Telecentric camera calibration with virtual patterns

A telecentric camera is generally described by an orthographic projection model. Based on the model, an arbitrary physical point (x_w, y_w, z_w ) on the 3D object and its projection (u, v) on the imaging plane can be formulated as [20]:

$$\begin{equation}\,\,\left[ {\begin{array}{*{20}{c}} u \\ v \\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \alpha &0&0&{{u_0}} \\ 0&\beta &0&{{v_0}} \\ 0&0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{R_{3 \times 3}}}&{{T_{3 \times 1}}} \\ 0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ {{z_w}} \\ 1 \end{array}} \right]\end{equation} \tag{ 1 }$$

where R_3×3 and T_3×1 = (t₁, t₂, t₃)^T represent the 3 × 3 rotation matrix and the 3 × 1 translation vector, respectively; α and β are the magnification radio along the U and V axes of the imaging plane; and (u₀, v₀) is the coordinate of the center of the imaging plane.

We can denote the ith row and jth column of matrix R by r_ij . Thus, we have

$$\begin{align}\left[ {\begin{array}{*{20}{c}} u \\ v \\ 1 \end{array}} \right]\,\, &= \,\underbrace {\left[ {\begin{array}{*{20}{c}} \alpha &{\text{0}}&{{u_{\text{0}}}} \\ {\text{0}}&{{\beta }}&{{v_{\text{0}}}} \\ {\text{0}}&{\text{0}}&{\text{1}} \end{array}} \right]}_{{A_s}}\,\,\underbrace {\left[ {\begin{array}{*{20}{c}} {{r_{{\text{11}}}}}&{{r_{{\text{12}}}}}&{{r_{{\text{13}}}}}&{{t_{\text{1}}}} \\ {{r_{{\text{21}}}}}&{{r_{{\text{22}}}}}&{{r_{{\text{23}}}}}&{{t_{\text{2}}}} \\ {\text{0}}&{\text{0}}&{\text{0}}&{\text{1}} \end{array}} \right]}_C\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ {{z_w}} \\ 1 \end{array}} \right]\,\nonumber\\& = \, \underbrace {\left[ {\begin{array}{*{20}{l}} {{m_{11}}}&{{m_{12}}}&{{m_{13}}}&{{m_{14}}} \\ {{m_{21}}}&{{m_{22}}}&{{m_{23}}}&{{m_{24}}} \\ 0&0&0&1 \end{array}} \right]}_M \,\,\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ {{z_w}} \\ 1 \end{array}} \right].\end{align} \tag{ 2 }$$

When a planar calibration target is used, Z_w of the world coordinate system is generally set as zero. Consequently, equation (2) can be written as

$$\begin{align}\,\,\left[ {\begin{array}{*{20}{c}} u \\ v \\ 1 \end{array}} \right]\, &= \,\left[ {\begin{array}{*{20}{c}} \alpha &0&{{u_0}} \\ 0&\beta &{{v_0}} \\ 0&0&1 \end{array}} \right]\,\underbrace {\left[ {\begin{array}{*{20}{l}} {{r_{11}}}&{{r_{12}}}&{{t_1}} \\ {{r_{21}}}&{{r_{22}}}&{{t_2}} \\ 0&0&1 \end{array}} \right]}_{{C_{s}}}\,\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ 1 \end{array}} \right]\,\nonumber\\ &= \,H\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ 1 \end{array}} \right]\,\end{align} \tag{ 3 }$$

where H is a 3 × 3 homography matrix, which can be decomposed for calculating all the parameters of telecentric camera.

Figure 1 shows the schematic diagram of the proposed telecentric camera calibration method. Specifically, a virtual target is first produced by preset parameters and orthogonally projected on an LCD screen. Then, the patterns displayed on the screen are acquired by a telecentric camera. Lastly, the intrinsic and extrinsic parameters of the camera are calculated according to the corresponding relationships between virtual world points and their projection points on the camera imaging plane.

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We assume that P_v on the virtual target is an arbitrary 3D feature point with coordinates (x_v, y_v , 0) in the virtual world coordinate system {O_v; X_v, Y_v, Z_v}. The coordinates of its projection in the screen coordinate system {O_s; X_s, Y_s } and in the camera pixel coordinate system {O; U, V} are P_s (x_s, y_s , 0) and P(u, v), respectively. The imaging process from the virtual world point P_v to image point P can be described as

$$\begin{equation}\left[ {\begin{array}{*{20}{c}} u \\ v \\ 1 \end{array}} \right] = \,{H^c}{H^v}\left[ {\begin{array}{*{20}{c}} {{x_v}} \\ {{y_v}} \\ 1 \end{array}} \right] = \,H\left[ {\begin{array}{*{20}{c}} {{x_v}} \\ {{y_v}} \\ 1 \end{array}} \right]\end{equation} \tag{ 4 }$$

where 3 × 3 homography matrix H^v denotes the preset orthogonal projection from the virtual calibration target to the screen. The 3 × 3 homography matrix H^c denotes the orthogonal projection from the screen to the camera imaging plane, which comprises the camera parameters to be solved. The 3 × 3 homography matrix H denotes orthogonal projection from the virtual world coordinate system to the camera pixel coordinate system, which can be directly calculated according to a set of corner points in virtual target and their 2D projections on the camera imaging plane.

We denote the ith row and jth column of matrix H by h_ij , the ith row and jth column of matrix H^c by $h_{ij}^c$, and the ith row and jth column of matrix H^v by $h_{ij}^v.$ Then equation (4) can be rewritten as

$$\begin{align}\left[ {\begin{array}{*{20}{c}} {{h_{11}}}&{{h_{12}}}&{{h_{13}}} \\ {{h_{21}}}&{{h_{22}}}&{{h_{{\text{23}}}}} \\ 0&0&1 \end{array}} \right] &= \left[ {\begin{array}{*{20}{c}} {h_{11}^c}&{h_{12}^c}&{h_{13}^c} \\ {h_{21}^c}&{h_{22}^c}&{h_{23}^c} \\ 0&0&1 \end{array}} \right]\nonumber\\&\quad\times\left[ {\begin{array}{*{20}{c}} {h_{11}^v}&{h_{12}^v}&{h_{13}^v} \\ {h_{21}^v}&{h_{22}^v}&{h_{23}^v} \\ 0&0&1 \end{array}} \right].\end{align} \tag{ 5 }$$

From equation (5), we have

$$\begin{equation}\left[ {\begin{array}{*{20}{c}} {h_{11}^c}&{h_{12}^c} \\ {h_{21}^c}&{h_{22}^c} \end{array}} \right] = \,K\left[ {\begin{array}{*{20}{c}} {{h_{11}}}&{{h_{12}}\,\,} \\ {{h_{21}}}&{{h_{22}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {h_{22}^v}&{ - h_{12}^v} \\ { - h_{21}^v}&{h_{11}^v} \end{array}} \right]\end{equation} \tag{ 6 }$$

with $K\, = \,\frac{1}{{h_{11}^vh_{22}^v - h_{21}^vh_{12}^v}}$.

According to equation (3), the following equation can be achieved

$$\begin{align}\left[ {\begin{array}{*{20}{c}} u \\ v \\ 1 \end{array}} \right] &= \left[ {\begin{array}{*{20}{c}} \alpha &0&{{u_0}} \\ 0&\beta &{{v_0}} \\ 0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{r_{11}}}&{{r_{12}}}&{{t_1}} \\ {{r_{21}}}&{{r_{22}}}&{{t_2}} \\ 0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ 1 \end{array}} \right]\nonumber\\ &= \;{H^c}\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ 1 \end{array}} \right].\end{align} \tag{ 7 }$$

Based on equation (7), we can get

$$\begin{equation}{R_l}\; = \left[ {\begin{array}{*{20}{c}} {{r_{11}}}&{{r_{12}}} \\ {{r_{21}}}&{{r_{22}}} \end{array}} \right] = \;\left[ {\begin{array}{*{20}{c}} {1/\alpha \;}&0 \\ 0&{1/\beta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {h_{11}^c}&{h_{12}^c} \\ {h_{21}^c}&{h_{22}^c} \end{array}} \right].\end{equation} \tag{ 8 }$$

Since the rotation matrix R is both unitary and orthogonal, we have the following three independent constraints

$$\begin{align}\left\{ {\begin{array}{*{20}{c}} {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\|{\boldsymbol{r}_1}\|\; = \;r_{11}^2\; + \;r_{12}^2\; + r_{13}^2\; = \;1} \\ {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\|{\boldsymbol{r}_2}\|\; = \;r_{21}^2\; + \;r_{22}^2\; + r_{23}^2\; = \;1} \\ {{\boldsymbol{r}_1} \cdot {\boldsymbol{r}_2}\; = \;{r_{11}}{r_{21}}\; + \;{r_{12}}{r_{22}}\; + \;{r_{13}}{r_{23}}\; = \;0} \end{array}} \right.\end{align} \tag{ 9 }$$

where ${\boldsymbol{r}_{\text{1}}}{\text{ and }}\,{\boldsymbol{r}_2}{ }$ are the row vectors of the matrix R. Substituting the first two equations in equation (9) into the last equation in equation (9) and eliminating ${r_{13}}$ and ${r_{23}}$, we have

$$\begin{equation}\|{R_l}\|_{F}^{2} - \,1\, = \,{\left| {{R_l}} \right|^2}\end{equation} \tag{ 10 }$$

where $\|{R_l}\|_{F}$ and $\left| {{R_l}} \right|$ are the Frobenius norm and determinant of ${R_l}$, respectively. After rearranging equation (10), we can derive a linear equation:

$$\begin{equation}{l_1} - \,l_2{F_1}\, - \,l_3{F_2}\, = \,Q\end{equation} \tag{ 11 }$$

where, $\,{l_1}\, = \,{\alpha ^2}{\beta ^2},\,{l_2}\, = \,{\alpha ^2},\,{l_3}\, = \,{\beta ^2},{F_1}\, = \,[{(h_{21}^c)^2} + \,{(h_{22}^c)^2}],$ ${F_2}\, = \,[{(h_{11}^c)^2} + \,{(h_{12}^c)^2}]$, $Q\, = \, - {\left[ {h_{11}^ch_{22}^c - h_{12}^ch_{21}^c} \right]^2}$.

If n virtual target images from different poses and orientations are captured, n corresponding homographic matrices H^v can be obtained. The element in the ith row and jth column of the nth homographic matrix H^{v (n)} is denoted as $h_{ij}^{v(n)}$. For the nth image, $F$ ₁, $F$ ₂, $Q$ are written as $F_1^{(n)}$, $F_2^{(n)}$, ${Q^{(n)}}$, respectively. Therefore, equation (11) can be stacked to be a n × 3 matrix as follows

$$\begin{equation}\;\underbrace {\left[ {\begin{array}{*{20}{c}} 1&{F_1^{(1)}}&{F_2^{(1)}} \\ 1&{F_1^{(2)}}&{F_2^{(2)}} \\ \vdots & \vdots & \vdots \\ 1&{F_1^{(n)}}&{F_2^{(n)}} \end{array}} \right]}_F\,\underbrace {\left[ {\begin{array}{*{20}{c}} {{l_1}} \\ {{l_2}} \\ {{l_3}} \end{array}} \right]}_L\, = \,\underbrace {\left[ {\begin{array}{*{20}{c}} {{Q^{(1)}}} \\ {{Q^{(2)}}} \\ \vdots \\ {{Q^{(n)}}} \end{array}} \right]}_Q.\end{equation} \tag{ 12 }$$

In equation (12), it should be mentioned that at least three images are required to calculate matrix L. The linear least-squares solution is given by

$$\begin{equation}L = \left({F}_{}^{T}F \right)^{ - 1}{F}^{T}Q.\end{equation} \tag{ 13 }$$

Therefore, the intrinsic parameters can be recovered by

$$\begin{equation}\alpha = \sqrt {{l_2}} ,{ }\beta = \sqrt {{l_3}} .\end{equation} \tag{ 14 }$$

Once α and β are known, according to equation (7), t₁ and t₂ can be calculated by

$$\begin{equation}\left\{ {\begin{array}{*{20}{c}} {\alpha {t_1}\; + \;{u_0}\; = \;h_{13}^c} \\ {\beta {t_2}\; + \;{v_0}\; = \;h_{23}^c} \end{array}} \right..\end{equation} \tag{ 15 }$$

Besides, the extrinsic parameters $r_{ij}^{ n}$ (i = 1, 2, j = 1, 2) for the nth images can be also acquired based on equation (8).

It is noteworthy that, at each target position, the rotation matrix C can be directly obtained except for the elements r₁₃ and r₂₃. According to the first two equation in equation (9), the parameters ${r_{13}}$ and ${r_{23}}$ can be calculated by

$$\begin{equation}{r_{13}}\, = \, \pm \,\sqrt {1\, - \,r_{11}^2\, - \,r_{12}^2} \end{equation} \tag{ 16 }$$

$$\begin{equation}{r_{23}} = \pm { }\sqrt {1 - r_{21}^2 - r_{22}^2} .\end{equation} \tag{ 17 }$$

Unfortunately, the signs of ${r_{13}}$ and ${r_{23}}$ cannot be directly determined from equations (16) and (17) since the orthogonality of R can only offer one constraint. To solve the problem, conventional methods provide a depth value for 2D physical calibration target by means of a translation stage, which makes calibration produce inflexible. Therefore, we use pre-defined parameters to locate a virtual calibration target rather than using a translation stage to move a physical calibration target. The original pattern in the screen coordinate system is displayed on the LCD screen and is converted into the local virtual world coordinate system. Then the original pattern in the virtual world coordinate system is used for identifying the signs of ${r_{13}}$ and ${r_{23}}$. To simplify the notation, the coordinate of a feature point in the screen is still (x_s, y_s ), then the point (x_w, y_w, z_w ) in the virtual world coordinate can be expressed by

$$\begin{equation}\left[ {\begin{array}{*{20}{c}} {{{{x}}_{{w}}}} \\ {{{{y}}_{{w}}}} \\ {{{{z}}_{{w}}}} \end{array}} \right] = \underbrace {\left[ {\begin{array}{*{20}{c}} {{{r}}_{{{11}}}^{{v}}}&{{{r}}_{{{12}}}^{{v}}}&{{{r}}_{{{13}}}^{{v}}} \\ {{{r}}_{{{21}}}^{{v}}}&{{{r}}_{{{22}}}^{{v}}}&{{{r}}_{{{23}}}^{{v}}} \\ {{{r}}_{{{31}}}^{{v}}}&{{{r}}_{{{32}}}^{{v}}}&{{{r}}_{{{33}}}^{{v}}} \end{array}} \right]}_{{{{R}}^{{v}}}}\left[ {\begin{array}{*{20}{c}} {{{{x}}_{{s}}}} \\ {{{{y}}_{{s}}}} \\ {{0}} \end{array}} \right] + \underbrace {\left[ {\begin{array}{*{20}{c}} {{{t}}_{{1}}^{{v}}} \\ {{{t}}_{{2}}^{{v}}} \\ {{{t}}_{{3}}^{{v}}} \end{array}} \right]}_{{{{T}}^{{v}}}}\end{equation} \tag{ 18 }$$

where R^v and T^v are 3 × 3 rotation matrix and 3 × 1 translation vector, respectively. And the pixel coordinates of this point on the camera imaging plane is (u_t, v_t ). Then, by combining equations (2) and (18), r₁₃ and r₂₃ meet the following criteria

$$\begin{align}\left[ {\begin{array}{*{20}{c}} {{r_{13}}} \\ {{r_{23}}} \\ 0 \end{array}} \right] &= \frac{1}{{r_{31}^v{x_s}\; + \;r_{32}^v{y_s}\; + \;t_3^v}}\left( {{\left[ {\begin{array}{*{20}{c}} \alpha &0&{{u_0}} \\ 0&\beta &{{v_0}} \\ 0&0&1 \end{array}} \right]}^{ - 1}}\left[ {\begin{array}{*{20}{c}} {{u_t}} \\ {{v_t}} \\ 1 \end{array}} \right]\right.\nonumber\\&\quad- \left.\left[ {\begin{array}{*{20}{c}} {{r_{11}}}&{{r_{12}}}&{{t_1}} \\ {{r_{21}}}&{{r_{22}}}&{{t_2}} \\ 0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {r_{11}^v}&{r_{12}^v}&{t_1^v} \\ {r_{21}^v}&{r_{22}^v}&{t_2^v} \\ 0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_s}} \\ {{y_s}} \\ 1 \end{array}} \right] \right).\end{align} \tag{ 19 }$$

Their signs can be well determined according to the values calculated by equation (19).

To reconstruct the 3D shape of an object, a single-camera telecentric pseudo-stereo vision system is established, as shown in figure 3(a). The proposed system is mainly composed of a telecentric lens, a digital camera, and a four-mirror adapter. The four planar mirrors (denoted as M₁, M₂, M₃, and M₄) are fixed in front of the telecentric camera. Note that the inside two mirrors M₃ and M₄ are perpendicularly fixed on the two sides of a triangular prism-shaped block and constitute a right angle, and the outside two mirrors M₁ and M₂ are mounted on a rotation stage. By properly adjusting the position of the tested object, its surface image can be clearly imaged by the camera through both the left and right optical paths in a single shot. As shown in figure 3(a), aided by the reflection of the mirror array, this system is equivalent to a pseudo-stereo vision system comprising two virtual telecentric cameras. The image recorded by each virtual telecentric camera corresponds to a half image captured by the real telecentric camera. Since the two virtual cameras are generated by the same camera, synchronization problem is avoided in the system.

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Figure 3(b) shows a photograph of the established experimental system. The system consists of a four-mirror adapter, a digital camera (TXG20, Baumer Electric AG, Switzerland) with a resolution of 1624 × 1236 pixels and a pixel size of 0.0044 mm, and a bilateral telecentric lens (Xenoplan 1: 5, Schneider Optics, Inc., Germany) with a fixed working distance of 268 mm and an FOV of 35.7 × 27.2 mm. Because the used telecentric lens has a distortion ratio less than 0.1%, lens distortion can be neglected for briefness. An iPad mini 5 with a physical resolution of 2048 × 1536 pixels and pixel size of 0.078 mm is used as a display screen for system calibration. Before experiments, the screen was placed in front of the established system to guarantee that the whole camera view is covered by the screen. By carefully adjusting the position of the telecentric camera and rotating the mirrors, patterns displayed on the screen can be simultaneously and clearly imaged on the left and right halves of the camera imaging plane.

To calibrate the developed system, a chessboard pattern with 5 × 8 corner points was firstly produced by computer software, as shown in figure 3. The distance between adjacent points is 1.56 mm. Then, by applying ten groups of preset parameters to the original pattern generated, ten virtual calibration targets which have different positions and directions in 3D space were created. Next, these virtual targets were orthogonally projected and displayed on the screen, as shown in figure 4. Afterwards, each image displayed on the screen was acquired for camera calibration. The feature points on all the captured images were extracted by using the method in [21] for calculating matrices A_s and C_s . Finally, the parameters r₁₃ and r₂₃ were recovered with the assistance of another virtual calibration target. The A_s and C of two virtual telecentric cameras expressed in equation (2) were calibrated as following:

$$\begin{align*}\left[ {\begin{array}{*{20}{c}} {{u_l}} \\ {{v_l}} \\ 1 \end{array}} \right] &= \left[ {\begin{array}{*{20}{c}} {45.522}&0&{406} \\ 0&{45.556}&{618} \\ 0&0&1 \end{array}} \right]\nonumber\\&\quad\times\left[ {\begin{array}{*{20}{c}} {0.917}&{ - 0.008}&{ - 0.399}&{ - 3.499} \\ { - 0.008}&{0.999}&{ - 0.054}&{ - 5.672} \\ 0&0&0&1 \end{array}} \right]\nonumber\\&\quad\times\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ {{z_w}} \\ 1 \end{array}} \right]\end{align*}$$

$$\begin{align*}\left[ {\begin{array}{*{20}{c}} {{u_r}} \\ {{v_r}} \\ 1 \end{array}} \right] &= \left[ {\begin{array}{*{20}{c}} {{\text{45}}{\text{.585}}}&{\text{0}}&{{\text{406}}} \\ {\text{0}}&{{\text{45}}{\text{.649}}}&{{\text{618}}} \\ {\text{0}}&{\text{0}}&{\text{1}} \end{array}} \right]\nonumber\\&\quad\times\left[ {\begin{array}{*{20}{c}} {{\text{0}}{\text{.837}}}&{ - {\text{0}}{\text{.042}}}&{{\text{0}}{\text{.546}}}&{ - {\text{5}}{\text{.725}}} \\ {{\text{0}}{\text{.051}}}&{{\text{0}}{\text{.998}}}&{ - {\text{0}}{\text{.049}}}&{ - {\text{5}}{\text{.152}}} \\ {\text{0}}&{\text{0}}&{\text{0}}&{\text{1}} \end{array}} \right]\nonumber\\&\quad\times\left[ {\begin{array}{*{20}{c}} {{x_w}} \\ {{y_w}} \\ {{z_w}} \\ 1 \end{array}} \right].\end{align*}$$

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Furthermore, the matrix G can be estimated as

$$\begin{align*}G = \left[ {\begin{array}{*{20}{c}} {{\text{41}}{\text{.729}}}&{ - {\text{0}}{\text{.377}}}&{ - {\text{18}}{\text{.188}}}&{{\text{246}}{\text{.742}}} \\ { - {\text{0}}{\text{.368}}}&{{\text{45}}{\text{.488}}}&{ - {\text{2}}{\text{.464}}}&{{\text{359}}{\text{.622}}} \\ {{\text{38}}{\text{.154}}}&{ - {\text{1}}{\text{.927}}}&{{\text{24}}{\text{.872}}}&{{\text{145}}{\text{.012}}} \\ {{\text{2}}{\text{.347}}}&{{\text{45}}{\text{.535}}}&{ - {\text{2}}{\text{.214}}}&{{\text{382}}{\text{.824}}} \end{array}} \right].\end{align*}$$

Based on the obtained camera parameters, the re-projection errors of the two virtual telecentric cameras were respectively calculated and shown in figure 5. Their standard deviations are (0.0700, 0.0778) pixels for the left virtual camera and (0.0751, 0.0752) pixels for virtual right camera, respectively, confirming the accuracy of the proposed calibration method.

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After calibrating the single-camera telecentric pseudo-stereo vision system, it can be used for 3D reconstruction of a test object surface. To quantitatively evaluate the performance of the calibrated system, two tests, i.e. 3D shape measurement of corner points on the LCD calibration board and a plate plane, were performed. After that, the proposed system was also applied to measure a key with complex geometry to validate its practicability in reconstructing complicated shape on small objects.

In the first test, the original pattern was first displayed on the screen and captured. Then, the pixel coordinates of all the corner points, where the black squares intersect, on the captured image were extracted, as shown by the red and green dots in figure 6(a). Finally, their 3D coordinates were reconstructed in accordance with the obtained system parameters, as shown in figure 6(b). The Euclidean distance between each two adjacent corner points in 3D space can be calculated. There are 35 and 32 pairs of adjacent corner points in X and Y directions, respectively. The measured distances among the reconstructed 3D coordinates have a mathematical expectation of 1.555 mm, which is close to real value (1.56 mm) of the chessboard pattern generated by computer software.

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In the second test, we reconstructed the 3D shape of the planar plane using the conventional method that requires human interaction to move physical target and the proposed method. Before experiment, black and white random speckles were sprayed on the plate surface for establishing the corresponding relationship between the left virtual camera and the right virtual camera. In experiment, the plate was placed in the appropriate position in front of the calibrated system, so that two views of the plate surface can be imaged on the camera imaging plane, as shown in figure 7(a). By correlating the left and the right virtual camera images positioned in a rectangular region of interest (ROI) using the advanced inverse-compositional Gauss-Newton (IC-GN) algorithm [22] with a subset size of 29 × 29 pixels, the 3D data of the ROI were calculated based on the calibrated extrinsic and intrinsic parameters. Also, we compare it with an ideal plane acquired through least-square fitting. Figures 7(b) and (c) show the 3D geometry of the plate plane and the color-coded error distribution map using the proposed method, respectively. The corresponding results using the conventional method are shown in figures 7(d) and (e). Clearly the proposed method has a little smaller error than the conventional method: 1.6 μm for our proposed method comparing against 1.8 μm for the conventional method. In the root mean square (RMS) error maps, the errors are appeared at boundary regions. The results may be caused by lens distortion. In future work, the distortion model with radial and tangential coefficients should be used for increasing measurement accuracy. The results from the two tests confirm that the established system and the proposed calibration method offer high measurement accuracy.

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To visually evaluate the performance of the calibrated system in reconstructing complex shape on small objects, a region on the surface of a key was measured by the calibrated system. Prior to the experiment, high contrast random speckles were sprayed on the region for finding the corresponding points between left and right sub-images. During the measurement, the key was placed in front of the system and clearly imaged in a single camera imaging plane via left and right optical reflection paths. As shown in figure 8(a), a circular ROI was first specified in the left sub-image, and the measured points within the ROI were searched in the right sub-image to determine their disparity data using the advanced IC-GN algorithm with a subset size of 29 × 29 pixels. Then, according to the known intrinsic and extrinsic parameters, the 3D shape of the specified ROI was reconstructed, as show in figure 8(b). It is clear that the 3D details of the specified ROI have been reconstructed. The results demonstrate that the proposed system is capable of reconstructing small objects with complex shape.

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