Automated optical inspection for the runout tolerance of circular saw blades

Abstract

Circular saw blades are fundamental cutting tools applied to cut off materials. Inspection of finished products of circular saw blades is important in order to ensure their manufacturing quality and sawing performance. Traditionally, a contact inspection method is adopted to measure the runout amounts of circular saw blades. In order to improve the quality of the runout inspection, a non-contact inspection method based on machine vision is required. In this paper, an automated optical inspection (AOI) system was developed exclusively for inspecting the runout tolerance of circular saw blades. Based on the integration of motion control and image processing techniques, calibration and automated inspection processes for the developed AOI system were then established. Experiments to inspect circular saw blade samples were also conducted in order to test the feasibility and reliability of the developed AOI system. From the experimental results, the developed AOI system, in combination with the automated inspection process, could achieve sufficient repeatability and was verified to be able to inspect the runout tolerance of certain circular saw blades.

Appendix

A general form of the applied edge detection method is described below. For a grayscale digital image consisting of an a×b array of pixels, its grayscale intensity function, G, can be represented by:

$$ G = G\left( {x,y} \right)\,{\text{for}}\,\,0 \leqslant x \leqslant \left( {a - {1}} \right){\text{ and }}0 \leqslant y \leqslant \left( {b - {1}} \right) $$

(A1)

The gradient of the grayscale intensity function is defined by:

$$ \nabla G = \left\{ {\matrix{ {{G_x}(x,{ }y)} \\ {{G_y}(x,{ }y)} \\ }<!end array> } \right\} = \left\{ {\matrix{ {\partial G(x,{ }y)/\partial x} \\ {\partial G(x,{ }y)/\partial y} \\ }<!end array> } \right\} $$

(A2)

By using the one-dimensional grayscale profile scanning [28, 29, 34] with y = b _i being assigned, an edge coordinate (x _e, b _i ) can be detected by finding the location where the locally extreme gradient along the x-direction exists and whose magnitude is larger than an assigned threshold value, t _e, that is:

$$ {G_x}\left( {{x_{\text{e}}},{ }{b_i}} \right) = \max \left( {\left| {{G_x}\left( {x,{ }{b_i}} \right)} \right|} \right) > {t_{\text{e}}}\,{\text{for}}\,{a_1} \leqslant x \leqslant {a_2},0 \leqslant {a_1},{a_2} \leqslant \left( {a - {1}} \right)\,\,{\text{and}}\,\,{t_{\text{e}}} > 0 $$

(A3)

which means that the edge coordinate (x _e, b _i ) exists when G _x (x _e, b _i ) is the extreme gradient in an assigned domain of a ₁ ≤ x ≤ a ₂ and whose magnitude is larger than t _e. Likewise, when x = a _i is assigned, an edge coordinate (a _i , y _e) can be detected by:

$$ {G_y}\left( {{a_i},{ }{y_{\text{e}}}} \right) = \max \left( {\left| {{G_y}({a_i},\,y)} \right|} \right) > {t_{\text{e}}}\,{\text{for}}\,{b_1} \leqslant y \leqslant {b_2},\,0 \leqslant {b_1},{b_2} \leqslant \left( {b - {1}} \right){\text{and}}\,\,{t_{\text{e}}} > 0 $$

(A4)

which means that the edge coordinate (a _i , y _e) exists when G _y (a _i , y _e) is the extreme gradient in an assigned domain of b ₁ ≤ y ≤ b ₂ and whose magnitude is larger than t _e. The detected edge coordinates (x _e, b _i ) and (a _i , y _e) can merely achieve full-pixel accuracy. For each case, a threshold grayscale value, g _ex or g _ey, corresponding to the detected edge can be obtained by:

$$ {g_{\text{ex}}} = {g_{\text{ex}}}\left( {{b_i}} \right) = G\left( {{x_{\text{e}}},{ }{b_i}} \right) $$

(A5)

or

$$ {g_{\text{ey}}} = {g_{\text{ey}}}\left( {{a_i}} \right) = G\left( {{a_i},{ }{y_{\text{e}}}} \right) $$

(A6)

Considering that continuous edges in any assigned domain have been detected, an averaged threshold grayscale value, g _ex(avg) or g _ey(avg), can be defined by:

$$ {{{{g_{{\text{ex(avg)}}}} = \int {{g_{\text{ex}}}dy} }} \left/ {{\int {dy} }} \right.} $$

(A7)

or

$$ {g_{{\text{ey(avg)}}}}{{{ = \int {{g_{\text{ey}}}dx} }} \left/ {{\int {dx} }} \right.} $$

(A8)

The averaged threshold grayscale value, g _ex(avg) or g _ey(avg), can represent a balanced threshold for re-detecting edges in any assigned domain when considering the effects of noise interference and non-uniform illumination on pixel intensity variation. In addition, by using the so-called curve fitting methods [35–37], a fitted one-dimensional grayscale profile for the neighborhood of (x _e, b _i ), Ĝ, can be obtained by:

$$ \widehat{G} = \widehat{G}\left( {x,\,{b_i}} \right) \,{\text{for}}\,\left( {{x_{\text{e}}} - {\text{D}}x} \right) \leqslant x \leqslant \left( {{x_{\text{e}}} + {\text{D}}x} \right) $$

(A9)

in which Δx is an assigned small pixel amount for controlling the range of the neighborhood. Then, a refined edge coordinate $(x_{\text{e}}^*,{ }{b_i})$ can be obtained by solving the following equation:

$$ \widehat{G}\left( {x_{\text{e}}^*,{ }{b_i}} \right) - {g_{{\text{ex(avg)}}}} = 0 $$

(A10)

When y = b _i is assigned, the obtained location of $x_{\text{e}}^*$ can achieve sub-pixel accuracy. Similarly, a fitted one-dimensional grayscale profile for the neighborhood of (a _i , y _e ), Ĝ, can be obtained by:

$$ \widehat{G} = \widehat{G}\left( {{a_i},{ }y} \right)\,{\text{for}}\,\left( {{y_{\text{e}}} - {\text{D}}y} \right) \leqslant y \leqslant \left( {{y_{\text{e}}} + {\text{D}}y} \right) $$

(A11)

in which Δy is an assigned small pixel amount for controlling the range of the neighborhood. Then, a refined edge coordinate $({a_i},y_{\text{e}}^*)$ can be obtained by solving the following equation:

$$ \widehat{G}\left( {{a_i},y_{\text{e}}^*} \right) - {g_{{\text{ey(avg)}}}} = 0 $$

(A12)

When x = a _i is assigned, the obtained location of $y_{\text{e}}^*$ can also achieve sub-pixel accuracy.