Abstract

Calculating through-focus characteristics of the human eye from a single objective measurement of wavefront aberration can be accomplished through a range of methods that are inherently computationally cumbersome. A simple yet accurate and computationally efficient method is developed, which combines the philosophy of the extended Nijboer–Zernike approach with the radial-basis-function-based approximation of the complex pupil function. The main advantage of the proposed technique is that the increase of the computational cost for a vector-valued defocus parameter is practically negligible in comparison to the corresponding scalar-valued defocus parameter.

© 2014 Optical Society of America

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2011 (1)

A. Martínez-Finkelshtein, D. Ramos-López, G. M. Castro-Luna, and J. L. Alió, “Adaptive corneal modeling from keratometric data,” Investig. Ophthalmol. Vis. Sci. 52, 4963–4970 (2011).
[CrossRef]

2010 (2)

J. Tarrant, A. Roorda, and C. F. Wildsoet, “Determining the accommodative response from wavefront aberrations,” J. Vis. 10(5):4 (2010).
[CrossRef]

F. Yi, D. R. Iskander, and M. J. Collins, “Estimation of the depth of focus from wavefront measurements,” J. Vis. 10(4):3 (2010).
[CrossRef]

2009 (3)

J. Nam, L. N. Thibos, and D. R. Iskander, “Zernike radial slope polynomials for wavefront reconstruction and refraction,” J. Opt. Soc. Am. A 26, 1035–1048 (2009).
[CrossRef]

A. Martínez-Finkelshtein, A. M. Delgado, G. M. Castro-Luna, A. Zarzo, and J. L. Alió, “Comparative analysis of some modal reconstruction methods of the shape of the cornea from corneal elevation data,” Investig. Ophthalmol. Vis. Sci. 50, 5639–5645 (2009).
[CrossRef]

E. Candès, L. Demanet, and L. Ying, “A fast butterfly algorithm for the computation of Fourier integral operators,” Multiscale Model. Simul. 7, 1727–1750 (2009).
[CrossRef]

2008 (1)

J. Schwiegerling and J. Choi, “Application of the polychromatic defocus transfer function to multifocal lenses,” J. Refract. Surg. 24, 965–969 (2008

2007 (5)

J. Schwiegerling, “Analysis of the optical performance of presbyopia treatments with the defocus transfer function,” J. Refract. Surg. 23, 965–971 (2007).

D. R. Iskander, B. A. Davis, M. J. Collins, and R. Franklin, “Objective refraction from monochromatic wavefront aberrations via Zernike power polynomials,” Ophthalmic Physiolog. Opt. 27, 245–255 (2007).

M. Gai and R. Cancelliere, “An efficient point spread function construction method,” Mon. Not. R. Astron. Soc. 377, 1337–1342 (2007).
[CrossRef]

B. Vasudevan, J. K. Ciuffreda, and B. Wang, “Subjective and objective depth-of-focus,” J. Mod. Opt. 54, 1307–1316 (2007).
[CrossRef]

P. A. Piers, H. A. Weeber, P. Artal, and S. Norrby, “Theoretical comparison of aberration-correcting customized and aspheric intraocular lenses,” J. Refract. Surg. 23, 374–384 (2007).

2006 (2)

B. Wang and K. J. Ciuffreda, “Depth-of-focus of the human eye: theory and clinical implications,” Surv. Ophthalmol. 51, 75–85 (2006).
[CrossRef]

M. J. Collins, T. Buehren, and D. R. Iskander, “Retinal image quality, reading and myopia,” Vis. Res. 46, 196–215 (2006).
[CrossRef]

2005 (1)

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 1–13 (2005).

2004 (3)

L. Llorente, S. Barbero, J. Merayo, and S. Marcos, “Total and corneal optical aberrations induced by laser in situ keratomileusis for hyperopia,” J. Refract. Surg. 20, 203–216 (2004).

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9 (2004).
[CrossRef]

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7 (2004).
[CrossRef]

2002 (4)

1999 (2)

M. Montoya-Hernández, M. Servín, D. Malacara-Hernández, and G. Paez, “Wavefront fitting using Gaussian functions,” Opt. Commun. 163, 259–269 (1999).
[CrossRef]

S. Marcos, E. Moreno, and R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vis. Res. 39, 2039–2049 (1999).
[CrossRef]

1998 (1)

N. M. Jansonius and A. C. Kooijman, “The effect of spherical and other aberrations upon the modulation transfer of the defocused human eye,” Ophthalmic Physiolog. Opt. 18, 504–513 (1998).

1997 (1)

1995 (1)

A. Artal, S. Marcos, I. Miranda, and M. Ferro, “Through focus image quality of eyes implanted with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[CrossRef]

1994 (1)

D. H. Bailey and P. N. Swartztrauber, “A fast method for the numerical evaluation of continuous Fourier and Laplace transforms,” SIAM J. Sci. Comput. 15, 1105–1110 (1994).
[CrossRef]

1993 (1)

1987 (1)

Alió, J. L.

A. Martínez-Finkelshtein, D. Ramos-López, G. M. Castro-Luna, and J. L. Alió, “Adaptive corneal modeling from keratometric data,” Investig. Ophthalmol. Vis. Sci. 52, 4963–4970 (2011).
[CrossRef]

A. Martínez-Finkelshtein, A. M. Delgado, G. M. Castro-Luna, A. Zarzo, and J. L. Alió, “Comparative analysis of some modal reconstruction methods of the shape of the cornea from corneal elevation data,” Investig. Ophthalmol. Vis. Sci. 50, 5639–5645 (2009).
[CrossRef]

Applegate, R. A.

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9 (2004).
[CrossRef]

Artal, A.

A. Artal, S. Marcos, I. Miranda, and M. Ferro, “Through focus image quality of eyes implanted with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[CrossRef]

Artal, P.

P. A. Piers, H. A. Weeber, P. Artal, and S. Norrby, “Theoretical comparison of aberration-correcting customized and aspheric intraocular lenses,” J. Refract. Surg. 23, 374–384 (2007).

Atchison, D. A.

D. A. Atchison, “Depth of focus of the human eye,” in Presbyopia: Origins, Effects and Treatment, I. Pallikaris, S. Plainis, and W. N. Charman, eds. (Slack Incorporated, 2012).

Bailey, D. H.

D. H. Bailey and P. N. Swartztrauber, “A fast method for the numerical evaluation of continuous Fourier and Laplace transforms,” SIAM J. Sci. Comput. 15, 1105–1110 (1994).
[CrossRef]

Barbero, S.

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 1–13 (2005).

L. Llorente, S. Barbero, J. Merayo, and S. Marcos, “Total and corneal optical aberrations induced by laser in situ keratomileusis for hyperopia,” J. Refract. Surg. 20, 203–216 (2004).

Braat, J. J. M.

Bradley, A.

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9 (2004).
[CrossRef]

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7 (2004).
[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]

Buehren, T.

M. J. Collins, T. Buehren, and D. R. Iskander, “Retinal image quality, reading and myopia,” Vis. Res. 46, 196–215 (2006).
[CrossRef]

Campbell, F. W.

Cancelliere, R.

M. Gai and R. Cancelliere, “An efficient point spread function construction method,” Mon. Not. R. Astron. Soc. 377, 1337–1342 (2007).
[CrossRef]

Candès, E.

E. Candès, L. Demanet, and L. Ying, “A fast butterfly algorithm for the computation of Fourier integral operators,” Multiscale Model. Simul. 7, 1727–1750 (2009).
[CrossRef]

Castro-Luna, G. M.

A. Martínez-Finkelshtein, D. Ramos-López, G. M. Castro-Luna, and J. L. Alió, “Adaptive corneal modeling from keratometric data,” Investig. Ophthalmol. Vis. Sci. 52, 4963–4970 (2011).
[CrossRef]

A. Martínez-Finkelshtein, A. M. Delgado, G. M. Castro-Luna, A. Zarzo, and J. L. Alió, “Comparative analysis of some modal reconstruction methods of the shape of the cornea from corneal elevation data,” Investig. Ophthalmol. Vis. Sci. 50, 5639–5645 (2009).
[CrossRef]

Cathey, W. T.

Cheng, X.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7 (2004).
[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]

Choi, J.

J. Schwiegerling and J. Choi, “Application of the polychromatic defocus transfer function to multifocal lenses,” J. Refract. Surg. 24, 965–969 (2008

Ciuffreda, J. K.

B. Vasudevan, J. K. Ciuffreda, and B. Wang, “Subjective and objective depth-of-focus,” J. Mod. Opt. 54, 1307–1316 (2007).
[CrossRef]

Ciuffreda, K. J.

B. Wang and K. J. Ciuffreda, “Depth-of-focus of the human eye: theory and clinical implications,” Surv. Ophthalmol. 51, 75–85 (2006).
[CrossRef]

Collins, M. J.

F. Yi, D. R. Iskander, and M. J. Collins, “Estimation of the depth of focus from wavefront measurements,” J. Vis. 10(4):3 (2010).
[CrossRef]

D. R. Iskander, B. A. Davis, M. J. Collins, and R. Franklin, “Objective refraction from monochromatic wavefront aberrations via Zernike power polynomials,” Ophthalmic Physiolog. Opt. 27, 245–255 (2007).

M. J. Collins, T. Buehren, and D. R. Iskander, “Retinal image quality, reading and myopia,” Vis. Res. 46, 196–215 (2006).
[CrossRef]

Cox, I. G.

Davis, B. A.

D. R. Iskander, B. A. Davis, M. J. Collins, and R. Franklin, “Objective refraction from monochromatic wavefront aberrations via Zernike power polynomials,” Ophthalmic Physiolog. Opt. 27, 245–255 (2007).

Delgado, A. M.

A. Martínez-Finkelshtein, A. M. Delgado, G. M. Castro-Luna, A. Zarzo, and J. L. Alió, “Comparative analysis of some modal reconstruction methods of the shape of the cornea from corneal elevation data,” Investig. Ophthalmol. Vis. Sci. 50, 5639–5645 (2009).
[CrossRef]

Demanet, L.

E. Candès, L. Demanet, and L. Ying, “A fast butterfly algorithm for the computation of Fourier integral operators,” Multiscale Model. Simul. 7, 1727–1750 (2009).
[CrossRef]

Dirksen, P.

Dowski, E. R.

Fasshauer, G. E.

G. E. Fasshauer, Meshfree Approximation Methods with Matlab (World Scientific, 2007).

Ferro, M.

A. Artal, S. Marcos, I. Miranda, and M. Ferro, “Through focus image quality of eyes implanted with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[CrossRef]

FitzGerrell, A. R.

Franklin, R.

D. R. Iskander, B. A. Davis, M. J. Collins, and R. Franklin, “Objective refraction from monochromatic wavefront aberrations via Zernike power polynomials,” Ophthalmic Physiolog. Opt. 27, 245–255 (2007).

Gai, M.

M. Gai and R. Cancelliere, “An efficient point spread function construction method,” Mon. Not. R. Astron. Soc. 377, 1337–1342 (2007).
[CrossRef]

Guirao, A.

Hansen, P. C.

P. C. Hansen, V. Pereyra, and G. Scherer, Least Squares Data Fitting with Applications (Johns Hopkins University, 2013).

Hong, X.

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9 (2004).
[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]

Iskander, D. R.

F. Yi, D. R. Iskander, and M. J. Collins, “Estimation of the depth of focus from wavefront measurements,” J. Vis. 10(4):3 (2010).
[CrossRef]

J. Nam, L. N. Thibos, and D. R. Iskander, “Zernike radial slope polynomials for wavefront reconstruction and refraction,” J. Opt. Soc. Am. A 26, 1035–1048 (2009).
[CrossRef]

D. R. Iskander, B. A. Davis, M. J. Collins, and R. Franklin, “Objective refraction from monochromatic wavefront aberrations via Zernike power polynomials,” Ophthalmic Physiolog. Opt. 27, 245–255 (2007).

M. J. Collins, T. Buehren, and D. R. Iskander, “Retinal image quality, reading and myopia,” Vis. Res. 46, 196–215 (2006).
[CrossRef]

Jansonius, N. M.

N. M. Jansonius and A. C. Kooijman, “The effect of spherical and other aberrations upon the modulation transfer of the defocused human eye,” Ophthalmic Physiolog. Opt. 18, 504–513 (1998).

Janssen, A. J. E. M.

Jimenez-Alfaro, I.

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 1–13 (2005).

Kooijman, A. C.

N. M. Jansonius and A. C. Kooijman, “The effect of spherical and other aberrations upon the modulation transfer of the defocused human eye,” Ophthalmic Physiolog. Opt. 18, 504–513 (1998).

Lakshminarayanan, V.

Lang, A. J.

Legge, G. E.

Llorente, L.

L. Llorente, S. Barbero, J. Merayo, and S. Marcos, “Total and corneal optical aberrations induced by laser in situ keratomileusis for hyperopia,” J. Refract. Surg. 20, 203–216 (2004).

Malacara-Hernández, D.

M. Montoya-Hernández, M. Servín, D. Malacara-Hernández, and G. Paez, “Wavefront fitting using Gaussian functions,” Opt. Commun. 163, 259–269 (1999).
[CrossRef]

Marcos, S.

S. Marcos, S. Barbero, and I. Jimenez-Alfaro, “Optical quality and depth-of-field of eyes implanted with spherical and aspheric intraocular lenses,” J. Refract. Surg. 21, 1–13 (2005).

L. Llorente, S. Barbero, J. Merayo, and S. Marcos, “Total and corneal optical aberrations induced by laser in situ keratomileusis for hyperopia,” J. Refract. Surg. 20, 203–216 (2004).

S. Marcos, E. Moreno, and R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vis. Res. 39, 2039–2049 (1999).
[CrossRef]

A. Artal, S. Marcos, I. Miranda, and M. Ferro, “Through focus image quality of eyes implanted with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[CrossRef]

Martínez-Finkelshtein, A.

A. Martínez-Finkelshtein, D. Ramos-López, G. M. Castro-Luna, and J. L. Alió, “Adaptive corneal modeling from keratometric data,” Investig. Ophthalmol. Vis. Sci. 52, 4963–4970 (2011).
[CrossRef]

A. Martínez-Finkelshtein, A. M. Delgado, G. M. Castro-Luna, A. Zarzo, and J. L. Alió, “Comparative analysis of some modal reconstruction methods of the shape of the cornea from corneal elevation data,” Investig. Ophthalmol. Vis. Sci. 50, 5639–5645 (2009).
[CrossRef]

Merayo, J.

L. Llorente, S. Barbero, J. Merayo, and S. Marcos, “Total and corneal optical aberrations induced by laser in situ keratomileusis for hyperopia,” J. Refract. Surg. 20, 203–216 (2004).

Miranda, I.

A. Artal, S. Marcos, I. Miranda, and M. Ferro, “Through focus image quality of eyes implanted with monofocal and multifocal intraocular lenses,” Opt. Eng. 34, 772–779 (1995).
[CrossRef]

Montoya-Hernández, M.

M. Montoya-Hernández, M. Servín, D. Malacara-Hernández, and G. Paez, “Wavefront fitting using Gaussian functions,” Opt. Commun. 163, 259–269 (1999).
[CrossRef]

Moreno, E.

S. Marcos, E. Moreno, and R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vis. Res. 39, 2039–2049 (1999).
[CrossRef]

Mullen, K. T.

Nam, J.

Navarro, R.

S. Marcos, E. Moreno, and R. Navarro, “The depth-of-field of the human eye from objective and subjective measurements,” Vis. Res. 39, 2039–2049 (1999).
[CrossRef]

Norrby, S.

P. A. Piers, H. A. Weeber, P. Artal, and S. Norrby, “Theoretical comparison of aberration-correcting customized and aspheric intraocular lenses,” J. Refract. Surg. 23, 374–384 (2007).

Paez, G.

M. Montoya-Hernández, M. Servín, D. Malacara-Hernández, and G. Paez, “Wavefront fitting using Gaussian functions,” Opt. Commun. 163, 259–269 (1999).
[CrossRef]

Pereyra, V.

P. C. Hansen, V. Pereyra, and G. Scherer, Least Squares Data Fitting with Applications (Johns Hopkins University, 2013).

Piers, P. A.

P. A. Piers, H. A. Weeber, P. Artal, and S. Norrby, “Theoretical comparison of aberration-correcting customized and aspheric intraocular lenses,” J. Refract. Surg. 23, 374–384 (2007).

Porter, J.

Portney, V.

Ramos-López, D.

A. Martínez-Finkelshtein, D. Ramos-López, G. M. Castro-Luna, and J. L. Alió, “Adaptive corneal modeling from keratometric data,” Investig. Ophthalmol. Vis. Sci. 52, 4963–4970 (2011).
[CrossRef]

Roorda, A.

J. Tarrant, A. Roorda, and C. F. Wildsoet, “Determining the accommodative response from wavefront aberrations,” J. Vis. 10(5):4 (2010).
[CrossRef]

Scherer, G.

P. C. Hansen, V. Pereyra, and G. Scherer, Least Squares Data Fitting with Applications (Johns Hopkins University, 2013).

Schwiegerling, J.

J. Schwiegerling and J. Choi, “Application of the polychromatic defocus transfer function to multifocal lenses,” J. Refract. Surg. 24, 965–969 (2008

J. Schwiegerling, “Analysis of the optical performance of presbyopia treatments with the defocus transfer function,” J. Refract. Surg. 23, 965–971 (2007).

Servín, M.

M. Montoya-Hernández, M. Servín, D. Malacara-Hernández, and G. Paez, “Wavefront fitting using Gaussian functions,” Opt. Commun. 163, 259–269 (1999).
[CrossRef]

Swartztrauber, P. N.

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[CrossRef]

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J. Tarrant, A. Roorda, and C. F. Wildsoet, “Determining the accommodative response from wavefront aberrations,” J. Vis. 10(5):4 (2010).
[CrossRef]

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J. Nam, L. N. Thibos, and D. R. Iskander, “Zernike radial slope polynomials for wavefront reconstruction and refraction,” J. Opt. Soc. Am. A 26, 1035–1048 (2009).
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L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9 (2004).
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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]

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[CrossRef]

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B. Vasudevan, J. K. Ciuffreda, and B. Wang, “Subjective and objective depth-of-focus,” J. Mod. Opt. 54, 1307–1316 (2007).
[CrossRef]

B. Wang and K. J. Ciuffreda, “Depth-of-focus of the human eye: theory and clinical implications,” Surv. Ophthalmol. 51, 75–85 (2006).
[CrossRef]

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P. A. Piers, H. A. Weeber, P. Artal, and S. Norrby, “Theoretical comparison of aberration-correcting customized and aspheric intraocular lenses,” J. Refract. Surg. 23, 374–384 (2007).

Wildsoet, C. F.

J. Tarrant, A. Roorda, and C. F. Wildsoet, “Determining the accommodative response from wavefront aberrations,” J. Vis. 10(5):4 (2010).
[CrossRef]

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[CrossRef]

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Appl. Opt. (1)

Investig. Ophthalmol. Vis. Sci. (2)

A. Martínez-Finkelshtein, A. M. Delgado, G. M. Castro-Luna, A. Zarzo, and J. L. Alió, “Comparative analysis of some modal reconstruction methods of the shape of the cornea from corneal elevation data,” Investig. Ophthalmol. Vis. Sci. 50, 5639–5645 (2009).
[CrossRef]

A. Martínez-Finkelshtein, D. Ramos-López, G. M. Castro-Luna, and J. L. Alió, “Adaptive corneal modeling from keratometric data,” Investig. Ophthalmol. Vis. Sci. 52, 4963–4970 (2011).
[CrossRef]

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B. Vasudevan, J. K. Ciuffreda, and B. Wang, “Subjective and objective depth-of-focus,” J. Mod. Opt. 54, 1307–1316 (2007).
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[CrossRef]

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[CrossRef]

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[CrossRef]

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7 (2004).
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B. Wang and K. J. Ciuffreda, “Depth-of-focus of the human eye: theory and clinical implications,” Surv. Ophthalmol. 51, 75–85 (2006).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

PSF with perfect wavefront, given by Eq. (10), calculated by each method.

Fig. 2.
Fig. 2.

Dependence of the execution time on the number of functions used in the description of the complex pupil function.

Fig. 3.
Fig. 3.

Execution time according to the number of different defocus parameters used in the complex pupil function. A fixed number of N functions (N=400 for GRBF and N=45 for ENZ) was used for each value of f.

Fig. 4.
Fig. 4.

3D plot of the synthetic wavefront function defined in Eq. (11).

Fig. 5.
Fig. 5.

PSF of the wavefront in Eq. (11) with f=0, calculated by each method. In the first row, from left to right, using GRBF and ENZ. In the second row, from left to right, by FFT2 and by quadrature.

Fig. 6.
Fig. 6.

Absolute error in the PSFs of Fig. 5 for each method, with respect to the solution by quadrature.

Fig. 7.
Fig. 7.

Values of the normalized PSFs of Fig. 5 along the horizontal diameter of the unit disk, calculated for each method, for f=1 (top), f=3 (middle), and f=10 (bottom).

Tables (2)

Tables Icon

Table 1. Estimates of the Minimal Computational Complexity of the Methodsa

Tables Icon

Table 2. Maximum Absolute Error and Root Mean Square Error Corresponding to the PSF Residual Distributions Shown in Fig. 6

Equations (19)

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

P(ρ,θ)=A(ρ,θ)exp(iΦ(ρ,θ)),
U(r,ϕ;f)=1π0102π[exp(ifρ2)P(ρ,θ)×exp((2πiρrcos(θϕ))]ρdθdρ,
GRBF(ρ,θ)=exp{L(q2+ρ22ρqcos(θα))}.
P(ρ,θ)=k=1Nckexp{L(qk2+ρ22ρqkcos(θαk))},
U(r,ϕ;f)=k=1NckUk(r,ϕ;f),
Uk(r,ϕ;f)=exp{Lqk2}s=0Ωs(s!)2ms(ifL),
Ω=ΩL(r,ϕ,qk,αk)=L2qk2+2πirLqkcos(ϕαk)π2r2,
ms(ξ)=01ρseξρdρ,s=0,1,.
m0(ξ)=eξ1ξ,ms+1(ξ)=eξ(s+1)ms(ξ)ξ.
U(r,ϕ;f)=k=1Nckexp{Lqk2}×s=0ms(ifL)(s!)2(L2qk2+2πirLqkcos(ϕαk)π2r2)s.
U(r,ϕ;0)=J1(2πr)πr,
Φ(ρ,θ)=0.5Z44(ρ,θ)+0.3Z53(ρ,θ)+0.3Z55(ρ,θ)+3g(ρcosθ,ρsinθ;0.5,0.3,5)+3g(ρcosθ,ρsinθ;0.5,0.3,5)+0.2g(ρcosθ,ρsinθ;0.3,0,15),
g(x,y;a,b,L)=exp{L[(xa)2+(yb)2]}.
gk(ρ,θ)=exp{Lk(qk2+ρ22ρqkcos(θαk))},
k=1Nckgk(xj,yj)P(xj,yj)=A(xj,yj)exp{iwj}.
P(ρ,θ)=exp{L(q2+ρ22ρqcos(θα))}.
I=02πexp(2Lρqcos(θα))exp(2πiρrcos(θϕ))dθ,
I=2πI0(2Ωρ),
Ω=Lk2qk2+2πirLkqkcos(ϕαk)π2r2.

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