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Figure 6 (a). Surface profile computed by the network from figure 5(a).
Figure 6 (b). Surface profile computed by the network from figure 5(b).
4. Parameter of the Vision System
In optical metrology, the object shape is reconstructed based on the parameters of the camera and the setup. Usually, these parameters are computed by external procedure to the reconstruction system. The camera parameters include focal distance, image center coordinates, pixel dimension, distortion and camera orientation. In the proposed binocular system, the camera parameters are determined based on the data provided by the network and image processing.
The camera parameters are determined based on the pinhole camera model, which is shown in figure 7. In the binocular system, the optical axis of the cameras is perpendicular to the reference plane. The camera orientation in the x- axis is determined by means of the geometry figure 8(a). In this geometry, the line
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projected on the reference plane and the object is indicated by ao and ap at the image plane, respectively. The distance between the image center and the laser
stripe in the x-axis is indicated by Aa. The object dimension is indicated by hi and
D = zi + hi. For this geometry, the camera orientation is performed based on si and hi from the network. According to the perpendicular optical axis, the object depth hi has a projection ki in the reference plane. From figure 8(a), the displacement is defined as si = (xc - ap) - (xc - ao). Thus, the projection ki at the reference plane is computed by
ki = |
F hi |
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From Eq.(16) F, xc, ao are constants and hi is computed by the network based on si. In this case, ki is a linear function. Therefore, the derivative ki respect to si dk/ds is a constant. Other configuration is an optical axis not perpendicular to the reference plane. In this case, si does not produce a linear function ki. Also, the derivative dk/ds is not a constant. The orientation of the camera in y-axis is performed based on the geometry of figure 8(b). In this case, the object is moved in y-axis over the line stripe. When the object is moved, the pattern position changes from ayp to ayi in the laser stripe. In this case, t = (ayc-ayp) - (ayc-ayi). For an optical axis perpendicular to the reference plane y-axis, a linear q produces a linear t at the image plane. Therefore, the derivative dt/dq is a constant. Thus, the orientation camera is performed by means of dk/ds = constant for the x-axis and dt/dq = constant for the y-axis. Based on these criterions, the optical axis is aligned perpendicular to x-axis and y-axis. For the orientation in x-axis, ki is computed from hi provided by the network. Due to the distortion, the derivative dk/ds is slightly different to a constant. But, this derivative is the more similar to a constant, which is shown in figure 8(c). In this figure, the dash line is dk/ds for β minor than 90° and the dot line is dk/ds for β major than 90°. Thus, the generated network corresponds to an optical axis aligned perpendicularly to the x-axis. For the orientation in y-axis, qi is provided by the electromechanical device and t is obtained by image processing. In this process, the object position is detected in the line in each movement. Due to the distortion, the derivative dt/dq is not exactly a constant. But, this derivative is the more similar to a constant. Thus, the optical axis is aligned perpendicular to the y-axis. In this manner, the network and image processing provide an optical axis aligned perpendicularly to the reference plane. Based on the optical axis perpendicular to reference plane, the camera
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parameters are obtained. To carry it out, the network produces the depth hi based on si for the calibration. The geometry of the setup figure 8(a) is described by
zi |
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zi + F |
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(17) |
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Aa |
+ Aa |
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From this equation η is scale factor to convert the pixels to millimeters. Using D = zi + hi and ηsi = η (xc - ap) - η (xc - ao), Eq.(17) is rewritten as
D − hi |
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D − hi + F |
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(18) |
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η(si + xc − ao ) |
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Aa |
+ Aa |
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Where D is the distance from the lens to the reference plane. From Eq.(18) the constants D, Aa, F, η, xc and ao should be determined. To carry it out, Eq.(18) is rewritten as equation system
h1 |
= D − |
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F Aa |
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η(s1 |
+ xc − ao ) |
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h2 |
= D − |
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F Aa |
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η(s2 |
+ xc − ao ) |
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h3 |
= D − |
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(19) |
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η(s3 |
+ xc − ao ) |
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h4 |
= D − |
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F Aa |
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η(s4 |
+ xc − ao ) |
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h5 |
= D − |
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η(s5 |
+ xc − ao ) |
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= D − |
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η(s6 |
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The values h1, h2,…., h6, are computed by the network according to s1, s2,…., s6. These values are substituted in Eq.(19) and the equation system is solved. Thus,
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the constants D, Aa, F, η, xc and ao are determined. The coordinate ayc is computed from the geometry figure 8(c) described by
ti = (ayc − ayp ) − |
F(D − hi ) |
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(20) |
η(Ab − qi−1 ) |
From Eq.(20) the constants D, F, η, qi, ti, hi are known and ayc, Ab, ayp should be
determined. To carry it out, Eq.(20) is rewritten as equation system for an hi constant by
t1 = (ayc − ay p ) − |
F(D − h1 ) |
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η(g − q ) |
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t2 |
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(21) |
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t3 |
= (ayc − ayp ) − |
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The values t1, t2, t3, are taken from the orientation in y-axis, q0= 0 and the values q1, q2, are provided by the electromechanical device. These values are substituted
in Eq.(21) and the equation system is solved. Thus, the constants ayc, ayp and Ab
are determined. In this manner the camera parameters are calibrated based on the network and image processing of the laser line.
The distortion is observed by means of the line position ap in the image plane, which is described by
ap = |
F Aa |
+ xc |
(22) |
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Based on Eq.(22), the behavior of ap respect to hi is a linear function. However, due to the distortion, the real data ap are not linear. The network is constructed by means of the real data using the displacement si =(ac- ap) - (xc- ao). Therefore, the network produces a non linear data h, which is shown in figure 8(d). Thus, the distortion is included in the network, which computes the object depth in the imaging system.
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Figure 7. Geometry of the pinhole camera model.
Figure 8 (a). Geometry of an optical axis perpendicular to x-axis.
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Figure 8 (b). Geometry of an optical axis perpendicular to y-axis.
Figure 8 (c). Derivative dk/ds for an optical axis perpendicular to x-axis and for an optical axis not perpendicular to x-axis.