Abstract

We describe the operation of a pyramid wavefront sensor used to measure and correct aberrations of the human eye. The system is designed for maximum speed when running in closed loop but can also provide calibrated open-loop measurements of aberrations with a range of sampling options. A detailed characterization of the system was performed to ensure measurement accuracy. Ocular aberrations after correction had root-mean-square errors consistently less than 0.1μm over a 6mm pupil for all subjects tested. The system frame rate is 83Hz in both open- and closed-loop modes.

© 2010 Optical Society of America

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2009 (1)

2008 (2)

2006 (3)

2004 (1)

2003 (2)

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

2002 (5)

2001 (3)

2000 (1)

S. Esposito, O. Feeney, and A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” Proc. SPIE 4007, 416–422 (2000).
[CrossRef]

1999 (1)

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).

1998 (1)

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, “Laboratory characterization of a “Foucault-like” wavefront sensor for adaptive optics,” Proc. SPIE 3353, 941–951 (1998).
[CrossRef]

1997 (1)

1996 (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

1994 (2)

1982 (1)

J. D. Scargle, “Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data,” Astrophys. J. 263, 835–853 (1982).
[CrossRef]

1979 (1)

1859 (1)

L. M. Foucault, “Mémoire sur la construction des télescopes en verre arganté,” Ann. Obs. Imp. Paris 5, 197–237 (1859).

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series, V.Lakshminarayanan, ed. (Optical Society of America, 2000), pp. 233–244.

Aragón, J. L.

Arines, J.

P. Rodríguez, R. Navarro, J. Arines, and S. Bará, “A new calibration set of phase plates for ocular aberrometers,” J. Refractive Surg. 22, 275–284 (2006).

Artal, P.

Bará, S.

P. Rodríguez, R. Navarro, J. Arines, and S. Bará, “A new calibration set of phase plates for ocular aberrometers,” J. Refractive Surg. 22, 275–284 (2006).

Baruffolo, A.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Bille, J.

Bindi, N.

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, “Laboratory characterization of a “Foucault-like” wavefront sensor for adaptive optics,” Proc. SPIE 3353, 941–951 (1998).
[CrossRef]

Bradley, A.

L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

Bradley, C.

Burvall, A.

Campbell, M. C. W.

Carbillet, M.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

Cecconi, M.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Chamot, S. R.

Chen, L.

Cheng, X.

Coburn, D.

Costa, J. B.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

Crimi, G.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Cubalchini, R.

Dainty, C.

Dalimier, E.

Daly, E.

Devaney, N.

Diolaiti, E.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Donnelly, W. J.

Drexler, W.

Esposito, S.

S. R. Chamot, C. Dainty, and S. Esposito, “Adaptive optics for ophthalmic applications using a pyramid wavefront sensor,” Opt. Express 14, 518–526 (2006).
[CrossRef] [PubMed]

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

S. Esposito and A. Riccardi, “Pyramid wavefront sensor behaviour in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

S. Esposito, O. Feeney, and A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” Proc. SPIE 4007, 416–422 (2000).
[CrossRef]

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, “Laboratory characterization of a “Foucault-like” wavefront sensor for adaptive optics,” Proc. SPIE 3353, 941–951 (1998).
[CrossRef]

Farinato, J.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).

Farrell, T.

Feeney, O.

S. Esposito, O. Feeney, and A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” Proc. SPIE 4007, 416–422 (2000).
[CrossRef]

O. Feeney, “Theory and laboratory characterization of a novel wavefront sensor for adaptive optics,” PhD thesis (National University of Ireland, Galway, 2001).

Feldy, M.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

Fercher, A. F.

Fernández, E. J.

Fini, L.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Foucault, L. M.

L. M. Foucault, “Mémoire sur la construction des télescopes en verre arganté,” Ann. Obs. Imp. Paris 5, 197–237 (1859).

Ghedina, A.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Ghigo, M.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Goelz, S.

Grimm, B.

Hampson, K. M.

K. M. Hampson, “Adaptive optics and vision,” J. Mod. Opt. 55, 3425–3467 (2008).
[CrossRef]

Hebert, T. J.

Hermann, B.

Hofer, H.

Hong, X.

L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

Iglesias, I.

Jolissaint, L.

Julien, Y.

LeDue, J.

Liang, J.

Mackey, D.

Mackey, R.

Marchetti, E.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Miller, D. T.

Navarro, R.

P. Rodríguez, R. Navarro, J. Arines, and S. Bará, “A new calibration set of phase plates for ocular aberrometers,” J. Refractive Surg. 22, 275–284 (2006).

Niero, T.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Platt, B.

B. Platt and R. V. Shack, “Lenticular Hartmann screen,” Opt. Sci. Center Newsl. (University of Arizona) 5, 15–16 (1994).

Prieto, P. M.

Puga, E.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

Puglisi, A.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

Queener, H.

Ragazzoni, R.

A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On sky test of the pyramid wavefront sensor,” Proc. SPIE 4839, 869–877 (2003).
[CrossRef]

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

I. Iglesias, R. Ragazzoni, Y. Julien, and P. Artal, “Extended source pyramid wave-front sensor for the human eye,” Opt. Express 10, 419–428 (2002).
[PubMed]

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, “Laboratory characterization of a “Foucault-like” wavefront sensor for adaptive optics,” Proc. SPIE 3353, 941–951 (1998).
[CrossRef]

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

Riccardi, A.

S. Esposito and A. Riccardi, “Pyramid wavefront sensor behaviour in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

S. Esposito, O. Feeney, and A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” Proc. SPIE 4007, 416–422 (2000).
[CrossRef]

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, “Laboratory characterization of a “Foucault-like” wavefront sensor for adaptive optics,” Proc. SPIE 3353, 941–951 (1998).
[CrossRef]

Rodríguez, P.

P. Rodríguez, R. Navarro, J. Arines, and S. Bará, “A new calibration set of phase plates for ocular aberrometers,” J. Refractive Surg. 22, 275–284 (2006).

Romero-Borja, F.

Roorda, A.

Sattmann, H.

Scargle, J. D.

J. D. Scargle, “Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data,” Astrophys. J. 263, 835–853 (1982).
[CrossRef]

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series, V.Lakshminarayanan, ed. (Optical Society of America, 2000), pp. 233–244.

Shack, R. V.

B. Platt and R. V. Shack, “Lenticular Hartmann screen,” Opt. Sci. Center Newsl. (University of Arizona) 5, 15–16 (1994).

Singer, B.

Stefanini, P.

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, “Laboratory characterization of a “Foucault-like” wavefront sensor for adaptive optics,” Proc. SPIE 3353, 941–951 (1998).
[CrossRef]

Thibos, L. N.

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, well-corrected eyes,” Ophthalmic Physiolog. Opt. 22, 427–433 (2002).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series, V.Lakshminarayanan, ed. (Optical Society of America, 2000), pp. 233–244.

Unterhuber, A.

Véran, J.-P.

Verinaud, C.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldy, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003).
[CrossRef]

Vernet, E.

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[CrossRef]

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, Vol. 35 of Trends in Optics and Photonics Series, V.Lakshminarayanan, ed. (Optical Society of America, 2000), pp. 233–244.

Williams, D. R.

Yamauchi, Y.

Yoon, G. Y.

Ann. Obs. Imp. Paris (1)

L. M. Foucault, “Mémoire sur la construction des télescopes en verre arganté,” Ann. Obs. Imp. Paris 5, 197–237 (1859).

Appl. Opt. (1)

Astron. Astrophys. (2)

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).

S. Esposito and A. Riccardi, “Pyramid wavefront sensor behaviour in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

Astrophys. J. (1)

J. D. Scargle, “Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data,” Astrophys. J. 263, 835–853 (1982).
[CrossRef]

J. Mod. Opt. (2)

K. M. Hampson, “Adaptive optics and vision,” J. Mod. Opt. 55, 3425–3467 (2008).
[CrossRef]

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Schematic optical setup showing probe, wavefront sensing (WFS), and science arms: BS1 and BS2, pellicle beam splitters; 1–14, lenses; DM, deformable mirror; SM, steering mirror; WFS, wavefront sensing camera; S, science camera. Retinal plane and conjugates are marked with dots, pupil plane and conjugates with open circles. S x and S y are the unscaled x and y gradients obtained from the pupils 1 to 4.

Fig. 2
Fig. 2

System speed as a function of CCD pixel binning.

Fig. 3
Fig. 3

(a) Experimental wavefront reconstruction error as a function of steering mirror modulation and CCD pixel binning. The applied wavefront has an RMS error of 0.13 μm . (b) Variation of the wavefront reconstruction error for B1 as a function of steering mirror modulation for two different applied wavefronts. Error bars are ± 1 standard deviation obtained from the 30 wavefront sensor frames.

Fig. 4
Fig. 4

(a) Predicted open-loop bandwidth obtained from the simple model. (b) Modeled error transfer function with loop gain g increasing in the direction of the arrow, taking values [0.1, 0.25, 0.5, 0.75, 1].

Fig. 5
Fig. 5

Time taken to reduce the RMS error to a stable closed-loop value on applying a static aberration ( 0.340 μm RMS) at B8M5, g = 0.5 . Filled circles are experimental data points, with error bars given by the sampling accuracy ( ± 0.012 s at 83 fps ). Filled squares are estimates for bandwidth using the inverse of the time values.

Fig. 6
Fig. 6

(a) Experimental trace showing reduction of the RMS error on closing the loop on an artificial phase plate. Loop gain was g = 0.5 and steering mirror modulation = M 5 . Because this was a static aberration, there was no need for speed; therefore, CCD binning was B1 for maximum spatial resolution. The inset shows the initial phase map overlaid with a map of the DM showing actuator placement. (b) Comparison of actual DM commands after correction with those predicted by our model.

Fig. 7
Fig. 7

RMS values for the eight subjects measured in open loop at B8M20 (open circles) and closed loop at B8M5 (filled circles). Also included are the initial RMS values at B8M5 before the loop was closed (crosses). Error bars are 0.100 μm long for open-loop data and 0.030 μm long for closed-loop and initial data. Subject 9 is the average subject with error bars in each series given by ± one standard deviation of the eight samples.

Fig. 8
Fig. 8

Open-loop (OL) and closed-loop (CL) double-pass ocular point spread functions for subjects 2, 3, and 6. The probe beam width at the retina was 16 μm under diffraction-limited conditions. Magnification to the science camera was 24 × . The white scale bar is equivalent to 50 μm on the retina.

Fig. 9
Fig. 9

Solid lines: spectral power averaged over all subjects in open and closed loop. The open-loop data were recorded at B8M20, the closed-loop data at B8M5. Dashed lines: power spectra obtained from an artificial eye under the same conditions. The displacement of the noise floor with steering mirror modulation is clear for both artificial and real eyes. Note the small rise in power ( 5 10 Hz ) in the artificial eye CL spectrum at B8M5—possibly associated with the rise in power expected at frequencies greater than the closed-loop bandwidth as discussed in the text?

Tables (1)

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Table 1 Radius of the Modulated Circular Path at the Pyramid Vertex in Multiples of the Aberration-Free Point Spread Function a with Corresponding Tilt Values b

Equations (9)

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S x ( x , y ) = [ I 1 ( x , y ) + I 3 ( x , y ) ] [ I 2 ( x , y ) + I 4 ( x , y ) ] i = 1 4 I i ( x , y ) , S y ( x , y ) = [ I 1 ( x , y ) + I 2 ( x , y ) ] [ I 3 ( x , y ) + I 4 ( x , y ) ] i = 1 4 I i ( x , y ) ,
W ( x , y ) x = R mod f sin [ π 2 S x ( x , y ) ] R mod f [ π 2 S x ( x , y ) ] , W ( x , y ) y = R mod f sin [ π 2 S y ( x , y ) ] R mod f [ π 2 S y ( x , y ) ] .
σ rec = [ j [ ψ int ( j ) ψ pws ( j ) ] 2 ¯ ] 1 / 2 ,
σ th 2 = k K σ k 2 = k K σ s 2 n k 2 = 1 N k K n k 2 ,
G = ( I F int ) ( I F pws ) T D T .
C i + 1 = g C i ( S X Y i S X Y CL ref ) + C i ,
G OL ( s ) = ( 1 e T s T s ) ( e τ s ) ( g 1 e T f s ) ( 1 e T f s T f s ) .
G CL ( s ) = ( G OL ( s ) 1 + G OL ( s ) ) ,
E ( s ) = ( 1 1 + G OL ( s ) ) .

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