Single-lens multi-mirror laser stereo vision-based system for measuring internal thread geometrical parameters

Xiao Li; Wei Li; Haijun Xin; Jingyu Zhou; Huaiyuan Chen; Xingpei Chen; Xiaokang Yin; Xin’an Yuan; Xin Ma; Xin Ma

doi:10.1364/OE.476796

1. Introduction

The geometric parameters of the internal thread of an oilfield pipe are an important standard for measuring the production performance. The American Petroleum Institute (API) clearly stipulates that the geometric parameters (i.e., pitch, tooth height, and taper) of manufactured internal threads should be measured [1]. The internal threads have the characteristics of small space, a complex profile, and numerous parameters. It is difficult for the existing methods to provide the accuracy and efficiency required for inspection, because they have low measurement efficiency (e.g., plug gauge [2], thread micrometer [3], taper gauge, and height gauge [4]) and poor accuracy (e.g., optical fiber [5], and grating [6]), and they are not capable for reproducing rough outlines of tooth profiles (e.g., position sensitive device (PSD) [7]), prone to wear (e.g., probe profiler [8]), and expensive (e.g., X-ray computed tomography (CT) [9]). Vision-based metrology possesses the advantages of being non-contact and having high accuracy, high efficiency, and low cost, and they present great potential for the measurement of internal thread geometric parameters.

Two-dimensional (2D) vision-based measurement methods consisting of a flat backlight, a telecentric lens, and a monocular camera are often used for external thread inspection. However, the geometric parameters of internal threads can only be measured by a 3D vision method. By arranging four plane-mirrors symmetrically in front of a monocular camera, the monocular camera can be turned into a monocular stereo vision (MSV), which effectively solves the aforementioned problems associated with occlusion, cost, and 3D measurement. For the special measurement scenarios of internal threads with MSV method, the research mainly focuses on optical path analysis and structural design. Pan et al. [10] clarified an optical path model (OPM) of a MSV DIC system, after analyzing the structural parameters (e.g., baseline distance and the effective FoV), both the target shape and 3D shape deformation are accurately measured by introducing a blue LED light and a dual bandpass optical filter into the system [11]. Other researchers have designed more compact and portable integrated MSV system to achieve the measurement of high-speed 3D displacement [12] and 3D full-field deformation [13]. However, in terms of OPM, the research has been mostly limited to two 2D dimensions [14–17] and has not yet analyzed relationships between imaging parameters (e.g., DoF, FoV, mirror size, and baseline distance) and the structural parameters on the 3D level. In addition, the existing 3D OPM model has not considered complete independent variables (e.g., focal length and image sensor size) [18], and the models have only been expressed implicitly.

Because the tooth profiles are not noticeable, point, line, or circular lasers are generally used to illuminate the internal threads. For the laser displacement and spectral confocal measurement methods based on point laser illumination, the lasers are bulky and poor in measurement efficiency. For the measurement method with a circular laser [19,20], a camera and a laser need to be connected coaxially through a glass cover to implement the measurement so that the imaging optical path will be refracted twice, which reduces the measurement accuracy. Additionally, with the effect of the lighting form and complex thread profile, the measurement methods based on a circular laser and a point laser [21,22] have the problems of low accuracy and incompleteness for measuring the tooth profiles caused by laser misalignment distortion and measurement inaccessibility. The vision-based measurement system of line laser illumination along the longitudinal cross-section of the internal thread can solve the above problems [23,24]. However, the camera and the line laser need to be arranged at an angle during measurement, it is difficult to place the system in the oilfield pipe, both the illumination and the imaging light paths are easily blocked. An endoscope can be inserted entirely into an oilfield pipe, but the measurement accuracy will be reduced due to its poor optical performance. Arranging mirrors inside the tubing improves the accessibility but increases the complexity of the optical path. While, the designs of the vision system in the current related research only rely on trial, which cannot provide effective theoretical design basis.

To solve the above problems, we propose a single-lens multi-mirror laser stereo vision-based system for measuring internal thread geometric parameters. By introducing multiple plane mirrors arranged appropriately, a single camera can be expanded into an enhanced vision system that takes into account both turning viewing and stereo-based measurement. The system is able to achieve the comprehensive, efficient, and high-accuracy measurement of multiple geometric parameters of internal threads with a single shot. The rest of the paper is organized as follows. In Section 2, we introduce the measurement principle, the measurement system, and the vision system calibration. In Section 3, a complete and explicit expression between the imaging parameters and the structural parameters of MSV without a reflecting prism is established. In Section 4, we propose the calculation model for imaging parameters of the MSV with a reflecting prism, and give the simulation results. In Section 5, the accuracy verification and the measurement experiments of the internal thread geometric parameters of two types of oilfield pipes are presented. Section 6 gives the conclusion of the paper.

2. Measurement system and method

2.1 Measurement system

The accuracy of manufactured threads is consistent, namely, if the profile of a certain longitudinal cross-section of an internal thread is out of tolerance, there will be consistent dimensional anomalies on each longitudinal cross-section. Thus, measuring only the geometric parameters of the tooth profiles on several longitudinal cross-sections along the axis can meet the evaluation requirements of the internal thread quality.

Therefore, in this paper, we propose a single-lens multi-mirror laser stereo-vision system and method for measuring internal thread geometric parameters. As shown in Fig. 1, the measurement system consists of a MSV, a reflecting prism, a cross-line laser, etc. The MSV is composed of a monocular camera, inner mirrors ${M_1}$ and ${M_2}$, and outer mirrors ${P_1}$ and ${P_2}$. With two reflections of the optical path by the inner mirrors and the outer mirrors, the image sensor is divided into two parts (i.e., left and right virtual cameras), thus the single camera is capable of 3D measurement. The reflecting prism (Fig. 1) is surrounded by four congruent triangular mirrors that can achieve the turning of the imaging angles. For the internal thread is a sparse 3D structure, the cross-line laser (four mutually perpendicular laser lines) is used to reproduce the complete tooth profiles. The region between the two inner mirrors provides the space for the coaxial installation of the laser and camera optical axis, thereby solving the problem of laser projection occlusion. In the process of measurement, the laser is first reflected to the surface of the internal thread along the longitudinal cross-section through the reflecting prism. Then, the laser profile is reflected by the reflecting prism, the inner mirrors, and the outer mirrors, and is then imaged at the ${K_1}$ and ${K_2}$ (Fig. 2) respectively. Thereafter, the features are extracted and undergo the stereo-matching process, and then a 3D point cloud of four laser profiles on two longitudinal cross-sections can be reconstructed. Finally, combined with the specifications of the API standard, the geometric parameters of the internal thread can be calculated from the 3D data.

Fig. 1. Vision-based system for measuring internal thread geometric parameters.

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Fig. 2. Schematic diagram of 3D measurement based on multiple mirror reflections.

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2.2 Measurement principle

A laser point in the world coordinate system (WCS) is denoted by ${\bf P} = {({x_w},\textrm{ }{y_w},\textrm{ }{z_w})^T}$. The space point obtained after being successively reflected by the reflecting prism, outer mirrors, and inner mirrors is:

(1)$$\left[ {\begin{array}{{c}} {{{x^{\prime}}_w}}\\ {{{y^{\prime}}_w}}\\ {{{z^{\prime}}_w}}\\ 1 \end{array}} \right]\textrm{ = }\prod\limits_{k = 1}^N {\left[ {\begin{array}{{cc}} {{\mathbf I} - 2{{\mathbf n}_k} \cdot {{\mathbf n}_k}^\textrm{T}}&{2{d_k}{{\mathbf n}_k}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right]} \cdot \left[ {\begin{array}{{c}} {{x_w}}\\ {{y_w}}\\ {{z_w}}\\ 1 \end{array}} \right]$$

where $N = 3$, ${{\mathbf M}_k} = \left[ {\begin{array}{{cc}} {{\mathbf I} - 2{{\mathbf n}_k} \cdot {{\mathbf n}_k}^\textrm{T}}&{2{d_k}{{\mathbf n}_k}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right]$ and ${{\mathbf n}_k} = {\textrm{(}{n_{kx}}\textrm{, }{n_{ky}},\textrm{ }{n_{kz}}\textrm{)}^T}$ ($n_{kx}^{}n_{kx}^{} + n_{ky}^{}n_{ky}^{} + n_{kz}^{}n_{kz}^{} = 1$) are the mapping matrix and the unit normal vector of the $k$- th mirror, respectively. ${d_k}$ describes the sign distance from the coordinate origin to the $k$-th mirror. The transformation matrixes between the WCS and the left and right virtual camera frames are denoted by $\left[ {\begin{array}{{cc}} {{{{\mathbf R^{\prime}}}_l}}&{{{{\mathbf t^{\prime}}}_l}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right]$ and $\left[ {\begin{array}{{cc}} {{{{\mathbf R^{\prime}}}_r}}&{{{{\mathbf t^{\prime}}}_r}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right]$. Then the camera model that characterizes the relationship between the ${\mathbf P}$ and its 2D pixel points ${{\mathbf p}_r}$ and ${{\mathbf p}_l}$ can be expressed as:

(2)$$\left\{ \begin{array}{l} {s_r}\left[ {\begin{array}{{c}} {{u_r}}\\ {{v_r}}\\ 1 \end{array}} \right] = \underbrace{{\left[ {\begin{array}{{cccc}} {{\alpha_x}}&0&{{u_0}}&0\\ 0&{{\alpha_y}}&{{v_0}}&0\\ 0&0&1&0 \end{array}} \right]}}_{{{{\mathbf M}_I}}}\underbrace{{ \cdot \left[ {\begin{array}{{cc}} {{{{\mathbf R^{\prime}}}_r}}&{{{{\mathbf t^{\prime}}}_r}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right] \cdot \prod\limits_{k = 1}^3 {\left[ {\begin{array}{{cc}} {{\mathbf I} - 2{{\mathbf n}_k} \cdot {{\mathbf n}_k}^\textrm{T}}&{2{d_k}{{\mathbf n}_k}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right]} }}_{{{{\mathbf M}_r}}} \cdot \left[ {\begin{array}{{c}} {{x_w}}\\ {{y_w}}\\ {{z_w}}\\ 1 \end{array}} \right]\\ {s_l}\left[ {\begin{array}{{c}} {{u_l}}\\ {{v_l}}\\ 1 \end{array}} \right] = \underbrace{{\left[ {\begin{array}{{cccc}} {{\alpha_x}}&0&{{u_0}}&0\\ 0&{{\alpha_y}}&{{v_0}}&0\\ 0&0&1&0 \end{array}} \right]}}_{{{{\mathbf M}_I}}}\underbrace{{ \cdot \left[ {\begin{array}{{cc}} {{{{\mathbf R^{\prime}}}_l}}&{{{{\mathbf t^{\prime}}}_l}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right] \cdot \prod\limits_{k = 1}^3 {\left[ {\begin{array}{{cc}} {{\mathbf I} - 2{{{\mathbf n^{\prime}}}_k} \cdot {{{\mathbf n^{\prime}}}_k}^\textrm{T}}&{2{d_k}{{{\mathbf n^{\prime}}}_k}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right]} }}_{{{{\mathbf M}_l}}} \cdot \left[ {\begin{array}{{c}} {{x_w}}\\ {{y_w}}\\ {{z_w}}\\ 1 \end{array}} \right] \end{array} \right.$$

where ${{\mathbf p}_r}\textrm{ = }{({u_r},\textrm{ }{v_r})^T}$, ${{\mathbf p}_l}\textrm{ = }{({u_l},\textrm{ }{v_l})^T}$, ${{\mathbf n}_1} = {{\mathbf n^{\prime}}_1}$. ${{\mathbf M}_I}$ denotes the intrinsic parameter matrix. ${\alpha _x}$ and ${\alpha _y}$ are the normalized focal lengths in the two-axis directions of the image coordinate system. ${({u_0},\textrm{ }{v_0})^T}$ represents the image distortion center. ${s_l}$ and ${s_r}$ denote the scale factors of the two virtual cameras. Letting ${{\mathbf M}_r} = \left[ {\begin{array}{{cc}} {{{\mathbf R}_r}}&{{{\mathbf t}_r}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right]$ and ${{\mathbf M}_l} = \left[ {\begin{array}{{cc}} {{{\mathbf R}_l}}&{{{\mathbf t}_l}}\\ {{{\mathbf 0}^\textrm{T}}}&1 \end{array}} \right]$ be the extrinsic parameter matrixes, respectively. Then, the transformation matrix between the two cameras is $\left[ {\begin{array}{{llll}} {{r_1}}&{{r_2}}&{{r_3}}&{{t_x}}\\ {{r_4}}&{{r_5}}&{{r_6}}&{{t_y}}\\ {{r_7}}&{{r_8}}&{{r_9}}&{{t_z}} \end{array}} \right] = \left[ {\begin{array}{{cc}} {{{\mathbf R}_l} \cdot {\mathbf R}_r^{\textrm{ - }1}}&{{{\mathbf t}_l} - {\mathbf R}_r^{\textrm{ - }1} \cdot {{\mathbf t}_r}} \end{array}} \right]$. Finally, the 3D coordinates of point ${\mathbf P}$ can be derived:

(3)$$\left\{ \begin{array}{l} {x_l} = \frac{{{z_l}({{u_l} - {u_0}} )}}{{{\alpha_l}}}\\ {y_l} = \frac{{{z_l}({{v_l} - {v_0}} )}}{{{\beta_l}}}\\ {z_l} = \frac{{{\alpha_x}{\beta_l}{\alpha_r}{t_x} - {\alpha_x}{\beta_l}({{u_r} - {u_0}} ){t_z}}}{{{\beta_\textrm{l}}({{u_l} - {u_0}} )[{({{u_r} - {u_0}} ){r_7} - {\alpha_r}{r_1}} ]+ {\alpha_l}({{v_l} - {v_0}} )[{({{u_r} - {u_0}} ){r_8} - {\alpha_r}{r_2}} ]+ {\alpha_l}{\beta_l}[{({{u_r} - {u_0}} ){r_9} - {\alpha_r}{r_3}} ]}} \end{array} \right.$$

A mirror can form a perfect image. However, manufacturing and assembly errors can cause the lens to produce radial and tangential imaging distortions and affect the measurement accuracy [25]. The lens distortions are closely related to the DoF [26], especially in the close-range photogrammetry. To improve the calibration accuracy, the equal-partition-based DoF-distortion calibration method [18] is used to solve the parameters of the camera model.

3. Explicit 3D OPM without the reflecting prism

The effective geometric dimensions of the four mirrors and the FoV parameters of MSV are closely related to the structural parameters. To solve the limitations of existing low dimensional 2D OPM and incomplete non-explicit 3D OPM, an explicit 3D OPM with complete structural parameters is proposed.

As shown in Fig. 3, first, a coordinate system is established according to the right-hand rule, where the intersection of the camera optical axis and the right inner mirror is the origin, the optical axis is the Z axis, and the horizontal axis is the X axis. Then, the angle between an inner mirror and the X axis is denoted as $\alpha $, the angle between an outer mirror and the X axis is denoted as $\beta $, the distance from the intersection of the two inner mirrors to the camera optical center ${{\mathbf O}_c}$ is defined as d, the horizontal distance from the intersection of the two inner mirrors to the intersection of an outer mirror and the X axis is denoted as L, the image sensor size in X- and Y-axis directions are denoted as S_x and S_y, respectively, and the focal length is defined as f.

Fig. 3. Schematic diagram of 3D OPM of the MSV without the reflecting prism.

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3.1 FoV parameters of MSV without the reflecting prism

The left and right image sensors are expressed as ${{\mathbf l}_i}$ and ${{\mathbf r}_i}$ ($i = 1,\textrm{ }2,\textrm{ } \cdots ,\textrm{ }4$), in which ${{\mathbf l}_i} = {{\mathbf r}_i}$ ($i = 1,\textrm{ }2$). According to Eq. (1), the mapping matrix with the combined contributions of the inner and outer mirrors can be obtained with:

(4)$${\mathbf M} = \left[ {\begin{array}{{llll}} {\cos (2\alpha - 2\beta )}&0&{\sin (2\beta - 2\alpha )}&{ - 2L\textrm{ }{{\sin }^2}\beta }\\ 0&1&0&0\\ {\sin (2\alpha - 2\beta )}&0&{\cos (2\alpha - 2\beta )}&{ - L\textrm{ }\sin 2\beta }\\ 0&0&0&1 \end{array}} \right]$$

Based on the parameter definitions, ${{\mathbf O}_c} = \textrm{ }{(0,\textrm{ }0,\textrm{ } - d)^T}$, ${{\mathbf r}_1}\textrm{ = }{{\mathbf l}_1} = {(0,\textrm{ }{{ - {S_y}} / 2},\textrm{ } - d - f)^T}$, ${{\mathbf r}_2} = {{\mathbf l}_2} = \textrm{ }{(0,\textrm{ }{{{S_y}} / 2},\textrm{ } - d - f\textrm{ })^T}$, ${{\mathbf r}_3} = \textrm{ }{({{ - {S_x}} / 2},\textrm{ }{{{S_y}} / 2},\textrm{ } - d - f)^T}$, ${{\mathbf r}_4} = \textrm{ }{({{ - {S_x}} / 2},\textrm{ }{{ - {S_y}} / 2},\textrm{ } - d - f)^T}$, ${{\mathbf l}_3} = \textrm{ }{({{{S_x}} / 2},\textrm{ }{{{S_y}} / 2},\textrm{ } - d - f)^T}$, and ${{\mathbf l}_4} = \textrm{ }{({{{S_x}} / 2},\textrm{ }{{ - {S_y}} / 2},\textrm{ } - d - f)^T}$. Then ${{\mathbf O}_r}$ and ${{\mathbf r^{\prime}}_i}$ ($i = 1,\textrm{ }2\textrm{ } \cdots \textrm{ }4$) in Fig. 3 can be obtained by the reflections of the inner and outer mirrors, in which ${{\mathbf O}_r} = {\left[ {\begin{array}{{cccc}} { - d\sin (2\beta - 2\alpha ) - 2L\textrm{ }{{\sin }^2}\beta }&0&{ - d\cos (2\alpha - 2\beta ) - L\textrm{ }\sin 2\beta } \end{array}} \right]^T}$. Further calculation yields the vector ${{\mathbf O}_r}{{\mathbf r^{\prime}}_i}$ after ${{\mathbf O}_c}{{\mathbf l}_i}$ is operated by ${\mathbf M}$, namely, ${{\mathbf O}_r}{{\mathbf r^{\prime}}_1} = [ { - f\sin (2\beta - 2\alpha )} { - {S_y}/2}\quad { - f\cos (2\alpha - 2\beta )} ]^T$, ${{\mathbf O}_r}{{\mathbf r^{\prime}}_2} = {\left[ {\begin{array}{{@{}ccc@{}}} { - f\sin (2\beta - 2\alpha )}&{{S_y}/2}&{ - f\cos (2\alpha - 2\beta )} \end{array}} \right]^T}$, ${{\mathbf O}_r}{{\mathbf r^{\prime}}_3} =$ ${\left[ {\begin{array}{{@{}ccc@{}}} {\frac{1}{2}{S_x}\cos (2\alpha - 2\beta ) - f\sin (2\beta - 2\alpha )}&{\frac{1}{2}{S_y}}&{\frac{1}{2}{S_x}\sin (2\alpha - 2\beta ) - f\cos (2\alpha - 2\beta )} \end{array}} \right]^T}$, and ${{\mathbf O}_r}{{\mathbf r^{\prime}}_4} = {\left[ {\begin{array}{{ccc}} {\frac{1}{2}{S_x}\cos (2\alpha - 2\beta ) - f\sin (2\beta - 2\alpha )}&{ - \frac{1}{2}{S_y}}&{\frac{1}{2}{S_x}\sin (2\alpha - 2\beta ) - f\cos (2\alpha - 2\beta )} \end{array}} \right]^T}$.

For a known direction vector ${({a\textrm{, }b,\textrm{ }c} )^T}$ along a line passing through the point ${({x\textrm{, }y,\textrm{ }z} )^T}$, the points on the plane and the normal vector of the plane are ${({{x_0}\textrm{, }{y_0},\textrm{ }{z_0}} )^T}$ and ${({m\textrm{, }n,\textrm{ }p} )^T}$, respectively. Letting $F = ma + nb + pc$ and $F \ne 0$, the line-plane intersection is:

(5)$${{\mathbf S}_e} = \left[ {\begin{array}{{c}} {{{(x(nb + pc) + a(m{x_0} + n({y_0} - y) + p({z_0} - z)))} / F}}\\ {{{(y(ma + pc) + b(n{y_0} + m({x_0} - x) + p({z_0} - z)))} / F}}\\ {{{(z(ma + nb) + c(p{z_0} + m({x_0} - x) + n({y_0} - y)))} / F}} \end{array}} \right]$$

The intersections of ${{\mathbf O}_r}{{\mathbf r^{\prime}}_1}$, ${{\mathbf O}_r}{{\mathbf r^{\prime}}_2}$, ${{\mathbf O}_r}{{\mathbf r^{\prime}}_3}$ and ${{\mathbf O}_r}{{\mathbf r^{\prime}}_4}$ with the plane $X = 0$ are denoted as ${{\mathbf F}_1}\textrm{ = }{({{F_{1x}}\textrm{, }{F_{1y}},\textrm{ }{F_{1z}}} )^T}$, ${{\mathbf F}_2} = {({{F_{1x}}\textrm{, } - {F_{1y}},\textrm{ }{F_{1z}}} )^T}$, ${{\mathbf F}_3}\textrm{ = }{({{F_{3x}}\textrm{, }{F_{3y}},\textrm{ }{F_{3z}}} )^T}$, and ${{\mathbf F}_4}\textrm{ = }{({{F_{3x}}\textrm{,} - {F_{3y}},\textrm{ }{F_{3z}}} )^T}$, respectively. According to Eq. (5) the ${{\mathbf F}_i}$ can be expressed as:

(6)$${{\mathbf F}_i} = {(0,\textrm{ } - \frac{{{b_i}x}}{{{a_i}}},\textrm{ }z - \frac{{{c_i}x}}{{{a_i}}})^T}\textrm{ }$$

Substituting ${{\mathbf O}_r}{{\mathbf r}_i}^\prime $ into Eq. (6) yields

{{\mathbf F}_1} = {\left[ {\begin{array}{{ccc}} 0&{\frac{{{S_y}(d\sin (2\beta - 2\alpha ) + 2L\textrm{ }{{\sin }^2}\beta )}}{{2f\sin (2\beta - 2\alpha )}}}&{ - d\cos (2\alpha - 2\beta ) - L\textrm{ }\sin 2\beta + \frac{{d\sin (2\beta - 2\alpha ) + 2L\textrm{ }{{\sin }^2}\beta }}{{\tan (2\beta - 2\alpha )}}} \end{array}} \right]^T},\, {\textrm{and}}

{{\mathbf F}_3} = \left[ {\begin{array}{{c}} 0\\ {\frac{{{S_y}(d\sin (2\beta - 2\alpha ) + 2L\textrm{ }{{\sin }^2}\beta )}}{{{S_x}\cos (2\alpha - 2\beta ) - 2f\sin (2\beta - 2\alpha )}}}\\ { - d\cos (2\alpha - 2\beta ) - L\textrm{ }\sin 2\beta + \frac{{({S_x}\sin (2\alpha - 2\beta ) - 2f\cos (2\alpha - 2\beta ))(d\sin (2\beta - 2\alpha ) + 2L\textrm{ }{{\sin }^2}\beta )}}{{{S_x}\cos (2\alpha - 2\beta ) - 2f\sin (2\beta - 2\alpha )}}} \end{array}} \right]

Letting ${{\mathbf P}_1}$ and ${{\mathbf P}_2}$ be the points on the two lines, and ${{\mathbf L}_1}$ and ${{\mathbf L}_2}$ be the unit direction vectors of the lines, then the intersection of the two lines is:

(7)$${{\mathbf K}_1} = {{\mathbf P}_2} \pm \frac{{|{{{\mathbf P}_1}{{\mathbf P}_2} \cdot {{\mathbf L}_1}} |}}{{{{\mathbf L}_1} \cdot {{\mathbf L}_2}}}{{\mathbf L}_2}$$

In Fig. 3, letting ${{\mathbf F}_{M1}}\textrm{ = }{({{F_{M1x}}\textrm{, }{F_{M1y}},\textrm{ }{F_{M1z}}} )^T}$ and ${{\mathbf F}_{M2}}\textrm{ = }{({{F_{M1x}}\textrm{, } - {F_{M1y}},\textrm{ }{F_{M1z}}} )^T}$, then according to Eq. (7), ${{\mathbf F}_{M1}}\textrm{ = }\left[ {\begin{array}{{c}} {( - d\sin (2\beta - 2\alpha ) - 2L\textrm{ }{{\sin }^2}\beta ) + ( - f\sin (2\beta - 2\alpha )){t_1}}\\ { - \frac{{{S_y}}}{2} \cdot {t_1}}\\ {\; ( - d\cos (2\alpha - 2\beta ) - L\textrm{ }\sin 2\beta ) + ( - f\cos (2\alpha - 2\beta )){t_1}} \end{array}} \right]$, where ${t_1} ={-} \frac{{\,(4f\,\cos (2\,\alpha - 2\,\beta ) - 2{S_x}\,\sin (2\,\alpha - 2\,\beta ))\,(d\,\sin (2\,\alpha - 2\,\beta ) - L + L\,\cos 2\beta )}}{{f\,\,({S_x}\cos (4\,\alpha - 4\,\beta ) + 2\,f\,\sin (4\,\alpha - 4\,\beta ))}}$.

With the above parameters, the imaging parameters of MSV are calculated as:

(8)$$\left\{ {\begin{array}{{l}} {FovM = 2|{{F_{M1x}}} |}\\ {{H_d}\textrm{ = }\frac{{2{S_y}(d\sin (2\beta - 2\alpha ) + 2L\textrm{ }{{\sin }^2}\beta )}}{{{S_x}\cos (2\alpha - 2\beta ) - 2f\sin (2\beta - 2\alpha )}}}\\ {{L_d}\textrm{ = }\frac{{{S_y}(d\sin (2\beta - 2\alpha ) + 2L\textrm{ }{{\sin }^2}\beta )}}{{f\sin (2\beta - 2\alpha )}}}\\ {{M_d} ={-} {S_y}{t_1}}\\ \begin{array}{l} Dof = \frac{{({S_x}\sin (2\alpha - 2\beta ) - 2f\cos (2\alpha - 2\beta ))(d\sin (2\beta - 2\alpha ) + 2L\textrm{ }{{\sin }^2}\beta )}}{{{S_x}\cos (2\alpha - 2\beta ) - 2f\sin (2\beta - 2\alpha )}}\\ \textrm{ } - \frac{{d\sin (2\beta - 2\alpha ) + 2L\textrm{ }{{\sin }^2}\beta }}{{\tan (2\beta - 2\alpha )}} \end{array}\\ \begin{array}{l} {B_a} = 2d\sin (2\beta - 2\alpha ) - 4L\textrm{ }{\sin^2}\beta \\ ObD = |{{F_{M1z}}} |\end{array}\\ {Phi = 2\alpha + 2\beta - 90} \end{array}} \right.$$

where $FovM$ is the maximum horizontal FoV. ${H_d}$ denotes the maximum vertical FoV. ${L_d}$ describes the minimum vertical FoV. ${M_d}$ represents the middle vertical FoV. $Dof$ is the DoF. ${B_a}$ and $ObD$ denote the baseline distance and object distance, respectively. $Phi$ is the angle between the optical axis of the right virtual camera and the baseline. Eq. (8) establishes the explicit relationship between the structural parameters (i.e., $\alpha $, $\beta $, d, L, ${S_x}$, ${S_y}$, and $f$) and the imaging parameters, which involves more comprehensive structural parameters and clearer characterization.

3.2 Effective sizes of inner and outer mirrors

The biggest problem in the measurements of the geometric parameters of the internal threads is the determination of whether the vision system can move into the pipes. Generally, to form the MSV, the overall structural sizes of the four mirrors are larger than the lens diameter, which directly determines the compactness of the system. Currently, the mirror sizes are determined empirically, which may not provide a theoretical basis for the system design. Thus, in this section we present a calculation model for the effective size of the four mirrors.

As shown in Fig. 3, for an inner mirror, the vertex ${{\mathbf S}_{Ii}}$ of its effective area is the intersection of ${{\mathbf O}_c}{{\mathbf l}_i}$ ($i = 1,2 \cdots 4$) and the inner mirror. According to Eq. (5), the line-plane intersection can be obtained:

(9)$${{\mathbf S}_{Ii}} = {(\frac{{{a_i}pd}}{{m{a_i} + p{c_i}}},\textrm{ }\frac{{{b_i}pd}}{{m{a_i} + p{c_i}}},\textrm{ } - \frac{{m{a_i}d}}{{m{a_i} + p{c_i}}})^T}$$

Letting ${{\mathbf O}_c}{{\mathbf l}_i}\textrm{ = }{({{a_i}\textrm{, }{b_i},\textrm{ }{c_i}} )^T}$, then the short side length, long side length, and height of the effective area can be obtained as $InS = {\raise0.7ex\hbox{${4fd}$} \!\mathord{/ {\vphantom {{4fd} {{S_y}}}}}\!\lower0.7ex\hbox{${{S_y}}$}}$, $InL\textrm{ = }\frac{{ - 2{S_y}d\cos \alpha }}{{{S_x}\sin \alpha - 2f\cos \alpha }}$, and $InH\textrm{ = }\frac{{{S_x}d}}{{{S_x}\sin \alpha - 2f\cos \alpha }}$ respectively by substituting ${{\mathbf O}_c}{{\mathbf l}_i}$ into Eq. (9), whereas $\tan \alpha > \frac{{2f}}{{{S_x}}}$.

For an outer mirror, the vertex ${{\mathbf S}_{{E_i}}}$ of the effective area is the intersection of ${{\mathbf O}_r}{{\mathbf r}_i}^\prime $($i = 1,2 \cdots ,4$) and the outer mirror. It is known that ${({x_0},\textrm{ }{y_0},\textrm{ }{z_0})^T} = {( - L,\textrm{ }0,\textrm{ }0)^T}$, ${(m,\textrm{ }n,\textrm{ }p)^T} = {(\sin \beta ,\textrm{ }0,\textrm{ }\cos \beta )^T}$, ${({x\textrm{, }y,\textrm{ }z} )^T} = {( - d\sin (2\beta - 2\alpha ) - 2L\textrm{ }{\sin ^2}\beta ,\textrm{ }0,\textrm{ } - d\cos (2\alpha - 2\beta ) - L\textrm{ }\sin 2\beta )^T}$. Substituting the above parameters into Eq. (10) yields:

(10)$${{\mathbf S}_{{E_i}}} = {(\frac{{{c_i}px + {a_i}( - mL - pz)}}{{m{a_i} + p{c_i}}},\textrm{ }\frac{{{b_i}(m( - L - x) - pz)}}{{m{a_i} + p{c_i}}},\textrm{ } - \frac{{m{a_i}z + {c_i}m( - L - x)}}{{m{a_i} + p{c_i}}})^T}$$

Letting ${{\mathbf O}_r}{{\mathbf r}_i}^\prime \textrm{ = }{({{a_i}\textrm{, }{b_i},\textrm{ }{c_i}} )^T}$ and the short side length, long side length and height of the effective area be $ExS = |{{{\mathbf S}_{E1}} - {{\mathbf S}_{E2}}} |$, $ExL = |{{{\mathbf S}_{E3}} - {{\mathbf S}_{E4}}} |$, and $ExH = \left|{\frac{{{{\mathbf S}_{E1}} + {{\mathbf S}_{E2}}}}{2} - \frac{{{{\mathbf S}_{E3}} + {{\mathbf S}_{E4}}}}{2}} \right|$ respectively. The calculation of ${{\mathbf S}_{E1}}$ and ${{\mathbf S}_{E2}}$ requires the ${{\mathbf O}_r}{{\mathbf r^{\prime}}_1}$ and ${{\mathbf O}_r}{{\mathbf r^{\prime}}_2}$. Since ${a_1}\textrm{ = }{a_3}$ and ${c_1}\textrm{ = }{c_3}$, according to Eq. 10, $ExS\textrm{ = }\left|{\frac{{{S_y}(d\cos (\beta - 2\alpha ) + L\textrm{ }\sin \beta )}}{{f\cos (\beta - 2\alpha )}}} \right|$. Similarly, $ExL\textrm{ = }\frac{{2{S_y}(d\cos (\beta - 2\alpha ) + L\textrm{ }\sin \beta )}}{{{S_x}\sin (\beta - 2\alpha ) + 2f\cos (\beta - 2\alpha )}}$, and $ExH = \frac{{{S_x}d\cos (\beta - 2\alpha ) + {S_x}L\sin \beta }}{{{S_x}\sin (\beta - 2\alpha )\cos (\beta - 2\alpha ) + 2f{{\cos }^2}(\beta - 2\alpha )}}$. For the equations above, $\tan (\beta - 2\alpha ) > \frac{{2f}}{{{S_x}}}$.

3.3 Simulation analysis of 3D optical path of MSV without the reflecting prism

In this section, we analyzed the effects of structural parameters (i.e., $\alpha $, $\beta $, d, L, f, ${S_x}$, and S_y) on the imaging parameters and effective mirror size of the MSV without the reflecting prism. The constant values of the structural parameters are set to α = 45°, β = 51°, d = 8 mm, L = 35 mm, f = 24 mm, S_x = 12 mm, and S_y = 12 mm. The variation ranges of each structural parameter are α = [35° 46°], β = [47° 59°], d = [8 mm 60 mm], L = [15 mm 60 mm], f = [15 mm 60 mm], S_x = [2 mm 16 mm], and S_y = [2 mm 16 mm], respectively. Figure 4 shows the simulation results of 3D optical path of MSV with several parameter configurations. It can be seen from the figure that the effective areas of the inner and outer mirrors occupied by the optical path are isosceles trapezoids, and with the change of structural parameters, the imaging parameters are also constantly changing.

Fig. 4. 3D light path of the MSV without the reflecting prism. (a) α = 45°, β = 55°, d = 20 mm, L = 40 mm, S_x=12 mm, S_y= 12 mm, and f = 20 mm. (b) α = 44°, β = 52°, d = 10 mm, L = 40 mm, S_x= 8 mm, S_y= 12 mm, and f = 20 mm. (c) α = 44°, β = 52°, d = 10 mm, L = 40 mm, S_x = 12 mm, S_y= 12 mm, and f = 40 mm.

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Figure 5 (a) - (g) is obtained by plotting change rate curves of imaging parameters with respect to each structural parameter. For each graph in Fig. 5, the arithmetic mean of the absolute values of each curve is calculated and sorted. The results are shown in Table 1.

Fig. 5. Changes in imaging parameters with respect to each structural parameter (without the reflecting prism). (a) α. (b) β. (c) d. (d) L. (e) S_x. (f) S_y. (g) f.

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Table 1. Sensitivity of the structural parameters to the imaging parameters (without the reflecting prism)

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In addition, as shown in Fig. 6, for each imaging parameter, its change rate curves with respect to all structural parameters are plotted on a single graph (e.g., Fig. 6 (a)). Also, the arithmetic mean of the absolute values of each curve is calculated. In this way, we can find the row where the structural parameter is when the value is maximized. Then, the imaging parameter is given in red in the row. In addition, some structural parameters have no effect on the imaging parameters, which are marked in blue in the Table 1. $\alpha $ and $\beta $ have the same effects on $Phi$, thus $Phi$ is marked in green. It can be seen from the table that the structural parameters that have an important influence on the imaging parameters are $\beta $, ${S_y}$, d and f, where $\beta $ has the greatest impact on the measurement accuracy and measurement accessibility, and ${S_y}$, d and f have a greater impact on the effective inner mirror size, these parameters can be adjusted preferentially during system design.

Fig. 6. Changes in each imaging parameter with respect to all structural parameters (without the reflecting prism). (a) H_d. (b) M_d. (c) L_d. (d) Dof. (e) FovM. (f) B_a. (g) InL. (h) InS. (i) InH. (j) ExL. (k) ExS. (l) ExH. (m) ObD. (n) Phi.

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4. OPM of MSV with the reflecting prism

Adding a reflecting prism in front of the MSV can allow for turning viewing and solve the problem of measurement inaccessibility. Letting the base of the prism be a square, half of the base length is denoted as $w = {\raise0.7ex\hbox{$l$} \!\mathord{/ {\vphantom {l 2}}}\!\lower0.7ex\hbox{$2$}}$, the angle between the reflecting face and the base of the prism is denoted as $\gamma$, and the base height is defined as h. From the calculation, the vertex of the regular quadrangular pyramid is ${\mathbf S}\textrm{ = }{({0\textrm{, }0,\textrm{ }g} )^T}$, where $g = h - w\tan \gamma$. Restricted by the FoV of MSV and the space of the internal thread, the minimum allowable size of the prism should be determined so that the prism can reach the interior of the thread. Besides, the FoV parameters of the MSV with the reflecting prism should be calculated to ensure the visual accessibility. To this end, we present a calculation model for the suitable size of the prism and the imaging parameters of the MSV with the reflecting prism.

4.1 Suitable size of the reflecting prism

As shown in Fig. 7, the suitable size of the prism base is determined by the four intersections between the $Z = h$ plane and the boundary lines of the FoV of the MSV without the reflecting prism, which can be determined by the following two cases:

1) ${F_{4z}} \le h {F_{M1z}}$

Fig. 7. Schematic diagram of the MSV with the reflecting prism.

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Fig. 8. Imaging parameters of the MSV with reflecting prism. (a) ${\bf S}{{\bf A}_\theta }{{\bf B}_\theta }$ and ${\bf S}{{\bf C}_\theta }{{\bf D}_\theta }$. (b) ${\bf S}{{\bf B}_\theta }{{\bf C}_\theta }$ and ${\bf S}{{\bf D}_\theta }{{\bf A}_\theta }$.

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Fig. 9. Visible range of the vision-based system for the conical face. (a) ${\bf S}{{\bf A}_\theta }{{\bf B}_\theta }$ and ${\bf S}{{\bf C}_\theta }{{\bf D}_\theta }$. (b) ${\bf S}{{\bf B}_\theta }{{\bf C}_\theta }$ and ${\bf S}{{\bf D}_\theta }{{\bf A}_\theta }$.

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According to Eq. (5), the intersection ${\mathbf H}\textrm{ = }{({{H_x}\textrm{, }{H_y},\textrm{ }{H_z}} )^T}$ of ${{\mathbf O}_r}{{\mathbf r^{\prime}}_4}$ and the $Z = h$ plane can be written as:

(11)$${\mathbf H} = {(x + \frac{{a(h - z)}}{c},\textrm{ }y + \frac{{b(h - z)}}{c},\textrm{ }h)^T}$$

Substituting ${{\mathbf O}_r}{{\mathbf r^{\prime}}_4}\textrm{ = }{({a\textrm{, }b,\textrm{ }c} )^T}\textrm{ = }\left[ {\begin{array}{{c}} {\frac{1}{2}{S_x}\cos (2\alpha - 2\beta ) - f\sin (2\beta - 2\alpha )}\\ { - \frac{1}{2}{S_y}}\\ {\frac{1}{2}{S_x}\sin (2\alpha - 2\beta ) - f\cos (2\alpha - 2\beta )} \end{array}} \right]$ into Eq. (11) yields

{\mathbf H} = \left[ {\begin{array}{{c}} { - d\sin (2\beta - 2\alpha ) - 2L\textrm{ }{{\sin }^2}\beta + \frac{{({S_x}\cos (2\alpha - 2\beta ) - 2f\sin (2\beta - 2\alpha ))(h + d\cos (2\alpha - 2\beta ) + L\textrm{ }\sin 2\beta )}}{{{S_x}\sin (2\alpha - 2\beta ) - 2f\cos (2\alpha - 2\beta )}}}\\ {\frac{{ - {S_y}(h + d\cos (2\alpha - 2\beta ) + L\textrm{ }\sin 2\beta )}}{{{S_x}\sin (2\alpha - 2\beta ) - 2f\cos (2\alpha - 2\beta )}}}\\ h \end{array}} \right]

2) ${F_{M1z}} \le h \le {F_{1z}}$

According to Eq. (5), the intersection ${\mathbf H}\textrm{ = }{({{H_x}\textrm{, }{H_y},\textrm{ }{H_z}} )^T}$ of ${{\mathbf O}_r}{{\mathbf r^{\prime}}_1}$ and the $Z = h$ plane can be expressed by ${\mathbf H} = \left[ {\begin{array}{{@{}c@{}}} { - d\sin (2\beta - 2\alpha ) - 2L\textrm{ }{{\sin }^2}\beta + \tan (2\beta - 2\alpha )(h + d\cos (2\alpha - 2\beta ) + L\textrm{ }\sin 2\beta )}\\ {\frac{{{S_y}(h + d\cos (2\alpha - 2\beta ) + L\textrm{ }\sin 2\beta )}}{{2f\cos (2\alpha - 2\beta )}}}\\ h \end{array}} \right]$.

From the above two cases, it can be obtained that the suitable sizes of the prism base along $X$- and $Y$-axis directions are $2|{{H_x}} |$ and $2|{{H_y}} |$, respectively. If a regular quadrangular pyramid mirror is used, the maximum sizes of the prism base is $\min (|{{H_x}} |,\textrm{ }|{{H_y}} |)$.

4.2 Imaging parameters of MSV with the reflecting prism

By definition, the four vertices of the prism base are ${\mathbf A} = {({ - w\textrm{, } - w,\textrm{ }h} )^T}$, ${\mathbf B} = {({ - w\textrm{, }w,\textrm{ }h} )^T}$, ${\mathbf C} = {({w\textrm{, }w,\textrm{ }h} )^T}$, and ${\mathbf D} = {({w\textrm{, } - w,\textrm{ }h} )^T}$.

When the laser plane is coplanar with the line formed by the two optical centers, the laser strip image collected by the camera cannot represent the thread information. The reflecting prism is mounted in such a way that the prism axis coincides with the optical axis of the real camera but it can rotate around the Z axis. It is assumed that the prism vertices are ${{\mathbf A}_\theta }$, ${{\mathbf B}_\theta }$, ${{\mathbf C}_\theta }$ and ${{\mathbf D}_\theta }$ after rotating $\theta $ around the Z axis. Then ${{\mathbf A}_\theta }\textrm{ = }{({ - w(\cos \theta - \sin \theta )\textrm{, } - w(\sin \theta + \cos \theta ),\textrm{ }h} )^T}$, ${{\mathbf B}_\theta } = {({ - w(\cos \theta + \sin \theta )\textrm{, }w(\cos \theta - \sin \theta ),\textrm{ }h} )^T}$, ${{\mathbf C}_\theta }\textrm{ = }{({w(\cos \theta - \sin \theta )\textrm{, }w(\sin \theta + \cos \theta ),\textrm{ }h} )^T}$, and ${{\mathbf D}_\theta }\textrm{ = }{({w(\cos \theta + \sin \theta )\textrm{, } - w( - \sin \theta + \cos \theta ),\textrm{ }h} )^T}$.

As seen in Fig. 8, the prism is surrounded by four reflecting faces in a clockwise direction, which are ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$, ${\mathbf S}{{\mathbf B}_\theta }{{\mathbf C}_\theta }$, ${\mathbf S}{{\mathbf C}_\theta }{{\mathbf D}_\theta }$, and ${\mathbf S}{{\mathbf D}_\theta }{{\mathbf A}_\theta }$, respectively. The mirror matrix of ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$ is ${{\mathbf M}_{f1}}= \left[ {\begin{array}{{cccc}} {1 - 2{{\sin }^2}\gamma {{\cos }^2}\theta }&{ - {{\sin }^2}\gamma \sin 2\theta }&{ - \sin 2\gamma \cos \theta }&{g\sin 2\gamma \cos \theta }\\ { - {{\sin }^2}\gamma \sin 2\theta }&{1 - 2{{\sin }^2}\gamma {{\sin }^2}\theta }&{ - \sin 2\gamma \sin \theta }&{g\sin 2\gamma \sin \theta }\\ { - \sin 2\gamma \cos \theta }&{ - \sin 2\gamma \sin \theta }&{ - \cos 2\gamma }&{2g{{\cos }^2}\gamma }\\ 0&0&0&1 \end{array}} \right]$. For imaging parameters of the MSV considering ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$, letting ${{\mathbf O}_r}\textrm{ = }{({x\textrm{, }y,\textrm{ }z} )^T}$ and ${{\mathbf O}_l}\textrm{ = }{({ - x\textrm{, }y,\textrm{ }z} )^T}$, then the corresponding points mirrored by ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$ are ${{\mathbf O}_{r1}} = \left[ {\begin{array}{{c}} {x(1 - 2{{\sin }^2}\gamma {{\cos }^2}\theta ) + (g - z)\sin 2\gamma \cos \theta }\\ { - x{{\sin }^2}\gamma \sin 2\theta + (g - z)\sin 2\gamma \sin \theta }\\ { - x\sin 2\gamma \cos \theta + (g - z)\cos 2\gamma + g} \end{array}} \right]$, and ${{\mathbf O}_{l1}} = \left[ {\begin{array}{{c}} {x(2{{\sin }^2}\gamma {{\cos }^2}\theta - 1) + (g - z)\sin 2\gamma \cos \theta }\\ {x{{\sin }^2}\gamma \sin 2\theta + (g - z)\sin 2\gamma \sin \theta }\\ {x\sin 2\gamma \cos \theta + (g - z)\cos 2\gamma + g} \end{array}} \right]$.

Then the intersection region ${\Pi _1}$ of the region ${\Omega _{r1}}$ (enclosed by the rays ${{\mathbf O}_{r1}}{\mathbf S}$, ${{\mathbf O}_{r1}}{{\mathbf A}_\theta }$, and ${{\mathbf O}_{r1}}{{\mathbf B}_\theta }$) with the region ${\Omega _{l1}}$ (enclosed by the rays ${{\mathbf O}_{l1}}{\mathbf S}$, ${{\mathbf O}_{l1}}{{\mathbf A}_\theta }$, and ${{\mathbf O}_{l1}}{{\mathbf B}_\theta }$) is solved. ${\Pi _1}$ is a triangular pyramid. Its base vertices are ${\mathbf S}$, ${{\mathbf A}_\theta }$ and ${{\mathbf B}_\theta }$, and its apex ${{\mathbf E}_1} = {({E_{x1}},\textrm{ }{E_{y1}},\textrm{ }{E_{z1}})^T}$ is the intersection of ${{\mathbf O}_{l1}}{\mathbf S}$ and ${{\mathbf O}_{r1}}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$. The imaging parameters of the MSV mirrored by ${\mathbf S}{{\mathbf C}_\theta }{{\mathbf D}_\theta }$ are consistent with those of ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$, thus the intersection region ${\Pi _3} = {{\mathbf E}_3}\textrm{ - }{\mathbf S}{{\mathbf C}_\theta }{{\mathbf D}_\theta }$, and ${{\mathbf E}_3} = {( - {E_{x1}},\textrm{ } - {E_{y1}},\textrm{ }{E_{z1}})^T}$. Given ${{\mathbf O}_{l1}}{\mathbf S}\textrm{ = }\left\lfloor {\begin{array}{{c}} {x(1 - 2{{\sin }^2}\gamma {{\cos }^2}\theta ) - (g - z)\sin 2\gamma \cos \theta }\\ { - x{{\sin }^2}\gamma \sin 2\theta - (g - z)\sin 2\gamma \sin \theta }\\ { - x\sin 2\gamma \cos \theta - (g - z)\cos 2\gamma } \end{array}} \right\rfloor$, ${\mathbf n} = {{\mathbf O}_{r1}}{{\mathbf A}_\theta } \times {{\mathbf A}_\theta }{{\mathbf B}_\theta } = {(m,\textrm{ }n,\textrm{ }p)^T} = \left[ {\begin{array}{{c}} {x(2{{\sin }^2}\gamma {{\cos }^2}\theta - 1) - (g - z)\sin 2\gamma \cos \theta - w(\cos \theta - \sin \theta )}\\ {x{{\sin }^2}\gamma \sin 2\theta - (g - z)\sin 2\gamma \sin \theta - w(\cos \theta + \sin \theta )}\\ {(x\cos \theta + w)\sin 2\gamma - (h - z)\cos 2\gamma } \end{array}} \right] \times \left[ {\begin{array}{{c}} { - 2w\cos \theta }\\ {2w\cos \theta }\\ 0 \end{array}} \right]$, and ${({a\textrm{, }b,\textrm{ }c} )^T} = \left[ {\begin{array}{{c}} {x(1 - 2{{\sin }^2}\gamma {{\cos }^2}\theta ) - (z + g)\sin 2\gamma \cos \theta }\\ { - x{{\sin }^2}\gamma \sin 2\theta - (z + g)\sin 2\gamma \sin \theta }\\ { - x\sin 2\gamma \cos \theta - (z + g)\cos 2\gamma } \end{array}} \right]$, then according to Eq. (5), ${{\mathbf E}_1} = {({E_{x1}},\textrm{ }{E_{y1}},\textrm{ }{E_{z1}})^T} = \left[ {\begin{array}{{c}} {\frac{{a[mw(\sin \theta - \cos \theta ) - nw(\cos \theta + \sin \theta ) + p(h - g)]}}{{ma + nb + pc}}}\\ {\frac{{b[ - nw(\cos \theta + \sin \theta ) + mw(\sin \theta - \cos \theta ) + p(h - g)]}}{{ma + nb + pc}}}\\ {\frac{{g(ma + nb) + c[ph + mw(\sin \theta - \cos \theta ) - nw(\cos \theta + \sin \theta )]}}{{ma + nb + pc}}} \end{array}} \right]$.

Finally, the visible depth of the MSV mirrored by ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$ and ${\mathbf S}{{\mathbf C}_\theta }{{\mathbf D}_\theta }$ is $MirVL\textrm{1} = \sqrt {{E_{x1}}^2 + {E_{y1}}^2}$.

Similarly, the intersection region ${\Pi _2}$ is a triangular pyramid ${{\mathbf E}_2}\textrm{ - }{\mathbf S}{{\mathbf B}_\theta }{{\mathbf C}_\theta }$, and its apex ${{\mathbf E}_2} = {({E_{2x}},\textrm{ }{E_{2y}},\textrm{ }{E_{2z}})^T}$ (${{\mathbf E}_4} = {( - {E_{2x}},\textrm{ } - {E_{2y}},\textrm{ }{E_{2z}})^T}$) is the intersection of the ${{\mathbf O}_{r2}}{{\mathbf B}_\theta }$ and ${{\mathbf O}_{l2}}{{\mathbf C}_\theta }{\mathbf S}$. Given ${{\mathbf O}_{r2}}{{\mathbf B}_\theta } = \left[ {\begin{array}{{c}} {x(2{{\sin }^2}\gamma {{\sin }^2}\theta - 1) + (z - g)\sin 2\gamma \sin \theta - w(\cos \theta + \sin \theta )}\\ { - x{{\sin }^2}\gamma \sin 2\theta + (g - z)\sin 2\gamma \cos \theta + w(\cos \theta - \sin \theta )}\\ {x\sin 2\gamma \sin \theta + (z - g)\cos 2\gamma - w\tan \gamma } \end{array}} \right]$, the direction vector of the straight line ${({a\textrm{, }b,\textrm{ }c} )^T} = \left[ {\begin{array}{{c}} {x(2{{\sin }^2}\gamma {{\cos }^2}\theta - 1) + (z - g)\sin 2\gamma \cos \theta - w(\cos \theta + \sin \theta )}\\ { - x{{\sin }^2}\gamma \sin 2\theta + (g - z)\sin 2\gamma \sin \theta + w(\cos \theta - \sin \theta )}\\ {x\sin 2\gamma \cos \theta + (z - g)\cos 2\gamma - w\tan \gamma } \end{array}} \right]$, the normal vector of the ${\mathbf S}{{\mathbf B}_\theta }{{\mathbf C}_\theta }$ ${\mathbf n} = {{\mathbf O}_{l2}}{\mathbf S} \times {\mathbf S}{{\mathbf C}_{\boldsymbol \theta }} = {(m,\textrm{ }n,\textrm{ }p)^T} = \left[ {\scriptsize\begin{array}{{@{}c@{}}} {x(1 - 2{{\sin }^2}\gamma {{\sin }^2}\theta ) + (z - g)\sin 2\gamma \sin \theta }\\ {x{{\sin }^2}\gamma \sin 2\theta + (g - z)\sin 2\gamma \cos \theta }\\ { - x\sin 2\gamma \sin \theta + (z - g)\cos 2\gamma } \end{array}} \right] \times \left[ {\begin{array}{{c}} {w(\cos \theta - \sin \theta )}\\ {w(\cos \theta + \sin \theta )}\\ {h - g} \end{array}} \right]$. then ${{\mathbf E}_2} = {({E_{2x}},\textrm{ }{E_{2y}},\textrm{ }{E_{2z}})^T} = \left[ {\begin{array}{{c}} {\frac{{ - w(\cos \theta + \sin \theta )(nb + pc) + a[ - nw(\cos \theta - \sin \theta ) + p(g - h)]}}{{ma + nb + pc}}}\\ {\frac{{w(\cos \theta - \sin \theta )(ma + pc) + b[mw(\cos \theta + \sin \theta ) + p(g - h)]}}{{ma + nb + pc}}}\\ {\frac{{h(ma + nb) + c[pg + mw(\cos \theta + \sin \theta ) - nw(\cos \theta - \sin \theta )]}}{{ma + nb + pc}}} \end{array}} \right]$.

Therefore, the visible depth of the MSV mirrored by ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$ and ${\mathbf S}{{\mathbf C}_\theta }{{\mathbf D}_\theta }$ is $MirVL\textrm{2} = \sqrt {{E_{x2}}^2 + {E_{y2}}^2}$.

To calculate the visible range of the vision-based system for the conical face $\mathrm{\Xi }$ of the internal thread, las seen in Fig, 9 letting R be the thread top diameter, $\delta $ be the thread taper, the direction vector of the $j$-th straight line ${\mathbf L}_i^j$ on the $i$-th mirror of the prism be ${\mathbf v}_i^j$, and the point on ${\mathbf L}_i^j$ be ${\mathbf E}_i^j$, then the intersection ${\mathbf P}_i^j\textrm{ = (}x_i^j,\textrm{ }y_i^j,\textrm{ }z_i^j{\textrm{)}^T}$ of ${\mathbf L}_i^j$ and $\mathrm{\Xi }$ can be expressed as:

(12)$${\mathbf P}_i^j = {\mathbf E}_i^j + k_i^j{\mathbf v}_i^j\textrm{ }i = 1,2, \ldots ,4\textrm{ }j = 1,2,3\textrm{ }$$

where $k_i^j = \frac{{ - b_i^j \pm \sqrt {{{(b_i^j)}^2} - 4a_i^jc_i^j} }}{{2a_i^j}}$, $a_i^j = 1 - {\mathbf v}_i^j \cdot {\mathbf n}_i^j \times {\mathbf v}_i^j \cdot {\mathbf n}_i^j - \frac{{{R^2}}}{{{{h^{\prime}}^2}}} \times {\mathbf v}_i^j \cdot {\mathbf n}_i^j \times {\mathbf v}_i^j \cdot {\mathbf n}_i^j$, $t^{\prime} = \frac{R}{{\tan \frac{\delta }{2}}} - g$, $h^{\prime} = \frac{R}{{\tan \frac{\delta }{2}}}$, ${\mathbf n} = {(0,\textrm{ }0,\textrm{ }1)^T}$, $b_i^j = 2{\mathbf E}_i^j \cdot {\mathbf v}_i^j - 2{\mathbf E}_i^j \cdot {\mathbf n}_i^j \times {\mathbf v}_i^j \cdot {\mathbf n}_i^j - \frac{{2{R^2}}}{{{{h^{\prime}}^2}}}t_i^j \times {\mathbf v}_i^j \cdot {\mathbf n}_i^j - \frac{{2{R^2}}}{{{{h^{\prime}}^2}}} \times {\mathbf E}_i^j \cdot {\mathbf n}_i^j \times {\mathbf v}_i^j \cdot {\mathbf n}_i^j$, $c_i^j = {\bf E}_i^j \cdot {\bf E}_i^j - {\bf E}_i^j \cdot {\bf n} \times {\bf E}_i^j \cdot {\bf n} - \frac{{{R^2}}}{{{{h'}^2}}}{t'^2} - \frac{{2{R^2}}}{{{{h'}^2}}}t' \times {\bf E}_i^j \cdot {\bf n} - \frac{{{R^2}}}{{{{h'}^2}}} \times {\bf E}_i^j \cdot {\bf n} \times {\bf E}_i^j \cdot {\bf n}$.

For the ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$, according to Eq. (12), the intersections ${\mathbf P}_1^j$ of ${{\mathbf E}_1}{{\mathbf A}_\theta }$, ${{\mathbf E}_1}{{\mathbf B}_\theta }$, and ${{\mathbf E}_1}{\mathbf S}$ with $\mathrm{\Xi }$ can be solved. The horizontal and vertical visible range of the MSV mirrored by ${\mathbf S}{{\mathbf A}_\theta }{{\mathbf B}_\theta }$ (or ${\mathbf S}{{\mathbf C}_\theta }{{\mathbf D}_\theta }$) to the $\mathrm{\Xi }$ are $MirFovL\textrm{1} = |{{\mathbf P}_1^1 - {\mathbf P}_1^2} |$ and $MirFovH\textrm{1} = \left|{\frac{1}{2}\textrm{(}{\mathbf P}_1^1\textrm{ + }{\mathbf P}_1^2\textrm{)} - {\mathbf P}_1^3} \right|$, respectively. Similarly, the horizontal and vertical visible range of the MSV mirrored by ${\mathbf S}{{\mathbf B}_\theta }{{\mathbf C}_\theta }$ (or ${\mathbf S}{{\mathbf D}_\theta }{{\mathbf A}_\theta }$) to $\mathrm{\Xi }$ are $MirFovL\textrm{2} = |{{\mathbf P}_2^1 - {\mathbf P}_2^2} |$ and $MirFovH\textrm{2} = \left|{\frac{1}{2}\textrm{(}{\mathbf P}_2^1\textrm{ + }{\mathbf P}_2^2\textrm{)} - {\mathbf P}_2^3} \right|$, respectively.

4.3 Simulation analysis of 3D optical path of MSV with the reflecting prism

In this section, the effects of structural parameters (i.e., $\alpha $, $\beta $, d, L, ${S_x}$, ${S_y}$, f, w, h, $\gamma $, $\theta $, $\delta $, and $R$) on the imaging parameters of the MSV with the reflecting prism are analyzed. Considering the fact that the crest of an internal thread is perpendicular to the axis, we set $\gamma = 45^\circ$. Besides, the prism and the laser are arranged symmetrically, so that the laser is reflected by the prism and hits the internal thread vertically to ensure the imaging accuracy. In the analysis, the constants $\alpha $, $\beta $, d, L, ${S_x}$, ${S_y}$ and f are the same as those listed in Section 3. With the premise of ensuring that the size of $l = 2w$ is within the FoV of the MSV, the variation ranges of these parameters are set to $l = \left[ {\begin{array}{{cc}} {5\textrm{mm}}&{45\textrm{mm}} \end{array}} \right]$, $h = \left[ {\begin{array}{{cc}} {60\textrm{mm}}&{120\textrm{mm}} \end{array}} \right]$, $\gamma = \left[ {\begin{array}{{cc}} {20^\circ }&{80^\circ } \end{array}} \right]$, $\theta = \left[ {\begin{array}{{cc}} {0^\circ }&{48^\circ } \end{array}} \right]$, $\delta = \left[ {\begin{array}{{cc}} {2^\circ }&{45^\circ } \end{array}} \right]$, $R = \left[ {\begin{array}{{cc}} {26\textrm{mm}}&{50\textrm{mm}} \end{array}} \right]$. Figure 10 shows the 3D light path diagrams of the vision system at several prism parameter configurations. It can be seen from the figure that with the change of the structural parameters of the reflecting prism, the imaging parameters of the vision system also change. Figure 11(a)–(f) is obtained by plotting the change rate curves of the imaging parameters with respect to the structural parameters of the reflecting prism.

Fig. 10. 3D optical path diagrams of MSV with the reflecting prism. (a) l = 40 mm, h = 64 mm, γ = 45°, θ = 0°, δ = 4°, R = 55mm. (b) l = 40 mm, h = 64 mm, γ = 35°, θ = 0°, δ = 4°, R = 55 mm. (c) l = 40 mm, h = 64 mm, γ = 45°, θ = 30°, δ = 4°, R = 55 mm.

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Fig. 11. Changes in imaging parameters with respect to each structural parameter (with the reflecting prism). (a) l. (b) h. (c) γ. (d) θ. (e) δ. (f) R.

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For each graph in Fig. 11, the arithmetic mean of the absolute values of each curve is calculated and sorted. The results are shown in Table 2.

Table 2. Sensitivity of the structural parameters to the imaging parameters (with the reflecting prism)

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Additionally, as shown in Fig. 12(a)–(f), for each imaging parameter, its change rate curves with respect to all structural parameters are plotted on a single graph, and the arithmetic mean of the absolute values of each curve is calculated. In this way, we can find the row where the structural parameter of the prism is when the value is maximized. And then the imaging parameter is given in red in the row. Similarly, some structural parameters have no effect on the imaging parameters, and they are marked in blue in the Table 2. It can be seen from Table 2 that the prism structural parameters that have an important influence on the imaging parameters are l and $\gamma $. Finally, taking the internal thread space, the sensitivity of the structural parameters, and the measurement accuracy as the constraints, the structural parameters of the vision-based system for measuring geometric parameters of the internal thread can be optimized by LM algorithm.

Fig. 12. Changes in each imaging parameter with respect to all structural parameters of the prism. (a) MirVL1. (b) MirFovL1. (c) MirFovH1. (d) MirVL2. (e) MirFovL2. (f) MirFovH2.

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Fig. 13. Experimental setup for geometric parameters of the internal threads.

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Fig. 14. Internal threads to be measured. (a) 3^1/2in. female pipe. (b) 2^7/8in. female pipe.

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5. Accuracy verification and experiments

5.1 Experimental system

As shown in Fig. 13, an experimental system for measuring the geometric parameters of internal threads is built, the vision system is composed of an MSV, a regular quadrangular pyramid prism, a cross-line laser, etc. A 1.6 million-pixel camera equipped with a 12 mm focal length lens is used to collect line laser images with a wavelength of 450 nm. The parameters are calculated according to the programmed image processing algorithm. Other relevant parameters of the vision system are shown in Table 3. According to the API standard, the geometric parameters of the internal threads of 3^1/2 in. and 2^7/8 in. (1 in. = 25.4 mm) in female pipe are tested. The internal threads to be measured are shown in Fig. 14. To use the same system to complete the inspection of the two specifications of oilfield pipes, the internal space of the smaller pipe is used to limit the size of the mirrors to ensure the penetrability of the vision-based measurement system. To ensure measurement accessibility, the radius of the larger pipe and the axial thread distribution size of smaller pipe are utilized to constrain the object distance and the FoV, respectively. In this paper, the Levenberg-Marquardt (LM) algorithm [27,28] is used to optimize the structural parameters of the vision-based measurement system, and the results are shown in Table 4.

Table 3. Key parameters of the vision-based measurement system

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Table 4. Structural parameters of the optimized vision system

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5.2 System calibration and accuracy verification

Before the measurement, a high-accuracy checkerboard is used to calibrate the vision-based measurement system. Using the precise distances (with an accuracy of 1.5 µm) between markers on a dot calibration target as the reference, the calibration accuracy of the vision system is assessed by comparing the difference between the reconstructed dot-dot distances and the standard ones. A total of 10 images of the dot calibration target are taken in the experiment, and a total of 490 distances are reconstructed. It can be seen from Fig. 15 that the maximum reconstructed distance error is 5 µm, the mean is 2.5 µm, and the root mean square value is 1.2 µm, indicating that the vision system has sufficient accuracy to detect the geometric parameters of the internal thread.

Fig. 15. Reconstruction errors of the distances between the markers by the vision system.

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5.3 Measurement experiment of internal thread geometric parameters

In oil and gas field, 3^1/2 in. and 2^7/8 in. female pipes are the most widely used. Hence, the pitches, tooth heights, and tapers of two types of internal threads within one inch are inspected. Figure 16 shows the image of the laser profiles in the longitudinal cross-sections. It can be seen from the image that the optimized structural parameters ensure the measurement accessibility of the internal thread. Besides, the laser profiles are clear, sharp, and continuous, and the high imaging quality allows for the complete reproduction of the tooth shapes. For the acquired image, first, the ROI (regions of interest) of laser profiles are obtained by region segmentation, and then the images are binarized and morphologically dilated in turn. Subsequently, the pure laser profile images are obtained by the area filtering. Thereafter, the sub-pixel laser profiles are extracted using the Steger algorithm [29]. Thereafter, the laser profiles in the left and right view are matched based on the epipolar constraint [30], and the 3D profiles are reconstructed after the stereo-matching. According to API specifications: ① Pitch is defined as the distance between corresponding tooth points per inch (ten teeth) measured parallel to the axis. ② Tooth height is defined as the distance between the crest envelope and the root envelope perpendicular to the axis of the internal thread. ③ The taper is defined as half of the angle between the meridians of the left and right internal threads on the longitudinal cross-section. The dimensional tolerance of inner thread parameters is shown in Table 5. As shown in Fig. 16, using the extracted 3D laser profiles, the tooth crest and the tooth root of the internal thread, we can determine the root envelope, the crest envelope, and the middle meridian of the internal thread by linear regression. Finally, the three parameters of the internal thread can be measured according to the above regulations.

Table 5. Dimensional tolerance of inner thread parameters

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Fig. 16. Sub-pixel extraction steps for laser profiles. (a) Original image. (b) Image denoising. (c) Sub-pixel edge extraction. (d) Locally enlarged image. (e) Four reconstructed 3D internal thread profiles (five teeth). (f) Four 3D internal thread profiles (ten teeth).

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Fig. 17. Reference system. (a) Coordinate Measurement Machine (CMM) (taper). (b) Profilometer (height and pitch).

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Additionally, under the same conditions, the pitches, tooth heights (1 in.), and tapers of the internal threads corresponding to the two cross-sections are measured with traditional system using a pitch gauge, a tooth height gauge, and a taper gauge, respectively. The measurement results are shown in the fifth column of Table 6. Furthermore, the reference system, the CMM NC 8107 (Leader Metrology Inc, Maryland, USA) and profilometer SP2030 (Meider, Wuxi, China), is used to measure the three parameters of the internal threads. The measurement results of the three methods show that the geometric parameters on the two longitudinal cross-sections agree well. The real thread sizes measured with the three systems are within the tolerance ranges, indicating that all the pipes are qualified. Furthermore, based on the measurement results of the reference systems shown in Fig. 17, the measurement accuracy of vision system is compared with that of traditional system. The results show that the average measurement errors for pitch, taper, and tooth height of the two types of internal threads with the proposed vision system are 0.0051 mm, 0.6055 mm/m, and 0.0071 mm, which are all less than 1/3 of the measured errors. While, the measurement errors of the three parameters with the traditional system are 0.0359 mm, 3.5460 mm/m and 0.0382 mm, which are 5 times more than that of the vision system. Besides, the time taken by the vision system, the traditional system and the reference system to measure the three parameters of two types of internal threads are 0.5 s, 60 s, 100 s, respectively. The above results verify the high accuracy and high efficiency of the proposed vision system.

Table 6. Measurement results of thread geometric parameters

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6. Conclusion

To solve the problems of accessibility, accuracy and efficiency of the existing methods for measuring internal thread geometric parameters, we present a single-lens multi-mirror laser stereo vision system. The major contributions of this work are the following. (1) A complete explicit 3D OPM of the MSV considering both the image sensor size and the focal length is proposed in this paper, solving the limitations of the incomplete representation of imaging parameters caused by the low dimension and incomplete consideration of variables of the existing OPMs. (2) The calculation model for the imaging parameters of the MSV with the reflecting prism is proposed, which provides a theoretical basis for the system design and ensuring high measurement accessibility and accuracy. (3) According to the optimized structural parameters, a vision system is built for measuring the internal thread geometric parameters of two types of oilfield pipes, the comparison results verify the high accuracy and high efficiency of the proposed system. The vision system proposed in this paper allows the measurement of three parameters in one setup with high accuracy and high efficiency. The disadvantage is that when more cross-section data are obtained by increasing the number of reflecting surfaces of the prism, the design and manufacture of the prism will be difficult. Future research will focus on improving the optical elements and projection patterns, and studying the corresponding optical path analysis and camera calibration methods, so as to ensure the accessibility and accuracy of vision measurement of complex profiles in limited space.

Funding

National Natural Science Foundation of China (52005513); Fundamental Research Funds for the Central Universities (22CX01003A-6, 22CX06019A).

Acknowledgments

The authors would like to acknowledge funding support from the National Natural Science Foundation of China and the Fundamental Research Funds for the Central Universities.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Indicators	Values
Laser diameter	12 mm
Laser divergence angle	30°
Laser pattern	Cross-line
Lens aperture	F16
Lens mount	C-mount
Frame rate	20 fps
Camera mount	USB

Specifications (inch)	Taper (mm/m)	Pitch (mm/in)	Tooth height (mm)
2-7/8	${62.5}_{- 2.6}^{+ 5.2}$	$25.4 \pm 0.076$	${1.412}_{- 0.102}^{+ 0.051}$
3-1/2	${62.5}_{- 2.6}^{+ 5.2}$	$25.4 \pm 0.076$	${1.412}_{- 0.102}^{+ 0.051}$

Specifications (in.)	Thread parameters	Cross-section	Measurement results of reference system	Measurement results of traditional system	Measurement errors of traditional system	Measurement results of vision system	Measurement errors of vision system
3^1/2	Pitch (mm)	Cross-section 1	25.371	25.339	0.0317	25.367	0.0044
	Pitch (mm)	Cross-section 2	25.394	25.360	0.0343	25.388	0.0057
	Taper (mm/m)	Cross-section 1	61.186	64.757	3.5713	61.742	0.5561
	Taper (mm/m)	Cross-section 2	61.774	65.312	3.5381	62.532	0.7582
	Height of ten teeth (mm)	Cross-section 1	1.397	1.438	0.0406	1.404	0.0068
			1.407	1.439	0.0322	1.418	0.0114
			1.411	1.454	0.0430	1.421	0.0103
			1.387	1.348	0.0388	1.393	0.0056
			1.402	1.352	0.0496	1.414	0.0115
			1.415	1.446	0.0311	1.420	0.0046
			1.396	1.347	0.0495	1.404	0.0056
			1.403	1.438	0.0348	1.418	0.0086
			1.387	1.434	0.0473	1.421	0.0059
			1.374	1.408	0.0342	1.393	0.0105
		Cross-section 2	1.418	1.465	0.0469	1.407	0.0112
			1.401	1.358	0.0427	1.394	0.0070
			1.386	1.347	0.0392	1.375	0.0114
			1.421	1.462	0.0412	1.426	0.0047
			1.394	1.439	0.0451	1.404	0.0096
			1.372	1.402	0.0302	1.366	0.0058
			1.410	1.365	0.0453	1.404	0.0064
			1.387	1.340	0.0471	1.376	0.0106
			1.423	1.470	0.0474	1.432	0.0088
			1.412	1.448	0.0359	1.417	0.0051
2^7/8	Pitch (mm)	Cross-section 1	25.416	25.379	0.0374	25.411	0.0050
	Pitch (mm)	Cross-section 2	25.424	25.384	0.0401	25.419	0.0054
	Taper (mm/m)	Cross-section 1	62.205	66.067	3.8622	61.695	0.5105
	Taper (mm/m)	Cross-section 2	62.473	65.685	3.2121	61.876	0.5973
	Height of ten teeth (mm)	Cross-section 1	1.380	1.409	0.0284	1.391	0.0109
			1.365	1.399	0.0340	1.368	0.0031
			1.407	1.454	0.0466	1.415	0.0074
			1.379	1.341	0.0379	1.385	0.0060
			1.403	1.376	0.0272	1.399	0.0044
			1.386	1.341	0.0451	1.383	0.0030
			1.399	1.352	0.0465	1.392	0.0067
			1.387	1.420	0.0333	1.379	0.0083
			1.363	1.408	0.0453	1.367	0.0041
			1.414	1.390	0.0239	1.420	0.0058
		Cross-section 2	1.392	1.361	0.0313	1.384	0.0084
			1.417	1.387	0.0297	1.413	0.0036
			1.402	1.358	0.0442	1.410	0.0074
			1.394	1.428	0.0342	1.402	0.0079
			1.385	1.422	0.0367	1.392	0.0071
			1.397	1.351	0.0459	1.400	0.0035
			1.404	1.376	0.0286	1.398	0.0063
			1.413	1.452	0.0384	1.407	0.0067
			1.385	1.414	0.0283	1.377	0.0081
			1.425	1.404	0.0217	1.430	0.0043


Indicators	Values
Laser diameter	12 mm
Laser divergence angle	30°
Laser pattern	Cross-line
Lens aperture	F16
Lens mount	C-mount
Frame rate	20 fps
Camera mount	USB

Specifications (inch)	Taper (mm/m)	Pitch (mm/in)	Tooth height (mm)
2-7/8	${62.5}_{- 2.6}^{+ 5.2}$	$25.4 \pm 0.076$	${1.412}_{- 0.102}^{+ 0.051}$
3-1/2	${62.5}_{- 2.6}^{+ 5.2}$	$25.4 \pm 0.076$	${1.412}_{- 0.102}^{+ 0.051}$

Single-lens multi-mirror laser stereo vision-based system for measuring internal thread geometrical parameters

Abstract

1. Introduction

2. Measurement system and method

2.1 Measurement system

2.2 Measurement principle

3. Explicit 3D OPM without the reflecting prism

3.1 FoV parameters of MSV without the reflecting prism

3.2 Effective sizes of inner and outer mirrors

3.3 Simulation analysis of 3D optical path of MSV without the reflecting prism

4. OPM of MSV with the reflecting prism

4.1 Suitable size of the reflecting prism

4.2 Imaging parameters of MSV with the reflecting prism

4.3 Simulation analysis of 3D optical path of MSV with the reflecting prism

5. Accuracy verification and experiments

5.1 Experimental system

5.2 System calibration and accuracy verification

5.3 Measurement experiment of internal thread geometric parameters

6. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (17)

Tables (6)

Equations (15)

Optics Express

Structural parameters	Imaging parameters
α	ObD	Dof	Hd	Md	*Phi*	Ld	FovM	Ba	ExL	ExH	InH	InL	ExS	*InS*
β	*Dof*	Hd	*ObD*	Md	Ba	*FovM*	Ld	*Phi*	ExS	ExH	ExL	*InL*	*InS*	*InH*
d	ObD	InL	Dof	*InH*	ExL	*InS*	Hd	Ba	ExS	ExH	Md	FovM	Ld	*Phi*
L	Dof	Hd	Ba	ObD	Md	FovM	Ld	ExL	ExS	ExH	*InL*	*InS*	*InH*	*Phi*
S_x	Dof	ObD	Ld	FovM	Md	ExH	InH	ExS	InL	Hd	Ba	*InS*	*ExL*	*Phi*
S_y	Hd	Md	Ld	*ExL*	*ExS*	InL	InS	FovM	*Dof*	Ba	*InH*	*ExH*	*ObD*	*Phi*
f	Hd	Dof	ObD	Md	ExL	FovM	Ld	ExS	*InL*	InH	*ExH*	InS	Ba	*Phi*

Structural parameters	Imaging parameters
l	MirFovL1	MirFovL2	MirFovH2	MirFovH1	MirVL2	MirVL1
h	MirVL2	MirFovL1	MirVL1	MirFovL2	MirFovH1	MirFovH2
γ	MirFovL1	MirVL2	MirVL1	MirFovL2	MirFovH1	MirFovH2
θ	MirVL2	MirFovL1	MirVL1	MirFovH1	MirFovL2	MirFovH2
δ	MirFovH2	MirFovH1	MirFovL1	MirFovL2	MirVL1	MirVL2
R	MirFovL2	MirVL2	MirFovH2	MirFovH1	MirVL1	MirVL2


Indicators	Values
α	45°
β	51°
d	8 mm
L	18 mm
S_x	3.6 mm
S_y	3.6 mm
l	20 mm
h	150 mm
γ	45°
θ	45°