Fast path planning algorithm for large-aperture aspheric optical
elements based on minimum object depth and a self-optimized overlap
coefficient

Fanyi Wang; Yongying Yang; Weiming Lou

doi:10.1364/AO.450995

Applied Optics
Vol. 61,
Issue 11,
pp. 3123-3133
(2022)
•https://doi.org/10.1364/AO.450995

Fast path planning algorithm for large-aperture aspheric optical elements based on minimum object depth and a self-optimized overlap coefficient

Fanyi Wang, Yongying Yang, and Weiming Lou

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Fanyi Wang, Yongying Yang, and Weiming Lou, "Fast path planning algorithm for large-aperture aspheric optical elements based on minimum object depth and a self-optimized overlap coefficient," Appl. Opt. 61, 3123-3133 (2022)

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Abstract

In the procedure of surface defects detection of large-aperture aspheric optical elements, it is necessary to scan the surface of the element to achieve full coverage inspection. Since the curvature of the aspherical element is constantly changing from the center to the edge, it is of great difficulty to carry out efficient path planning. In addition, the machine vision system is a microscopic system with limited depth of field, and the sub-aperture imaging of aspherical elements has a visual depth along the object side. When the object depth is greater than the depth of field, out-of-focus blur will generate, so the object depth needs to be as small as possible. In response to these problems, this paper proposes a fast path planning algorithm based on the minimum object depth of a sub-aperture. To ensure minimum object depth, the machine vision system collects images along the normal direction of the sub-aperture plane. To address the problem of the surface curvatures of aspheric elements being different and the overlap coefficient difficult to determine, this paper proposes an image processing based overlap coefficient self-optimization algorithm. When scanning with full coverage of elements, there is only one connected domain in the horizontal projection image of all sub-apertures. According to this premise, the overlap coefficient is optimized through an image processing method to obtain a local optimal path planning strategy. According to the obtained path planning strategy, combining the component parameters and mechanical structure, the mapping matrix of the path planning algorithm transplanted to the detection system is calculated. Through computer programming, automatic sub-aperture acquisition is realized, and the self-edited sub-aperture stitching program is applied to reconstruct the collected sub-apertures. Our algorithm can complete path planning within 5 s, and the experimental results show that the maximum stitching misalignment error of the collected sub-apertures is no more than four pixels, and the average is one pixel. The reconstruction accuracy satisfies the needs of subsequent image processing and digital quantization.

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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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System Parameters			Component Parameters				Simulation Parameters
Field of View	Depth of Field	Swing Distance	Aperture	Thickness	Vertex Radius	Conic Constant	Overlap Coefficient
$2 r_{s}$	$Δ L$	$H$	$D_{a}$	$h^{'}$	$r$	$k$	$η, 0 < η < 1$

Symbol	Introduction
$S_{i}^{j}$	Sub-aperture of $i$ th longitude layer and $j$ th latitude layer
$a_{2 i - 1}, a_{2 i}$	Intersection point of $S_{i}^{1}$ and $y - z$ plane
$(y_{2 i - 1}, z_{2 i - 1}), (y_{2 i}, z_{2 i})$	Coordinates of $a_{2 i - 1}, a_{2 i}$
$c_{i}^{j}$	Center of $S_{i}^{j}$
$(x_{i}^{j}, y_{i}^{j}, z_{i}^{j})$	Center coordinate of $S_{i}^{j}$
$\vec{n_{i}^{j}}$	Normal vector of $S_{i}^{j}$
$l_{i}$	Projection line segment of $S_{i}^{1}$ on the $y - z$ plane
$K_{i}$	Slope of $l_{i}$
$l_{i}^{'}, (l_{i} ⊥ l_{i}^{'})$	Overlap factor auxiliary line
$K_{i}^{'}$	Slope of $l_{i}^{'}$
$P_{i}$	Point farthest from $S_{i}^{j}$ on the sub-aperture of the aspheric surface
$h_{i}$	Object depth of the $i$ th longitude layer
$N_{i}$	Number of sub-apertures of the $i$ th latitude layer
$O_{c}$	Center of the apex of the aspherical element (origin for simulation)
$O_{m}$	Mechanical swing center (origin for the mechanical system)
$β_{i}$	With $O_{m}$ as the swing center, the swing angle from $S_{1}^{1}$ to $S_{i}^{1}$
$α_{i}, α_{i} = 2 π / N_{i}$	With spin axis as the center, the spin step angle of the $i$ th latitude layer
$O_{i}$	Intersection point of $\vec{n_{i}^{j}}$ and the spin axis (refer to Fig. 7)
${\overset{\to j}{C}}_{i}$	Vector from $O_{m}$ to $c_{i}^{j}$ (refer to Fig. 7)

Name	Stroke Range	Absolute Positioning Accuracy	Repeat Positioning Accuracy
$X$	0–800µm	0.9 µm	0.3 µm
$Y$	0–250µm	0.3 µm	1.0 µm
$Z$	0–150µm	2.4 µm	1.7 µm
$α$	$- \infty \sim + \infty µ m$	6.012 $^{''}$	—
$β$	$- 30^{\circ} \sim {+ 30}^{\circ}$	1.296 $^{''}$	—

Model Number of Lens	$r$	$k$	$D_{a}$	$h^{'}$	High-Order Item	$η$	Time/s
#1 (ellipsoid)	60.145	−0.7	80	35.6000	0	0.28	2.4
#2 (paraboloid)	110.000	−1.0	60	20.0000	0	0.36	1.8
#3 (hyperboloid)	64.2945	−1.434	75	13.0000	$A_{4} = 4.02 E - 07$	0.36	2.2
#4 (hyperboloid)	123.655	−3.5	150	19.0689	0	0.28	4.6

Model Number of Lens	$h_{1}$	$h_{2}$	$h_{3}$	$h_{4}$	$h_{5}$	$h_{6}$	$h_{7}$	$h_{8}$
#1	0.5135	0.4880	0.4386	0.3810
#2	0.2804	0.2748	0.2634
#3	0.4784	0.4410	0.3778	0.3091
#4	0.2488	0.2342	0.2052	0.1705	0.1371	0.1087	0.0858	0.0680

Model Number	Latitude Layer	$N_{i}$	$(x_{i}^{1}, y_{i}^{1}, z_{i}^{1})$	$\vec{n_{i}^{1}}$
#3	1	$N_{1} = 1$	$(x_{1}^{1}, y_{1}^{1}, z_{1}^{1}) = (0 .0, 0 .0, 0 .478448)$	$\vec{n_{1}^{1}} = (0.0, 0.0, 1.0)$
	2	$N_{2} = 8$	$(x_{2}^{1}, y_{2}^{1}, z_{2}^{1}) = (0 .0, 12 .727847, 1 .703884)$	$\vec{n_{2}^{1}} = (0.0, 0.188619, 0.981382)$
	3	$N_{3} = 14$	$(x_{3}^{1}, y_{3}^{1}, z_{3}^{1}) = (0 .0, 22 .240645, 4 .196645)$	$\vec{n_{3}^{1}} = (0.0, 0.303767, 0.947811)$

Abstract

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Applied Optics