Global Calibration of Multiple Cameras Based on Sphere Targets Junhua Sun 1,† , Huabin He 1,† and Debing Zeng 2, * Received: 1 December 2015; Accepted: 3 January 2016; Published: 8 January 2016 Academic Editor: Vittorio M.N. Passaro 1 2

* †

Ministry of Education Key Laboratory of Precision Opto-mechatronics Technology, Beihang University, Beijing 100191, China; [email protected] (J.S.); [email protected] (H.H.) Department of Applied Science and Technology and Center for Microplasma Science and Technology, Saint Peter’s University, Jersey City, NJ 07036, USA Correspondence: [email protected]; Tel.: +1-201-761-7922 These authors contributed equally to this work.

Abstract: Global calibration methods for multi-camera system are critical to the accuracy of vision measurement. Proposed in this paper is such a method based on several groups of sphere targets and a precision auxiliary camera. Each camera to be calibrated observes a group of spheres (at least three), while the auxiliary camera observes all the spheres. The global calibration can be achieved after each camera reconstructs the sphere centers in its field of view. In the process of reconstructing a sphere center, a parameter equation is used to describe the sphere projection model. Theoretical analysis and computer simulation are carried out to analyze the factors that affect the calibration accuracy. Simulation results show that the parameter equation can largely improve the reconstruction accuracy. In the experiments, a two-camera system calibrated by our method is used to measure a distance about 578 mm, and the root mean squared error is within 0.14 mm. Furthermore, the experiments indicate that the method has simple operation and good flexibility, especially for the onsite multiple cameras without common field of view. Keywords: multiple vision sensors; global calibration; sphere targets; sphere center reconstruction

1. Introduction Three-dimensional (3D) vision systems have the advantages of high precision and good flexibility, so they are widely applied in various fields. The multi-sensor vision system (MVS) is always used, because it has larger measurement range than a single sensor. The MVS needs onsite global calibration after being installed. As one of the most important technical indices of MVS, the measurement accuracy is directly influenced by the global calibration accuracy. In most practical applications of MVS, the structures of the measured objects and the environments are complex, and always lead to a complex distribution of the vision sensors. Vision sensors even have the feature of non-overlapping field of view (FOV), which requires that the global calibration methods have high precision and good flexibility. Most classical global calibration methods [1–5] rely on matching features in the common FOV of all the sensors and are not applicable in the case of non-overlapping FOV. To overcome the limitations of non-overlapping FOV, Luo [6] used a two-theodolite system, and Kitahara et al. [7] used a 3D laser-surveying instrument to accomplish their global calibrations. They used precision auxiliary instruments to reconstruct the feature points in the world coordinate system (WCS) to acquire the transformation matrix between the camera coordinate system (CCS) and the WCS. However, when the on-site space is narrow, the auxiliary instruments have blind zones, and there is even might not be room for them. In the areas of video surveillance and motion tracking, Heng et al. [8], Pflugfelder et al. [9], Carrera et al. [10] and Esquivel et al. [11] used self-calibration methods. The cameras are globally Sensors 2016, 16, 77; doi:10.3390/s16010077

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Heng et al. [8], Pflugfelder et al. [9], Carrera et al. [10] and Esquivel et al. [11] used self-calibration methods. The cameras are globally calibrated through observing objects with specific structures in their FOV. In industrial measurement, there is little satisfactory scene information for the self-calibration process, and its accuracy usually cannot meet the requirements. Agrawal et al. [12], Sensors 2016, 16, [13] 77 2 ofcamera 14 Takahashi et al. and Kumar et al. [14] achieved the global calibration by making each view the targets in a mirror. However, no clear target is guaranteed to be observed by every camera in the complexthrough MVS. observing The fixedobjects constraints of multiple targets are FOV. usedInbyindustrial Liu et al.measurement, [15] to calibrate calibrated with specific structures in their multiple High accuracy can be achieved with this method, repeated pair-wise there cameras. is little satisfactory scene information for the self-calibration process,but and the its accuracy usually operations would reduce the calibration accuracy of MVS. Liu et al. [16] proposed a global calibration cannot meet the requirements. Agrawal et al. [12], Takahashi et al. [13] and Kumar et al. [14] achieved the global making each camera view the targets inand a mirror. clear target iswith method basedcalibration on skewbylaser lines. This method is flexible can However, deal withnothe cameras guaranteed to be observed by every camera in the complex MVS. The fixed constraints of multiple different viewing directions, but there are operational difficulties when we use it to calibrate targets are used by Liu et al. et [15] calibrate multiple cameras. accuracy can be target achieved multiple on-site cameras. Liu al.to[17] and De et al. [18] usedHigh a one-dimensional to with calibrate this method, but the repeated pair-wise operations would reduce the calibration accuracy of MVS. at a MVS, but it is hard to process a long one-dimensional target to calibrate vision sensors et al. [16] proposed a global calibration method based on skew laser lines. This method is longLiu distance. flexible and can deal with the cameras with different viewing directions, but there are operational Focusing on complexly distributed multi-camera systems with non-overlapping FOV, we difficulties when we use it to calibrate multiple on-site cameras. Liu et al. [17] and De et al. [18] used present a global calibration method. It is based on several groups of spherical targets. A group of a one-dimensional target to calibrate MVS, but it is hard to process a long one-dimensional target to spheres is made up of at at least three spheres without constraint. Each camera observes a group of calibrate vision sensors a long distance. spheres. Focusing At the same time, andistributed auxiliary precision camera views the spheres. The coincides on complexly multi-camera systems withall non-overlapping FOV,WCS we present withathe coordinate system of the auxiliary camera. Every camera to be calibrated and the auxiliary global calibration method. It is based on several groups of spherical targets. A group of spheres is camera reconstruct thethree sphere centers of a constraint. group of spheres, so the transformation made up of at least spheres without Each camera observes a group of matrix spheres.from At theevery time, an auxiliary camera all the camera spheres. The WCS and coincides withand the coordinate CCSsame to the WCS can be precision calculated. Theviews auxiliary is light handy, can be easily system of the auxiliary camera. Every camera to be calibrated and the auxiliary camera reconstruct theblind operated. Moreover, the sphere targets can be observed from different directions, so that any sphere centers of a group of spheres, so the transformation matrix from every CCS to the WCS can be zones would be greatly reduced. Besides, this global calibration method can be realized through a calculated. The auxiliary camera is light and handy, and can be easily operated. Moreover, the sphere one-time operation. It avoids the heavy workloads and accuracy losses caused by other targets can be observed from different directions, so that any blind zones would be greatly reduced. repeated operations. Besides, this global calibration method can be realized through a one-time operation. It avoids the The paper is organized as follows: in Section 2, the calculation method of the sphere center heavy workloads and accuracy losses caused by other repeated operations. reconstruction is first proved inasdetail. Then the calculation method method of the transformation matrix is The paper is organized follows: in Section 2, the calculation of the sphere center given, followed by the proved nonlinear optimization. Section 3method provides thetransformation accuracy analysis reconstruction is first in detail. Then the calculation of the matrix and experimental results. Both and experimental data are provided to testanalysis and verify is given, followed by thesimulation nonlinear optimization. Section 3 provides the accuracy and the presented technique. TheBoth conclusions areand stated in Sectiondata 4. are provided to test and verify the experimental results. simulation experimental presented technique. The conclusions are stated in Section 4.

2. Global Calibration Principle

2. Global Calibration Principle

The principle of global calibration for multiple cameras is shown in Figure 1. Only two groups The principle of global calibration for multiple cameras is shown in Figure 1. Only two groups of of cameras are drawn for the sake of discussion. There is no common FOV between the two groups cameras are drawn for the sake of discussion. There is no common FOV between the two groups of of cameras becalibrated, calibrated, of them can observe at least threetargets. spherical targets. The cameras to to be andand eacheach of them can observe at least three spherical The auxiliary auxiliary camera canallview of thetargets sphere (atIfleast If there are three orofmore groups camera can view of theallsphere (attargets least six). theresix). are three or more groups cameras to of cameras to be calibrated, all the sphere targets should to camera. the auxiliary camera. be calibrated, all the sphere targets should be visible to be thevisible auxiliary

Figure1.1.The Theprinciple principle of Figure of global globalcalibration. calibration.

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The global calibration process works as follows: 1.

2. 3.

Install multiple cameras whose intrinsic parameters have been obtained. Then in the FOV of each camera, place at least three sphere targets. They should not be shaded by each other. Fix a precision auxiliary camera. It can view all the targets, and its coordinate system is regarded as the WCS. Each camera takes a few images of the targets in its FOV. Then move the targets several times and repeat taking images. Reconstruct the sphere centers of each group of spheres in the corresponding CCSs and the WCS. Then calculate the transformation matrix from every CCS to the WCS using nonlinear optimization. Thus the global calibration is completed.

2.1. Sphere Center Reconstruction The general equation of ellipse includes five degrees of freedom. However, the ellipse in the sphere projection has two constraints, so it just has three degrees of freedom. Hence it can be represented by a parameter equation with only three parameters. The synthetic data show that the fitting accuracy of parameter equation is obviously higher than that of general equation. If appropriate parameters are chosen, the parameter equation is a linear expression of the three parameters. They can be acquired by linear least squares method. This calculation method is simple and has high accuracy. In the following paragraphs, firstly, the equation F(x,y,z) = 0 of the conic surface with three parameters λ, µ and σ is established. Next, the equation f(x,y) = 0 of the ellipse curve is given according to the geometric relationship of the sphere projection. Then we calculate the three parameters λ, µ and σ through fitting a group of sampling points on the ellipse curve. Finally, the 3D coordinates of the sphere center in the CCS can be acquired from the three parameters. 2.1.1. Sphere Projection Model In the following paragraphs, we propose a parameter equation F(x,y,z) = 0 to describe the sphere projection model. The geometric relationship of the sphere projection is shown in Figure 2. Point O is the center of the camera, and O-xyz is the CCS (in millimeters). Point C is the principal point of the camera and the origin of the image coordinate system (in millimeters). Its x-axis and y-axis are respectively in the same direction with the x-axis and y-axis of the CCS. Point OS is the sphere center, and the point O1s is an intersection point of the line OOS and the image plane. The line OD and the sphere is tangent to the point D. Point A is an intersection point of the line OD and the image plane. The length of OC is the focal length f, and the sphere radius is known as R. The sphere surface and the origin of the CCS, point O, can determine a conic surface. Its vertex is O. Each element of the conic surface is tangent to the sphere, and line OOS is the symmetry axis of the conic surface. Let =DOOS be θ, and the unit direction vector along OOS be S0 “ rcosα, cosβ, cosγsT They satisfies the constraint cos2 α ` cos2 β ` cos2 γ “ 1. Let P(x,y,z) be any point of the conic surface, so that we have the equation of the conic surface as: ´

¯ OP ¨ S0 { |OP| “ cosθ

(1)

Transforming Equation (1) to coordinate form and simplifying it, we have: cosα cosβ cosθ x` y`z´ cosγ cosγ cosγ

b x 2 ` y2 ` z2 “ 0

(2)

Substituting three parameters: λ = cosα/cosγ, µ = cosβ/cosγ and σ = cosθ/cosγ into Equation (2), we have the parameter equation of the conic surface in the CCS: b Fpx, y, zq “ λx ` µy ` z ´ σ

x 2 ` y2 ` z2 “ 0

(3)

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Figure 2. The sphere projection projection model. model.

2.1.2. Ellipse Curve Fitting The intersection intersection of of the the conic conic surface surfaceand andthe theimage imageplane planezz== ff generates the ellipse curve, so that The equation in in the the CCS CCS is: is: its equation # a λx µy ` z ´ σ x22 ` y22 ` z2 2 “ 0 λ ` (4) x + μ y + z −σ x + y + z = 0 z“ f (4)

z= f

The origin of the image coordinate system is C and its x-axis and y-axis are respectively in the originwith of the image is C and its x-axisthe andz y-axis are respectively in the the sameThe direction the x-axiscoordinate and y-axissystem of the CCS. Eliminating in Equation (4), we have same direction x-axis and y-axissystem: of the CCS. Eliminating the z in Equation (4), we have the ellipse equationwith in thethe image coordinate ellipse equation in the image coordinate system: b fpx, yq “ λx ` µy ´ σ

x 2 ` y2 ` f 2 ` f “ 0

f(x, y) = λ x + μ y −σ x2 + y2 + f 2 + f = 0

(5) (5)

If the normalized focal lengths of the camera are known as fx and fy , and its principal point is If the normalized focal lengths of the camera are known as fx and fy, and its principal point is known as (u0 ,v0 ), we have: known as (u0,v0), we have: x u ´ u0 y v ´ v0 “ and “ (6) f x u f−x u f y vf−y v 0 0 and = = (6) f same f y point on the image plane. Substituting where (x,y) (in millimeters) and (u,v) (inf pixels)f x express the Equation (6) into Equation (5), we have the ellipse equation (in pixels): where (x,y) (in millimeters) and (u,v) (in pixels) express the same point on the image plane. d Substituting Equation (6) into Equation (5), we have the 2ellipse equation (in pixels): u ´ u0 u ´ u0 v ´ v0 v ´ v0 2 λp q ` µp q´σ p q `p q `1`1 “ 0 (7) fx u − u f yv − v u f−x u0 2 v − vf y0 2 0 0 λ( ) + μ( ) −σ ( ) +( ) +1 +1 = 0 (7) fx f fx f If {(ui ,vi )| i = 1,2,3 . . . ,n} are known yas the coordinates (in ypixels) of a group of sampling points of theIf ellipse on an are image, we as can linear least method tooffitsampling the ellipse curve, {(ui,vi)⏐icurve = 1,2,3…,n} known theuse coordinates (insquares pixels) of a group points of Equation (7). Then we acquire the optimal solutions of parameters λ, µ and σ. the ellipse curve on an image, we can use linear least squares method to fit the ellipse curve,

Equation (7). Center Then we acquire the optimal solutions of parameters λ, μ and σ. 2.1.3. Sphere Coordinate Calculation the right triangle ODOSCalculation , we have |OOs | “ R{sinθ. Moreover, S0 “ rcosα, cosβ, cosγsT is 2.1.3.InSphere Center Coordinate defined as a unit direction vector along line OOS , so we have: T 0 In the right triangle ODOS, we have O O s = R / sin θ . Moreover, S = [ cosα ,cos β ,cos γ ] is „ Rcosα Rcosβ Rcosγ T 0 lineROO0S, so we defined as a unit direction along have: | OOvector “ |OO s “ s “ , , (8) s s sinθ sinθ sinθ sinθ T R 0 R cos α R cos β R cos γ s = , , ΟΟ s = ΟΟ s s 0 = (8) sin θ sin θ sin θ sin θ

It is not hard to have the equation set:

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2 2 2 2 2 It is not hardto have the equation set: cos 2 α + cos 2 β + cos 2 γ = 1 sin θ = 1 + λ + μ − σ / 1 + λ + μ 2 $ $ a α = λ / 1 + λ 2 + μa cos λ 2= cos α / 2cos γ 2α ` cos 2 ` µ2 ´ σ 2 { 1 ` λ2 ` µ2 ’ ’ 1 ` λ cos β ` cos γ “ 1 sinθ “ (9) ’ ’ a ’ ’ 2 & & = cos β / cos γ 2 β =“ μ λ{ cos / 1 +1λ`2 +λ2μ` λ “μcosα{cosγ cosα µ a (9) ñ = cos θ / cos γ 2 λ2 2` µ2 ’ ’ cosβ µ “σcosβ{cosγ γ =“ + 1λ ` +μ cos 1 /µ{1a ’ ’ ’ ’ % % cosγ “ 1{ 1 ` λ2 ` µ2 σ “ cosθ{cosγ Substituting equation set Equation (9) into Equation (8), we know OOS is T Substituting equation set Equation (9) into Equation (8), we know OOS is Rλ Rμ R ffT we know the coordinates of the « , , . Then 2 2 2 R 2 a1 + λ 2 +Rλ μ 2 − σ 2 ,1a 1 +, λ μ σ + λ 2 + μRµ −σ 2 + − a . Then we know the coordinates 1 ` λ2 ` µ2 ´ σ 2 1 ` λ2 ` µ2 ´ σ 2 1 ` λ2 ` µ2 ´ σ 2 ˜ Rλ ¸ Rμ R . R sphere center OS in the CCS is , ,Rµ Rλ 1is+ λ 2a a2 − σ 2 1 + λ 2 + μ ,2 a of the sphere center OS in the CCS . + μ 2 − σ2 2 12 + λ 22+, μ − σ 2 2 1`λ `µ ´σ 1 ` λ2 ` µ2 ´ σ 2 1 ` λ ` µ2 ´ σ 2 If the theintrinsic intrinsicparameters parametersofof camera sphere radius are known, both known, the sphere If thethe camera andand the the sphere radius are both the sphere center center can be reconstructed from aimage. singleThe image. Thecenter sphere center coordinates reconstructed by can be reconstructed from a single sphere coordinates reconstructed by multiple multiple images can also be averaged to reduce the reconstruction error. images can also be averaged to reduce the reconstruction error.

Transformation Matrix Matrix Calculation Calculation 2.2. Transformation be distributed distributed to form an asymmetric 3D structure in As shown in Figure 3, all the spheres can be procedure. The distances between any sphere center and the others can form a the global calibration procedure. vector. It can can be be used used to to distinguish distinguish this this sphere sphere center center from from the the others. others. Through Through matching matching signature vector. signature vectors, vectors, we we match match the the sphere sphere centers centers reconstructed reconstructedin inthe theCCS CCSand andthe theWCS. WCS. signature

Figure 3. 3. The The sphere sphere centers centers matching matching method. method. Figure

In 3D space, the same sphere center can be described by two vectors, such as vector P in the CCS In 3D space, the same sphere center can be described by two vectors, such as vector P in the and vector Q in the WCS. In such a case, P and Q are called a pair of homonymous vectors. If three CCS and vector Q in the WCS. In such a case, P and Q are called a pair of homonymous vectors. non-collinear sphere centers are reconstructed in both the CCS and the WCS, we get three pairs of If three non-collinear sphere centers are reconstructed in both the CCS and the WCS, we get three homonymyous vectors. The transformation matrix between the CCS and the WCS can be calculated pairs of homonymyous vectors. The transformation matrix between the CCS and the WCS can be through them. Let three non-collinear sphere centers be described in the CCS as P1, P2 and P3, calculated through them. Let three non-collinear sphere centers be described in the CCS as P1 , P2 respectively, and Q1, Q2 and Q3 in the WCS. The transformation from the CCS to the WCS is defined and P3 , respectively, and Q1 , Q2 and Q3 in the WCS. The transformation from the « CCS to ff the WCS is R T . as {Qi = R·Pi + T⏐ i = 1,2,3…,n}, so the transformation matrix is H = R T 0 is1H “ defined as {Qi = R¨ Pi + T| i = 1,2,3 . . . ,n}, so the transformation matrix . 0 1 The rotation matrix is calculated by: The rotation matrix is calculated by:

R = [Q1Q2 , Q2Q3 , Q1Q2 × Q2Q3 ] ⋅[ PP 1 2 , P2 P3 , PP 1 2 × P2 P3 ]

−1

(10) R “ rQ1 Q2 , Q2 Q3 , Q1 Q2 ˆ Q2 Q3 s ¨ rP1 P2 , P2 P3 , P1 P2 ˆ P2 P3 s´1 (10) with Q1Q2 = Q2 − Q1 , Q 2 Q 3 = Q 3 − Q 2 , P1P2 = P2 − P1 and P2 P3 = P3 − P2 . The translation vector with Q1 Q2 “ Q2 ´ Q1 , Q2 Q3 “ Q3 ´ Q2 , P1 P2 “ P2 ´ P1 and P2 P3 “ P3 ´ P2 . The translation vector T = (Q1 + Q 2 + Q3 − RP1 − RP2 − RP3 ) / 3 (11) T “ pQ1 ` Q2 ` Q3 ´ RP1 ´ RP2 ´ RP3 q{3 (11) Consequently, the transformation matrix from the CCS to the WCS is acquired. Consequently, the transformation matrix from the CCS to the WCS is acquired. 2.3. Nonlinear Optimization

In the practical calibration procedure, we acquire n pairs of homonymous vectors by fixing the cameras and placing the targets many times. Then the optimal solution of the transformation matrix H is calculated by using nonlinear optimization.

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2.3. Nonlinear Optimization In the practical calibration procedure, we acquire n pairs of homonymous vectors by fixing the cameras and placing the targets many times. Then the optimal solution of the transformation matrix H is calculated by using nonlinear optimization. The objective function is: 2 n ÿ „ „ min F “ ||Qi ´ HPi || (12) i“1

„

„

„

„

with Qi “ rxq , yq , zq , 1sT and Pi “ rx p , y p , z p , 1sT , where » Qi and Pi are fi respectively the homogeneous r1 r2 r3 — ffi coordinates of Qi and Pi . Let the rotation matrix R be – r4 r5 r6 fl, and it must satisfy orthogonal r7 r8 r9 constraint. We have an equation set: $ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ %

h1 “ r12 ` r22 ` r32 ´ 1 h2 “ r42 ` r52 ` r62 ´ 1 h3 “ r72 ` r82 ` r92 ´ 1 h4 “ r1 r4 ` r2 r5 ` r3 r6 h5 “ r1 r7 ` r2 r8 ` r3 r9 h6 “ r4 r7 ` r5 r8 ` r6 r9

(13)

Using the method of penalty function, from Equations (12) and (13) we get the unconstrained optimal objective function: 2 n 6 ÿ ÿ „ „ minF “ ||Qi ´ HPi || ` M h2j (14) i “1

j “1

where the penalty factor M determines the orthogonal error of the rotation matrix R. Here M takes a values of 10 considering the error distribution. Equation (14) is solved by the Levenberg-Marquardt method. The transformation matrix calculated from Equations (10) and (11) is used as the initial value of the iterations to solve Equation (14). Then we get the optimal value of the transformation matrix. 3. Analysis and Experiment This section first discusses the factors which affect the calibration accuracy. The mathematical description and computer simulation are both given to analysis the calibration errors. Finally real data are given to evaluate the calibration accuracy. 3.1. Accuracy Analysis The factors affecting the sphere center reconstruction are discussed here. Section 2.1.3, we have: « oos “ x ¨ i ` y ¨ j ` z ¨ k “

Rλ

Rµ

R

According to

a ,a ,a 1 ` λ2 ` µ2 ´ σ 2 1 ` λ2 ` µ2 ´ σ 2 1 ` λ2 ` µ2 ´ σ 2

ffT (15)

According to Equation (15), the partial derivative is: Boos Bx By Bz “ ¨i` ¨j` ¨k Bλ Bλ Bλ Bλ

(16)

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Let ∆λ be the errors caused by noise, and ∆oos be the variation of the sphere center reconstruction, so we get: Bx By Bz Boos ¨ ∆λ “ ∆λ ¨ i ` ∆λ ¨ j ` ∆λ ¨ k (17) ∆oos “ Bλ Bλ Bλ Bλ Let |∆oos | be the error of sphere center reconstruction, from Equation (17) we have: «ˆ |∆oos | “

Bx ∆λ Bλ

˙2

ˆ `

By ∆λ Bλ

˙2

ˆ `

Bz ∆λ Bλ

˙2 ff0.5 (18)

From Equation (15), we the partial derivatives: Bz x2 z By xyz xz2 Bx “ z´ 2 , “ ´ 2 and “´ 2 Bλ Bλ R Bλ R R

(19)

Substituting Equation (19) into Equation (18), we have: ¯0.5 z2 x 2 ´ 2 2 |∆oos | “ z ` 4 L ´ 2R ∆λ R „

2

(20)

` ˘0.5 where L “ x2 ` y2 ` z2 “ |oos |, which means the distance between the sphere center and the camera center. When considering all the errors of ∆λ, ∆µ and ∆σ, through the same proof procedure as above, we get: „ „ „ 2 2´ ¯0.5 ¯0.5 ¯0.5 z2 x 2 ´ z2 y2 ´ z L 2 ´ R2 |∆oos | “ z2 ` 4 L2 ´ 2R2 ¨ ∆λ ` z2 ` 4 L2 ´ 2R2 ¨ ∆µ ` L ¨ ∆σ R R R4

(21)

From Equation (21), we conclude that the larger R, the smaller |∆oos |. The smaller L, the smaller |∆oos |. Therefore, the calibration accuracy can be improved by enlarging the radius of the sphere targets or putting the sphere targets closer to the camera. 3.2. Synthetic Data In this section, we first analyzed some factors that affect the sphere center reconstruction accuracy. Then we analyzed the influence that the nonlinear optimization makes on the calibration accuracy. 3.2.1. Factors That Affect the Sphere Center Reconstruction Accuracy To verify the effectiveness of the sphere center reconstruction method, we use the software MATLAB to simulate it. The five factors include the ellipse fitting method, image noise level, sphere radius, sampling point number and distance between the sphere and camera. Their influences on the reconstruction error are studied. In the experiments, “general equation fitting” means using the general equation of the ellipse to fit it, and “parameter equation fitting” means using the parameter equation of the ellipse, Equation (7), to fit it. The experiment for each factor is repeated for 1000 times, and the reconstruction accuracy is evaluated by the root mean squared (RMS) error of the sphere center. The sphere radius is 40.000 mm in Figure 4a,c,d. The number of sampling points in Figure 4a,b,d is 600. Gaussian noise with 0 mean and 0.5 standard deviation is added to the image points in Figure 4b,c,d. The focal length is 24 mm and the image resolution is 1024 ˆ 1024. As shown in Figure 4a–d, improving the imaging quality, increasing the sphere radius, increasing the sampling point number and decreasing the distance between the sphere and camera can all improve the reconstruction accuracy. However, when the radius is more than 35 mm, the number of sampling points is more than 600 or the distance between the sphere and camera is less than 1000 mm, their influences are very small. Comparing the two fitting methods, we can conclude that the fitting accuracy of the parameter equation is higher than that of the general equation.

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(a)

(b)

(c)

(d)

(e) Figure4.4.The Theeffects effectsofofseveral severalmajor majorfactors factorson onthe thereconstruction reconstructionerror. error.(a) (a)The Thenoise noiselevel levelof ofimage image Figure points; (b) The sphere radius; (c) The number of sampling points; (d) The distance between the points; (b) The sphere radius; (c) The number of sampling points; (d) The distance between the sphere sphere and camera; (e) Multiple images and aimage. single image. and camera; (e) Multiple images and a single

Finally, two spheres with constant distance are projected from four different directions to Finally, two spheres with constant distance are projected from four different directions to generate generate four images. The same Gaussian noise is added to each one. Each image is used to four images. The same Gaussian noise is added to each one. Each image is used to reconstruct the reconstruct the two sphere centers to calculate their distance, and the four distances are averaged as two sphere centers to calculate their distance, and the four distances are averaged as the simulation the simulation result of multiple images. The experiment is repeated 1000 times, and the result of multiple images. The experiment is repeated 1000 times, and the reconstruction accuracy is reconstruction accuracy is evaluated by the RMS error of the sphere center distance. The sphere evaluated by the RMS error of the sphere center distance. The sphere radius is 40.000 mm, and the radius is 40.000 mm, and the number of sampling points is 600. The focal length is 24 mm, and the number of sampling points is 600. The focal length is 24 mm, and the real distance between the two real distance between the two sphere centers is 400.000 mm. As shown in Figure 4e, the sphere centers is 400.000 mm. As shown in Figure 4e, the reconstruction accuracy of four images is reconstruction accuracy of four images is higher than that of a single image. higher than that of a single image. 3.2.2. Effect of Nonlinear Optimization on Calibration Accuracy 3.2.2. Effect of Nonlinear Optimization on Calibration Accuracy In this subsection, we analyze the influence of the nonlinear optimization on the calibration In this subsection, we analyze the influence of the nonlinear optimization on the calibration accuracy through computer simulation. First, two cameras are globally calibrated by viewing three accuracy through computer simulation. First, two cameras are globally calibrated by viewing three spherical targets at the same time, and the calibration results are calculated without optimization. Then, the two cameras are globally calibrated by viewing 15 sphere targets at the same time, and the calibration results are calculated through nonlinear optimization. The two kinds of results are

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Sensors 2016, 16, 77 9 of 14 spherical targets at the same time, and the calibration results are calculated without optimization. Then, the two cameras are globally calibrated by viewing 15 sphere targets at the same time, and compared to show the effect of optimization on the calibration accuracy. The intrinsic parameters of the calibration results are calculated through nonlinear optimization. The two kinds of results are the two cameras are both set as: compared to show the effect of optimization on the calibration accuracy. The intrinsic parameters of the two cameras are both set as: 0 800fi » 2000 0 0 800 K—= 2000 2000 600ffi (22) K“– 0 (22) 2000 600 fl 0 0 1 0 0 1

The position relationship of the two cameras are set as Rlr = [0.8, 0.02, 0.1]T and The position relationship of the two cameras are set as Rlr = [0.8, 0.02, 0.1]T and Tlr = [–50, –400, Tlr = [–50, –400, –100]. The rotation vector is expressed as a Rodriguez vector here. The radius of the –100]. The rotation vector is expressed as a Rodriguez vector here. The radius of the sphere targets is 40 sphere targets is 40 mm, and the distance between the camera and the sphere center is about mm, and the distance between the camera and the sphere center is about 1000 mm. The calibration 1000 mm. The calibration accuracy is expressed by the relative error of Rlr and Tlr. That is accuracy is expressed by the relative error of Rlr and Tlr . That is |∆Rlr |/|Rlr | and |∆Tlr |/|Tlr |. ⏐ΔRlr⏐/⏐Rlr⏐ and ⏐ΔTlr⏐/⏐Tlr⏐. Gaussian noise with 0 mean and standard deviation (0.05–0.5) is Gaussian noise with 0 mean and standard deviation (0.05–0.5) is added to the image points. The added to the image points. The experiment is repeated for 1000 times, and the average of the relative experiment is repeated for 1000 times, and the average of the relative error is regarded as the calibration error is regarded as the calibration error. As shown in Figure 5, the calibration results calculated error. As shown in Figure 5, the calibration results calculated through nonlinear optimization are more through nonlinear optimization are more accurate than those without optimization. accurate than those without optimization.

(a)

(b)

Figure5.5. The Theeffects effectsof ofnonlinear nonlinearoptimization optimizationon onthe thecalibration calibrationaccuracy. accuracy. (a) (a) The The effects effects on onthe the Figure rotation vector; (b) The effects on the translation vector. rotation vector; (b) The effects on the translation vector.

3.3.Real RealData Data 3.3. 3.3.1.Sphere SphereCenter CenterDistance DistanceMeasurement Measurement 3.3.1. Inthis thisexperiment, experiment, sphere and distances between thecenters spherehave centers been In the the sphere radii radii and distances between the sphere been have accurately accurately measured in advance as real values. A single camera is used to reconstruct two sphere measured in advance as real values. A single camera is used to reconstruct two sphere centers to centers totheir calculate theirIndistance. In Section 2.1.3, we a single can the reconstruct calculate distance. Section 2.1.3, we proved thatproved a singlethat camera can camera reconstruct sphere the sphere center through a single image. two FOV of we a camera, wetake use center through a single image. When two When spheres arespheres both inare theboth FOVinofthe a camera, use it to it to take an image of the two spheres, then the two sphere centers can be reconstructed through an image of the two spheres, then the two sphere centers can be reconstructed through the image.the If image. If we get the 3D coordinates of thecenters two sphere thecalculate CCS, wetheir candistance calculateastheir we get the 3D coordinates of the two sphere in the centers CCS, weincan the distance as the measurement result. Ten measurement results are used totocalculate error to measurement result. Ten measurement results are used to calculate RMS error evaluateRMS the accuracy evaluate the accuracy of sphere center reconstruction. of sphere center reconstruction. Thesphere spheretargets targetsand andtheir theirserial serialnumbers numbersare areshown shownininFigure Figure6a. 6a.The Theprecise precisemeasurement measurement The shows that the diameters of sphere targets 1 and 2 are 40.325 mm and 40.298 mm, respectively, shows that the diameters of sphere targets 1 and 2 are 40.325 mm and 40.298 mm, respectively, and and the thecoordinates 3D coordinates of their centers are respectively (0.010, and (113.219, So 3D of their centers are respectively (0.010, 0.019,0.019, 0.000)0.000) and (113.219, 0.022,0.022, 0.000).0.000). So that that the standard value of the distance between the centers of spheres 1 and 2 is 113.229 mm. the standard value of the distance between the centers of spheres 1 and 2 is 113.229 mm.

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(a)

(b)

Figure 6. The experiment of sphere center distance measurement. (a) The positions and serial

Figure 6. The experiment of sphere center distance measurement. numbers of the targets; (b) A sample image for distance measurement. (a) The positions and serial numbers of the targets; (b) A sample image for distance measurement. Sensors 2016, 16, 77 10 of 14 The resolution ratio of the high-precision camera used in the experiment is 4256 × 2832, and its focal length is 24 mm, and its angle of view is 74° × 53°. The intrinsic parameters of this camera are:

The resolution ratio of the high-precision camera used in the experiment is 4256 ˆ 2832, and its 0 2135.46395 2893.64571 ˝ ˆ 53˝ . The intrinsic focal length is 24 mm, and its angle of view is 74 parameters of(23) this camera are: K= 0 2900.50269 1401.48662 »

0

0

1

2893.64571 2135.46395 and “Plumb Bob” distortion model [19] is used to get 0 its distortion coefficients: — K“–

0 0

2900.50269 0

fi

ffi 1401.48662 fl 1

kc = 0.09616, −0.07878, −0.00039, 0.00006, −0.00896

and

(24)

Forty images are captured and one of them is shown in Figure 6b. All the images are divided (a) (b) into 10 groups, and every group includes four images. Every image is used to calculate the distance “Plumbbetween Bob” distortion [19] isThe used get its distortion coefficients: Figure 6. The experiment sphere center distance measurement. (a) every The positions and serial are the centers ofmodel sphere 1of and 2. fourto values calculated from group of images numbers the average targets; (b) A sample image foras distance measurement. averaged, andofthe value is regarded a measurement result. All the ten measurement results are shown 1. kcin“Table r0.09616, ´0.07878, ´0.00039, 0.00006, ´0.00896s The resolution ratio of the high-precision camera used in the experiment is 4256 × 2832, and its focal length is 24 mm, Table and its1. angle of viewresults is 74° of × 53°. The intrinsic parameters of this camera are: Measurement the sphere center distance.

(23)

(24)

Forty images are captured and one of them is shown in Figure 6b. All the images are divided 113.282 113.246 113.413 113.312 0 2135.46395 2893.64571 113.329 Ten measurement resultsincludes (mm) into 10 groups, and every group four images. Every image is used 113.298 to calculate the distance 113.348 113.129 113.267 113.175 K= 0 2900.50269 1401.48662 (23) Average Value (mm) 113.280 between the centers of sphere 1 and 2. The four values calculated from every group of images are 0 0 1 Real value (mm) 113.229 averaged, and the average value regarded a to measurement result. All the ten measurement results RMS (mm)is model 0.09 and “Plumb Bob”error distortion [19] is as used get its distortion coefficients: are shown in Table 1. c = 0.09616, −0.07878, −0.00039, 0.00006, −0.00896 3.3.2. Global CalibrationkResults

(24)

Forty images are 1. captured and oneofofresults them targets isofshown inused Figure 6b. All the arewithout divided Table Measurement theare sphere center distance. As shown in Figure 7a, two groups sphere to calibrate twoimages cameras into 10 groups, every includes images. Every image is used to groups calculate distance common FOV to and verify the group effectiveness of four the global calibration method. Two of the targets are between theplaced centersinofthe sphere 1 of and Thecameras. four values from every group of are respectively views the2.two Eachcalculated camera 113.413 observes the targets in images their own 113.282 113.246 113.329 113.312 averaged, the average value is regarded measurement result. All the ten measurement Ten measurement (mm) FOV, andresults anand auxiliary precision camera observesas allathe sphere targets. results are shown in Table 1. 113.348 113.129 113.267 113.175 113.298

Average Value (mm) 113.280 Table 1. Measurement results of the sphere center distance. Real value (mm) 113.229 113.282 113.246 113.413 113.329 Ten measurement results (mm) RMS error (mm) 113.348 113.129 113.2670.09 113.175

3.3.2. Global

Average Value (mm) Real value (mm) Calibration RMSResults error (mm)

113.312 113.298

113.280 113.229 0.09

(a)7a, two (c) two cameras without As shown inGlobal Figure groups of sphere(b) targets are used to calibrate 3.3.2. Calibration Results common FOV to verify the effectiveness of the global calibration method. Two groups Figure 7. The global calibration experiment. (a) the physical system; (b) serial numbers of the targets As shown in Figure 7a, two groups of sphere targets are used to calibrate two cameras withoutof targets are viewedFOV by the the camera; (c) serial numbers of the targets viewed bymethod. the right camera. to left verify the of effectiveness ofcameras. the global calibration Two groups of targets arein their own respectively common placed in views the two Each camera observes the targets respectively placed in the views of the two cameras. Each camera observes the targets in their own FOV, and an auxiliary precision camera observes all the sphere targets. FOV, and an auxiliary precision camera observes all the sphere targets.

(a)

(b)

(c)

Figure 7. The global calibration experiment. (a) the physical system; (b) serial numbers of the targets

Figure 7. Theviewed global (a)ofthe physical system; (b) camera. serial numbers of the targets by calibration the left camera;experiment. (c) serial numbers the targets viewed by the right viewed by the left camera; (c) serial numbers of the targets viewed by the right camera.

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The targets in the experiment are white matte ceramic spheres. Their serial numbers are shown in Figure The of theare spheres precisely measured advance, and the results are The 7b,c. targets in diameters the experiment whiteare matte ceramic spheres. in Their serial numbers are shown shown in Table in Figure 7b,c. 2. The diameters of the spheres are precisely measured in advance, and the results are

shown in Table 2.

Table 2. Diameters of the sphere targets.

Table 2. Diameters of the sphere targets. Serial Number 1 2 3 4 5

Serial Number Diameter (mm)1 50.700 250.720 Diameter (mm) 50.700 50.720

6

3 5 50.715 50.708 4 50.710 50.723 50.715 50.708 50.710

6 50.723

The auxiliary camera used in the experiment is the camera used in Section 3.3.1 above, and its The auxiliary camera used in the experiment is the camera used in Section 3.3.1 above, and its intrinsic parameters are shown as Equations (23) and (24). The cameras to be calibrated are common intrinsic parameters are shown as Equations (23) and (24). The cameras to be calibrated are common industrial cameras, whose resolution ratio is 1360 ˆ 1024. Their intrinsic parameters are: industrial cameras, whose resolution ratio is 1360 × 1024. Their intrinsic parameters are: » fi 1974.52417 728.88468 00 728.88468 1974.52417 — ffi Kleft K “ –= 00 1974.65442 549.29770 549.29770 fl 1974.65442 left 0 0 1 0 0 1

and: and:

»

fi 1977.18335 0 674.41864 1977.18335 0 674.41864 — ffi Kright “ – 0 1976.94817 514.46720 fl 0 1976.94817 514.46720 K right = 0 0 1 0 0 1 respectively, respectively,and and“Plumb “PlumbBob” Bob”distortion distortionmodel model[19] [19]isisused usedtotoget gettheir theirdistortion distortioncoefficients coefficients kkc_left = [–0.13109, 0.25232, –0.00007, 0.00018, 0.00000] and k = [–0.12748, 0.21361, c_left = [–0.13109, 0.25232, –0.00007, 0.00018, 0.00000] and c_right kc_right = [–0.12748, 0.21361, –0.00000, –0.00000, 0.00002, 0.00000]. 0.00002, 0.00000]. All Allthe thecameras camerasare arefixed, fixed,and andthe thetwo twogroups groupsofofsphere spheretargets targetsare areplaced placedfor forten tentimes timesinin appropriate positions. Ten groups of images are captured and one of them is shown in Figure appropriate positions. Ten groups of images are captured and one of them is shown in Figure8.8.

(a)

(b)

(c)

Figure8.8.A A group of sample images forglobal the global calibration. An from image the auxiliary Figure group of sample images for the calibration. (a) An (a) image thefrom auxiliary camera; camera; (b) An image from the left camera; (c) An image from the right camera. (b) An image from the left camera; (c) An image from the right camera.

The transformation matrices from the left and right CCS to the WCS are respectively calculated The transformation matrices from the left and right CCS to the WCS are respectively calculated using Equation (14). The results are: using Equation (14). The results are: »

0.9319 — ´0.2239 — Hlw “ — – ´0.2855 0

0.3497 0.7641 0.5421 0

25.057 0.9319 0.3497 0.0968 − 0.2239 0.7641 − 0.6050 −180.185 and fi » H lw = − 0.2855 ´0.2788 0.0968 25.057 0.5421 0.7903 0.9581 389.945´0.0653 ffi — 0.2442 0.6949 ´0.6050 ´180.185 0.6763 0 0 — 1 ffi0

0.7903 0

ffi and Hrw “ — – 0.1496 389.945 ´0.7161 −11.453 0.9581 fl−0.0653 −0.2788 0 0 1

0.2442 0.6949 Hrw = 0.1496 −0.7161 0 0

0.6763

0.6818 0

204.524 413.723 1

0.6818 0

´11.453 204.524 413.723 1

fi ffi ffi ffi fl

(25) (25)

Then the transformation matrix from the left CCS to the right CCS can be calculated from Equation (25). The global calibration result of the experiment is:

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Then the transformation matrix from the left CCS to the right CCS can be calculated from Equation (25). 12 of 14 result of the experiment is:

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0.7955 0.7955 0.6027 0.60270.0632 0.0632 − 62.5153 ´62.5153 — − − − 0.0120 0.1200 0.9927 252.7238 ´0.0120 0.1200 ´0.9927 ´252.7238 = — Hlrlr “ H — –−0.6059 ´0.60590.7889 0.78890.1026 0.1026−286.5962 ´286.5962 0 0 0 1 0 0 0 1

»

fi ffi ffi ffi fl

(26) (26)

3.3.3. Global Calibration Accuracy Evaluation 3.3.3. Global Calibration Accuracy Evaluation verify accuracy global calibration Section 3.3.2 above, thetwo twocameras camerasmake makeup up a To To verify thethe accuracy of of thethe global calibration inin Section 3.3.2 above, the binocular vision system as shown in Figure 9a. Its global calibration result is shown in Equation a binocular vision system as shown in Figure 9a. Its global calibration result is shown in(26). Sphere(26). target 1 andtarget sphere target 2 aretarget fixed 2byare a rigid respectively laid in thelaid views Equation Sphere 1 and sphere fixed rod, by a and rigidare rod, and are respectively in of left of and right diameters are shownare in Table 2. in The auxiliary measures thethe views the left cameras. and rightTheir cameras. Their diameters shown Table 2. Thecamera auxiliary camerathe distance of the two sphere centers, and the average value of the results is regarded as the standard measures the distance of the two sphere centers, and the average value of the results is regarded as of the distance. two sphere centers arecenters respectively reconstructed by the two thevalue standard value of theThe distance. The two sphere are respectively reconstructed bycameras, the two so that their distance can be calculated based onbased the global results. results. The distances measured cameras, so that their distance can be calculated on thecalibration global calibration The distances by the binocular vision system are compared with the standard value to evaluate the accuracy of the measured by the binocular vision system are compared with the standard value to evaluate the global of calibration. accuracy the global calibration.

(a)

(b)

(c)

(d)

Figure 9. 9.The global calibration calibration accuracy accuracyevaluation. evaluation.(a)(a) physical system; Figure Theexperiment experimentof of global TheThe physical system; (b) An (b) image An image from the auxiliary camera; (c) An image from the left camera; (d) An image from the from the auxiliary camera; (c) An image from the left camera; (d) An image from the right camera. right camera.

One of the images captured by the auxiliary camera is shown in Figure 9b. The ten measurement One of the images captured by the auxiliary camera is shown in Figure 9b. The ten results are shown in Table 3. measurement results are shown in Table 3. Table 3. Distances measured by the auxiliary camera. Table 3. Distances measured by the auxiliary camera. 578.004

578.154

578.004 578.154 Ten results measurement Ten measurement (mm)results (mm) 578.158 578.228 578.158 578.228 value L0 (mm) Average value Average L 0 (mm)

578.091

578.043

578.291

578.091 578.043 578.291 577.982 578.143 578.301 577.982 578.143 578.301 578.140 578.140

two spheretargets targetsare areplaced placed in the times, andand oneone group of the TheThe two sphere the right rightpositions positionsfor forten ten times, group ofimages the captured by the binocular vision system is shown in Figure 9c,d. The ten distances measured by images captured by the binocular vision system is shown in Figure 9c,d. The ten distances measuredthe vision system and and the EMS errorerror are shown in Table 4. 4. by binocular the binocular vision system the EMS are shown in Table

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Table 4. Distances measured by the binocular vision system. Distance (mm)

578.068

578.191

578.227

578.254

578.264

578.407

577.919

578.238

578.019

578.023

Real value L0 (mm) Absolute error (mm)

578.140 ´0.072

0.051

0.087

0.114

0.124

0.267

´0.221

0.098

´0.121

´0.117

RMS error (mm)

0.14

4. Conclusions In this paper, we have developed a new global calibration method. In the calibration process, an isotropic sphere target can be simultaneously observed by different cameras from different directions, so the blind zones are reduced. There is no restriction on the position relationship between any two spheres, so the method is flexible in complex on-site environments. Moreover, a one-time operation can globally calibrate all the cameras without common FOV. This avoids the heavy workloads and accuracy loss caused by other repeated operations. A parameter equation is also used to fit the ellipse curve to improve the global calibration accuracy. Our experiments show that the proposed method has the advantages of simple operation, high accuracy and good flexibility. It can conveniently realize the global calibration of complexly distributed cameras without common FOV. In the practical application of this method, images with high-quality ellipse contours are necessary to the high accuracy of calibration. Acknowledgments: This project is supported by National Natural Science Foundation of China under grant (Nos. 61275162 and 61127009). Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund (ACS PRF #53486-UNI6) for partial support of this research. Author Contributions: The work presented here was carried out in collaboration among all authors. All authors have contributed to, seen and approved the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations The following abbreviations are used in this manuscript: 3D MVS FOV CCS WCS RMS

three-dimensional multi-sensor vision system field of view camera coordinate system world coordinate system root mean squared

References 1. 2. 3. 4. 5.

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Luo, M. Mutli-Sensors Vision Measurement System And Applications. Ph.D. Thesis, Tianjin University, Tianjin, China, 1996. Kitahara, I.; Saito, H.; Akimichi, S.; Onno, T.; Ohta, Y.; Kanade, T. Large-scale virtualized reality. In Proceedings of the IEEE Computer Vision and Pattern Recognition(CVPR), Technical Sketches, Kauai, HI, USA, 8–14 December 2001; pp. 312–315. Heng, L.; Furgale, P.; Pollefeys, M. Leveraging Image-based Localization for Infrastructure-based Calibration of a Multi-camera Rig. J. Field Robot. 2014, 32, 4912–4919. [CrossRef] Pflugfelder, R.; Bischof, H. Localization and trajectory reconstruction in surveillance cameras with nonoverlapping views. IEEE Trans. Pattern Anal. Mach. Intell. 2010, 32, 709–721. [CrossRef] [PubMed] Carrera, G.; Angeli, A.; Davison, A.J. SLAM-based automatic extrinsic calibration of a multi-camera rig. In Proceedings of the 2011 IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China, 9–13 May 2011; pp. 2652–2659. Esquivel, S.; Woelk, F.; Koch, R. Calibration of a multi-camera rig from non-overlapping views. Lect. Notes Comput. Sci. 2007, 4713, 82–91. Agrawal, A. Extrinsic camera calibration without a direct view using spherical mirror. In Proceedings of the 2013 IEEE International Conference on Computer Vision (ICCV), Sydney, Australia, 1–8 December 2013; pp. 2368–2375. Takahashi, K.; Nobuhara, S.; Matsuyama, T. A new mirror-based extrinsic camera calibration using an orthogonality constraint. In Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Providence, RI, USA, 16–21 June 2012; pp. 1051–1058. Kumar, R.K.; Ilie, A.; Frahm, J.M.; Pollefeys, M. Simple calibration of non-overlapping cameras with a mirror. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Anchorage, AK, USA, 23–28 June 2008; pp. 1–7. Liu, Z.; Zhang, G.; Wei, Z.; Sun, J. A global calibration method for multiple vision sensors based on multiple targets. Meas. Sci. Technol. 2011, 22, 102–125. [CrossRef] Liu, Q.; Sun, J.; Liu, Z.; Zhang, G. Global calibration method of multi-sensor vision system using skew laser lines. Chin. J. Mech. Eng. 2012, 25, 405–410. [CrossRef] Liu, Z.; Zhang, G.; Wei, Z.; Sun, J. Novel calibration method for non-overlapping multiple vision sensors based on 1D target. Opt. Lasers Eng. 2011, 49, 570–577. [CrossRef] De Franç, J.A.; Stemmer, M.R.; França, M.B.M.; Piai, J.C. A new robust algorithmic for multi-camera calibration with a 1D object under general motions without prior knowledge of any camera intrinsic parameter. Pattern Recognit. 2012, 45, 3636–3647. [CrossRef] Brown, D.C. Decentering distortion of lenses. Photom. Eng. 1966, 32, 444–462. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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