Design of an optical linear-discriminant filter: optimization for enhancement of filter transmittance and discrimination accuracy

Jun-Ichiro Sugisaka; Shingo Shimada; Koichi Hirayama; Takashi Yasui

doi:10.1364/JOSAA.506713

Journal of the Optical Society of America A
Vol. 41,
Issue 1,
pp. 139-146
(2024)
•https://doi.org/10.1364/JOSAA.506713

Design of an optical linear-discriminant filter: optimization for enhancement of filter transmittance and discrimination accuracy

Jun-Ichiro Sugisaka, Shingo Shimada, Koichi Hirayama, and Takashi Yasui

Open Access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Jun-Ichiro Sugisaka, Shingo Shimada, Koichi Hirayama, and Takashi Yasui, "Design of an optical linear-discriminant filter: optimization for enhancement of filter transmittance and discrimination accuracy," J. Opt. Soc. Am. A 41, 139-146 (2024)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article
Editors' Pick

Check for updates

Related Topics
Table of Contents Category
- Computational Sensing and Imaging
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: September 22, 2023
- Revised Manuscript: November 30, 2023
- Manuscript Accepted: December 1, 2023
- Published: December 20, 2023

Abstract

To discriminate fine concave and convex defects on a dielectric substrate, an optical machine learning system is proposed. This system comprises an optical linear-discriminant filter (OLDF) that performs linear discriminant analysis (LDA) of the scattered-wave distribution from target samples. However, the filter output from the OLDF is considerably weak and cannot be measured experimentally. Therefore, an algorithm is also proposed to improve the discrimination accuracy and filter transmittance. The designed filter is validated using a rigorous optical simulator based on vector diffraction theory. We also analyze and discuss a mechanism that provides high transmittance with high discrimination accuracy.

Full Article | PDF Article

More Like This

Design of an optical linear discriminant filter for classification of subwavelength concave and convex defects on dielectric substrates

Jun-ichiro Sugisaka, Takashi Yasui, and Koichi Hirayama
J. Opt. Soc. Am. A 39(3) 342-351 (2022)

Efficient surface defect identification for optical components via multi-scale mixed Kernels and structural re-parameterization

Xiao Liang, Hancen Zhen, Xuewei Wang, Jie Li, Yanjun Han, and Jingbo Guo
J. Opt. Soc. Am. A 40(6) 1107-1115 (2023)

Automated discrimination between digs and dust particles on optical surfaces with dark-field scattering microscopy

Lu Li, Dong Liu, Pin Cao, Shibin Xie, Yang Li, Yangjie Chen, and Yongying Yang
Appl. Opt. 53(23) 5131-5140 (2014)

Data availability

The 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.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Fig. 1. Optical system of the defect discrimination system.

${\rm S}$

, sample with defect;

${{\rm L}_1}$

, objective lens (focal length of

${{\rm F}_1}$

);

${{\rm L}_2}$

, imaging lens (focal length of

${{\rm F}_2}$

); P, output plane on which the observation point exists.

Download Full Size | PDF

Fig. 2. Geometric interpretation of the filter vector

${\boldsymbol v} \in {\mathbb{R}^2}$

. The direction of

${\boldsymbol v}$

represents the weight vector (transmittance distribution of the OLDF) and the norm represents the threshold value. The dashed lines

${\rm D}$

and

${{\rm D}^\prime}$

are decision boundaries.

Download Full Size | PDF

Fig. 3. Cross section of (a) concave and (b) convex defects.

Download Full Size | PDF

Fig. 4. (a)

${f_1}$

, (b)

${f_2}$

, and (c) discrimination error for each iteration step for

$\beta = 0$

, 0.05, and 0.5.

Download Full Size | PDF

Fig. 5. Discrimination error of the validation dataset.

Download Full Size | PDF

Fig. 6. Irradiance distribution of the concave (

$(d,h) = (0.5350\lambda , - 0.5656\lambda)$

) and convex defects (

$(d,h) = (0.5110\lambda ,0.4674\lambda)$

) for the OLDF with (a)

$\beta = 0$

, (b) 0.05, and (c) 0.5.

Download Full Size | PDF

Fig. 7. Irradiance distribution of the concave (

$(d,h) = (0.5350\lambda , - 0.5656\lambda)$

) and convex defects (

$(d,h) = (0.5110\lambda ,0.4674\lambda)$

) for the OLDF based on Fisher’s LDA.

Download Full Size | PDF

Fig. 8. Relationship between the defect height and irradiance at (a)

$\xi _{\rm o}^\prime $

and (b)

$\xi _{\rm o}^\prime + 20\lambda$

Download Full Size | PDF

Fig. 9. Irradiance distribution of the concave (

$(d,h) = (0.5350\lambda , - 0.5656\lambda)$

) and convex defects (

$(d,h) = (0.5110\lambda ,0.4674\lambda)$

) for the SDF.

Download Full Size | PDF

Fig. 10. Simple dataset of

${\boldsymbol x} \in {\mathbb{R}^2}$

and ideal decision boundary (dashed line) determined by Fisher’s LDA. The vector

${\boldsymbol v}$

is the filter vector corresponding to the ideal decision boundary.

Download Full Size | PDF

Fig. 11. Filter vectors (blue arrows) and decision boundaries (dashed line) for (a)

$\beta = 0$

and (b)

$\beta = 0.5$

. A white circle with a number

$m$

represents

${{\boldsymbol v}_m}$

Download Full Size | PDF

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

C \exp (j k \frac{ξ}{F_{2}} ξ^{'}),

\begin{aligned} f_{n} (ξ^{'}) & ≃ x_{n} \int_{c_{n} - h_{n} / 2}^{c_{n} + h_{n} / 2} C \exp (j k \frac{ξ}{F_{2}} ξ^{'}) d ξ \\ = C x_{n} \frac{\sin (\frac{k h_{n}}{2 F_{2}} ξ^{'})}{\frac{k h_{n}}{2 F_{2}} ξ^{'}} h_{n} \exp (j k \frac{c_{n}}{F_{2}} ξ^{'}), \end{aligned}

h_{n} = \frac{Δ ξ}{π} \arcsin (| w_{n} |) (n = 1, 2, \dots, N),

c_{n} = Δ ξ [(n - \frac{N}{2}) - \frac{\arg (w_{n})}{2 π}] (n = 1, 2, \dots, N),

y (ξ^{'}) = \sum_{n = 1}^{N} f_{n} (ξ^{'}) .

y_{o} = \sum_{n = 1}^{N} f_{n} (\frac{2 π F_{2}}{k Δ ξ}) = \sum_{n = 1}^{N} w_{n}^{*} x_{n} .

x = [x_{1}, x_{2}, \dots, x_{N}]^{T},

w = [w_{1}, w_{2}, \dots, w_{N}]^{T},

w = \frac{v}{| v |},

w_{0} = | v |^{2} .

f_{1} (v) = \sum_{n = 1}^{N_{A}} g_{A n} (v) + \sum_{n = 1}^{N_{B}} g_{B n} (v),

g_{A n} (v) = \log {1 + \exp [a (\frac{| v^{T} x_{A n} |^{2}}{| v |^{2}} - | v |^{2})]},

g_{B n} (v) = \log {1 + \exp [- a (\frac{| v^{T} x_{B n} |^{2}}{| v |^{2}} - | v |^{2})]},

f_{2} (v) = 1 - \frac{{| v^{H} {\bar{x}}_{B}^{(s)} |}^{2}}{| {\bar{x}}_{B}^{(s)} |^{2} | v |^{2}} .

v_{m + 1, i} = v_{m, i} - \frac{α}{\sqrt{h_{m + 1, i}}} g_{i},

h_{m + 1, i} = h_{m, i} + {| g_{i} |}^{2},

g_{i} = (1 - β) \frac{e}{e + f_{2}} \frac{\partial f_{1}}{\partial v_{i}} + β \frac{f_{2}}{e + f_{2}} \frac{\partial f_{2}}{\partial v_{i}} .

e = \frac{N {x_{A n} | {| w^{H} x_{A n} |}^{2} < w_{0}} + N {x_{B n} | {| w^{H} x_{B n} |}^{2} > w_{0}}}{N {x_{A n}} + N {x_{B n}}},

\frac{\partial f_{1} (v)}{\partial v} = \sum_{n = 1}^{N_{A}} \frac{\partial y_{A n} (v)}{\partial v} + \sum_{n = 1}^{N_{B}} \frac{\partial y_{B n} (v)}{\partial v},

\begin{aligned} \frac{\partial y_{A n} (v)}{\partial v} & = - 2 a \frac{1}{1 + \exp [- a (\frac{| v^{T} x_{A n} |^{2}}{| v |^{2}} - | v |^{2})]} \\ \times (v + \frac{| v^{T} x_{A n} |^{2}}{| v |^{4}} v - \frac{{(v^{T} x_{A n})}^{*}}{| v |^{2}} x_{A n}), \end{aligned}

\begin{aligned} \frac{\partial y_{B n} (v)}{\partial v} & = 2 a \frac{1}{1 + \exp [a (\frac{| v^{T} x_{B n} |^{2}}{| v |^{2}} - | v |^{2})]} \\ \times (v + \frac{| v^{T} x_{B n} |^{2}}{| v |^{4}} v - \frac{{(v^{T} x_{B n})}^{*}}{| v |^{2}} x_{B n}), \end{aligned}

\frac{\partial f_{2} (v)}{\partial v} = - 2 \frac{{(v^{T} {\bar{x}}_{B})}^{*}}{| {\bar{x}}_{B} |^{2} | v |^{2}} {\bar{x}}_{B} + 2 \frac{| v^{T} {\bar{x}}_{B} |^{2}}{| {\bar{x}}_{B} |^{2} | v |^{4}} v .

v_{0} = \sqrt{\frac{{\bar{x}}_{B}^{(s) H}}{| {\bar{x}}_{B}^{(s)} |} \frac{{\bar{x}}_{A} + {\bar{x}}_{B}}{2}} \frac{{\bar{x}}_{B}^{(s)}}{| {\bar{x}}_{B}^{(s)} |} .

Abstract

Data availability

Cited By

Figures (11)

Equations (23)

Journal of the Optical Society of America A