeneous coordinatesThen the transformed distortedimage point 1006704P119889 can be represented as

1006704P119889 = (100670411988311988910067041198841198891 ) = (119883119889 minus 1198891199060119884119889 minus 119889V01 )= (1 0 minus11988911990600 1 minus119889V00 0 1 )(1198831198891198841198891 ) = (1 0 minus11988911990600 1 minus119889V00 0 1 )P119889

(10)

Similarly the transformed undistorted image point 1006704P119906and the original undistorted image point P119906 are described as

1006704P119906 = (1 0 minus11988911990600 1 minus119889V00 0 1 )P119906 (11)

When the coordinate origin is converted to the center ofdistortion the homography matrix is redefined as 1006704H and thecorresponding radial fundamental matrix is defined as 1006704F119903 =[1006704e]times1006704H Then from (8) we can get[1006704P119889]119879 1006704F119903P119908 = 0 (12)

Using (10) into (12) we can get

[P119889]119879(1 0 minus11988911990600 1 minus119889V00 0 1 )119879 1006704F119903P119908 = 0 (13)

Besides from (8) and (13) we can get

F119903 = (1 0 minus11988911990600 1 minus119889V00 0 1 )119879 1006704F119903 (14)

1006704F119903 can be calculated as

1006704F119903 = ( 1 0 00 1 01198891199060 119889V0 1) F119903 (15)

Note that 1006704F119903 = [1006704e]times1006704H and 1006704e = [0 0 1]119879 we have1006704F119903 = [1006704e]times 1006704H = (0 minus1 01 0 00 0 0) 1006704H (16)

Let 1006704F119903 = [1006704f1198791 1006704f1198792 1006704f1198793 ] and 1006704H = [1006704h1198791 1006704h1198792 1006704h1198793 ] Here 1006704f119879119894 =[ 10067041198911198941 10067041198911198942 10067041198911198943] is the 119894th row of the radial fundamental matrix1006704F119903 and 1006704h119879119894 = [1006704ℎ1198941 1006704ℎ1198942 1006704ℎ1198943] is the 119894th row of matrix 1006704H Due tothe fact that final row of 1006704F119903 is zero so we only need to solvethe first two rows of 1006704H from (16) that is1006704h1198791 = 1006704f1198792 1006704h1198792 = minus1006704f1198791 (17)

Then by calculating the homography 1006704H present up toonly three unknown parameters 1006704h1198793 = [1006704ℎ31 1006704ℎ32 1006704ℎ33] As thehomography 1006704H relates themodel pointP119908 and its undistortedimage point 1006704P119906 1199041006704P119906 = 1006704HP119908 (18)

Note that 1006704P119906 = [1006704119883119906 1006704119884119906 1] is the undistorted image pointafter the origin of the image coordinates is transferred to theCOD Considering the division model (DM) we have1006704119883119906 = 1006704119883119889119871 (1006704119903119889 119896) 1006704119884119906 = 1006704119884119889119871 (1006704119903119889 119896) (19)

where 119871(1006704119903119889 119896) is the same as in (4) that is119871 (1006704119903119889 119896) = (1 + 119896110067041199032119889 + 119896210067041199034119889 + sdot sdot sdot) (20)

We substitute (19) into (18) then we can get

119904 [[[[[[[[[1006704119883119889119871 (1006704119903119889 119896)1006704119884119889119871 (1006704119903119889 119896)1

]]]]]]]]]= [[[[

1006704h11987911006704h11987921006704h1198793]]]]P119908 (21)


So each point can be given the two equations

[[1006704119883119889 (1006704h1198793P119908) minus (1006704h1198791P119908) sdot (1198961 [1006704119903119889]2 + 1198962 [1006704119903119889]4 + sdot sdot sdot )1006704119884119889 (1006704h1198793P119908) minus (1006704h1198792P119908) sdot (1198961 [1006704119903119889]2 + 1198962 [1006704119903119889]4 + sdot sdot sdot )]]= [1006704h1198791P1199081006704h1198792P119908]

(22)

Using (17) in (22) we can get

[[1006704119883119889 [P119908]119879 (minus1006704f1198792 P119908) [[1006704119903119889]2 [1006704119903119889]4 sdot sdot sdot]1006704119884119889 [P119908]119879 (1006704f1198791 P119908) [[1006704119903119889]2 [1006704119903119889]4 sdot sdot sdot] ]]

[[[[[[[[1006704h311989611198962]]]]]]]]= [ 1006704f1198792 P119908minus1006704f1198791 P119908]

(23)

Given 119873 corresponding point and that it satisfies that 2119873 ge119899+3 where 119899 is the number of distortion parameters we cansolve the system of equations in the least squares sense Thenthe third row of 1006704H (ie 1006704h1198793 ) is solvedMoreover the distortioncoefficients 1198961 1198962 are computed at the same time

33 Linear Solution of Intrinsic Parameters of the CameraAfter 1006704H is obtained the homographymatrixH for the originalimage coordinates can be computed very easily From (3) (11)and (18) we can get

H = (1 0 minus11988911990600 1 minus119889V00 0 1 )minus1 1006704H = (1 0 11988911990600 1 119889V00 0 1 ) 1006704H (24)

Let us denote homography matrix byH = [h1198881 h1198882 h1198883](where h119888119894 is the 119894th column of H) According to theorthogonality of the rotation matrix R (r1198791 r2 = 0 r1198791 r1 =r1198792 r2) we get

h1198791198881Kminus119879Kminus1h1198882 = 0

h1198791198881Kminus119879Kminus1h1198881 = h1198791198882K

minus119879Kminus1h1198882 (25)

Note that K is the camera intrinsic matrix Becausea homography has 8 degrees of freedom and there are 6extrinsic parameters (3 for rotation and 3 for translation)we can only obtain 2 constraints on the intrinsic parametersTherefore we need at least three images to calculate allthe parameters After the homography H is obtained theanalytical solution of the intrinsic parameters of the cameracan be solved in the literature [11]

34 Maximum Likelihood Estimation The intrinsic parame-ters obtained in the previous section are not accurate enoughdue to image noise We can refine it through maximumlikelihood estimation Ahmed and Farag [13] have proved

that including the distortion center and the decenteringcoefficients in the nonlinear optimization step may lead toinstability of the estimation algorithm Therefore we do notneed to optimize the distortion coefficient and distortioncenter Meanwhile it can reduce the search space of thecalibration problem without sacrificing the accuracy andproduce more stable and noise-robust results Here we give119899 images of a model plane and there are 119898 points on themodel plane Assume that the image points are corruptedby independent and identically distributed noise Then themaximum likelihood estimate can be obtained byminimizingthe following function

119899sum119894=1

119898sum119895=1

10038171003817100381710038171003817P119894119895 minus 1006704P (KR119894 t119894P119895)100381710038171003817100381710038172 (26)

where 1006704P(KR119894 t119894P119895) is the projection of point P119895 in image 119894according to (3) A rotation R is parameterized by a vector of3 parameters denoted by r which is parallel to the rotationaxis and whose magnitude is equal to the rotation angleMinimizing (26) is a nonlinearminimization problem whichis solved by the Levenberg-Marquardt algorithm [27] Itrequires an initial value of K and R119894 119905119894 | 119894 = 1 119899 whichcan be obtained using the technique described in the previoussubsection

4 Experimental Results

In this section the proposed calibration method has beentested on both computer simulated data and real data Exper-iments mainly consider two calibration methods Zhangrsquoscalibration method based on planar target [28] and ourmethod (before-optimization and after-optimization)

41 Simulated Experiments The simulated camera hasthe following property 119891119906 = 850 119891V = 850 1199060 = 512V0 = 384 The image size is 1024 times 768 pixels The skewfactor is set to zero Besides a second-order radial distortionis simulated with the coefficients 1198961 = minus609 times 10minus7pixel-2 1198962 = minus197 times 10minus13 pixel-4 and the COD isset to (500 366) The plane is a checkerboard image with70 corners (7 times 10) evenly distributed and the minimumpoint interval is set to 23mm which is the same asthe real data experiments The direction of the plane isdenoted by a 3-dimensional vector r which is parallel tothe rotation axis and whose magnitude is the same as theangle of rotation In addition the position of the planeis denoted by a 3-dimensional vector t We employ fourplanes with 1199031 = [20∘ 0∘ 0∘]119879 1199051 = [minus80 minus60 200]119879 1199032 =[0∘ 0∘ 20∘]119879 1199052 = [minus110 minus80 250]119879 1199033 = [minus40∘ 0∘ 20∘]1198791199053 = [minus100 minus40 330]119879 1199034 = [minus10∘ 0∘ 20∘]119879 1199054 = [minus100minus60 280]119879 in the experiment Figure 2 presents thesimulation of the images which show the true projectionof the image point and the distorted point The Gauss noisewith a mean of 0 and a standard deviation of 120590 is added tothe distorted image

Then the estimated camera parameters and the actualvalue of the real were compared We calculated the relative


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error for 119891119906 and 119891V and the absolute errors for 1199060 and V0Besides the mean value of the reprojection error 119864rms is alsoused as an evaluation index It is calculated by the discrepancybetween the ground true pointP119905119894 and the reprojection imagepoint P119903119894

119864rms = (sum119899119894=1radic1003817100381710038171003817P119903119894 minus P11990511989410038171003817100381710038172)119899 (27)

Noise is added to the projected image points with 120590ranging from 01 pixels to 20 pixels For each noise level50 independent repeated trials were performed and theresults were shown to be average The relative error andcalibration precision of the camera are shown in Figures 3and 4 respectively From the two figures we can see that theerrors increase linearly with the noise level It is worth notingthat the error of the two methods is relatively low when thenoise is low For example for 120590 = 05 (it is larger than thenormal noise in practical calibration) the absolute errors in1199060 and V0 are around 1 pixel the relative errors in 119891119906 119891V areless than 03 and the mean calibration errors are around02 pixels Besides the error in V0 is larger than that in 1199060Themain reason is that there are less data in the V0 direction thanin the 1199060 direction42 Real Data Experiments For the real data experimentsthe calibration images provided by The Robotics Instituteof Carnegie Mellon University [28] are used to test ourapproach The calibration template is a planar checkerboardpattern with 70 corners (7 times 10) evenly distributed and theminimumpoint interval is 23mm in both the vertical and thehorizontal directions Ten images of the plane under differentorientations were taken as shown in Figure 5 (It is worthnoting that the algorithm proposed in this paper is suitablefor the calibration of all kinds of image of the board includingall the pictures provided by Carnegie Mellon University Dueto the limitation of space this paper only randomly selectedten pictures as our experimental object) We can observe anobvious lens distortion in the images particularly in Image 1Image 3 Image 5 and Image 6 In addition the resolution ofthe image is 1024 times 768 pixels

First we use the corner detection method to get subpixelprecision of checkerboard angular point position Next wecompare our proposed calibration method with Zhangrsquosmethod In the experiment we utilize the first three images toobtain the camera parameters which are displayed in Table 1In Zhangrsquosmethod he does not consider the distortion center(1198891199060 119889V0) while our method does It is necessary to explainthat (1198891199060 119889V0) are the mean of the first three images In facttheCODof the three images are 507368 508087 and 506139respectively In addition our approach adopts the DM torepresent lens distortion andZhangrsquosmethod depends on thePM so the generating distortion coefficients are different Atlast it is worth noting that the before-optimization methodalso computes theCODHowever the resulting119891119906 119891V 1199060 andV0 are almost the same as Zhangrsquos We will see later that theaccuracy of the before-optimization is not high

To investigate the effectiveness of the proposed methodthe other seven test images are used to evaluate the calibration

Table 1 Comparative result of intrinsic parameters and distortioncoefficients

Method 119891119906 119891V 1199060 V0Zhang 83950 83856 50789 36702Before-optimization 83979 83887 50617 36714

After-optimization 83823 83727 50599 36621

Method 1198961 1198962 1198891199060 119889V0Zhang minus042mm-2 020mm-4 mdash mdashBefore-optimization

minus609119890 minus 07pixel-2

minus197119890 minus 13pixel-4 50720 36759

After-optimization



accuracy Here we once again put the calculated reprojectionerror 119864119903-rms as the evaluation index Unlike the simulationtest the ground true image points are unknown here so weuse the undistorted pointP119906119889119894 insteadThe reprojection error119864119903-rms is defined as

119864119903-rms = (sum119899119894=1radic10038171003817100381710038171003817P119903119901119894 minus P119906119889119894100381710038171003817100381710038172)119899 (28)

Table 2 shows the comparison of the reprojection errorof the test data and the distribution is shown in Figure 6As shown in Table 2 in both the average accuracy leveland standard error the after-optimization method is betterthan Zhangrsquos method and the before-optimization methodFrom Figure 6 we can see that the distribution of the after-optimization method is more concentrated in the near zero

Besides in order to further study the stability of the pro-posed (after-optimization) algorithm we have also appliedit to some combinations of 4 images from the fifth to theninth images The results are shown in Table 3 where thesecond column (5678) for example displays the result withthe quadruple of the fifth sixth seventh and eighth imageThe last two columns display the mean and deviations Thedeviations for all parameters are very small whichmeans thatthe proposed algorithm is quite stable

Finally for a few images with a larger distortion weconducted an experiment separately We utilize four images(Image 1 Image 3 Image 5 and Image 6) to calibrate theinternal parameters The results are shown in Table 4 Ascan be seen from the results we obtained similar resultscompared to Zhang

5 Conclusion

In this paper we proposed a flexible and high-accuracycamera calibration method Compared with the traditionalmethod this method overcomes a lot of problems Firstlyit decouples the estimation of the distortion coefficientsand intrinsic parameters producing more stable and reliable


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Figure 3 Effects of pixel coordinates noise on intrinsic parameters using the proposed method (after-optimization (a) the relative error for119891119906 and 119891V and (b) the absolute error for 1199060 and V0 before-optimization (c) the relative error for 119891119906 and 119891V and (d) the absolute error for 1199060and V0)


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Figure 4 Effects of pixel coordinates noise on calibration accuracy (a) after-optimization (b) before-optimization

Table 2 Comparative result of calibration error 119864119903-rms evaluated by testing data

Method Image 4 Image 5 Image 6 Image 7 Image 8

Zhang 01683 01763 01319 01380 01658Before-optimization

01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609

Method Image 9 Image 10 Average Standard error mdash

Zhang 01321 01580 01529 00186 mdashBefore-optimization

01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash

results Then the distortion of the center is accurately esti-mated and it is important to obtain the best resultsMoreoverwhether it is for a relatively small image distortion ordistortion larger image utilizing our method can get a goodresult Finally the robustness and accuracy of the proposedmethod are verified by simulation and real data experimentsand the experimental results show that this method has theadvantages of simple operation high accuracy and betterflexibility

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the support from the NationalNatural Science Foundation of China (no 61501429) Thanks


Table 3 Variation of the calibration results among all quadruples of images

Quadruples (5678) (5689) (5789) (6789)119891119906 84339 84326 84331 84570119891V 84260 84244 84265 844711199060 50450 50479 50473 50492V0 36568 36548 36575 366511198961 minus60822119890 minus 07 minus61079119890 minus 07 minus60732119890 minus 07 minus60985119890 minus 071198962 minus22510119890 minus 13 minus20867119890 minus 13 minus22243119890 minus 13 minus19719119890 minus 13Quadruples (5679) Mean Var mdash119891119906 84307 8437460 12070 mdash119891V 84223 8429260 10214 mdash1199060 50453 5046940 00315 mdashV0 36541 3657660 01924 mdash1198961 minus61882119890 minus 07 minus61100119890 minus 07 20947119890 minus 17 mdash1198962 minus12948119890 minus 13 minus19657119890 minus 13 15326119890 minus 27 mdash

Image 4Image 3Image 2Image 1


Image 10Image 9

Figure 5 Images used for calibration


04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)


Image 8Image 9Image 10

04020minus02minus04minus06

(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)



0604020minus02minus04

(b)

Figure 6 Reprojection error (a) Zhangrsquos method (b) proposed method (after-optimization) Each point marked as ldquo+rdquo in the figure denotesthe reprojection error of corresponding chessboard corner in the testing images







After-optimization



are due to the accompaniers working with them in theDepartment of the Laboratory of Photoelectric Sensor Tech-nology Institute of Optics and Electronics Chinese Academyof Sciences Thanks are also due to Yuzhen Hong and DanZuo for giving advice and checking the English and toThe Robotics Institute of Carnegie Mellon University forproviding the real testing images

References

[1] E Peng and L Li ldquoCamera calibration using one-dimensionalinformation and its applications in both controlled and uncon-trolled environmentsrdquo Pattern Recognition vol 43 no 3 pp1188ndash1198 2010

[2] Y Cui F Zhou YWang L Liu andHGao ldquoPrecise calibrationof binocular vision system used for visionmeasurementrdquoOpticsExpress vol 22 no 8 pp 9134ndash9149 2014

[3] E L Hall J B K Tio C A McPherson and F A SadjadildquoMeasuring curved surfaces for robot visionrdquoComputer vol 15no 12 pp 42ndash54 1982

[4] J Heikkila ldquoGeometric camera calibration using circular con-trol pointsrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 22 no 10 pp 1066ndash1077 2000

[5] J Wang F Shi J Zhang and Y Liu ldquoA new calibration modelof camera lens distortionrdquo Pattern Recognition vol 41 no 2 pp607ndash615 2008

[6] R Y Tsai ldquoA versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TVcameras and lensesrdquo IEEE Journal on Robotics and Automationvol 3 no 4 pp 323ndash344 1987

[7] J Weng P Cohen and M Herniou ldquoCamera calibration withdistortion models and accuracy evaluationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 14 no 10 pp965ndash980 1992


[8] Z Zhang D Zhu J Zhang and Z Peng ldquoImproved robust andaccurate camera calibration method used for machine visionapplicationrdquo Optical Engineering vol 47 no 11 Article ID117201 2008

[9] Y Hong G Ren and E Liu ldquoNon-iterative method for cameracalibrationrdquo Optics Express vol 23 no 18 pp 23992ndash240032015

[10] J Liu Y Li and S Chen ldquoRobust camera calibration by optimallocalization of spatial control pointsrdquo IEEE Transactions onInstrumentation and Measurement vol 63 no 12 pp 3076ndash3087 2014

[11] Z Zhang ldquoA flexible new technique for camera calibrationrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 22 no 11 pp 1330ndash1334 2000


[13] M Ahmed and A Farag ldquoNonmetric calibration of camera lensdistortion differential methods and robust estimationrdquo IEEETransactions on Image Processing vol 14 no 8 pp 1215ndash12302005

[14] C Ricolfe-Viala and A-J Sanchez-Salmeron ldquoRobust metriccalibration of non-linear camera lens distortionrdquo Pattern Recog-nition vol 43 no 4 pp 1688ndash1699 2010

[15] C Ricolfe-Viala and A-J Sanchez-Salmeron ldquoCamera calibra-tion under optimal conditionsrdquoOptics Express vol 19 no 11 pp10769ndash10775 2011

[16] A Datta J-S Kim and T Kanade ldquoAccurate camera calibrationusing iterative refinement of control pointsrdquo in Proceedingsof the IEEE 12th International Conference on Computer VisionWorkshops (ICCV rsquo09) pp 1201ndash1208 Kyoto Japan October2009

[17] D Douxchamps and K Chihara ldquoHigh-accuracy and robustlocalization of large control markers for geometric cameracalibrationrdquo IEEETransactions on PatternAnalysis andMachineIntelligence vol 31 no 2 pp 376ndash383 2009

[18] T Rahman and N Krouglicof ldquoAn efficient camera calibrationtechnique offering robustness and accuracy over a wide rangeof lens distortionrdquo IEEE Transactions on Image Processing vol21 no 2 pp 626ndash637 2012

[19] F Zhou Y Cui B Peng and Y Wang ldquoA novel optimizationmethod of camera parameters used for vision measurementrdquoOptics and Laser Technology vol 44 no 6 pp 1840ndash1849 2012

[20] H Li andRHartley ldquoAnon-iterativemethod for correcting lensdistortion from nine-point correspondencesrdquo in Proceedings ofthe ICCV-Workshop (OmniVision rsquo05) 2005

[21] A W Fitzgibbon ldquoSimultaneous linear estimation of multipleview geometry and lens distortionrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition pp I125ndashI132 IEEE Kauai Hawaii USA Decem-ber 2001

[22] D C Brown ldquoDecentering distortion of lensesrdquo Photogrammet-ric Engineering and Remote Sensing vol 24 pp 555ndash566 1966

[23] S Thirthala and M Pollefeys ldquoThe radial trifocal tensor a toolfor calibrating the radial distortion of wide-angle camerasrdquo inProceedings of the IEEE Computer Society Conference on Com-puter Vision and Pattern Recognition (CVPR rsquo05) vol 1 pp 321ndash328 IEEE San Diego Calif USA June 2005

[24] R I Hartley and S B Kang ldquoParameter-free radial distortioncorrection with centre of distortion estimationrdquo in Proceedingsof the IEEE International Conference on Computer Vision (ICCVrsquo05) pp 1834ndash1841 Beijing China October 2005

[25] R Hartley and A ZissermanMultiple View Geometry in Com-puter Vision Cambridge University Press Cambridge UK 2ndedition 2003

[26] G Q Wei and S D Ma ldquoImplicit and explicit camera calibra-tion theory and experimentsrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 16 no 5 pp 469ndash4801994

[27] J J More ldquoThe Levenberg-Marquardt algorithm implemen-tation and theoryrdquo in Numerical Analysis G A Watson Edvol 630 of Lecture Notes in Mathematics pp 105ndash116 SpringerBerlin Germany 1977

[28] M Higuchi ldquoPrecise camera calibrationrdquo httpwwwricmueduresearch project detailhtmlproject id=617ampmenu id=261

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


to real sensor array plane Ahmed and Farag [13] proposeda robust approach to distortion calibration by using a least-median-of-squares estimator based on the analysis of dis-torted straight lines in the images Later Ricolfe-Viala andSanchez-Salmeron [14] presented a robust metric calibrationmethod with nonlinear camera lens distortion which com-putes the camera lens distortion isolated from the cameracalibration process under stable conditions independentlyof the computed lens distortion model or the number ofparameters Many other researchers [15ndash19] also proposedthe nonlinear objective function in three-dimensional spaceor distortion free space to minimize the calibration error toimprove the accuracy of the calibration

However in the above literatures most of the calibrationmethods employ the same nonlinear optimization methodsAnd almost completely the lens distortion coefficients andother intrinsic and extrinsic camera parameters are estimatedin an optimization framework at the same time Accordingto the report of Hartley [20] such nonlinear iteration can betroublesome and may converge to the local minima withoutselecting a good initial value Moreover due to the distortionoften coupling parameters in the camerarsquos internal parametersand external parameters methods which extend the calibra-tion of the pinhole model to obtain the camera distortionparameters lead to high internal parameters miscalculation[7] In addition the coupling between the different param-eters can make the estimation results quite unreliable [713] Therefore it is important to use a different method toestimate the camera distortion coefficients apart from thepinhole model In [14] Ricolfe-Viala and Sanchez-Salmeroncomputed lens distortion model or the number of distortionparameters independently But they solved the distortion cen-ter by a nonlinear minimization method Although Ahmedand Farag [13] successfully decoupled intrinsic parametersand distortion coefficient they relied on the fact that straightlines in the scenemust always perspectively project to straightlines in the imageMoreover they assumed that the distortioncenter is known In [21] Fitzgibbon proposed a noniterativemethod to estimate radial distortion by introducing thedivision model He also successfully decoupled intrinsicparameters and distortion coefficient However the radialdistortion includes only one parameter in his work

In this paper a high-accuracy calibration method isproposed Different from [13 14 21] this method computesthe center of radial distortion firstly which is important toobtain optimal results Afterwards based on the divisionmodel the interior parameter and distortion coefficients ofthe camera are estimated by linear method which providea good initial value for the subsequent optimization More-over our method can calculate camera intrinsic parametersand any number of distortion coefficients In addition theintrinsic parameters of the camera are optimized via theLevenberg-Marquardt algorithm and then we obtain moreaccurate results Experimental results show that the proposedmethod can be obtaining a higher accuracy than the classicalmethods

This paper is organized as follows Section 2 gives a briefdescription of the camera model and the lens distortionmodel Section 3 describes the detail procedure of the

Xc

Zc

Yc

xOxy

y OdOu u

Yw

Ow Xw

Zw

Oc

Pd Pu

Pw(Xw Yw Zw)

Figure 1 The model of camera

proposed camera calibration method Section 4 verifies theproposed calibration method by simulation experiments andreal data experiments At last this paper ends with severalimportant conclusions in Section 5

2 Camera Model

The camera model is shown in Figure 1 Given one homoge-neous coordinates point P119908 = (119883119908 119884119908 119885119908 1)119879 in 3D spaceP119906 = (119883119906 119884119906 1)119879 is the perspective projection of P119908 in theimage plane based on pinhole model without lens distortionP119889 = (119883119889 119884119889 1)119879 is the real projection of P119908 in the imageplane of camera considering the lens distortion

Without consideration of lens distortion the mappingbetween the 3D point P119908 and 2D image point P119906 is given by

119904P119906 = K [R t]P119908K = [[[

119891119906 119904119896 11990600 119891V V00 0 1 ]]] (1)

where 119904 represents the nonzero scale factors K is the cameraintrinsic matrix with (1199060 V0) being the coordinates of theprincipal point 119891119906 and 119891V being the effective focal lengthin pixels and 119904119896 being the parameter describing the skewof the two image axes [R t] represent the rotation andtranslation vector from the world coordinate system to thecamera coordinate system respectively



119904(1198831199061198841199061 ) = K ( r1 r2 r3 t)(11988311990811988411990801 )= K (r1 r2 t)(1198831199081198841199081 )

(2)


H = K (r1 r2 t) (3)





P119906 minus e = (P119889 minus e)119871 (119903119889 119896) (5)











e119879F119903 = e119879 [e]timesH = 0 (9)





(10)


1006704P119906 = (1 0 minus11988911990600 1 minus119889V00 0 1 )P119906 (11)



[P119889]119879(1 0 minus11988911990600 1 minus119889V00 0 1 )119879 1006704F119903P119908 = 0 (13)


F119903 = (1 0 minus11988911990600 1 minus119889V00 0 1 )119879 1006704F119903 (14)


1006704F119903 = ( 1 0 00 1 01198891199060 119889V0 1) F119903 (15)







119904 [[[[[[[[[1006704119883119889119871 (1006704119903119889 119896)1006704119884119889119871 (1006704119903119889 119896)1

]]]]]]]]]= [[[[

1006704h11987911006704h11987921006704h1198793]]]]P119908 (21)




(22)



[[[[[[[[1006704h311989611198962]]]]]]]]= [ 1006704f1198792 P119908minus1006704f1198791 P119908]

(23)





h1198791198881Kminus119879Kminus1h1198882 = 0


minus119879Kminus1h1198882 (25)




119899sum119894=1

119898sum119895=1

10038171003817100381710038171003817P119894119895 minus 1006704P (KR119894 t119894P119895)100381710038171003817100381710038172 (26)







700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000














After-optimization








5 Conclusion



16

14

12

1

08

06

04

02

018 2

Relat

ive e

rror

()


fu

f

1614121080604020

(a)

18 2


7

6

5

4

3

2

1

0Ab

solu

te er

ror (

pixe

ls)

u0

0

(b)

Relat

ive e

rror

()

fu

f

18 2


15

1

05

0

(c)

7

6

5

4

3

2

1

0

Abso

lute

erro

r (pi

xels)

u0

0

18 2


(d)



1

09

08

07

05

06

04

03

02

01

0

Repr

ojec

tion

erro

r (pi

xels)


Image 1Image 2

Image 3Image 4

2181614121080604020

(a)

Repr

ojec

tion

erro

r (pi

xels)


1

04

06

08

12

14

02

0

Image 1Image 2

Image 3Image 4

(b)





01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609



01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash


Competing Interests


Acknowledgments







Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904(1198831199061198841199061 ) = K ( r1 r2 r3 t)(11988311990811988411990801 )= K (r1 r2 t)(1198831199081198841199081 )

(2)


H = K (r1 r2 t) (3)





P119906 minus e = (P119889 minus e)119871 (119903119889 119896) (5)











e119879F119903 = e119879 [e]timesH = 0 (9)





(10)


1006704P119906 = (1 0 minus11988911990600 1 minus119889V00 0 1 )P119906 (11)



[P119889]119879(1 0 minus11988911990600 1 minus119889V00 0 1 )119879 1006704F119903P119908 = 0 (13)


F119903 = (1 0 minus11988911990600 1 minus119889V00 0 1 )119879 1006704F119903 (14)


1006704F119903 = ( 1 0 00 1 01198891199060 119889V0 1) F119903 (15)







119904 [[[[[[[[[1006704119883119889119871 (1006704119903119889 119896)1006704119884119889119871 (1006704119903119889 119896)1

]]]]]]]]]= [[[[

1006704h11987911006704h11987921006704h1198793]]]]P119908 (21)




(22)



[[[[[[[[1006704h311989611198962]]]]]]]]= [ 1006704f1198792 P119908minus1006704f1198791 P119908]

(23)





h1198791198881Kminus119879Kminus1h1198882 = 0


minus119879Kminus1h1198882 (25)




119899sum119894=1

119898sum119895=1

10038171003817100381710038171003817P119894119895 minus 1006704P (KR119894 t119894P119895)100381710038171003817100381710038172 (26)







700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000














After-optimization








5 Conclusion



16

14

12

1

08

06

04

02

018 2

Relat

ive e

rror

()


fu

f

1614121080604020

(a)

18 2


7

6

5

4

3

2

1

0Ab

solu

te er

ror (

pixe

ls)

u0

0

(b)

Relat

ive e

rror

()

fu

f

18 2


15

1

05

0

(c)

7

6

5

4

3

2

1

0

Abso

lute

erro

r (pi

xels)

u0

0

18 2


(d)



1

09

08

07

05

06

04

03

02

01

0

Repr

ojec

tion

erro

r (pi

xels)


Image 1Image 2

Image 3Image 4

2181614121080604020

(a)

Repr

ojec

tion

erro

r (pi

xels)


1

04

06

08

12

14

02

0

Image 1Image 2

Image 3Image 4

(b)





01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609



01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash


Competing Interests


Acknowledgments







Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











e119879F119903 = e119879 [e]timesH = 0 (9)





(10)


1006704P119906 = (1 0 minus11988911990600 1 minus119889V00 0 1 )P119906 (11)



[P119889]119879(1 0 minus11988911990600 1 minus119889V00 0 1 )119879 1006704F119903P119908 = 0 (13)


F119903 = (1 0 minus11988911990600 1 minus119889V00 0 1 )119879 1006704F119903 (14)


1006704F119903 = ( 1 0 00 1 01198891199060 119889V0 1) F119903 (15)







119904 [[[[[[[[[1006704119883119889119871 (1006704119903119889 119896)1006704119884119889119871 (1006704119903119889 119896)1

]]]]]]]]]= [[[[

1006704h11987911006704h11987921006704h1198793]]]]P119908 (21)




(22)



[[[[[[[[1006704h311989611198962]]]]]]]]= [ 1006704f1198792 P119908minus1006704f1198791 P119908]

(23)





h1198791198881Kminus119879Kminus1h1198882 = 0


minus119879Kminus1h1198882 (25)




119899sum119894=1

119898sum119895=1

10038171003817100381710038171003817P119894119895 minus 1006704P (KR119894 t119894P119895)100381710038171003817100381710038172 (26)







700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000














After-optimization








5 Conclusion



16

14

12

1

08

06

04

02

018 2

Relat

ive e

rror

()


fu

f

1614121080604020

(a)

18 2


7

6

5

4

3

2

1

0Ab

solu

te er

ror (

pixe

ls)

u0

0

(b)

Relat

ive e

rror

()

fu

f

18 2


15

1

05

0

(c)

7

6

5

4

3

2

1

0

Abso

lute

erro

r (pi

xels)

u0

0

18 2


(d)



1

09

08

07

05

06

04

03

02

01

0

Repr

ojec

tion

erro

r (pi

xels)


Image 1Image 2

Image 3Image 4

2181614121080604020

(a)

Repr

ojec

tion

erro

r (pi

xels)


1

04

06

08

12

14

02

0

Image 1Image 2

Image 3Image 4

(b)





01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609



01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash


Competing Interests


Acknowledgments







Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(22)



[[[[[[[[1006704h311989611198962]]]]]]]]= [ 1006704f1198792 P119908minus1006704f1198791 P119908]

(23)





h1198791198881Kminus119879Kminus1h1198882 = 0


minus119879Kminus1h1198882 (25)




119899sum119894=1

119898sum119895=1

10038171003817100381710038171003817P119894119895 minus 1006704P (KR119894 t119894P119895)100381710038171003817100381710038172 (26)







700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000














After-optimization








5 Conclusion



16

14

12

1

08

06

04

02

018 2

Relat

ive e

rror

()


fu

f

1614121080604020

(a)

18 2


7

6

5

4

3

2

1

0Ab

solu

te er

ror (

pixe

ls)

u0

0

(b)

Relat

ive e

rror

()

fu

f

18 2


15

1

05

0

(c)

7

6

5

4

3

2

1

0

Abso

lute

erro

r (pi

xels)

u0

0

18 2


(d)



1

09

08

07

05

06

04

03

02

01

0

Repr

ojec

tion

erro

r (pi

xels)


Image 1Image 2

Image 3Image 4

2181614121080604020

(a)

Repr

ojec

tion

erro

r (pi

xels)


1

04

06

08

12

14

02

0

Image 1Image 2

Image 3Image 4

(b)





01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609



01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash


Competing Interests


Acknowledgments







Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000

700

600

500

400

300

200

100

0

(p

ixel

)

u (pixel)


10009008007006005004003002001000














After-optimization








5 Conclusion



16

14

12

1

08

06

04

02

018 2

Relat

ive e

rror

()


fu

f

1614121080604020

(a)

18 2


7

6

5

4

3

2

1

0Ab

solu

te er

ror (

pixe

ls)

u0

0

(b)

Relat

ive e

rror

()

fu

f

18 2


15

1

05

0

(c)

7

6

5

4

3

2

1

0

Abso

lute

erro

r (pi

xels)

u0

0

18 2


(d)



1

09

08

07

05

06

04

03

02

01

0

Repr

ojec

tion

erro

r (pi

xels)


Image 1Image 2

Image 3Image 4

2181614121080604020

(a)

Repr

ojec

tion

erro

r (pi

xels)


1

04

06

08

12

14

02

0

Image 1Image 2

Image 3Image 4

(b)





01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609



01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash


Competing Interests


Acknowledgments







Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















After-optimization








5 Conclusion



16

14

12

1

08

06

04

02

018 2

Relat

ive e

rror

()


fu

f

1614121080604020

(a)

18 2


7

6

5

4

3

2

1

0Ab

solu

te er

ror (

pixe

ls)

u0

0

(b)

Relat

ive e

rror

()

fu

f

18 2


15

1

05

0

(c)

7

6

5

4

3

2

1

0

Abso

lute

erro

r (pi

xels)

u0

0

18 2


(d)



1

09

08

07

05

06

04

03

02

01

0

Repr

ojec

tion

erro

r (pi

xels)


Image 1Image 2

Image 3Image 4

2181614121080604020

(a)

Repr

ojec

tion

erro

r (pi

xels)


1

04

06

08

12

14

02

0

Image 1Image 2

Image 3Image 4

(b)





01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609



01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash


Competing Interests


Acknowledgments







Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










16

14

12

1

08

06

04

02

018 2

Relat

ive e

rror

()


fu

f

1614121080604020

(a)

18 2


7

6

5

4

3

2

1

0Ab

solu

te er

ror (

pixe

ls)

u0

0

(b)

Relat

ive e

rror

()

fu

f

18 2


15

1

05

0

(c)

7

6

5

4

3

2

1

0

Abso

lute

erro

r (pi

xels)

u0

0

18 2


(d)



1

09

08

07

05

06

04

03

02

01

0

Repr

ojec

tion

erro

r (pi

xels)


Image 1Image 2

Image 3Image 4

2181614121080604020

(a)

Repr

ojec

tion

erro

r (pi

xels)


1

04

06

08

12

14

02

0

Image 1Image 2

Image 3Image 4

(b)





01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609



01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash


Competing Interests


Acknowledgments







Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

09

08

07

05

06

04

03

02

01

0

Repr

ojec

tion

erro

r (pi

xels)


Image 1Image 2

Image 3Image 4

2181614121080604020

(a)

Repr

ojec

tion

erro

r (pi

xels)


1

04

06

08

12

14

02

0

Image 1Image 2

Image 3Image 4

(b)





01613 01716 01389 01421 01746

After-optimization

01534 01670 01374 01310 01609



01347 01372 01515 00172 mdash

After-optimization

01212 01383 01442 00167 mdash


Competing Interests


Acknowledgments







Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Image 10Image 9



04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










04

03

02

01

0

minus01

minus02

minus03

minus04

minus05

Reprojection error

X (pixels)

Y(p

ixel

s)




(a)

Reprojection error

X (pixels)

04

03

02

01

0

minus01

minus02

minus03

minus04

minus05Y

(pix

els)




(b)








After-optimization




References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Decoupled Calibration Method for...

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