Abstract

Scratches on the surface of optical components have serious impacts on optical system such as imaging quality of lens and/or mirrors in optical imaging systems, light-collecting abilities of laser fusion and solar concentrator systems. The size of the scratches is a key issue for analyzing and assessing the impacts quantitatively. Most of the available testing methods for scratches depend on human visual inspection (HVI) with naked eyes by workers, which leads to low efficiency and accuracy. This paper presents an automatic detecting method for the scratches on optical surface with machine vision inspection (MVI) method. The microscopic dark-field scattering imaging system is used as the front end of the detection system. A dedicated algorithm is designed for non-closing scratch detection. The core merits of this algorithm lies in three folds: 1) automatic processing capabilities, which includes positioning, clustering, and precise estimation of the length of the scratches; 2) high efficiency, which is characterized by a short time interval, i.e., about 0.138 second per binary image with 2724 × 2724 pixels in our experiments; 3) high accuracy, where the error rate of the total length of the scratches detected is less than 5% when compared with the nominal visual measurement result obtained via HVI method. The proposed scratch detecting algorithm can be used for non-destructive testing (NDT) of the glass-like surfaces.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2016 (1)

X. Tao, D. Xu, Z. T. Zhang, F. Zhang, X. L. Liu, and D. P. Zhang, “Weak scratch detection and defect classification methods for a large-aperture optical element,” Opt. Commun. 387, 390–400 (2016).
[Crossref]

2015 (3)

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

J.-W. Dong, Q.-J. Wu, W.-W. Jiang, and Q. Xu, “The review of new NDT methods of metal material fatigue monitoring,” Int. J. Hybrid Inf. Technol. 8, 225–232 (2015).
[Crossref]

X. Tao, Z.-T. Zhang, F. Zhang, and D. Xu, “A novel and effective surface flaw inspection instrument for large-aperture optical elements,” IEEE Trans. Instrum. Meas. 64, 2530–2540 (2015).
[Crossref]

2014 (2)

G.-H. Hu, G.-H. Zhang, and Q.-H. Wang, “Automated defect detection in textured materials using wavelet-domain hidden markov models,” Opt. Eng. 53, 093107 (2014).
[Crossref]

L. Li, D. Liu, P. Cao, S. Xie, Y. Li, Y. Chen, and Y. Yang, “Automated discrimination between digs and dust particles on optical surfaces with dark-field scattering microscopy,” Appl. Opt. 53, 5131–5140 (2014).
[Crossref] [PubMed]

2013 (1)

2011 (2)

C. Akinlar and C. Topal, “EDLines: A real-time line segment detector with a false detection control,” Pattern Recogn. Lett. 32, 1633–1642 (2011).
[Crossref]

H.-H. Jiang and G.-F. Yin, “Surface defect inspection and classification of segment magnet by using machine vision technique,” Adv. Mat. Res. 339, 32–35 (2011).

2010 (1)

R. G. von Gioi, J. Jakubowicz, J.-M. Morel, and G. Randall, “LSD: a fast line segment detector with a false detection control,” IEEE Trans. PAMI 32, 722–732 (2010).
[Crossref]

2007 (1)

D. Liu, Y.-Y. Yang, L. Wang, Y.-M. Zhuo, C.-H. Lu, L.-M. Yang, and R.-J. Li, “Microscopic scattering imaging measurement and digital evaluation system of defects for fine optical surface,” Opt. Commun. 278, 240–246 (2007).
[Crossref]

2000 (1)

1998 (1)

1997 (1)

1995 (1)

N. Guil, J. Villalba, and E. L. Zapata, “A fast Hough transform for segment detection,” IEEE Trans. IP 4, 1541–1548 (1995).

1991 (1)

N. Kiryati, Y. Eldar, and A. M. Bruckstein, “A probabilistic Hough transform,” Pattern Recogn. 24, 303–316 (1991).
[Crossref]

1987 (1)

J. Illingworth and J. Kittler, “The adaptive Hough transform,” IEEE Trans. PAMI 9, 690–698 (1987).
[Crossref]

1981 (1)

M. A. Fischer and R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24, 381–395 (1981).
[Crossref]

Akinlar, C.

C. Akinlar and C. Topal, “EDLines: A real-time line segment detector with a false detection control,” Pattern Recogn. Lett. 32, 1633–1642 (2011).
[Crossref]

Bai, J.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

Bernabeu, E.

Blois, M. S.

E. H. Shortliffe and M. S. Blois, “The computer meets medicine and biology: emergence of a discipline,” Biomed. Infor. pp. 3–45 (2006).
[Crossref]

Bolles, R. C.

M. A. Fischer and R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24, 381–395 (1981).
[Crossref]

Bourgeade, A.

P. Cormont, A. Bourgeade, S. Cavaro, T. Doualle, G. Gaborit, L. Gallais, J.-L. Rullier, and D. Taroux, “Process for repairing large scratches on fused silica optics,” in Proc. SPIE, vol. 9633 (2015), p. 9633A.

Bradski, G.

A. Kaehler and G. Bradski, Learning OpenCV 3: Computer Vision in C++ with the OpenCV Library (O’Reilly Media, 2017).

Bruckstein, A. M.

N. Kiryati, Y. Eldar, and A. M. Bruckstein, “A probabilistic Hough transform,” Pattern Recogn. 24, 303–316 (1991).
[Crossref]

Cao, P.

Cavaro, S.

P. Cormont, A. Bourgeade, S. Cavaro, T. Doualle, G. Gaborit, L. Gallais, J.-L. Rullier, and D. Taroux, “Process for repairing large scratches on fused silica optics,” in Proc. SPIE, vol. 9633 (2015), p. 9633A.

Chen, Y.

Cheng, Z.

Cormont, P.

P. Cormont, A. Bourgeade, S. Cavaro, T. Doualle, G. Gaborit, L. Gallais, J.-L. Rullier, and D. Taroux, “Process for repairing large scratches on fused silica optics,” in Proc. SPIE, vol. 9633 (2015), p. 9633A.

Dong, J.-W.

J.-W. Dong, Q.-J. Wu, W.-W. Jiang, and Q. Xu, “The review of new NDT methods of metal material fatigue monitoring,” Int. J. Hybrid Inf. Technol. 8, 225–232 (2015).
[Crossref]

Doualle, T.

P. Cormont, A. Bourgeade, S. Cavaro, T. Doualle, G. Gaborit, L. Gallais, J.-L. Rullier, and D. Taroux, “Process for repairing large scratches on fused silica optics,” in Proc. SPIE, vol. 9633 (2015), p. 9633A.

Eldar, Y.

N. Kiryati, Y. Eldar, and A. M. Bruckstein, “A probabilistic Hough transform,” Pattern Recogn. 24, 303–316 (1991).
[Crossref]

Fischer, M. A.

M. A. Fischer and R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24, 381–395 (1981).
[Crossref]

Gaborit, G.

P. Cormont, A. Bourgeade, S. Cavaro, T. Doualle, G. Gaborit, L. Gallais, J.-L. Rullier, and D. Taroux, “Process for repairing large scratches on fused silica optics,” in Proc. SPIE, vol. 9633 (2015), p. 9633A.

Gahagan, K. T.

V. M. Schneider, M. Meljnek, and K. T. Gahagan, “Fast detection of single sided diffracted defects in display glass,” (OSA, 2009), pp. 638–644.

Gallais, L.

P. Cormont, A. Bourgeade, S. Cavaro, T. Doualle, G. Gaborit, L. Gallais, J.-L. Rullier, and D. Taroux, “Process for repairing large scratches on fused silica optics,” in Proc. SPIE, vol. 9633 (2015), p. 9633A.

Gao, X.

George, P.

S. Radovan, P. George, M. Panagiotis, and G. Manos, “An approach for automated inspection of wood boards,” in Proc. 2001 ICIP (Cat. No.01CH37205), vol. 1 (2001), pp. 798–801.

Germer, T. A.

Gu, Z.-H.

Guil, N.

N. Guil, J. Villalba, and E. L. Zapata, “A fast Hough transform for segment detection,” IEEE Trans. IP 4, 1541–1548 (1995).

Hu, G.-H.

G.-H. Hu, G.-H. Zhang, and Q.-H. Wang, “Automated defect detection in textured materials using wavelet-domain hidden markov models,” Opt. Eng. 53, 093107 (2014).
[Crossref]

Illingworth, J.

J. Illingworth and J. Kittler, “The adaptive Hough transform,” IEEE Trans. PAMI 9, 690–698 (1987).
[Crossref]

Jakubowicz, J.

R. G. von Gioi, J. Jakubowicz, J.-M. Morel, and G. Randall, “LSD: a fast line segment detector with a false detection control,” IEEE Trans. PAMI 32, 722–732 (2010).
[Crossref]

Jiang, H.-H.

H.-H. Jiang and G.-F. Yin, “Surface defect inspection and classification of segment magnet by using machine vision technique,” Adv. Mat. Res. 339, 32–35 (2011).

Jiang, H.-Z.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

Jiang, W.-W.

J.-W. Dong, Q.-J. Wu, W.-W. Jiang, and Q. Xu, “The review of new NDT methods of metal material fatigue monitoring,” Int. J. Hybrid Inf. Technol. 8, 225–232 (2015).
[Crossref]

Kaehler, A.

A. Kaehler and G. Bradski, Learning OpenCV 3: Computer Vision in C++ with the OpenCV Library (O’Reilly Media, 2017).

Kiryati, N.

N. Kiryati, Y. Eldar, and A. M. Bruckstein, “A probabilistic Hough transform,” Pattern Recogn. 24, 303–316 (1991).
[Crossref]

Kittler, J.

J. Illingworth and J. Kittler, “The adaptive Hough transform,” IEEE Trans. PAMI 9, 690–698 (1987).
[Crossref]

Li, C.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

Li, L.

Li, R.-J.

D. Liu, Y.-Y. Yang, L. Wang, Y.-M. Zhuo, C.-H. Lu, L.-M. Yang, and R.-J. Li, “Microscopic scattering imaging measurement and digital evaluation system of defects for fine optical surface,” Opt. Commun. 278, 240–246 (2007).
[Crossref]

Li, Y.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

L. Li, D. Liu, P. Cao, S. Xie, Y. Li, Y. Chen, and Y. Yang, “Automated discrimination between digs and dust particles on optical surfaces with dark-field scattering microscopy,” Appl. Opt. 53, 5131–5140 (2014).
[Crossref] [PubMed]

Liu, D.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

L. Li, D. Liu, P. Cao, S. Xie, Y. Li, Y. Chen, and Y. Yang, “Automated discrimination between digs and dust particles on optical surfaces with dark-field scattering microscopy,” Appl. Opt. 53, 5131–5140 (2014).
[Crossref] [PubMed]

D. Liu, S. Wang, P. Cao, L. Li, Z. Cheng, X. Gao, and Y. Yang, “Dark-field microscopic image stitching method for surface defects evaluation of large fine optics,” Opt. Express 21, 5974–5987 (2013).
[Crossref] [PubMed]

D. Liu, Y.-Y. Yang, L. Wang, Y.-M. Zhuo, C.-H. Lu, L.-M. Yang, and R.-J. Li, “Microscopic scattering imaging measurement and digital evaluation system of defects for fine optical surface,” Opt. Commun. 278, 240–246 (2007).
[Crossref]

Liu, X.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

Liu, X. L.

X. Tao, D. Xu, Z. T. Zhang, F. Zhang, X. L. Liu, and D. P. Zhang, “Weak scratch detection and defect classification methods for a large-aperture optical element,” Opt. Commun. 387, 390–400 (2016).
[Crossref]

Lu, C.-H.

D. Liu, Y.-Y. Yang, L. Wang, Y.-M. Zhuo, C.-H. Lu, L.-M. Yang, and R.-J. Li, “Microscopic scattering imaging measurement and digital evaluation system of defects for fine optical surface,” Opt. Commun. 278, 240–246 (2007).
[Crossref]

Manos, G.

S. Radovan, P. George, M. Panagiotis, and G. Manos, “An approach for automated inspection of wood boards,” in Proc. 2001 ICIP (Cat. No.01CH37205), vol. 1 (2001), pp. 798–801.

Mariño, P.

P. Mariño, V. Pastoriza, and M. Santamaría, “Machine vision method for online surface inspection of easy open can ends,” in Proc. SPIE, vol. 6382 (2006), pp. 638206.
[Crossref]

Meljnek, M.

V. M. Schneider, M. Meljnek, and K. T. Gahagan, “Fast detection of single sided diffracted defects in display glass,” (OSA, 2009), pp. 638–644.

Morel, J.-M.

R. G. von Gioi, J. Jakubowicz, J.-M. Morel, and G. Randall, “LSD: a fast line segment detector with a false detection control,” IEEE Trans. PAMI 32, 722–732 (2010).
[Crossref]

Panagiotis, M.

S. Radovan, P. George, M. Panagiotis, and G. Manos, “An approach for automated inspection of wood boards,” in Proc. 2001 ICIP (Cat. No.01CH37205), vol. 1 (2001), pp. 798–801.

Pastoriza, V.

P. Mariño, V. Pastoriza, and M. Santamaría, “Machine vision method for online surface inspection of easy open can ends,” in Proc. SPIE, vol. 6382 (2006), pp. 638206.
[Crossref]

Radovan, S.

S. Radovan, P. George, M. Panagiotis, and G. Manos, “An approach for automated inspection of wood boards,” in Proc. 2001 ICIP (Cat. No.01CH37205), vol. 1 (2001), pp. 798–801.

Randall, G.

R. G. von Gioi, J. Jakubowicz, J.-M. Morel, and G. Randall, “LSD: a fast line segment detector with a false detection control,” IEEE Trans. PAMI 32, 722–732 (2010).
[Crossref]

Rebollo, M. A.

Rullier, J.-L.

P. Cormont, A. Bourgeade, S. Cavaro, T. Doualle, G. Gaborit, L. Gallais, J.-L. Rullier, and D. Taroux, “Process for repairing large scratches on fused silica optics,” in Proc. SPIE, vol. 9633 (2015), p. 9633A.

Sanchez-Brea, L. M.

Santamaría, M.

P. Mariño, V. Pastoriza, and M. Santamaría, “Machine vision method for online surface inspection of easy open can ends,” in Proc. SPIE, vol. 6382 (2006), pp. 638206.
[Crossref]

Schneider, V. M.

V. M. Schneider, M. Meljnek, and K. T. Gahagan, “Fast detection of single sided diffracted defects in display glass,” (OSA, 2009), pp. 638–644.

Shen, Y.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

Shortliffe, E. H.

E. H. Shortliffe and M. S. Blois, “The computer meets medicine and biology: emergence of a discipline,” Biomed. Infor. pp. 3–45 (2006).
[Crossref]

Siegmann, P.

Tao, X.

X. Tao, D. Xu, Z. T. Zhang, F. Zhang, X. L. Liu, and D. P. Zhang, “Weak scratch detection and defect classification methods for a large-aperture optical element,” Opt. Commun. 387, 390–400 (2016).
[Crossref]

X. Tao, Z.-T. Zhang, F. Zhang, and D. Xu, “A novel and effective surface flaw inspection instrument for large-aperture optical elements,” IEEE Trans. Instrum. Meas. 64, 2530–2540 (2015).
[Crossref]

L.-X. Yuan, Z.-T. Zhang, and X. Tao, “The development and prospect of surface defect detection based on vision measurement method,” in World Congress on Intelligent Control and Automation, (2016), pp. 1382–1387.

Taroux, D.

P. Cormont, A. Bourgeade, S. Cavaro, T. Doualle, G. Gaborit, L. Gallais, J.-L. Rullier, and D. Taroux, “Process for repairing large scratches on fused silica optics,” in Proc. SPIE, vol. 9633 (2015), p. 9633A.

Topal, C.

C. Akinlar and C. Topal, “EDLines: A real-time line segment detector with a false detection control,” Pattern Recogn. Lett. 32, 1633–1642 (2011).
[Crossref]

Villalba, J.

N. Guil, J. Villalba, and E. L. Zapata, “A fast Hough transform for segment detection,” IEEE Trans. IP 4, 1541–1548 (1995).

von Gioi, R. G.

R. G. von Gioi, J. Jakubowicz, J.-M. Morel, and G. Randall, “LSD: a fast line segment detector with a false detection control,” IEEE Trans. PAMI 32, 722–732 (2010).
[Crossref]

Wang, L.

D. Liu, Y.-Y. Yang, L. Wang, Y.-M. Zhuo, C.-H. Lu, L.-M. Yang, and R.-J. Li, “Microscopic scattering imaging measurement and digital evaluation system of defects for fine optical surface,” Opt. Commun. 278, 240–246 (2007).
[Crossref]

Wang, Q.-H.

G.-H. Hu, G.-H. Zhang, and Q.-H. Wang, “Automated defect detection in textured materials using wavelet-domain hidden markov models,” Opt. Eng. 53, 093107 (2014).
[Crossref]

Wang, S.

Wu, Q.-J.

J.-W. Dong, Q.-J. Wu, W.-W. Jiang, and Q. Xu, “The review of new NDT methods of metal material fatigue monitoring,” Int. J. Hybrid Inf. Technol. 8, 225–232 (2015).
[Crossref]

Xie, S.

Xie, S.-B.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

Xiong, H.-L.

C. Li, Y.-Y. Yang, H.-L. Xiong, D. Liu, S.-B. Xie, Y. Li, J. Bai, Y. Shen, H.-Z. Jiang, and X. Liu, “Dual-threshold algorithm study of weak-scratch extraction based on the filter and difference,” High Power Laser Part. Beams 27, 1–8 (2015). In Chinese.

Xu, D.

X. Tao, D. Xu, Z. T. Zhang, F. Zhang, X. L. Liu, and D. P. Zhang, “Weak scratch detection and defect classification methods for a large-aperture optical element,” Opt. Commun. 387, 390–400 (2016).
[Crossref]

X. Tao, Z.-T. Zhang, F. Zhang, and D. Xu, “A novel and effective surface flaw inspection instrument for large-aperture optical elements,” IEEE Trans. Instrum. Meas. 64, 2530–2540 (2015).
[Crossref]

Xu, Q.

J.-W. Dong, Q.-J. Wu, W.-W. Jiang, and Q. Xu, “The review of new NDT methods of metal material fatigue monitoring,” Int. J. Hybrid Inf. Technol. 8, 225–232 (2015).
[Crossref]

Yang, L.-M.

D. Liu, Y.-Y. Yang, L. Wang, Y.-M. Zhuo, C.-H. Lu, L.-M. Yang, and R.-J. Li, “Microscopic scattering imaging measurement and digital evaluation system of defects for fine optical surface,” Opt. Commun. 278, 240–246 (2007).
[Crossref]

Yang, Y.

Yang, Y.-Y.

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[Crossref]

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Zhang, Z.-T.

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D. Liu, Y.-Y. Yang, L. Wang, Y.-M. Zhuo, C.-H. Lu, L.-M. Yang, and R.-J. Li, “Microscopic scattering imaging measurement and digital evaluation system of defects for fine optical surface,” Opt. Commun. 278, 240–246 (2007).
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Adv. Mat. Res. (1)

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X. Tao, Z.-T. Zhang, F. Zhang, and D. Xu, “A novel and effective surface flaw inspection instrument for large-aperture optical elements,” IEEE Trans. Instrum. Meas. 64, 2530–2540 (2015).
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Figures (14)

Fig. 1
Fig. 1 Original images from MDSI system (enhanced).
Fig. 2
Fig. 2 Binary images (enhanced).
Fig. 3
Fig. 3 Two line segments with a included angle.
Fig. 4
Fig. 4 Flowchart for detecting scratches.
Fig. 5
Fig. 5 Classification of the line segments by the included angle.
Fig. 6
Fig. 6 Classification of the line segments by the DOP.
Fig. 7
Fig. 7 Verification of scratches.
Fig. 8
Fig. 8 LSD detection result in red color.
Fig. 9
Fig. 9 Collecting points of straight scratches.
Fig. 10
Fig. 10 Collecting points of curved scratches.
Fig. 11
Fig. 11 Seeking for start endpoint.
Fig. 12
Fig. 12 Connection of scratches.
Fig. 13
Fig. 13 Find real endpoints of scratches.
Fig. 14
Fig. 14 Dealing with large curvature scratches.

Tables (5)

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Table 1 Data Structures for Line Segment L = 〈ps, peand Scratch S = 〈; ps, pe

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Algorithm 1 Line Segment Classification, {ctg, K} = LineSegmentClassifier(, θmax, DOPmax, rmin).

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Algorithm 2 Proper Endpoints Positioning, {ps, pe} = EndpointSelector(��Sn, Ln, dmax, min).

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Algorithm 3 Scratch Detection for Binary Image, �� = ScratchDetector(b, θmax, DOPmax, rmin, dmax, min).

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Table 2 Comparison of Performance for Different Scratch Detection Algorithms

Equations (31)

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: Ω { 0 , , 255 } [ x , y ] ( x , y )
Ω = { [ x , y ] : 0 x < W , 0 y < H } ,
t : { 0 , 1 , 2 , , 254 , 255 } { 0 , 255 }
b = t : Ω { 0 , 255 } , p b ( p ) = { 0 , for ( p ) t ; 255 , for ( p ) < t .
L = p s , p e
= l = p e p s = ( x s x e ) 2 + ( y s y e ) 2 ,
l = p e p s .
S = ; p s , p e
p e p s = l
sin θ i j = det ( [ l i , l j ] ) l i l j
DOP i j = DOP ( L i , L j ) = DOP ( p s i , p e i , p s j , p e j ) = min { p e i p e j , p e i p s j , p s i p e j , p s i p s j } ,
𝒟 = b 1 ( 0 ) = { p Ω : b ( p ) = 0 } .
𝒟 = { points of the scratches } { noise points } = 𝒟 S 𝒟 N
E = | mvi hvi | hvi × 100 % .
= i = 1 n L i = i = 1 n { p i , p i : p i , p i b } .
R i j = k = 1 n t R i j k
R i j k R i j q = , k q .
δ ( R i j k ) = { 1 , p 𝒟 i j k ; 0 , otherwise .
n s = k = 1 n t δ ( R i j k ) .
r i j = r ( R i j ) = n s n t × 100 % .
𝒟 Σ = Σ 𝒟 .
𝒟 Σ = ( Σ 𝒟 S ) ( Σ 𝒟 N ) = 𝒟 Σ S 𝒟 Σ N
x = k , k 1 < x k , k .
𝒟 S n = k = 1 n r 𝒟 S n k .
𝒟 S = n = 1 K 𝒟 S n = n = 1 K k = 1 n r 𝒟 S n k .
Order y : 𝒟 S n × 𝒟 S n { True , False } ( p i , p j ) { True , if p i . y < p j . y ; False , otherwise .
Order x : 𝒟 S n × 𝒟 S n { True , False } ( p i , p j ) { True , if p i . x < p j . x ; False , otherwise .
dist ( p i , p i + 1 ) = p i + 1 p i .
{ DOP max = α min d max = β min
α 3 , β 2 .
θ max arcsin ( w min ) .

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