Modeling and Calibration of a Structured-light Optical - Michigan ...

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Given a commercial laser stripe sensor, it is not only unnecessary but also impossible to perform intrinsic calibration. Therefore it is impossible to combine the intrinsic and extrinsic calibration. The sensor frame is regarded as stationary instead of moving when skewed frame representation is employed. The advantage of this modeling is that scan data can be pooled as they are collected, i.e., no postprocessing of the data is necessary before performing extrinsic calibration. This makes it possible to scan a target in the sensor field of view and extract desirable feature points directly from the scanned data for extrinsic calibration. The use of moving frame model prevented Chen and Kak (1987) from knowing where exactly the object points are located so that they had to scan at least six non-parallel lines on a polyhedron target for calibration. We present a new tetrahedrontarget-based extrinsic calibration approach without the use of pin-hole assumption. Two scan lines are all that are needed to calculate the four conjugate pairs, and the two lines can be parallel. The new calibration procedure becomes so simple that frequent on-line calibration becomes feasible whenever the system configuration is altered. 5.1 Calibration Algorithm Formulation. The same point Pt in a 3D Euclidean space can be represented by two sets of coordinate vectors, one in the non-Cartesian skewed sensor frame, Psi = (xsi, ysi, zst) T and the other in the Cartesian world frame, Pwi = (xni,ywi, zwd T • These two coordinate vectors are often called "conjugate pairs" because they describe the same point in two different coordinate systems. For extrinsic calibration, we need to find the matrix JT from n conjugate pairs. Then the absolute orientation solution becomes a constrained least squares optimization problem (Che, 1995). That is, we want to minimize the augmented objective function F = I HP* - JT-Psi || 2 + \,[r?, + t 2 2l + t 2 M - 1] i=l + X2-+[f?2 + t\2 + t\2 " 1] + \3[t\3 + th + tj3 - 1] t22' in + h2' hi ] (21) where the \,'s are Lagrange multipliers. Since there are four constraints among the twelve unknowns of T, only eight independent parameters exist. Therefore a minimum of three conjugate pairs are needed as they provide nine coordinates. However, this makes the algorithm nonlinear. For on-line use, a linear algorithm has to be found to provide an initial guess to the nonlinear algorithm. This is achieved by ignoring the four constraints in Eq. (21). The objective function then becomes F = y ii p *• A^ II * w sT-Psi (22) and a minimum of four conjugate pairs are needed. With four pairs of calibration points, an initial guess of $T can be determined by scan lines I and n (a) schematic of the target (b) actual target Fig. 9(a)(6) Tetrahedron target where and Fig. 10 Scanned image of a tetrahedron target W S = P. i 1 w-s~ Pwl Pw3 1 1 Psl Psi 1 1 Pw4 1 PsA 1 (23) (24) (25) If more than four conjugate pairs are used to achieve higher calibration accuracy, a least squares linear regression will give the initial guess for fT (Horn, 1986) where and W = S = fT= W-S T -(S-S T y PS2 1 and n is the number of conjugate pairs. (26) (27) (28) 5.2 Tetrahedron-target-based Calibration. From the calibration algorithm formulation' we know that at least four conjugate pairs are needed for a linear initial guess. So the calibration target should have at least four vertices to be used as the four conjugate pairs. A theodolite-based calibration approach was proposed (Dewar, 1988; Greer, 1988; Dewar and Greer, 1989). They use a four-wire optical fiber target to create the conjugate pairs. However, the accuracy of the theodolite is not satisfactory and the procedure is time consuming and involves human operator, which makes the automation of on-line calibration impossible. We propose to use a tetrahedron as one of the possible calibration targets. As can be seen from Fig. 9(a), there are four planes on the target. Four points A, B, C, and D can be uniquely defined as the intersection points of the four planes. The procedure for the identification of the four conjugate pairs is as follows: (1) Use a mechanical CMM (accuracy should be at least one order higher than the laser scanner) to sample several points (at least three) on each face of the target. (2) Apply a least squares plane fitting algorithm to the 600 / Vol. 118, NOVEMBER 1996 Transactions of the ASME Downloaded 26 Sep 2011 to 141.212.97.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Table 1 Comparative analysis of errors of the angles among the four planes of the tetrahedron target before and after calibration (unit: degree) True angles measured by CMM Angle error before calibration Angle error after calibration angle_l_2 angle_l_3 angle_l_4 angle_2_3 angle_2_4 50.9588 50.9382 50.9396 29.7757 29.7862 8.8664 9.0837 21.6584 -3.0446 13.0731 0.0025 0.0016 0.0084 -0.0028 0.0057 angle_3_4 29.7672 13.3489 0.0045 sampled data points to obtain the equation for the four planes. (3) Combine the four plane equations and solve for the coordinates of the four vertex points A, B, C and D of the target. (4) Use a multiple-axis structured-light laser stripe scanning system to scan the entire top surface of the target. A minimum of two scan lines are needed to cover all four surfaces, as shown in Fig. 9(b) by lines I and II. However, more scan lines are obtained to reduce sampling error. (5) Use an image segmentation algorithm to extract data clouds for each plane from the entire image into separate data files. (6) Find the coordinates of points A, B, C and D in the skewed sensor frame following procedures (2) and (3). For plane fitting on the CMM measurement data, the equation of the plane is of the form z = ax + by + c. For skewed coordinates of the laser scanner measurement, it can be proved that the plane equation is still of the form z = ax + by + c, as the nonhomogeneous transformation of a laser scanning process is still linear. 6 Experimental Studies A tetrahedron target in Fig. 9(b) is used to validate the proposed laser stripe scanning calibration algorithm. The material is a high density foam with white paint, which approximates a Lambertian surface. The target is first measured on a Sheffield RS-30 bridge-type CMM by sampling 14 points on each plane surface of the target. After measurement on the CMM, the (x, y,z) coordinates of both the vertices and the angles among the four planes on the target are known relative to a Cartesian world coordinate system (here we use the CMM coordinate system). Then, the target is scanned on the integrated multiple-axis laser stripe optical CMM when the sensor is rotated around the Ys axis about 37 deg. Under this rotation, the scanned image of the target is obtained in the non-Cartesian laser sensor coordinate system, as shown in Fig. 10. From these two sets of data, the coordinate transformation matrix ^T can be calculated as 1.0045 -0.0091 0.6073 595.0213 0.0050 -1.0160 -0.0148 26.6586 0.0014 -0.0040 0.7952 -139.7755 0.0000 0.0000 0.0000 0.0000 (29) Table 2 Comparative analysis of errors of the distances between vertices of the tetrahedron before and after calibration (unit: mm) True distance measured by CMM Distance error before calibration Distance error after calibration distance_l_2 76.9113 9.6711 0.0073 distance 1 3 distance_l_4 distance 2 3 distance_2_4 distance 3 4 76.9198 76.8763 128.0921 128.0579 128.0352 9.3278 -13.5267 -2.0328 -0.9545 -1.0657 0.0091 -0.0040 0.0067 0.0056 0.0057 Note: distanced j refers to the distance between the i" and the /* vertex of the target Journal of Manufacturing Science and Engineering Fig. 11 Top portion of the panel The nonhomogeneity can be clearly identified by looking at the non-orthogonality of the column vectors of JT, or more clearly, at the off-diagonal terms of R T -R, R T -R 1.0034 -0.0048 -0.0048 0.9990 -0.3419 -6.917x10" -0.3419 -6.917 X 10" 1.0001 (30) The nonzero inner products of the column vectors in (30) demonstrate that the coordinate transformation from a non- Cartesian laser sensor frame to a Cartesian frame (the CMM coordinate system in this case) is nonhomogeneous and the mapping of the whole image to a Cartesian world coordinate system is required. The nonlinear optimization search program we use is the "constr.m" in the Matlab Optimization Toolbox (Math Works, 1992). The implementation of the Sequential Quadratic Programming algorithm used in the constrained optimization routine "constr.m" is by updating the Hessian matrix. Usually it takes several hundreds function evaluation to converge. The iteration termination tolerances for independent variables and objective function have been set as 0.001 mm, and the termination criterion on constraint violation has also been set at 0.001 mm. Using the obtained matrix, all the data on the target are transformed from the skewed coordinate system to the CMM Cartesian coordinate system. Now, the real shape of the tetrahedron target is recovered, as seen from the comparison of the errors from Tables 1 and 2. From Table 1, we can see that the angle errors of the four planes before the nonhomogeneous coordinate transformation is much greater than that obtained after the transformation. Fig. 12 Bottom portion of the panel scanned at a different angle NOVEMBER 1996, Vol. 118/601 Downloaded 26 Sep 2011 to 141.212.97.74. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Page 1 and 2: Chenggang Che Graduate Research Ass
Page 3 and 4: In Figs. 3 and 4, the orientation o
Page 5: Fig. 6 Angular deformation as a fun
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