Abstract

The exact measurement of positions is of fundamental importance in a multitude of image-sensor based optical measurement systems. We propose a new method for enhancing the accuracy of image-sensor based optical measurement systems by using a computer-generated hologram in front of the imaging system. Thereby, the measurement spot is replicated to a predefined pattern. Given enough light to correctly expose the sensor, the position detection accuracy can be considerably improved compared to the conventional one-spot approach. For the evaluation of the spot position we used center-of-gravity based averaging. We present simulated as well as experimental results showing an improvement by a factor of 3.6 to a positioning accuracy of better than three thousandths of a pixel for a standard industrial CCD sensor.

© 2014 Optical Society of America

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    [CrossRef]
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2013 (1)

S. Wang, B. Yan, M. Dong, J. Wang, and P. Sun, “An improved centroid location algorithm for infrared LED feature points,” Proc. SPIE 8916, 891619 (2013).
[CrossRef]

2012 (2)

J. Yu, S. R. Kulkarni, and H. V. Poor, “Robust ellipse and spheroid fitting,” Pattern Recognition Letters 33, 492– 499 (2012).
[CrossRef]

S. Ma, J. Pang, and Q. Ma, “The systematic error in digital image correlation induced by self-heating of a digital camera,” Meas. Sci. Technol. 23, 025403 (2012).
[CrossRef]

2011 (1)

M. V. Konnik and J. S. Welsh, “On numerical simulation of high-speed CCD/CMOS-based wavefront sensors in adaptive optics,” Proc. SPIE 8149, 81490F (2011).
[CrossRef]

2010 (2)

L. Bo, D. Mingli, J. Wang, and Y. Bixi, “Sub-pixel location of center of target based on Zernike moment,” Proc. SPIE 7544, 75443A (2010).
[CrossRef]

Z. Xiao, J. Liang, D. Yu, Z. Tang, and A. Asundi, “An accurate stereo vision system using cross-shaped target self-calibration method based on photogrammetry,” Optics and Lasers in Engineering 48, 1252– 1261 (2010).
[CrossRef]

2009 (2)

Z. Jiandong, Z. Liyan, and D. Xiaoyu, “Accurate 3D target positioning in close range photogrammetry with implicit image correction,” Chinese Journal of Aeronautics 22, 649–657 (2009).
[CrossRef]

C. Li, M. Xia, Z. Liu, D. Li, and L. Xuan, “Optimization for high precision Shack-Hartmann wavefront sensor,” Opt. Commun. 282, 4333–4338 (2009).
[CrossRef]

2007 (1)

2004 (2)

K. Takita, M. Muquit, T. Aoki, and T. Higuchi, “A sub-pixel correspondence search technique for computer vision applications,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E87-A, 1913–1923 (2004).

S. Thomas, “Optimized centroid computing in a Shack-Hartmann sensor,” Proc. SPIE 5490, pp. 1238–1246 (2004).
[CrossRef]

2003 (3)

G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135– 147 (2003).
[CrossRef]

L. Seifert, J. Liesener, and H. Tiziani, “The adaptive Shack Hartmann sensor,” Opt. Commun. 216, 313– 319 (2003).
[CrossRef]

L. A. Poyneer, “Scene-based Shack-Hartmann wave-front sensing: Analysis and simulation,” Appl. Opt. 42, 5807–5815 (2003).
[CrossRef] [PubMed]

2002 (3)

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

W.-Y. Leung, M. Tallon, and R. Lane, “Centroid estimation by model-fitting from undersampled wavefront sensing images,” Opt. Commun. 201, 11– 20 (2002).
[CrossRef]

J. Arines and J. Ares, “Minimum variance centroid thresholding,” Opt. Lett. 27, 497–499 (2002).
[CrossRef]

2000 (1)

T. R. Rimmele, “Solar adaptive optics,” Proc. SPIE 4007, 218–231 (2000).
[CrossRef]

1998 (1)

1995 (2)

J. Trinder, J. Jansa, and Y. Huang, “An assessment of the precision and accuracy of methods of digital target location,” Journal of Photogrammetry and Remote Sensing 50, 12–20 (1995).
[CrossRef]

M. R. Shortis and T. A. Clarke, “Practical testing of the precision and accuracy of target image centring algorithms,” Proc. SPIE 2598, 65–76 (1995).
[CrossRef]

1994 (2)

T. A. Clarke, “Analysis of the properties of targets used in digital close-range photogrammetric measurement,” Proc. SPIE 2350, 251–262 (1994).
[CrossRef]

M. R. Shortis, T. A. Clarke, and T. Short, “Comparison of some techniques for the subpixel location of discrete target images,” Proc. SPIE 2350, 239–250 (1994).
[CrossRef]

1984 (1)

F. Ackermann, “Digital image correlation: Performance and potential application in photogrammetry,” The Photogrammetric Record 11, 429–439 (1984).
[CrossRef]

Accardo, D.

G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135– 147 (2003).
[CrossRef]

Ackermann, F.

F. Ackermann, “Digital image correlation: Performance and potential application in photogrammetry,” The Photogrammetric Record 11, 429–439 (1984).
[CrossRef]

Aoki, T.

K. Takita, M. Muquit, T. Aoki, and T. Higuchi, “A sub-pixel correspondence search technique for computer vision applications,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E87-A, 1913–1923 (2004).

Ares, J.

Arines, J.

Asundi, A.

Z. Xiao, J. Liang, D. Yu, Z. Tang, and A. Asundi, “An accurate stereo vision system using cross-shaped target self-calibration method based on photogrammetry,” Optics and Lasers in Engineering 48, 1252– 1261 (2010).
[CrossRef]

Baker, K. L.

Bixi, Y.

L. Bo, D. Mingli, J. Wang, and Y. Bixi, “Sub-pixel location of center of target based on Zernike moment,” Proc. SPIE 7544, 75443A (2010).
[CrossRef]

Bo, L.

L. Bo, D. Mingli, J. Wang, and Y. Bixi, “Sub-pixel location of center of target based on Zernike moment,” Proc. SPIE 7544, 75443A (2010).
[CrossRef]

Bradski, D. G. R.

D. G. R. Bradski and A. Kaehler, Learning Opencv (O’Reilly, 2008).

Clarke, T. A.

M. R. Shortis and T. A. Clarke, “Practical testing of the precision and accuracy of target image centring algorithms,” Proc. SPIE 2598, 65–76 (1995).
[CrossRef]

T. A. Clarke, “Analysis of the properties of targets used in digital close-range photogrammetric measurement,” Proc. SPIE 2350, 251–262 (1994).
[CrossRef]

M. R. Shortis, T. A. Clarke, and T. Short, “Comparison of some techniques for the subpixel location of discrete target images,” Proc. SPIE 2350, 239–250 (1994).
[CrossRef]

Copland, J.

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Dong, M.

S. Wang, B. Yan, M. Dong, J. Wang, and P. Sun, “An improved centroid location algorithm for infrared LED feature points,” Proc. SPIE 8916, 891619 (2013).
[CrossRef]

Fiete, R.

R. Fiete, Modeling the Imaging Chain of Digital Cameras, Tutorial Text Series (SPIE Press, 2010).

Gross, H.

H. Gross, Handbook of Optical Systems: Fundamentals of Technical Optics (Wiley-VCH, 2005).
[CrossRef]

He, M.

T. Li, M. He, N. Lei, C. Li, and Q. Wang, “TDI CCD non-uniformity correction algorithm,” in “4th IEEE Conference on Industrial Electronics and Applications, ICIEA,” (2009), pp. 1483–1487.

Higuchi, T.

K. Takita, M. Muquit, T. Aoki, and T. Higuchi, “A sub-pixel correspondence search technique for computer vision applications,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E87-A, 1913–1923 (2004).

Huang, Y.

J. Trinder, J. Jansa, and Y. Huang, “An assessment of the precision and accuracy of methods of digital target location,” Journal of Photogrammetry and Remote Sensing 50, 12–20 (1995).
[CrossRef]

Jähne, B.

B. Jähne, Practical Handbook on Image Processing for Scientific Applications (CRC Press, 1997).

Jansa, J.

J. Trinder, J. Jansa, and Y. Huang, “An assessment of the precision and accuracy of methods of digital target location,” Journal of Photogrammetry and Remote Sensing 50, 12–20 (1995).
[CrossRef]

Jiandong, Z.

Z. Jiandong, Z. Liyan, and D. Xiaoyu, “Accurate 3D target positioning in close range photogrammetry with implicit image correction,” Chinese Journal of Aeronautics 22, 649–657 (2009).
[CrossRef]

Kaehler, A.

D. G. R. Bradski and A. Kaehler, Learning Opencv (O’Reilly, 2008).

Konnik, M. V.

M. V. Konnik and J. S. Welsh, “On numerical simulation of high-speed CCD/CMOS-based wavefront sensors in adaptive optics,” Proc. SPIE 8149, 81490F (2011).
[CrossRef]

Kulkarni, S. R.

J. Yu, S. R. Kulkarni, and H. V. Poor, “Robust ellipse and spheroid fitting,” Pattern Recognition Letters 33, 492– 499 (2012).
[CrossRef]

Lane, R.

W.-Y. Leung, M. Tallon, and R. Lane, “Centroid estimation by model-fitting from undersampled wavefront sensing images,” Opt. Commun. 201, 11– 20 (2002).
[CrossRef]

Lei, N.

T. Li, M. He, N. Lei, C. Li, and Q. Wang, “TDI CCD non-uniformity correction algorithm,” in “4th IEEE Conference on Industrial Electronics and Applications, ICIEA,” (2009), pp. 1483–1487.

Leung, W.-Y.

W.-Y. Leung, M. Tallon, and R. Lane, “Centroid estimation by model-fitting from undersampled wavefront sensing images,” Opt. Commun. 201, 11– 20 (2002).
[CrossRef]

Li, C.

C. Li, M. Xia, Z. Liu, D. Li, and L. Xuan, “Optimization for high precision Shack-Hartmann wavefront sensor,” Opt. Commun. 282, 4333–4338 (2009).
[CrossRef]

T. Li, M. He, N. Lei, C. Li, and Q. Wang, “TDI CCD non-uniformity correction algorithm,” in “4th IEEE Conference on Industrial Electronics and Applications, ICIEA,” (2009), pp. 1483–1487.

Li, D.

C. Li, M. Xia, Z. Liu, D. Li, and L. Xuan, “Optimization for high precision Shack-Hartmann wavefront sensor,” Opt. Commun. 282, 4333–4338 (2009).
[CrossRef]

Li, T.

T. Li, M. He, N. Lei, C. Li, and Q. Wang, “TDI CCD non-uniformity correction algorithm,” in “4th IEEE Conference on Industrial Electronics and Applications, ICIEA,” (2009), pp. 1483–1487.

Liang, J.

Z. Xiao, J. Liang, D. Yu, Z. Tang, and A. Asundi, “An accurate stereo vision system using cross-shaped target self-calibration method based on photogrammetry,” Optics and Lasers in Engineering 48, 1252– 1261 (2010).
[CrossRef]

Liesener, J.

L. Seifert, J. Liesener, and H. Tiziani, “The adaptive Shack Hartmann sensor,” Opt. Commun. 216, 313– 319 (2003).
[CrossRef]

Lindlein, N.

Liu, Z.

C. Li, M. Xia, Z. Liu, D. Li, and L. Xuan, “Optimization for high precision Shack-Hartmann wavefront sensor,” Opt. Commun. 282, 4333–4338 (2009).
[CrossRef]

Liyan, Z.

Z. Jiandong, Z. Liyan, and D. Xiaoyu, “Accurate 3D target positioning in close range photogrammetry with implicit image correction,” Chinese Journal of Aeronautics 22, 649–657 (2009).
[CrossRef]

Ma, Q.

S. Ma, J. Pang, and Q. Ma, “The systematic error in digital image correlation induced by self-heating of a digital camera,” Meas. Sci. Technol. 23, 025403 (2012).
[CrossRef]

Ma, S.

S. Ma, J. Pang, and Q. Ma, “The systematic error in digital image correlation induced by self-heating of a digital camera,” Meas. Sci. Technol. 23, 025403 (2012).
[CrossRef]

Mingli, D.

L. Bo, D. Mingli, J. Wang, and Y. Bixi, “Sub-pixel location of center of target based on Zernike moment,” Proc. SPIE 7544, 75443A (2010).
[CrossRef]

Moallem, M. M.

Muquit, M.

K. Takita, M. Muquit, T. Aoki, and T. Higuchi, “A sub-pixel correspondence search technique for computer vision applications,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E87-A, 1913–1923 (2004).

Neal, D. A.

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Neal, D. R.

D. R. Neal, J. Copland, and D. A. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002).
[CrossRef]

Pang, J.

S. Ma, J. Pang, and Q. Ma, “The systematic error in digital image correlation induced by self-heating of a digital camera,” Meas. Sci. Technol. 23, 025403 (2012).
[CrossRef]

Pfund, J.

Poor, H. V.

J. Yu, S. R. Kulkarni, and H. V. Poor, “Robust ellipse and spheroid fitting,” Pattern Recognition Letters 33, 492– 499 (2012).
[CrossRef]

Poyneer, L. A.

Prasad, B.

A. Vyas, M. Roopashree, and B. Prasad, “Performance of centroiding algorithms at low light level conditions in adaptive optics,” 2009 International Conference on Advances in Recent Technologies in Communication and Computing pp. 366–369 (2009).

Rimmele, T. R.

T. R. Rimmele, “Solar adaptive optics,” Proc. SPIE 4007, 218–231 (2000).
[CrossRef]

Roopashree, M.

A. Vyas, M. Roopashree, and B. Prasad, “Performance of centroiding algorithms at low light level conditions in adaptive optics,” 2009 International Conference on Advances in Recent Technologies in Communication and Computing pp. 366–369 (2009).

Rufino, G.

G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135– 147 (2003).
[CrossRef]

Saleh, B.

B. Saleh and M. Teich, Fundamentals of Photonics, Wiley Series in Pure and Applied Optics (Wiley, 2007).

Schwider, J.

Seifert, L.

L. Seifert, J. Liesener, and H. Tiziani, “The adaptive Shack Hartmann sensor,” Opt. Commun. 216, 313– 319 (2003).
[CrossRef]

Short, T.

M. R. Shortis, T. A. Clarke, and T. Short, “Comparison of some techniques for the subpixel location of discrete target images,” Proc. SPIE 2350, 239–250 (1994).
[CrossRef]

Shortis, M. R.

M. R. Shortis and T. A. Clarke, “Practical testing of the precision and accuracy of target image centring algorithms,” Proc. SPIE 2598, 65–76 (1995).
[CrossRef]

M. R. Shortis, T. A. Clarke, and T. Short, “Comparison of some techniques for the subpixel location of discrete target images,” Proc. SPIE 2350, 239–250 (1994).
[CrossRef]

Sun, P.

S. Wang, B. Yan, M. Dong, J. Wang, and P. Sun, “An improved centroid location algorithm for infrared LED feature points,” Proc. SPIE 8916, 891619 (2013).
[CrossRef]

Takita, K.

K. Takita, M. Muquit, T. Aoki, and T. Higuchi, “A sub-pixel correspondence search technique for computer vision applications,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E87-A, 1913–1923 (2004).

Tallon, M.

W.-Y. Leung, M. Tallon, and R. Lane, “Centroid estimation by model-fitting from undersampled wavefront sensing images,” Opt. Commun. 201, 11– 20 (2002).
[CrossRef]

Tang, Z.

Z. Xiao, J. Liang, D. Yu, Z. Tang, and A. Asundi, “An accurate stereo vision system using cross-shaped target self-calibration method based on photogrammetry,” Optics and Lasers in Engineering 48, 1252– 1261 (2010).
[CrossRef]

Teich, M.

B. Saleh and M. Teich, Fundamentals of Photonics, Wiley Series in Pure and Applied Optics (Wiley, 2007).

Thomas, S.

S. Thomas, “Optimized centroid computing in a Shack-Hartmann sensor,” Proc. SPIE 5490, pp. 1238–1246 (2004).
[CrossRef]

Tiziani, H.

L. Seifert, J. Liesener, and H. Tiziani, “The adaptive Shack Hartmann sensor,” Opt. Commun. 216, 313– 319 (2003).
[CrossRef]

Trinder, J.

J. Trinder, J. Jansa, and Y. Huang, “An assessment of the precision and accuracy of methods of digital target location,” Journal of Photogrammetry and Remote Sensing 50, 12–20 (1995).
[CrossRef]

Vyas, A.

A. Vyas, M. Roopashree, and B. Prasad, “Performance of centroiding algorithms at low light level conditions in adaptive optics,” 2009 International Conference on Advances in Recent Technologies in Communication and Computing pp. 366–369 (2009).

Wang, J.

S. Wang, B. Yan, M. Dong, J. Wang, and P. Sun, “An improved centroid location algorithm for infrared LED feature points,” Proc. SPIE 8916, 891619 (2013).
[CrossRef]

L. Bo, D. Mingli, J. Wang, and Y. Bixi, “Sub-pixel location of center of target based on Zernike moment,” Proc. SPIE 7544, 75443A (2010).
[CrossRef]

Wang, Q.

T. Li, M. He, N. Lei, C. Li, and Q. Wang, “TDI CCD non-uniformity correction algorithm,” in “4th IEEE Conference on Industrial Electronics and Applications, ICIEA,” (2009), pp. 1483–1487.

Wang, S.

S. Wang, B. Yan, M. Dong, J. Wang, and P. Sun, “An improved centroid location algorithm for infrared LED feature points,” Proc. SPIE 8916, 891619 (2013).
[CrossRef]

Welsh, J. S.

M. V. Konnik and J. S. Welsh, “On numerical simulation of high-speed CCD/CMOS-based wavefront sensors in adaptive optics,” Proc. SPIE 8149, 81490F (2011).
[CrossRef]

Xia, M.

C. Li, M. Xia, Z. Liu, D. Li, and L. Xuan, “Optimization for high precision Shack-Hartmann wavefront sensor,” Opt. Commun. 282, 4333–4338 (2009).
[CrossRef]

Xiao, Z.

Z. Xiao, J. Liang, D. Yu, Z. Tang, and A. Asundi, “An accurate stereo vision system using cross-shaped target self-calibration method based on photogrammetry,” Optics and Lasers in Engineering 48, 1252– 1261 (2010).
[CrossRef]

Xiaoyu, D.

Z. Jiandong, Z. Liyan, and D. Xiaoyu, “Accurate 3D target positioning in close range photogrammetry with implicit image correction,” Chinese Journal of Aeronautics 22, 649–657 (2009).
[CrossRef]

Xuan, L.

C. Li, M. Xia, Z. Liu, D. Li, and L. Xuan, “Optimization for high precision Shack-Hartmann wavefront sensor,” Opt. Commun. 282, 4333–4338 (2009).
[CrossRef]

Yan, B.

S. Wang, B. Yan, M. Dong, J. Wang, and P. Sun, “An improved centroid location algorithm for infrared LED feature points,” Proc. SPIE 8916, 891619 (2013).
[CrossRef]

Yu, D.

Z. Xiao, J. Liang, D. Yu, Z. Tang, and A. Asundi, “An accurate stereo vision system using cross-shaped target self-calibration method based on photogrammetry,” Optics and Lasers in Engineering 48, 1252– 1261 (2010).
[CrossRef]

Yu, J.

J. Yu, S. R. Kulkarni, and H. V. Poor, “Robust ellipse and spheroid fitting,” Pattern Recognition Letters 33, 492– 499 (2012).
[CrossRef]

Acta Astronautica (1)

G. Rufino and D. Accardo, “Enhancement of the centroiding algorithm for star tracker measure refinement,” Acta Astronautica 53, 135– 147 (2003).
[CrossRef]

Appl. Opt. (2)

Chinese Journal of Aeronautics (1)

Z. Jiandong, Z. Liyan, and D. Xiaoyu, “Accurate 3D target positioning in close range photogrammetry with implicit image correction,” Chinese Journal of Aeronautics 22, 649–657 (2009).
[CrossRef]

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences (1)

K. Takita, M. Muquit, T. Aoki, and T. Higuchi, “A sub-pixel correspondence search technique for computer vision applications,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E87-A, 1913–1923 (2004).

Journal of Photogrammetry and Remote Sensing (1)

J. Trinder, J. Jansa, and Y. Huang, “An assessment of the precision and accuracy of methods of digital target location,” Journal of Photogrammetry and Remote Sensing 50, 12–20 (1995).
[CrossRef]

Meas. Sci. Technol. (1)

S. Ma, J. Pang, and Q. Ma, “The systematic error in digital image correlation induced by self-heating of a digital camera,” Meas. Sci. Technol. 23, 025403 (2012).
[CrossRef]

Opt. Commun. (3)

L. Seifert, J. Liesener, and H. Tiziani, “The adaptive Shack Hartmann sensor,” Opt. Commun. 216, 313– 319 (2003).
[CrossRef]

W.-Y. Leung, M. Tallon, and R. Lane, “Centroid estimation by model-fitting from undersampled wavefront sensing images,” Opt. Commun. 201, 11– 20 (2002).
[CrossRef]

C. Li, M. Xia, Z. Liu, D. Li, and L. Xuan, “Optimization for high precision Shack-Hartmann wavefront sensor,” Opt. Commun. 282, 4333–4338 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optics and Lasers in Engineering (1)

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Figures (8)

Fig. 1
Fig. 1

Typical imaging geometries for point measurements.

Fig. 2
Fig. 2

By use of a hologram, each image point is replicated to N points.

Fig. 3
Fig. 3

Simulated results for the position error, defined as the root mean squared deviation from the perfect position for a linear movement, using M number of spots. All errors are mean errors for 20 individual simulations and each simulation consists of the RMS errors for 40 different positions. The errorbars denote the standard deviation of the mean values. The theoretical curve shows the expected behavior proportional to 1 / M with a start value (1 point) of 0.018.

Fig. 4
Fig. 4

Setup for measurement of the subpixel shift centroid. A tilted parallel plate is used in transmission and in reflection to obtain very accurate shifts in the image sensor plane.

Fig. 5
Fig. 5

Typical experimental spot image. Left hand side: without processing, Right hand side: linear renormalization (oversaturation) to enhance the Airy structure.

Fig. 6
Fig. 6

Comparison of measured centroid positions for one spot and the average of 14 spots in the case for one single image. S1, S2, and S3 show the measured positions for one of the individual spots. The thick lines shows the average position based on 14 spots and the very thin straight line shows the linear regression.

Fig. 7
Fig. 7

Comparison of measured centroid positions for one spot and the average of 14 spots. All centroids have been computed based on a time-averaged image (30 recordings). S1, S2, and S3 show the measured positions for one of the individual spots. The thick lines shows the average position based on 14 spots and the very thin straight line shows the linear regression.

Fig. 8
Fig. 8

Repeatability measurement for one fixed position and one short exposure. S1, S2, and S3 show the measured positions for one of the individual spots. The thick lines shows the average position based on 14 spots.

Equations (10)

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σ = 1 L i = 1 L ( x i x c ) 2 ,
σ = σ A N ,
I ( x , y ) = I ( x , y ) T for I ( x , y ) > T
I ( x , y ) = 0 for I ( x , y ) T
δ x = d sin ( ϕ ) [ 1 cos ( ϕ ) n 2 sin 2 ( ϕ ) ]
tan ( ϕ ) = x s l 0
δ x = d x s [ 1 ( n 2 + n 2 x s 2 l 0 2 x s 2 l 0 2 ( 1 + x s 2 l 0 2 ) x s 4 l 0 4 ( 1 + x s 2 l 0 2 ) ) 1 / 2 ] l 0 1 + x s 2 / l 0 2
Δ x = Δ z f δ x ,
Δ x p p = d f x s ( 1 ( n 2 + n 2 x s 2 l 0 2 x s 2 l 0 2 + x s 2 x s 4 l 0 2 ( l 0 2 + x s 2 ) ) 1 / 2 ) ( a f ) l 0 p p 1 + x s 2 l 0 2
d f ( 1 1 / n ) ( a f ) l 0 p p x s

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