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This section explains the widely used camera calibration method published by Tsai [234] Let P~ be the location of the origin in the image plane, r~ be the vector from P~ to the image point P~ = (x~, yD, Pi = (Xi, Yi, Zi) be a calibration point, and ri be the vector from the point (0,0, Zi) on the optical axis to Pi If the difference between the uncorrected image coordinates (Xi, Yi) and the true image coordinates (x~, yD is due only to radial lens distortion, then r~ is parallel to rio The camera constant and translation in Z do not affect the direction of r~, since both image coordinates will be scaled by the same amount These constraints are sufficient to solve the exterior orientation problem [234] Assume that the calibration points lie in a plane with Z = and assume that the camera is placed relative to this plane to satisfy the following two crucial conditions:
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1 The origin in absolute coordinates is not in the field of view
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2 The origin in absolute coordinates does not project to a point in the image that is close to the y axis of the image plane coordinate system Condition 1 decouples the effects of radial lens distortion from the camera constant and distance to the calibration plane Condition 2 guarantees that
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CHAPTER 12 CALIBRATION
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the y component of the rigid body translation, which occurs in the denominator of many equations below, will not be close to zero These conditions are easy to satisfy in many imaging situations For example, suppose that the camera is placed above a table, looking down at the middle of the table The absolute coordinate system can be defined with z = 0 corresponding to the plane of the table, with the x and y axes running along the edges of the table, and with the corner of the table that is the origin in absolute coordinates outside of the field of view Suppose that there are n calibration points For each calibration point, we have the absolute coordinates of the point (Xi, Yi, Zi) and the uncorrected image coordinates (Xi, iii) Use these observations to form a matrix A with rows ai, (12132) Let U = (Ul, U2, U3, U4, U5) be a vector of unknown parameters that are related to the parameters of the rigid body transformation:
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rxx py r xy py ryx py ryy py Px py
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(12133) (12134) (12135) (12136) (12137)
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Form a vector b = (Xl, X2, ,xn ) from the n observations of the calibration points With more than five calibration points, we have an overdetermined system of linear equations, Au=b, (12138) for the parameter vector u Solve this linear system using singular value decomposition, and use the solution parameters, Ul, U2, U3, U4, and U5, to compute the rigid body transformation, except for Pz, which scales with the
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1210 CAMERA CALIBRATION
First, compute the magnitude of the Y component of translation If and U2 are not both zero and U3 and U4 are not both zero, then
(12139)
where U
ui + u~ + u~ + u~; otherwise, if Ul
are both zero, then
(12140)
otherwise, using
(12141)
Second, determine the sign of Py Pick the calibration point p = (x, y, z) that projects to an image point that is farthest from the center of the image (the scene point and corresponding image point that are farthest in the periphery of the field of view) Compute r xx , r xy , r yx , r yy , and Px from the solution vector obtained above:
rxx rxy ryx ryy Px UIPy U2Py U3Py U4Py U5Py'
(12142) (12143) (12144) (12145) (12146)
Let ~x = rxxx + rxyY + Px and ~y = ryxx + ryyY + Py If ~x and x have the same sign and ~y and y have the same sign, then py has the correct sign (positive); otherwise, negate Py Note that the parameters of the rigid body transformation computed above are correct, regardless of the sign of PY' and do not need to be changed Third, compute the remaining parameters of the rigid body transformation:
(12147)
(12148)
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