Abstract

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, some of the higher-order modes contain linear and quadratic terms. A new aberration series is proposed to better separate the low- versus higher-order aberration components. Because its higher-order modes are devoid of linear and quadratic terms, our new basis can be used to better fit the low- and higher-order components of the wavefront. This new basis may quantify the aberrations more accurately and provide clinicians with coefficient magnitudes which better underline the impact of clinically significant aberration modes.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, and VSIA Standards Taskforce Members: Vision Science and Its Applications, “Standards for reporting the optical aberrations of eyes,” J. Refractive Surg. 18, S652–S660 (2002).
  2. V. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
    [Crossref]
  3. F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
    [Crossref]
  4. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  5. A. Bradley, R. Xu, L. Thibos, G. Marin, and M. Hernandez, “Influence of spherical aberration, stimulus spatial frequency, and pupil apodisation on subjective refractions,” Ophthalmic Physiolog. Opt. 34, 309–320 (2014).
    [Crossref]
  6. X. Cheng, A. Bradley, S. Ravimukar, and L. Thibos, “Visual impact of Zernike and Seidel forms of monochromatic aberrations,” Optom. Vis. Sci. 87, 300–312 (2010).
    [Crossref]
  7. M. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 24, 766–795 (1961).
  8. J. Liang, B. Grimme, S. Goelz, and J. Bille, “Objective measurement of the wave aberration of the human eye using a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. 11, 1949–1957 (1994).
    [Crossref]
  9. P. Stephenson, “Optical aberrations described by an alternative series expansion,” J. Opt. Soc. Am. 26, 265–273 (2009).
    [Crossref]
  10. L. Thibos, X. Hong, A. Bradley, and R. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9, 329–351 (2004).
    [Crossref]
  11. X. Cheng, A. Bradley, and L. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7, 310–321 (2004).
    [Crossref]
  12. L. Chen, B. Singer, A. Guirao, J. Porter, and D. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82, 358–369 (2005).
    [Crossref]
  13. R. Applegate, C. Ballentine, B. Gross, E. Sarver, and C. Sarver, “Visual acuity as a function of Zernike mode and level of root mean square error,” Optom. Vis. Sci. 80, 97–105 (2003).
    [Crossref]
  14. A. Guirao and D. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vis. Sci. 80, 36–42 (2003).
    [Crossref]
  15. R. Xu, A. Bradley, N. L. Gil, and L. Thibos, “Modelling the effects of secondary spherical aberration on refractive error, image quality and depth of focus,” Ophthalmic Physiolog. Opt. 35, 28–38 (2015).
    [Crossref]
  16. A. Guirao, J. Porter, D. Williams, and I. Cox, “Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes,” J. Opt. Soc. Am. A 19, 1–9 (2002).
    [Crossref]
  17. F. Yi, D. Iskander, and M. Collins, “Depth of focus and visual acuity with primary and secondary spherical aberration,” Vis. Res. 51, 1648–1658 (2011).
    [Crossref]
  18. R. Legras, Y. Benard, and N. Lopez-Gil, “Effect of coma and spherical aberration on depth-of-focus measured using adaptive optics and computationally blurred images,” J. Cataract. Refract. Surg. 38, 458–469 (2012).
    [Crossref]
  19. L. Zheleznyak, R. Sabesan, J. Oh, S. MacRae, and G. Yoon, “Modified monovision with spherical aberration to improve presbyopic through-focus visual performance,” Invest. Ophthalmol. Vis. Sci. 54, 3157–3165 (2013).
    [Crossref]
  20. L. Zheleznyak, H. Jung, and G. Yoon, “Impact of pupil transmission apodization on presbyopic through-focus visual performance with spherical aberration,” Invest. Ophthalmol. Vis. Sci. 55, 70–77 (2014).
    [Crossref]

2015 (1)

R. Xu, A. Bradley, N. L. Gil, and L. Thibos, “Modelling the effects of secondary spherical aberration on refractive error, image quality and depth of focus,” Ophthalmic Physiolog. Opt. 35, 28–38 (2015).
[Crossref]

2014 (2)

A. Bradley, R. Xu, L. Thibos, G. Marin, and M. Hernandez, “Influence of spherical aberration, stimulus spatial frequency, and pupil apodisation on subjective refractions,” Ophthalmic Physiolog. Opt. 34, 309–320 (2014).
[Crossref]

L. Zheleznyak, H. Jung, and G. Yoon, “Impact of pupil transmission apodization on presbyopic through-focus visual performance with spherical aberration,” Invest. Ophthalmol. Vis. Sci. 55, 70–77 (2014).
[Crossref]

2013 (1)

L. Zheleznyak, R. Sabesan, J. Oh, S. MacRae, and G. Yoon, “Modified monovision with spherical aberration to improve presbyopic through-focus visual performance,” Invest. Ophthalmol. Vis. Sci. 54, 3157–3165 (2013).
[Crossref]

2012 (1)

R. Legras, Y. Benard, and N. Lopez-Gil, “Effect of coma and spherical aberration on depth-of-focus measured using adaptive optics and computationally blurred images,” J. Cataract. Refract. Surg. 38, 458–469 (2012).
[Crossref]

2011 (1)

F. Yi, D. Iskander, and M. Collins, “Depth of focus and visual acuity with primary and secondary spherical aberration,” Vis. Res. 51, 1648–1658 (2011).
[Crossref]

2010 (1)

X. Cheng, A. Bradley, S. Ravimukar, and L. Thibos, “Visual impact of Zernike and Seidel forms of monochromatic aberrations,” Optom. Vis. Sci. 87, 300–312 (2010).
[Crossref]

2009 (1)

P. Stephenson, “Optical aberrations described by an alternative series expansion,” J. Opt. Soc. Am. 26, 265–273 (2009).
[Crossref]

2005 (1)

L. Chen, B. Singer, A. Guirao, J. Porter, and D. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82, 358–369 (2005).
[Crossref]

2004 (2)

L. Thibos, X. Hong, A. Bradley, and R. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9, 329–351 (2004).
[Crossref]

X. Cheng, A. Bradley, and L. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7, 310–321 (2004).
[Crossref]

2003 (2)

R. Applegate, C. Ballentine, B. Gross, E. Sarver, and C. Sarver, “Visual acuity as a function of Zernike mode and level of root mean square error,” Optom. Vis. Sci. 80, 97–105 (2003).
[Crossref]

A. Guirao and D. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vis. Sci. 80, 36–42 (2003).
[Crossref]

2002 (2)

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, and VSIA Standards Taskforce Members: Vision Science and Its Applications, “Standards for reporting the optical aberrations of eyes,” J. Refractive Surg. 18, S652–S660 (2002).

A. Guirao, J. Porter, D. Williams, and I. Cox, “Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes,” J. Opt. Soc. Am. A 19, 1–9 (2002).
[Crossref]

1994 (2)

V. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
[Crossref]

J. Liang, B. Grimme, S. Goelz, and J. Bille, “Objective measurement of the wave aberration of the human eye using a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. 11, 1949–1957 (1994).
[Crossref]

1976 (1)

1961 (1)

M. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 24, 766–795 (1961).

1934 (1)

F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Applegate, R.

L. Thibos, X. Hong, A. Bradley, and R. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9, 329–351 (2004).
[Crossref]

R. Applegate, C. Ballentine, B. Gross, E. Sarver, and C. Sarver, “Visual acuity as a function of Zernike mode and level of root mean square error,” Optom. Vis. Sci. 80, 97–105 (2003).
[Crossref]

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, and VSIA Standards Taskforce Members: Vision Science and Its Applications, “Standards for reporting the optical aberrations of eyes,” J. Refractive Surg. 18, S652–S660 (2002).

Ballentine, C.

R. Applegate, C. Ballentine, B. Gross, E. Sarver, and C. Sarver, “Visual acuity as a function of Zernike mode and level of root mean square error,” Optom. Vis. Sci. 80, 97–105 (2003).
[Crossref]

Benard, Y.

R. Legras, Y. Benard, and N. Lopez-Gil, “Effect of coma and spherical aberration on depth-of-focus measured using adaptive optics and computationally blurred images,” J. Cataract. Refract. Surg. 38, 458–469 (2012).
[Crossref]

Bille, J.

J. Liang, B. Grimme, S. Goelz, and J. Bille, “Objective measurement of the wave aberration of the human eye using a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. 11, 1949–1957 (1994).
[Crossref]

Bradley, A.

R. Xu, A. Bradley, N. L. Gil, and L. Thibos, “Modelling the effects of secondary spherical aberration on refractive error, image quality and depth of focus,” Ophthalmic Physiolog. Opt. 35, 28–38 (2015).
[Crossref]

A. Bradley, R. Xu, L. Thibos, G. Marin, and M. Hernandez, “Influence of spherical aberration, stimulus spatial frequency, and pupil apodisation on subjective refractions,” Ophthalmic Physiolog. Opt. 34, 309–320 (2014).
[Crossref]

X. Cheng, A. Bradley, S. Ravimukar, and L. Thibos, “Visual impact of Zernike and Seidel forms of monochromatic aberrations,” Optom. Vis. Sci. 87, 300–312 (2010).
[Crossref]

L. Thibos, X. Hong, A. Bradley, and R. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9, 329–351 (2004).
[Crossref]

X. Cheng, A. Bradley, and L. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7, 310–321 (2004).
[Crossref]

Chen, L.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82, 358–369 (2005).
[Crossref]

Cheng, X.

X. Cheng, A. Bradley, S. Ravimukar, and L. Thibos, “Visual impact of Zernike and Seidel forms of monochromatic aberrations,” Optom. Vis. Sci. 87, 300–312 (2010).
[Crossref]

X. Cheng, A. Bradley, and L. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7, 310–321 (2004).
[Crossref]

Collins, M.

F. Yi, D. Iskander, and M. Collins, “Depth of focus and visual acuity with primary and secondary spherical aberration,” Vis. Res. 51, 1648–1658 (2011).
[Crossref]

Cox, I.

Gil, N. L.

R. Xu, A. Bradley, N. L. Gil, and L. Thibos, “Modelling the effects of secondary spherical aberration on refractive error, image quality and depth of focus,” Ophthalmic Physiolog. Opt. 35, 28–38 (2015).
[Crossref]

Goelz, S.

J. Liang, B. Grimme, S. Goelz, and J. Bille, “Objective measurement of the wave aberration of the human eye using a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. 11, 1949–1957 (1994).
[Crossref]

Grimme, B.

J. Liang, B. Grimme, S. Goelz, and J. Bille, “Objective measurement of the wave aberration of the human eye using a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. 11, 1949–1957 (1994).
[Crossref]

Gross, B.

R. Applegate, C. Ballentine, B. Gross, E. Sarver, and C. Sarver, “Visual acuity as a function of Zernike mode and level of root mean square error,” Optom. Vis. Sci. 80, 97–105 (2003).
[Crossref]

Guirao, A.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82, 358–369 (2005).
[Crossref]

A. Guirao and D. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vis. Sci. 80, 36–42 (2003).
[Crossref]

A. Guirao, J. Porter, D. Williams, and I. Cox, “Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes,” J. Opt. Soc. Am. A 19, 1–9 (2002).
[Crossref]

Hernandez, M.

A. Bradley, R. Xu, L. Thibos, G. Marin, and M. Hernandez, “Influence of spherical aberration, stimulus spatial frequency, and pupil apodisation on subjective refractions,” Ophthalmic Physiolog. Opt. 34, 309–320 (2014).
[Crossref]

Hong, X.

L. Thibos, X. Hong, A. Bradley, and R. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9, 329–351 (2004).
[Crossref]

Iskander, D.

F. Yi, D. Iskander, and M. Collins, “Depth of focus and visual acuity with primary and secondary spherical aberration,” Vis. Res. 51, 1648–1658 (2011).
[Crossref]

Jung, H.

L. Zheleznyak, H. Jung, and G. Yoon, “Impact of pupil transmission apodization on presbyopic through-focus visual performance with spherical aberration,” Invest. Ophthalmol. Vis. Sci. 55, 70–77 (2014).
[Crossref]

Legras, R.

R. Legras, Y. Benard, and N. Lopez-Gil, “Effect of coma and spherical aberration on depth-of-focus measured using adaptive optics and computationally blurred images,” J. Cataract. Refract. Surg. 38, 458–469 (2012).
[Crossref]

Liang, J.

J. Liang, B. Grimme, S. Goelz, and J. Bille, “Objective measurement of the wave aberration of the human eye using a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. 11, 1949–1957 (1994).
[Crossref]

Lopez-Gil, N.

R. Legras, Y. Benard, and N. Lopez-Gil, “Effect of coma and spherical aberration on depth-of-focus measured using adaptive optics and computationally blurred images,” J. Cataract. Refract. Surg. 38, 458–469 (2012).
[Crossref]

MacRae, S.

L. Zheleznyak, R. Sabesan, J. Oh, S. MacRae, and G. Yoon, “Modified monovision with spherical aberration to improve presbyopic through-focus visual performance,” Invest. Ophthalmol. Vis. Sci. 54, 3157–3165 (2013).
[Crossref]

Mahajan, V.

Marin, G.

A. Bradley, R. Xu, L. Thibos, G. Marin, and M. Hernandez, “Influence of spherical aberration, stimulus spatial frequency, and pupil apodisation on subjective refractions,” Ophthalmic Physiolog. Opt. 34, 309–320 (2014).
[Crossref]

Noll, R.

Oh, J.

L. Zheleznyak, R. Sabesan, J. Oh, S. MacRae, and G. Yoon, “Modified monovision with spherical aberration to improve presbyopic through-focus visual performance,” Invest. Ophthalmol. Vis. Sci. 54, 3157–3165 (2013).
[Crossref]

Porter, J.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82, 358–369 (2005).
[Crossref]

A. Guirao, J. Porter, D. Williams, and I. Cox, “Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes,” J. Opt. Soc. Am. A 19, 1–9 (2002).
[Crossref]

Ravimukar, S.

X. Cheng, A. Bradley, S. Ravimukar, and L. Thibos, “Visual impact of Zernike and Seidel forms of monochromatic aberrations,” Optom. Vis. Sci. 87, 300–312 (2010).
[Crossref]

Sabesan, R.

L. Zheleznyak, R. Sabesan, J. Oh, S. MacRae, and G. Yoon, “Modified monovision with spherical aberration to improve presbyopic through-focus visual performance,” Invest. Ophthalmol. Vis. Sci. 54, 3157–3165 (2013).
[Crossref]

Sarver, C.

R. Applegate, C. Ballentine, B. Gross, E. Sarver, and C. Sarver, “Visual acuity as a function of Zernike mode and level of root mean square error,” Optom. Vis. Sci. 80, 97–105 (2003).
[Crossref]

Sarver, E.

R. Applegate, C. Ballentine, B. Gross, E. Sarver, and C. Sarver, “Visual acuity as a function of Zernike mode and level of root mean square error,” Optom. Vis. Sci. 80, 97–105 (2003).
[Crossref]

Schwiegerling, J.

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, and VSIA Standards Taskforce Members: Vision Science and Its Applications, “Standards for reporting the optical aberrations of eyes,” J. Refractive Surg. 18, S652–S660 (2002).

Singer, B.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82, 358–369 (2005).
[Crossref]

Smirnov, M.

M. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 24, 766–795 (1961).

Stephenson, P.

P. Stephenson, “Optical aberrations described by an alternative series expansion,” J. Opt. Soc. Am. 26, 265–273 (2009).
[Crossref]

Thibos, L.

R. Xu, A. Bradley, N. L. Gil, and L. Thibos, “Modelling the effects of secondary spherical aberration on refractive error, image quality and depth of focus,” Ophthalmic Physiolog. Opt. 35, 28–38 (2015).
[Crossref]

A. Bradley, R. Xu, L. Thibos, G. Marin, and M. Hernandez, “Influence of spherical aberration, stimulus spatial frequency, and pupil apodisation on subjective refractions,” Ophthalmic Physiolog. Opt. 34, 309–320 (2014).
[Crossref]

X. Cheng, A. Bradley, S. Ravimukar, and L. Thibos, “Visual impact of Zernike and Seidel forms of monochromatic aberrations,” Optom. Vis. Sci. 87, 300–312 (2010).
[Crossref]

L. Thibos, X. Hong, A. Bradley, and R. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9, 329–351 (2004).
[Crossref]

X. Cheng, A. Bradley, and L. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7, 310–321 (2004).
[Crossref]

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, and VSIA Standards Taskforce Members: Vision Science and Its Applications, “Standards for reporting the optical aberrations of eyes,” J. Refractive Surg. 18, S652–S660 (2002).

Webb, R.

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, and VSIA Standards Taskforce Members: Vision Science and Its Applications, “Standards for reporting the optical aberrations of eyes,” J. Refractive Surg. 18, S652–S660 (2002).

Williams, D.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82, 358–369 (2005).
[Crossref]

A. Guirao and D. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vis. Sci. 80, 36–42 (2003).
[Crossref]

A. Guirao, J. Porter, D. Williams, and I. Cox, “Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes,” J. Opt. Soc. Am. A 19, 1–9 (2002).
[Crossref]

Xu, R.

R. Xu, A. Bradley, N. L. Gil, and L. Thibos, “Modelling the effects of secondary spherical aberration on refractive error, image quality and depth of focus,” Ophthalmic Physiolog. Opt. 35, 28–38 (2015).
[Crossref]

A. Bradley, R. Xu, L. Thibos, G. Marin, and M. Hernandez, “Influence of spherical aberration, stimulus spatial frequency, and pupil apodisation on subjective refractions,” Ophthalmic Physiolog. Opt. 34, 309–320 (2014).
[Crossref]

Yi, F.

F. Yi, D. Iskander, and M. Collins, “Depth of focus and visual acuity with primary and secondary spherical aberration,” Vis. Res. 51, 1648–1658 (2011).
[Crossref]

Yoon, G.

L. Zheleznyak, H. Jung, and G. Yoon, “Impact of pupil transmission apodization on presbyopic through-focus visual performance with spherical aberration,” Invest. Ophthalmol. Vis. Sci. 55, 70–77 (2014).
[Crossref]

L. Zheleznyak, R. Sabesan, J. Oh, S. MacRae, and G. Yoon, “Modified monovision with spherical aberration to improve presbyopic through-focus visual performance,” Invest. Ophthalmol. Vis. Sci. 54, 3157–3165 (2013).
[Crossref]

Zernike, F.

F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Zheleznyak, L.

L. Zheleznyak, H. Jung, and G. Yoon, “Impact of pupil transmission apodization on presbyopic through-focus visual performance with spherical aberration,” Invest. Ophthalmol. Vis. Sci. 55, 70–77 (2014).
[Crossref]

L. Zheleznyak, R. Sabesan, J. Oh, S. MacRae, and G. Yoon, “Modified monovision with spherical aberration to improve presbyopic through-focus visual performance,” Invest. Ophthalmol. Vis. Sci. 54, 3157–3165 (2013).
[Crossref]

Appl. Opt. (1)

Biophysics (1)

M. Smirnov, “Measurement of the wave aberration of the human eye,” Biophysics 24, 766–795 (1961).

Invest. Ophthalmol. Vis. Sci. (2)

L. Zheleznyak, R. Sabesan, J. Oh, S. MacRae, and G. Yoon, “Modified monovision with spherical aberration to improve presbyopic through-focus visual performance,” Invest. Ophthalmol. Vis. Sci. 54, 3157–3165 (2013).
[Crossref]

L. Zheleznyak, H. Jung, and G. Yoon, “Impact of pupil transmission apodization on presbyopic through-focus visual performance with spherical aberration,” Invest. Ophthalmol. Vis. Sci. 55, 70–77 (2014).
[Crossref]

J. Cataract. Refract. Surg. (1)

R. Legras, Y. Benard, and N. Lopez-Gil, “Effect of coma and spherical aberration on depth-of-focus measured using adaptive optics and computationally blurred images,” J. Cataract. Refract. Surg. 38, 458–469 (2012).
[Crossref]

J. Opt. Soc. Am. (3)

R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[Crossref]

J. Liang, B. Grimme, S. Goelz, and J. Bille, “Objective measurement of the wave aberration of the human eye using a Shack-Hartmann wavefront sensor,” J. Opt. Soc. Am. 11, 1949–1957 (1994).
[Crossref]

P. Stephenson, “Optical aberrations described by an alternative series expansion,” J. Opt. Soc. Am. 26, 265–273 (2009).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Refractive Surg. (1)

L. Thibos, R. Applegate, J. Schwiegerling, and R. Webb, and VSIA Standards Taskforce Members: Vision Science and Its Applications, “Standards for reporting the optical aberrations of eyes,” J. Refractive Surg. 18, S652–S660 (2002).

J. Vis. (2)

L. Thibos, X. Hong, A. Bradley, and R. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4):9, 329–351 (2004).
[Crossref]

X. Cheng, A. Bradley, and L. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4):7, 310–321 (2004).
[Crossref]

Ophthalmic Physiolog. Opt. (2)

A. Bradley, R. Xu, L. Thibos, G. Marin, and M. Hernandez, “Influence of spherical aberration, stimulus spatial frequency, and pupil apodisation on subjective refractions,” Ophthalmic Physiolog. Opt. 34, 309–320 (2014).
[Crossref]

R. Xu, A. Bradley, N. L. Gil, and L. Thibos, “Modelling the effects of secondary spherical aberration on refractive error, image quality and depth of focus,” Ophthalmic Physiolog. Opt. 35, 28–38 (2015).
[Crossref]

Optom. Vis. Sci. (4)

X. Cheng, A. Bradley, S. Ravimukar, and L. Thibos, “Visual impact of Zernike and Seidel forms of monochromatic aberrations,” Optom. Vis. Sci. 87, 300–312 (2010).
[Crossref]

L. Chen, B. Singer, A. Guirao, J. Porter, and D. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82, 358–369 (2005).
[Crossref]

R. Applegate, C. Ballentine, B. Gross, E. Sarver, and C. Sarver, “Visual acuity as a function of Zernike mode and level of root mean square error,” Optom. Vis. Sci. 80, 97–105 (2003).
[Crossref]

A. Guirao and D. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vis. Sci. 80, 36–42 (2003).
[Crossref]

Physica (1)

F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Vis. Res. (1)

F. Yi, D. Iskander, and M. Collins, “Depth of focus and visual acuity with primary and secondary spherical aberration,” Vis. Res. 51, 1648–1658 (2011).
[Crossref]

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Figures (14)

Fig. 1.
Fig. 1. Visualization of the differences between the Zernike and new LDHD modes of clinical importance.
Fig. 2.
Fig. 2. Pyramid of the new expansion mode envelopes.
Fig. 3.
Fig. 3. Visualization of the differences between the Zernike and new third-order coefficients.
Fig. 4.
Fig. 4. Visualization of the differences between the new LDHD and Zernike third-order coefficients.
Fig. 5.
Fig. 5. Visualization of the differences between the Zernike and new fourth-order coefficients.
Fig. 6.
Fig. 6. Visualization of the differences between the new LDHD and Zernike fourth-order coefficients.
Fig. 7.
Fig. 7. Comparison between the Zernike and new expansion coefficients of a myopic eye.
Fig. 8.
Fig. 8. Geometrical representations of the Zernike and GM expansions: (a) comparison of f z and f ; (b) comparison of the Zernike versus LDHD split between low ( f z versus f ) and high ( f z h versus f h ) components; (c),(d) comparison between the Zernike HD versus new HD expansion.
Fig. 9.
Fig. 9. Comparison between the Zernike and new expansion coefficients of an eye with keratoconus.
Fig. 10.
Fig. 10. Geometrical representations of the Zernike and GM expansions: (a) comparison of f z * and f * ; (b) comparison of the Zernike versus LDHD split between low ( f z * versus f * ) and high ( f z h * and f h * ) components; (c),(d) comparison between the Zernike HD versus new HD expansion.
Fig. 11.
Fig. 11. Comparison between the Zernike and new expansion coefficients of an eye corrected with LASIK for myopia.
Fig. 12.
Fig. 12. Geometrical representations of the Zernike and GM expansions: (a) comparison of f z * and f * ; (b) comparison of the Zernike versus LDHD split between low ( f z * versus f * ) and high ( f z h * and f z h * ) components; (c),(d) comparison between the Zernike HD versus new HD expansion.
Fig. 13.
Fig. 13. Comparison between the Zernike and new expansion coefficients of an eye operated with an aspheric photoablation for hyperopia and presbyopia—PresbyLASIK.
Fig. 14.
Fig. 14. Spatial representations of the Zernike and LDHD expansion: (a) comparison of f z * and f * ; (b) comparison of the Zernike versus LDHD split between low ( f z * versus f z * ) and high ( f z h * and f h * ) components; (c),(d) comparison between the Zernike HD versus new HD expansion.

Equations (55)

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D { ( x , y ) R 2 ; x 2 + y 2 1 } ,
D = { ( r cos ( t ) , r sin ( t ) ) ; 0 r 1 , π t + π } .
f : { D R ( x , y ) f ( x , y ) .
f , g D 1 π D f ( x , y ) g ( x , y ) d x d y .
d ( f , g ) ( 1 π D ( f ( x , y ) g ( x , y ) ) 2 d x d y ) 1 / 2 .
p 0 ( f ) 1 π D f ( x , y ) d x d y ( piston ) ,
RMS = d ( f , p 0 ( f ) ) = ( 1 π D ( f ( x , y ) p 0 ( f ) ) 2 d x d y ) 1 / 2 ·
f = n 0    m I n x n m ( f ) X n m ,
{ X n m ( x , y ) x n + m 2    y n m 2 x n m ( f ) 1 n !    n f x n + m 2 y n m 2 ( 0 , 0 ) ·
f = n 0 ( m I n a n m ( f ) A n m ) ,
f ( r cos ( t ) , r sin ( t ) ) = n 0 ( m I n a n m ( f ) R ( n ) ( r ) T m ( t ) ) ,
L D span { A n m ; n 2 , m I n } ,
A 0 0 = 1 , A 1 1 = 2 r sin ( t ) = 2 y , A 1 + 1 = 2 r cos ( t ) = 2 x , A 2 2 = r 2 6 sin ( 2 t ) = 2 6 x y , A 2 0 = 3 r 2 = 3 ( x 2 + y 2 ) , A 2 + 2 = r 2 6 cos ( 2 t ) = 6 ( x 2 y 2 ) .
P D = L D H D ,
H D span { A n m ; n 3 , m I n } .
f = f + f h .
f = n = 0 2 ( m I n a n m ( f ) A n m ) ,
f h = n 3 ( m I n a n m ( f ) A n m ) .
f = n 0 ( m I n z n m ( f ) Z n m ) ,
f ( r cos ( t ) , r sin ( t ) ) = n 0 ( m I n z n m ( f ) R n m ( r ) T m ( t ) ) ,
{ I n { n , n + 2 , , + n 2 , + n } T m ( t ) { 2 sin ( m t ) si m < 0 1 si m = 0 2 cos ( m t ) si m > 0 R n m ( r ) n + 1 0 k n | m | 2 ( 1 ) k ( n k ) ! k ! ( n m 2 k ) ! ( n + m 2 k ) !    r n 2 k .
R 0 0 ( r ) = 1 , R 1 1 ( r ) = R 1 1 ( r ) = 2 r , R 2 2 ( r ) = R 2 2 ( r ) = 3    r 2 , R 2 0 ( r ) = 3 ( 2 r 2 1 ) , R 3 3 ( r ) = R 3 3 ( r ) = 2 r 3 , R 3 1 ( r ) = R 3 1 ( r ) = 2 ( 3 r 3 2 r ) , R 4 4 ( r ) = R 4 4 ( r ) = 5    r 4 , R 4 2 ( r ) = R 4 2 ( r ) = 5 ( 4 r 4 3 r 2 ) , R 4 0 ( r ) = 5 ( 6 r 4 6 r 2 + 1 ) .
( U , V ) U , V azi 1 2 π π + π U ( t ) V ( t ) d t .
( R , S ) R , S rad 2 0 1 R ( r ) S ( r ) r d r .
Z n m = R n m T m ( r , t ) R n m ( r ) T m ( t )
G n m { R n 3 + 1 2 N ( n ) T m if    | m | 2 R n m T m otherwise ,
G 3 3 = Z 3 3 = 2 2 r 3 sin ( 3 t ) , G 3 1 = 2 2 r 3 sin ( t ) , Z 3 1 = 2 2 ( 3 r 3 2 r ) sin ( t ) , G 3 + 1 = 2 2 r 3 cos ( t ) , Z 3 + 1 = 2 2 ( 3 r 3 2 r ) cos ( t ) , G 3 + 3 = Z 3 33 = 2 2 r 3 cos ( 3 t ) , G 4 4 = Z 4 4 = 10 r 4 sin ( 4 t ) , G 4 2 = 10 r 4 sin ( 2 t ) , Z 4 2 = 10 ( 4 r 4 3 r 2 ) sin ( 2 t ) , G 4 0 = 5 r 4 , Z 4 0 = 5 ( 6 r 4 6 r 2 + 1 ) , G 4 2 = 10 r 4 cos ( 2 t ) , Z 4 2 = 10 ( 4 r 4 3 r 2 ) sin ( 2 t ) , G 4 + 4 = Z 4 + 4 = 10 r 4 cos ( 4 t ) , G 5 5 = Z 5 5 = 2 3 r 5 sin ( 5 t ) , G 5 3 = Z 5 3 = 2 3 ( 5 r 5 4 r 3 ) sin ( 3 t ) , G 5 1 = 2 3 ( 5 r 5 4 r 3 ) sin ( t ) , Z 5 1 = 2 3 ( 10 r 5 12 r 3 + 3 r ) sin ( t ) , G 5 + 1 = 2 3 ( 5 r 5 4 r 3 ) cos ( t ) , Z 5 + 1 = 2 3 ( 10 r 5 12 r 3 + 3 r ) cos ( t ) , G 5 + 3 = Z 5 + 3 = 2 3 ( 5 r 5 4 r 3 ) cos ( 3 t ) , G 5 + 5 = Z 5 + 5 = 2 3 r 5 cos ( 5 t ) , G 6 6 = Z 6 6 = 14 r 6 sin ( 6 t ) , G 6 4 = Z 6 4 = 14 ( 6 r 6 5 r 4 ) sin ( 4 t ) , G 6 2 = 14 ( 6 r 6 5 r 4 ) sin ( 2 t ) , Z 6 2 = 14 ( 15 r 6 20 r 4 + 6 r 2 ) sin ( 2 t ) , G 6 0 = 7 ( 6 r 6 5 r 4 ) , Z 6 0 = 7 ( 20 r 6 30 r 4 + 12 r 2 1 ) , G 6 + 2 = 14 ( 6 r 6 5 r 4 ) cos ( 2 t ) , Z 6 + 2 = 14 ( 15 r 6 20 r 4 + 6 r 2 ) cos ( 2 t ) , G 6 + 4 = Z 6 + 4 = 14 ( 6 r 6 5 r 4 ) cos ( 4 t ) , G 6 + 6 = Z 6 + 6 = 2 14 r 6 cos ( 6 t ) , G 7 7 = Z 7 7 = 4 r 7 sin ( 7 t ) , G 7 5 = Z 5 5 = 4 ( 7 r 7 6 r 5 ) sin ( 5 t ) , G 7 3 = Z 7 3 = 4 ( 21 r 7 30 r 5 + 10 r 3 ) sin ( 3 t ) , G 7 1 = 4 ( 21 r 7 30 r 5 + 10 r 3 ) sin ( t ) , Z 7 1 = 4 ( 35 r 7 60 r 5 + 30 r 3 4 r ) sin ( t ) , G 7 + 1 = 4 ( 21 r 7 30 r 5 + 10 r 3 ) cos ( t ) , Z 7 + 1 = 4 ( 35 r 7 60 r 5 + 30 r 3 4 r ) cos ( t ) , G 7 + 3 = Z 7 + 3 = 4 ( 21 r 7 30 r 5 + 10 r 3 ) cos ( 3 t ) , G 7 + 5 = Z 5 + 5 = 4 ( 7 r 7 6 r 5 ) cos ( 5 t ) , G 7 + 7 = Z 7 + 7 = 4 r 7 cos ( 7 t ) .
f = f + f h = n = 0 2 ( m I n g n m ( f ) Z n m ) + n 3 ( m I n g n m ( f ) G n m ) ,
{ g n m ( f ) = f , Z n m D if    n 2 g n m ( f ) = f h , G n m D if    n 3 ,
c n + 2 p m ( n ) = n + 2 p + 1    ( 1 ) p ( n + p ) ! p ! ( n m 2 ) ! ( n + m 2 ) ! ,
R n + 2 p m ( r ) = n + 2 p + 1 0 k n + 2 p | m | 2 ( 1 ) k ( n + 2 p k ) ! k ! ( n + 2 p m 2 k ) ! ( n + 2 p + m 2 k ) !    r n + 2 p 2 k .
a n m ( f ) = p 0 z n + 2 p m ( f ) c n + 2 p m ( n ) .
f = n = 0 2 ( m I n ( p 0 z n + 2 p m ( f ) c n + 2 p m ( n ) ) ) A n m = n = 0 2 ( m I n g n m ( f ) Z n m ) ,
f h = f f = n 3 ( m I n g n m ( f ) G n m ) .
f * = f z 0 0 ( f ) Z 0 0 z 1 1 ( f ) Z 1 1 z 1 + 1 ( f ) Z 1 + 1 = n 2 ( m I n z n m ( f ) Z n m ) .
V * ( f ) = f * D 2 = 1 π D ( f * ( x , y ) ) 2 d x d y = n 2 ( m I n ( z n m ( f ) ) 2 ) ,
RMS * ( f ) = V * ( f ) .
f * D = V * ( f ) .
C * ( f , g ) = f * , g * D = 1 π D f * ( x , y ) g * ( x , y ) d x d y = n 2 ( m I n z n m ( f ) z n m ( g ) ) ,
Angle * ( f , g ) = ArcCos ( C * ( f , g ) f * D g * D ) .
G ( f ) = ( V * ( f z ) 0 C * ( f z , f ) C * ( f z , f h ) 0 V * ( f z h ) 0 C * ( f z h , f h ) C * ( f , f z ) 0 V * ( f ) C * ( f , f h ) C * ( f h , f z ) C * ( f h , f z h ) C * ( f h , f ) V * ( f h ) ) ,
{ f z = n = 0 2 ( m I n z n m ( f ) Z n m ) f z h = n 3 ( m I n z n m ( f ) Z n m ) f = n = 0 2 ( m I n g n m ( f ) Z n m ) f h = n 3 ( m I n g n m ( f ) G n m ) ·
f * = f * , Z 2 2 D Z 2 2 + f * , Z 2 0 D Z 2 0 + f * , Z 2 + 2 D Z 2 + 2 ,
f z * = f z * , Z 2 2 D Z 2 2 + f z * , Z 2 0 D Z 2 0 + f z * , Z 2 + 2 D Z 2 + 2 .
f * = f z * + f z h * = f * + f h *
{ U f , 1 = 1 f z * D f z * U f , 2 = 1 f z h * D f z h * U f , 3 = 1 f z * D f z * × 1 f z h * D f z h *
f z * = f z * D U f , 1 , f z h * = f z h * D U f , 2 , f * = f * , U f , 1 D U f , 1 + f * , U f , 3 D U f , 3 , f h * = f h * , U f , 1 D U f , 1 + f h * , U f , 2 D U f , 2 + f h * , U f , 3 D U f , 3 .
f z h * = f z h 3 * + f z h 4 * + f z h 56 * ,
{ f z h 3 * = m I 3 z 3 m ( f ) Z 3 m f z h 4 * = m I 4 z 4 m ( f ) Z 4 m f z h 56 * = n 5 ( m I n z n m ( f ) Z n m ) ·
{ U f z h , 1 = 1 f z h 3 * D f z h 3 * U f z h , 2 = 1 f z h 4 * D f z h 4 * U f z h , 3 = 1 f z h 56 * D f z h 56 *
f z h * = f z h 3 * D U f z h , 1 + f z h 4 * D U f z h , 2 + f z h 56 * D U f z h , 3 .
f h * = f h 3 * + f h 4 * + f h 56 * ,
{ f h 3 * = m I 3 g 3 m ( f ) G 3 m f h 4 * = m I 4 g 4 m ( f ) G 4 m f h 56 * = n 5 ( m I n g n m ( f ) G n m ) .
{ U f h , 1 = 1 f h 3 * D f h 3 * U f h , 2 = 1 f h 4 * D f h 4 * U f h , 3 = 1 f h 56 * D f h 56 *
f h * = f h 3 * D U f h , 1 + f h 4 * D U f h , 2 + f h 56 * D U f h , 3 .