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X-axis andthe direction of an axis of maximum tangential distortion.The decentering distortion in the Cartesian coordinatescan be described as follows: δXPdδYPd = cos ϕ −sin ϕsin ϕ cos ϕ • δrdδtd . (9)Note that cos ϕ = XP /ρ and sin ϕ = YP /ρ.Let p1 =−j1 sin ϕ0 and p2 = j1 cos ϕ0, then δXPd =p1 3X2P + Y 2P +2p2XP YP + O (XP ,YP )4 δYPd =2p1XP YP + p2 X2P +3Y 2P + O (XP ,YP )4 (10)where p1 and p2 are the coefficients of decenteringdistortion.3) Thin prism distortion: This causes the amounts of bothradial and tangential distortions and can be expressed as δrp =(i1ρ2 + i2ρ4 + •••)sin(ϕ − ϕ1)δtp =(i1ρ2 + i2ρ4 + •••)cos(ϕ − ϕ1)(11)where ϕ1 is the angle between the positiveX-axis and thedirection of an axis of maximum tangential distortion. Lets1 = −i1 sin ϕ1 and s2 = i1 cos ϕ1, then the thin prismdistortion in the Cartesian coordinates can be describedas follows: δXPp = s1 X2P + Y 2P + O (XP ,YP )4 δYPp = s2 X2P + Y 2P + O (XP ,YP )4 (12)where s1 and s2 are the coefficients of thin prismdistortion.4) Total distortion: Combining (7), (10), and (12) and as-suming that the terms of order higher than 3 are negli-gible, then we can obtain the optical system distortionerrors (δXP ,δYP ) in the CCD image plane coordinatesystem in (13), shown at the bottom of the page. Let g1 =p1+s1, g2 =p2+s2, g3 =2p1, and g4 =2p2,(13) for each image point can be rewritten as follows:⎧⎪ ⎪ ⎨⎪ ⎪ ⎩δXPi = k1XPi X2Pi + Y 2Pi +(g1 + g3)X2Pi + g4XPiYPi + g1Y 2PiδYPi = k1YPi X2Pi + Y 2Pi +g2X2Pi + g3XPiYPi +(g2 + g4)Y 2Pi.(14)In (14), (XPi,YPi) do not have distortion errors and corre-spond to (XIi,YIi), which also do not have distortion errors.However, (XIi,YIi) cannot be obtained; thus, in the actualcalculation, (xIi,yIi), which contain distortion errors, are used.Let (δxPi ,δyPi ) denote the optical system distortion errorsin the CCD image plane coordinate system corresponding to(xPi,yPi). The optical system distortion errors (δxIi ,δyIi )in the computer image coordinate system corresponding to(xIi,yIi) can be denoted as follows according to (14): δxIi =k1xIi x2Ii+ y2Ii +(g1 + g3)x2Ii + g4xIiyIi + g1y2IiδyIi =k1yIi x2Ii+ y2Ii + g2x2Ii + g3xIiyIi +(g2 + g4)y2Ii.(15)(xPi,yPi) and (xIi,yIi) can be compensated by (δxPi ,δyPi )and (δxIi ,δyIi ) as follows: XPi = xPi + δxPiYPi = yPi + δyPi(16) XIi = xIi + δxIiYIi = yIi + δyIi .(17)Because the transfer frequency of the horizontal shift registerof the CCD camera is different from the horizontal samplingfrequency of the frame grabber, a horizontal direction scalefactor needs to be calibrated. Since the line numbers of CCDsensor elements in the Y direction on the CCD image plane isthe same as the sampling line numbers of the frame grabber,the vertical direction scale factor is 1 [27]. Let kX denote thehorizontal direction scale factor on the CCD image plane, letsX represent the center-to-center distance between adjacentCCD sensor elements in the X (scan line) direction, and let sYrepresent the center-to-center distance between adjacent CCDsensor elements in the
Y direction. Then, we have xPi = kX • sX • xIiyPi = sY • yIi.(18)According to the geometric model of camera calibration [25],[26], the calibration parameters of the camera can be pidedinto six categories that are classified into internal and externalparameters.1) Internal parameters: 1) Horizontal direction scale factorkX; 2) image distance v, which denotes the distancebetween the link line of the principle points of the lensof the left and right camera positions to the CCD im-age plane (image distance); 3) pixel position (xI0,yI0),which corresponds to the actual principle point on theCCD image plane; and 4) distortion coefficients k1, g1,g2, g3, and g4.2) External parameters: 1) Rotation matrix R and 2) trans-lation matrix T0.The calibration process can be fulfilled according to thefollowing steps.First, kX and the lateral magnification factor β canbe calibrated through a standard threading ruler, and vcan be calculated according to the lens formula (1/f)=(1/u)+(1/v).Second, the calibration parameter sets m and d,the3-D coordinate set ΩW of calibration characteristic pointsin the large-scale CMM coordinate system, and the cor-responding 2-D image coordinate set ΩI , which containslens distortion errors in the computer image coordinatesystem, are defined as follows:⎧⎪ ⎨⎪ ⎩m = {(xI0,yI0),R,T0}d = {k1,g1,g2,g3,g4}ΩW = {(xWi ,yWi ,zWi), 1 ≤ i ≤ N}ΩI = {(xIi ,yIi ), 1 ≤ i ≤ N} .(19)The coordinate transformation from the large-scale CMMcoordinate system to the camera coordinate system can berealized as follows:⎡⎣xCiyCizCi⎤⎦=R •⎡⎣xWiyWizWi⎤⎦+Ti =⎡⎣r11 r12 r13r21 r22 r23r31 r32 r33⎤⎦•⎡⎣xWiyWizWi⎤⎦+⎡⎣tXitYitZi⎤⎦.(20)Since the rotation matrix R is an orthogonal matrix, it has12 orthogonal constraints, i.e.,⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩r211 + r212 + r213 =1r221 + r222 + r223 =1r231 + r232 + r233 =1r211 + r221 + r231 =1r212 + r222 + r232 =1r213 + r223 + r233 =1r11 • r21 + r12 • r22 + r13 • r23 =0r21 • r31 + r22 • r32 + r23 • r33 =0r11 • r31 + r12 • r32 + r13 • r33 =0r11 • r12 + r21 • r22 + r31 • r32 =0r11 • r13 + r21 • r23 + r31 • r33 =0r12 • r13 + r22 • r23 + r32 • r33 =0.(21)Because the camera is usually fixed in its calibration,its translation matrix T has three time-invariant components[tX,tY ,tZ]. In the proposed measuring system, because thecamera replaces the touch-trigger probe of the large-scaleCMM and it can be moved in the XYZ directions of thelarge-scale CMM, the translation matrix Ti corresponding toeach measuring position of the camera is different. Then, Ticonsists of three time-variant components [tXi,tYi,tZi], andonly three components [tX0,tY 0,tZ0] corresponding to theoriginal measuring position T0 need to be calibrated. Ti, whichcorresponds to other camera measuring position, is equal tothe sum of T0 and the product of R with Δi, and Δi is thedisplacements of the camera in the XYZ directions relative tothe original camera measuring position in the large-scale CMM coordinate system. Δi can be read from the large-scale