Theoretical and experimental guideline of optimum design of defect-inspection apparatus for transparent material using phase-shift illumination approach

Abstract

Machine vision system has great significance for the automatic inspection to enhance unclear defects. For the purpose of the improvement of the recognition accuracy in the automation inspection, in addition to development of image-processing technology, the optical technology is also required to emphasize the contrast of defects difficult to find in a camera. Here, we develop both the optical-engineering and image-processing technology with high throughput to enhance defects in transparent material, using the phase-shift illumination method with striped structured illumination. We succeeded in the enhancement of shape defects difficult to visualize in a bright-field observation with our approach to construct the composite image from a few pictures. Our theoretical model clarified the relationship of the composite image to parameters about optical configuration and edge sharpness of surface structures in a target material, and reproduced dependencies of the index in the experiment. Based on the theoretical model, we proposed the optimum design parameters of the equipment for enhancing the scratches in our transmissive phase-shift-illumination method, so that we can maximize the magnitude of enhancement due to the refraction of light rays in the defect shape.

Appendix

1.1 Derivation of Eq. 6 ${\varvec{\Gamma}}\left({\varvec{x}},{\varvec{y}}\right)$

In Eqs. 2 and 3, $S\left(x,y\right)$ and $C\left(x,y\right)$ are newly defined by the following equations:

$$S\left( {x,y} \right) = \frac{{I_{0} \left( {x,y} \right) - I_{\pi } \left( {x,y} \right)}}{{2I_{{{\text{Off}}.}} \left( {x,y} \right)}},$$

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and

$${ }C\left( {x,y} \right) = \frac{{I_{{\frac{\pi }{2}}} \left( {x,y} \right) - I_{{\frac{3\pi }{2}}} \left( {x,y} \right)}}{{2I_{{{\text{Off}}.}} \left( {x,y} \right)}},$$

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respectively. Here, $\Gamma \left(x,y\right)$ can be represented with $\Gamma \left(x,y\right)=\sqrt{S{\left(x,y\right)}^{2}+C{\left(x,y\right)}^{2}}=\left|C\left(x,y\right)+iS\left(x,y\right)\right|$ in complex form.

Substituting Eq. 4 for ${I}_{\uppsi }\left(x,y\right)$, and then Eq. 5 for ${B}_{\uppsi }(x, y;\uptheta ,\upphi )$ in $S\left(x,y\right)$, we obtain $S\left(x,y\right)$. From

$$\begin{aligned} I_{0} \left( {x,y} \right) - I_{\pi } \left( {x,y} \right) & = T\left( {x,y} \right)\mathop \smallint \limits_{{\Omega_{0} }} d\Omega \left( {B_{0} \left( {x^{\prime},y^{\prime};\theta ,\phi } \right) - B_{\pi } \left( {x^{\prime},y^{\prime};\theta ,\phi } \right)} \right)\cos \theta \\ & = T\left( {x,y} \right)B_{Amp.} \mathop \smallint \limits_{{\Omega_{0} }} d\Omega \sin \left( {\frac{2\pi }{L}y^{\prime } } \right)g_{0} \left( {\theta , \phi } \right)\cos \theta \\ \end{aligned}$$

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and

$$I_{{{\text{Off}}{.}}} \left( {x,y} \right) = T\left( {x,y} \right)\frac{{B_{{{\text{Amp}}{.}}} }}{2}\mathop \smallint \limits_{{\Omega_{0} }} d\Omega g_{0} \left( {\theta , \phi } \right)\cos \theta ,$$

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$S\left(x,y\right)$ is given as

$$S\left( {x,y} \right) = \frac{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega \sin \left( {\frac{2\pi }{L}y^{\prime } } \right)g_{0} \left( {\theta , \phi } \right)\cos \theta }}{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega g_{0} \left( {\theta , \phi } \right)\cos \theta }}.$$

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Similarly, $C\left(x,y\right)$ is obtained as

$$C\left( {x,y} \right) = \frac{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega \cos \left( {\frac{2\pi }{L}y^{\prime } } \right)g_{0} \left( {\theta , \phi } \right)\cos \theta }}{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega g_{0} \left( {\theta , \phi } \right)\cos \theta }}.$$

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Hence, $\Gamma \left(x,y\right)$ is given as

$$\begin{aligned} {\Gamma }\left( {x,y} \right) & = \left| {\frac{{\mathop \smallint \nolimits_{{{\Omega }_{0} }} d{\Omega }\left[ {\cos \left( {\frac{2\pi }{L}y^{\prime}} \right) + i\sin \left( {\frac{2\pi }{L}y^{\prime}} \right)} \right]g_{0} \left( {{\uptheta },{ }\phi } \right)\cos \theta }}{{\mathop \smallint \nolimits_{{{\Omega }_{0} }} d{\Omega }g_{0} \left( {{\uptheta },{ }\phi } \right)\cos \theta }}} \right| \\ & = \left| {\frac{{\mathop \smallint \nolimits_{{{\Omega }_{0} }} d{\Omega }\exp \left( {i\frac{2\pi }{L}y^{\prime}} \right)g_{0} \left( {{\uptheta },{ }\phi } \right)\cos \theta }}{{\mathop \smallint \nolimits_{{{\Omega }_{0} }} d{\Omega }g_{0} \left( {{\uptheta },{ }\phi } \right)\cos \theta }}} \right|. \\ \end{aligned}$$

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Substituting ${y}^{^{\prime}}$ for ${y}^{^{\prime}}=y-d\mathrm{tan}\theta \mathrm{sin}\phi$, $\Gamma \left(x,y\right)$ is given as

$$\begin{aligned} \Gamma \left( {x,y} \right) & = \left| {\frac{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega \exp \left( {i\frac{2\pi }{L}\left( {y - d\tan \theta \sin \phi } \right)} \right)g_{0} \left( {\theta , \phi } \right)\cos \theta }}{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega g_{0} \left( {\theta , \phi } \right)\cos \theta }}} \right| \\ & = \left| {\frac{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega \exp \left( { - i\frac{2\pi d}{L}\tan \theta \sin \phi } \right)g_{0} \left( {\theta , \phi } \right)\cos \theta }}{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega g_{0} \left( {\theta , \phi } \right)\cos \theta }}} \right|. \\ \end{aligned}$$

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By the approximation that the numerical aperture of the telecentric optical system is small ($\left|\theta \right|\ll 1$) and the angular distribution ${g}_{0}\left(\uptheta ,\upphi \right)$ of rays is sufficiently close to constant within the integration interval, $\Gamma \left(x,y\right)$ is given as

$$\Gamma \left( {x,y} \right) = \left| {\frac{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega \exp \left( { - i\frac{2\pi d}{L}\tan \theta \sin \phi } \right)\cos \theta }}{{\mathop \smallint \nolimits_{{\Omega_{0} }} d\Omega \cos \theta }}} \right|.$$

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