Abstract

Ophthalmic wavefront sensors typically measure wavefront slope, from which wavefront phase is reconstructed. We show that ophthalmic prescriptions (in power-vector format) can be obtained directly from slope measurements without wavefront reconstruction. This is achieved by fitting the measurement data with a new set of orthonormal basis functions called Zernike radial slope polynomials. Coefficients of this expansion can be used to specify the ophthalmic power vector using explicit formulas derived by a variety of methods. Zernike coefficients for wavefront error can be recovered from the coefficients of radial slope polynomials, thereby offering an alternative way to perform wavefront reconstruction.

© 2009 Optical Society of America

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    [PubMed]
  2. G. Smith, “Refraction and visual acuity measurements: what are their measurement uncertainties?” Clin. Exp. Optom. 89, 66-72 (2006).
    [CrossRef] [PubMed]
  3. M. Berman, P. Nelson, and B. Caden, “Objective refraction: comparison of retinoscopy and automated techniques,” Am. J. Optom. Physiol. Opt. 61, 204-209 (1984).
    [PubMed]
  4. M. A. Bullimore, R. E. Fusaro, and C. W. Adams, “The repeatability of automated and clinician refraction,” Optom. Vision Sci. 75, 617-622 (1998).
    [CrossRef]
  5. K. Pesudovs and H. S. Weisinger, “A comparison of autorefractor performance,” Optom. Vision Sci. 81, 554-558 (2004).
    [CrossRef]
  6. G. Walsh, W. N. Charman, and H. C. Howland, “Objective technique for the determination of monochromatic aberrations of the human eye,” J. Opt. Soc. Am. A 1, 987-992 (1984).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  9. J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873-2883 (1997).
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    [CrossRef]
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    [CrossRef]
  28. V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 1998).
  29. H. Saunders, “A method for determining the mean value of refractive error,” Br. J. Physiol. Opt. 34, 1-11 (1980).
  30. R. Courant and D. Hilbert, Methods of Mathematical Physics, (Wiley, 1989).
  31. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).
  32. A. G. Bennett and R. B. Rabbetts, Bennett & Rabbetts' Clinical Visual Optics (Butterworth-Heinemann, 1998).
  33. Strictly speaking, the equivalent quadratic introduced by Thibos is a paraboloid described by W2. If a=b, the family of surfaces comprises circular paraboloids; if a and b have the same sign, it comprises elliptic paraboloids; and if a and b have opposite signs, it comprises hyperbolic paraboloids.

2008 (1)

A. B. Watson and A. J. Ahumada, Jr., “Predicting visual acuity from wavefront aberrations,” J. Vision 8, 11-19 (2008).
[CrossRef]

2007 (2)

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

D. R. Iskander, B. A. Davis, and M. J. Collins, “The skew ray ambiguity in the analysis of videokeratoscopic data,” Optom. Vision Sci. 84, 435-442 (2007).
[CrossRef]

2006 (1)

G. Smith, “Refraction and visual acuity measurements: what are their measurement uncertainties?” Clin. Exp. Optom. 89, 66-72 (2006).
[CrossRef] [PubMed]

2004 (3)

K. Pesudovs and H. S. Weisinger, “A comparison of autorefractor performance,” Optom. Vision Sci. 81, 554-558 (2004).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vision 4, 329-351 (2004).
[CrossRef]

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

2003 (1)

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

2002 (1)

2000 (1)

L. N. Thibos, “Principles of Hartmann-Shack aberrometry,” J. Refract. Surg. 16, 563-565 (2000).

1998 (1)

M. A. Bullimore, R. E. Fusaro, and C. W. Adams, “The repeatability of automated and clinician refraction,” Optom. Vision Sci. 75, 617-622 (1998).
[CrossRef]

1997 (2)

J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873-2883 (1997).
[CrossRef]

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

1996 (1)

D. A. Goss and T. Grosvenor, “Reliability of refraction--a literature review,” J. Am. Optom. Assoc. 67, 619-630 (1996).
[PubMed]

1995 (3)

M. J. Collins, C. F. Wildsoet, and D. A. Atchison, “Monochromatic aberrations and myopia,” Vision Res. 35, 1157-1163 (1995).
[CrossRef] [PubMed]

S. A. Klein and R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2096-2109 (1995).

S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155-2159 (1995).

1994 (2)

J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949-1957 (1994).
[CrossRef]

C. Roberts, “The accuracy of 'power' maps to display curvature data in corneal topography systems,” Invest. Ophthalmol. Visual Sci. 35, 3525-3532 (1994).

1993 (1)

R. K. Maloney, S. J. Bogan, and G. O. Waring, 3rd, “Determination of corneal image-forming properties from corneal topography,” Am. J. Ophthalmol. 115, 31-41 (1993).
[PubMed]

1984 (2)

G. Walsh, W. N. Charman, and H. C. Howland, “Objective technique for the determination of monochromatic aberrations of the human eye,” J. Opt. Soc. Am. A 1, 987-992 (1984).
[CrossRef] [PubMed]

M. Berman, P. Nelson, and B. Caden, “Objective refraction: comparison of retinoscopy and automated techniques,” Am. J. Optom. Physiol. Opt. 61, 204-209 (1984).
[PubMed]

1980 (1)

H. Saunders, “A method for determining the mean value of refractive error,” Br. J. Physiol. Opt. 34, 1-11 (1980).

1963 (1)

T. C. Jenkins, “Aberrations of the eye and their effects on vision: 1. Spherical aberration,” Br. J. Physiol. Opt. 20, 59-91 (1963).
[PubMed]

Adams, C. W.

M. A. Bullimore, R. E. Fusaro, and C. W. Adams, “The repeatability of automated and clinician refraction,” Optom. Vision Sci. 75, 617-622 (1998).
[CrossRef]

Ahumada, A. J.

A. B. Watson and A. J. Ahumada, Jr., “Predicting visual acuity from wavefront aberrations,” J. Vision 8, 11-19 (2008).
[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. Vision 4, 329-351 (2004).
[CrossRef]

Atchison, D. A.

M. J. Collins, C. F. Wildsoet, and D. A. Atchison, “Monochromatic aberrations and myopia,” Vision Res. 35, 1157-1163 (1995).
[CrossRef] [PubMed]

Bennett, A. G.

A. G. Bennett and R. B. Rabbetts, Bennett & Rabbetts' Clinical Visual Optics (Butterworth-Heinemann, 1998).

Berman, M.

M. Berman, P. Nelson, and B. Caden, “Objective refraction: comparison of retinoscopy and automated techniques,” Am. J. Optom. Physiol. Opt. 61, 204-209 (1984).
[PubMed]

Bille, J. F.

Bogan, S. J.

R. K. Maloney, S. J. Bogan, and G. O. Waring, 3rd, “Determination of corneal image-forming properties from corneal topography,” Am. J. Ophthalmol. 115, 31-41 (1993).
[PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Bradley, A.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vision 4, 329-351 (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]

Bullimore, M. A.

M. A. Bullimore, R. E. Fusaro, and C. W. Adams, “The repeatability of automated and clinician refraction,” Optom. Vision Sci. 75, 617-622 (1998).
[CrossRef]

Caden, B.

M. Berman, P. Nelson, and B. Caden, “Objective refraction: comparison of retinoscopy and automated techniques,” Am. J. Optom. Physiol. Opt. 61, 204-209 (1984).
[PubMed]

Charman, W. N.

Cheng, X.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (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]

Coe, C. D.

L. N. Thibos, N. L. Himebaugh, and C. D. Coe, “Wavefront refraction,” in Borish's Clinical Refraction, W.J.Benjamin, ed. (Butterworth-Heinemann, 2006), pp. 765-789.
[CrossRef]

Collins, M. J.

D. R. Iskander, B. A. Davis, and M. J. Collins, “The skew ray ambiguity in the analysis of videokeratoscopic data,” Optom. Vision Sci. 84, 435-442 (2007).
[CrossRef]

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

M. J. Collins, C. F. Wildsoet, and D. A. Atchison, “Monochromatic aberrations and myopia,” Vision Res. 35, 1157-1163 (1995).
[CrossRef] [PubMed]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics, (Wiley, 1989).

Davis, B. A.

D. R. Iskander, B. A. Davis, and M. J. Collins, “The skew ray ambiguity in the analysis of videokeratoscopic data,” Optom. Vision Sci. 84, 435-442 (2007).
[CrossRef]

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

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 Physiol. Opt. 27, 245-255 (2007).
[CrossRef]

Fusaro, R. E.

M. A. Bullimore, R. E. Fusaro, and C. W. Adams, “The repeatability of automated and clinician refraction,” Optom. Vision Sci. 75, 617-622 (1998).
[CrossRef]

Goelz, S.

Goss, D. A.

D. A. Goss and T. Grosvenor, “Reliability of refraction--a literature review,” J. Am. Optom. Assoc. 67, 619-630 (1996).
[PubMed]

Grimm, B.

Grosvenor, T.

D. A. Goss and T. Grosvenor, “Reliability of refraction--a literature review,” J. Am. Optom. Assoc. 67, 619-630 (1996).
[PubMed]

Guirao, A.

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

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics, (Wiley, 1989).

Himebaugh, N. L.

L. N. Thibos, N. L. Himebaugh, and C. D. Coe, “Wavefront refraction,” in Borish's Clinical Refraction, W.J.Benjamin, ed. (Butterworth-Heinemann, 2006), pp. 765-789.
[CrossRef]

Hong, X.

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vision 4, 329-351 (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]

Horner, D.

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

Howland, H. C.

Iskander, D. R.

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

D. R. Iskander, B. A. Davis, and M. J. Collins, “The skew ray ambiguity in the analysis of videokeratoscopic data,” Optom. Vision Sci. 84, 435-442 (2007).
[CrossRef]

Jenkins, T. C.

T. C. Jenkins, “Aberrations of the eye and their effects on vision: 1. Spherical aberration,” Br. J. Physiol. Opt. 20, 59-91 (1963).
[PubMed]

Klein, S. A.

S. A. Klein and R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2096-2109 (1995).

S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155-2159 (1995).

Liang, J.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 1998).

Maloney, R. K.

R. K. Maloney, S. J. Bogan, and G. O. Waring, 3rd, “Determination of corneal image-forming properties from corneal topography,” Am. J. Ophthalmol. 115, 31-41 (1993).
[PubMed]

Mandell, R. B.

S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155-2159 (1995).

S. A. Klein and R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2096-2109 (1995).

Nelson, P.

M. Berman, P. Nelson, and B. Caden, “Objective refraction: comparison of retinoscopy and automated techniques,” Am. J. Optom. Physiol. Opt. 61, 204-209 (1984).
[PubMed]

Pesudovs, K.

K. Pesudovs and H. S. Weisinger, “A comparison of autorefractor performance,” Optom. Vision Sci. 81, 554-558 (2004).
[CrossRef]

Rabbetts, R. B.

A. G. Bennett and R. B. Rabbetts, Bennett & Rabbetts' Clinical Visual Optics (Butterworth-Heinemann, 1998).

Roberts, C.

C. Roberts, “The accuracy of 'power' maps to display curvature data in corneal topography systems,” Invest. Ophthalmol. Visual Sci. 35, 3525-3532 (1994).

Saunders, H.

H. Saunders, “A method for determining the mean value of refractive error,” Br. J. Physiol. Opt. 34, 1-11 (1980).

Smith, G.

G. Smith, “Refraction and visual acuity measurements: what are their measurement uncertainties?” Clin. Exp. Optom. 89, 66-72 (2006).
[CrossRef] [PubMed]

Thibos,

Strictly speaking, the equivalent quadratic introduced by Thibos is a paraboloid described by W2. If a=b, the family of surfaces comprises circular paraboloids; if a and b have the same sign, it comprises elliptic paraboloids; and if a and b have opposite signs, it comprises hyperbolic paraboloids.

Thibos, L. N.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vision 4, 329-351 (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]

L. N. Thibos, “Principles of Hartmann-Shack aberrometry,” J. Refract. Surg. 16, 563-565 (2000).

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

L. N. Thibos, N. L. Himebaugh, and C. D. Coe, “Wavefront refraction,” in Borish's Clinical Refraction, W.J.Benjamin, ed. (Butterworth-Heinemann, 2006), pp. 765-789.
[CrossRef]

Walsh, G.

Waring, G. O.

R. K. Maloney, S. J. Bogan, and G. O. Waring, 3rd, “Determination of corneal image-forming properties from corneal topography,” Am. J. Ophthalmol. 115, 31-41 (1993).
[PubMed]

Watson, A. B.

A. B. Watson and A. J. Ahumada, Jr., “Predicting visual acuity from wavefront aberrations,” J. Vision 8, 11-19 (2008).
[CrossRef]

Weisinger, H. S.

K. Pesudovs and H. S. Weisinger, “A comparison of autorefractor performance,” Optom. Vision Sci. 81, 554-558 (2004).
[CrossRef]

Wheeler, W.

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

Wildsoet, C. F.

M. J. Collins, C. F. Wildsoet, and D. A. Atchison, “Monochromatic aberrations and myopia,” Vision Res. 35, 1157-1163 (1995).
[CrossRef] [PubMed]

Williams, D. R.

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

J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873-2883 (1997).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Am. J. Ophthalmol. (1)

R. K. Maloney, S. J. Bogan, and G. O. Waring, 3rd, “Determination of corneal image-forming properties from corneal topography,” Am. J. Ophthalmol. 115, 31-41 (1993).
[PubMed]

Am. J. Optom. Physiol. Opt. (1)

M. Berman, P. Nelson, and B. Caden, “Objective refraction: comparison of retinoscopy and automated techniques,” Am. J. Optom. Physiol. Opt. 61, 204-209 (1984).
[PubMed]

Br. J. Physiol. Opt. (2)

T. C. Jenkins, “Aberrations of the eye and their effects on vision: 1. Spherical aberration,” Br. J. Physiol. Opt. 20, 59-91 (1963).
[PubMed]

H. Saunders, “A method for determining the mean value of refractive error,” Br. J. Physiol. Opt. 34, 1-11 (1980).

Clin. Exp. Optom. (1)

G. Smith, “Refraction and visual acuity measurements: what are their measurement uncertainties?” Clin. Exp. Optom. 89, 66-72 (2006).
[CrossRef] [PubMed]

Invest. Ophthalmol. Visual Sci. (3)

C. Roberts, “The accuracy of 'power' maps to display curvature data in corneal topography systems,” Invest. Ophthalmol. Visual Sci. 35, 3525-3532 (1994).

S. A. Klein and R. B. Mandell, “Shape and refractive powers in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2096-2109 (1995).

S. A. Klein and R. B. Mandell, “Axial and instantaneous power conversion in corneal topography,” Invest. Ophthalmol. Visual Sci. 36, 2155-2159 (1995).

J. Am. Optom. Assoc. (1)

D. A. Goss and T. Grosvenor, “Reliability of refraction--a literature review,” J. Am. Optom. Assoc. 67, 619-630 (1996).
[PubMed]

J. Opt. Soc. Am. A (4)

J. Refract. Surg. (1)

L. N. Thibos, “Principles of Hartmann-Shack aberrometry,” J. Refract. Surg. 16, 563-565 (2000).

J. Vision (3)

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

A. B. Watson and A. J. Ahumada, Jr., “Predicting visual acuity from wavefront aberrations,” J. Vision 8, 11-19 (2008).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vision 4, 329-351 (2004).
[CrossRef]

Ophthalmic Physiol. Opt. (1)

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

Optom. Vision Sci. (5)

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

M. A. Bullimore, R. E. Fusaro, and C. W. Adams, “The repeatability of automated and clinician refraction,” Optom. Vision Sci. 75, 617-622 (1998).
[CrossRef]

K. Pesudovs and H. S. Weisinger, “A comparison of autorefractor performance,” Optom. Vision Sci. 81, 554-558 (2004).
[CrossRef]

D. R. Iskander, B. A. Davis, and M. J. Collins, “The skew ray ambiguity in the analysis of videokeratoscopic data,” Optom. Vision Sci. 84, 435-442 (2007).
[CrossRef]

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

Vision Res. (1)

M. J. Collins, C. F. Wildsoet, and D. A. Atchison, “Monochromatic aberrations and myopia,” Vision Res. 35, 1157-1163 (1995).
[CrossRef] [PubMed]

Other (9)

ANSI, “American National Standards for Ophthalmics--Methods for reporting optical aberrations of eyes,” American National Standards InstituteANSI Z80.28 (2004).

Online Etymology Dictionary (http://www.etymonline.com).

The American Heritage Dictionary of the English Language, 4th ed. (Houghton Mifflin, 2000).

L. N. Thibos, N. L. Himebaugh, and C. D. Coe, “Wavefront refraction,” in Borish's Clinical Refraction, W.J.Benjamin, ed. (Butterworth-Heinemann, 2006), pp. 765-789.
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 1998).

R. Courant and D. Hilbert, Methods of Mathematical Physics, (Wiley, 1989).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

A. G. Bennett and R. B. Rabbetts, Bennett & Rabbetts' Clinical Visual Optics (Butterworth-Heinemann, 1998).

Strictly speaking, the equivalent quadratic introduced by Thibos is a paraboloid described by W2. If a=b, the family of surfaces comprises circular paraboloids; if a and b have the same sign, it comprises elliptic paraboloids; and if a and b have opposite signs, it comprises hyperbolic paraboloids.

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Figures (5)

Fig. 1
Fig. 1

Geometry of wavefront vergence V ( r , θ ) definition. The ( x , y , z ) coordinate reference frame is constructed such that the z axis is the chief ray and the x, y plane is perpendicular to the z axis at the center of the exit pupil (gray bar). The plane of the diagram intersects the ( x , y ) plane in a line that becomes a radial r axis inclined at angle θ to the horizontal. The z axis is orthogonal to the wavefront but not necessarily orthogonal to the exit pupil.

Fig. 2
Fig. 2

Original wavefront W = Z 2 2 + 3 Z 2 0 + 2 Z 2 2 is identical to equivalent quadratics W e constructed with the power vectors from the three different methods. The physical units are mm for x and y and micrometers for W and W e .

Fig. 3
Fig. 3

Comparison of equivalent quadratic surfaces ( μ m ) . (a) The original wavefront W = Z 2 2 + 3 Z 2 0 + 2 Z 2 2 + 2 Z 4 0 0.5 Z 4 2 . (b) An equivalent quadric from method 1. (c) An equivalent quadric from method 2. (d) An equivalent quadric from method 3. x and y are in mm.

Fig. 4
Fig. 4

Contour maps of equivalent quadratic surfaces. (a) The original wavefront W = Z 2 2 + 3 Z 2 0 + 2 Z 2 2 + 2 Z 4 0 0.5 Z 4 2 . (b) An equivalent quadratic from method 1. (c) An equivalent quadratic from method 2. (d) An equivalent quadratic from method 3.

Fig. 5
Fig. 5

Comparison of equivalent quadratic surfaces ( μ m ) . (a) The original wavefront of Seidel aberration W = 10 ρ 4 + 2 ρ 2 cos 2 θ + 3 ρ 2 . (b) An equivalent quadratic from method 1. (c) An equivalent quadratic from method 2. (d) An equivalent quadratic from method 3. The fourth-order term ρ 4 has no effect at all on M with method 1. Method 2 is most influenced by higher-order terms. Method 3 has a moderate effect on M. x and y are in mm.

Tables (4)

Tables Icon

Table 1 List of the New Basis Function Y n m ( ρ , θ ) , n 4

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Table 2 Power Vectors (D) Obtained from the Zernike Radial Slope Coefficients B n m a

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Table 3 Power Vectors (D) Obtained from the Zernike Radial Slope Coefficients B n m a

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Table 4 Power Vectors (D) Obtained from the Zernike Radial Slope Coefficients B n m a

Equations (103)

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V ( r , θ ) = n d = n W r r .
V ( r = 0 , θ ) = [ W r r ] r = 0 = ( M + J 0 cos 2 θ + J 45 sin 2 θ ) .
M = 1 π r max 2 0 2 π 0 r max V ( r , θ ) r d r d θ ,
J 0 = 2 π r max 2 0 2 π 0 r max V ( r , θ ) cos ( 2 θ ) r d r d θ ,
J 45 = 2 π r max 2 0 2 π 0 r max V ( r , θ ) sin ( 2 θ ) r d r d θ .
M = 1 2 π r max 0 π r max r max V ( r , θ ) d r d θ ,
J 0 = 1 π r max 0 π r max r max V ( r , θ ) cos ( 2 θ ) d r d θ ,
J 45 = 1 π r max 0 π r max r max V ( r , θ ) sin ( 2 θ ) d r d θ .
M = C 1 0 π r max r max V ( r , θ ) w ( r , θ ) r d r d θ ,
J 0 = C 2 0 π r max r max V ( r , θ ) cos ( 2 θ ) w ( r , θ ) r d r d θ ,
J 45 = C 2 0 π r max r max V ( r , θ ) sin ( 2 θ ) w ( r , θ ) r d r d θ .
W ( r , θ ) = C 2 2 6 ( r r max ) 2 sin 2 θ + C 2 0 3 [ 2 ( r r max ) 2 1 ] + C 2 2 6 ( r r max ) 2 cos 2 θ ,
W ( r , θ ) r = C 2 2 6 ( 1 r max 2 ) 2 r sin 2 θ + C 2 0 3 ( 1 r max 2 ) 4 r + C 2 2 6 ( 1 r max 2 ) 2 r cos 2 θ ,
V ( r , θ ) = C 2 2 6 ( 2 r max 2 ) sin 2 θ + C 2 0 3 ( 4 r max 2 ) + C 2 2 6 ( 2 r max 2 ) cos 2 θ .
M = 4 3 C 2 0 r max 2 , J 0 = 2 6 C 2 2 r max 2 , J 45 = 2 6 C 2 2 r max 2 .
W ( r , θ ) = C 2 0 3 [ 2 ( r r max ) 2 1 ] + C 4 0 5 [ 6 ( r r max ) 4 6 ( r r max ) 2 + 1 ] .
W r = C 2 0 3 ( 2 r max 2 ) 2 r + C 4 0 5 [ ( 6 r max 4 ) 4 r 3 ( 6 r max 2 ) 2 r ] ,
V ( r , θ ) = C 2 0 3 ( 4 r max 2 ) + C 4 0 5 [ ( 24 r max 4 ) r 2 ( 12 r max 2 ) ] .
M = [ 4 3 C 2 0 12 5 C 4 0 r max 2 ] , J 0 = 0 , J 45 = 0 .
M = 4 3 C 2 0 r max 2 , J 0 = 0 , J 45 = 0 .
M = [ 4 3 C 2 0 4 5 C 4 0 r max 2 ] , J 0 = 0 , J 45 = 0 .
Method 1 : M = 0 , J 0 = [ 2 6 C 2 2 6 10 C 4 2 r max 2 ] , J 45 = 0 .
Method 2 : M = 0 , J 0 = [ 2 6 C 2 2 + 2 10 C 4 2 r max 2 ] , J 45 = 0 .
Method 3 : M = 0 , J 0 = [ 6 6 C 2 2 2 10 C 4 2 3 r max 2 ] , J 45 = 0 .
{ 1 ρ Z n m ( ρ , θ ) d ρ } ,
r V = W r .
Z n m ( ρ , θ ) ρ ,
Y n m ( ρ , θ ) = { N n 1 m R n 1 m 1 ( ρ ) cos m θ , if m 0 N n 1 m R n 1 m 1 ( ρ ) sin m θ , if m < 0 } ,
W ( ρ , θ ) ρ = ρ V ( ρ , θ ) = n , m B n m Y n m ( ρ , θ ) ,
= B 1 1 2 sin θ + B 1 1 2 cos θ + B 2 2 2 ρ sin 2 θ + B 2 0 2 ρ + B 2 2 2 ρ cos 2 θ + ,
B n m = 1 π 0 2 π 0 1 ρ V ( ρ , θ ) Y n m ( ρ , θ ) ρ d ρ d θ ,
= 1 π 0 2 π 0 1 V ( ρ , θ ) Y n m ( ρ , θ ) ρ 2 d ρ d θ ,
= 1 π 0 2 π 0 1 W ( ρ , θ ) ρ Y n m ( ρ , θ ) ρ d ρ d θ .
V ( ρ , θ ) = B 1 1 2 sin θ ρ + B 1 1 2 cos θ ρ + B 2 2 2 sin 2 θ + B 2 0 2 + B 2 2 2 cos 2 θ + .
W r ( r max ρ , θ ) = n , m B n m Y n m ( ρ , θ ) .
B n m = 1 π 0 2 π 0 1 W r ( r max ρ , θ ) Y n m ( ρ , θ ) ρ d ρ d θ .
V ( r , θ ) = n , m B n m Y n m ( r r max , θ ) 1 r .
V ( r , θ ) = n B n 0 Y n 0 ( r r max , θ ) 1 r + n B n 2 Y n 2 ( r r max , θ ) 1 r + n B n 2 Y n 2 ( r r max , θ ) 1 r + .
M = [ 2 B 2 0 4 B 4 0 + 3 6 B 6 0 8 2 B 8 0 + 5 10 B 10 0 + r max ] ,
J 0 = [ 2 B 2 2 4 2 B 4 2 + 6 3 B 6 2 16 B 8 2 + 10 5 B 10 2 + r max ] ,
J 45 = [ 2 B 2 2 4 2 B 4 2 + 6 3 B 6 2 16 B 8 2 + 10 5 B 10 2 + r max ] .
M = [ 2 B 2 0 B 4 0 + 1 3 6 B 6 0 1 2 2 B 8 0 + 1 5 10 B 10 0 + r max ] ,
J 0 = [ 2 B 2 2 2 B 4 2 + 2 3 3 B 6 2 B 8 2 + 2 5 5 B 10 2 + r max ] ,
J 45 = [ 2 B 2 2 2 B 4 2 + 2 3 3 B 6 2 B 8 2 + 2 5 5 B 10 2 + r max ] .
M = [ 2 B 2 0 2 B 4 0 + 6 B 6 0 2 2 B 8 0 + 10 B 10 0 + r max ] ,
J 0 = [ 2 B 2 2 2 2 B 4 2 + 2 3 B 6 2 4 B 8 2 + 2 5 B 10 2 + r max ] ,
J 45 = [ 2 B 2 2 2 2 B 4 2 + 2 3 B 6 2 4 B 8 2 + 2 5 B 10 2 + r max ] .
a x 2 + b y 2 = 2 R W P W 2 ,
r 2 ( a cos 2 θ + b sin 2 θ ) = 2 R W P W 2 .
r 2 = 2 R θ W P θ W 2 ,
W 2 ( r , θ ) = r 2 2 R θ ,
W 4 ( r , θ ) = r 2 2 R θ + P θ r 4 8 R θ 3 ,
W 6 ( r , θ ) = r 2 2 R θ + P θ r 4 8 R θ 3 + P θ 2 r 6 16 R θ 5 .
M = ( a + b 2 R ) , J 0 = ( a b 2 R ) , J 45 = 0 ,
M = [ a + b 2 R + ( 3 a 2 + 2 a b + 3 b 2 ) P r max 2 32 R 3 ] ,
J 0 = [ a b 2 R + ( a 2 b 2 ) P r max 2 8 R 3 ] ,
J 45 = 0 .
M = [ a + b 2 R + ( 3 a 2 + 2 a b + 3 b 2 ) P r max 2 32 R 3 + ( a + b ) ( 5 a 2 2 a b + 5 b 2 ) P 2 r max 4 128 R 5 ] ,
J 0 = [ a b 2 R + ( a 2 b 2 ) P r max 2 8 R 3 + 3 ( a b ) ( 5 a 2 + 6 a b + 5 b 2 ) P 2 r max 4 256 R 5 ] ,
J 45 = 0 .
M = [ a + b 2 R + ( 3 a 2 + 2 a b + 3 b 2 ) P r max 2 48 R 3 ] ,
J 0 = [ a b 2 R + ( a 2 b 2 ) P r max 2 12 R 3 ] ,
J 45 = 0 ,
M = [ a + b 2 R + ( 3 a 2 + 2 a b + 3 b 2 ) P r max 2 48 R 3 + 3 ( a + b ) ( 5 a 2 2 a b + 5 b 2 ) P 2 r max 4 640 R 5 ] ,
J 0 = [ a b 2 R + ( a 2 b 2 ) P r max 2 12 R 3 + 9 ( a b ) ( 5 a 2 + 6 a b + 5 b 2 ) P 2 r max 4 1280 R 5 ] ,
J 45 = 0 .
n = 2 , M = 4 3 C 2 0 r max 2 , J 0 = 2 6 C 2 2 r max 2 , J 45 = 2 6 C 2 2 r max 2 ,
n = 4 , M = 12 5 C 4 0 r max 2 , J 0 = 6 10 C 4 2 r max 2 , J 45 = 6 10 C 4 2 r max 2 ,
n = 6 , M = 24 7 C 6 0 r max 2 , J 0 = 12 14 C 6 2 r max 2 ,
J 45 = 12 14 C 6 2 r max 2 .
n = 2 , M = 4 3 C 2 0 r max 2 , J 0 = 2 6 C 2 2 r max 2 , J 45 = 2 6 C 2 2 r max 2 ,
n = 4 , M = 0 , J 0 = 2 10 C 4 2 r max 2 , J 45 = 2 10 C 4 2 r max 2 ,
n = 6 , M = 4 7 C 6 0 r max 2 , J 0 = 2 14 C 6 2 r max 2 ,
J 45 = 2 14 C 6 2 r max 2 .
n = 2 , M = 4 3 C 2 0 r max 2 , J 0 = 2 6 C 2 2 r max 2 , J 45 = 2 6 C 2 2 r max 2 ,
n = 4 , M = 4 5 C 4 0 r max 2 , J 0 = 2 3 10 C 4 2 r max 2 , J 45 = 2 3 10 C 4 2 r max 2 ,
n = 6 , M = 8 7 C 6 0 r max 2 , J 0 = 10 3 14 C 6 2 r max 2 ,
J 45 = 10 3 14 C 6 2 r max 2 .
Method 1 : M = 4 3 C 2 0 + 12 5 C 4 0 24 7 C 6 0 r max 2 ,
J 0 = 2 6 C 2 2 + 6 10 C 4 2 12 14 C 6 2 r max 2 ,
J 45 = 2 6 C 2 2 + 6 10 C 4 2 12 14 C 6 2 r max 2 .
Method 2 : M = 4 3 C 2 0 4 7 C 6 0 r max 2 ,
J 0 = 2 6 C 2 2 2 10 C 4 2 2 14 C 6 2 r max 2 ,
J 45 = 2 6 C 2 2 2 10 C 4 2 2 14 C 6 2 r max 2 .
Method 3 : M = 4 3 C 2 0 + 4 5 C 4 0 8 7 C 6 0 r max 2 ,
J 0 = 2 6 C 2 2 + 2 3 10 C 4 2 10 3 14 C 6 2 r max 2 ,
J 45 = 2 6 C 2 2 + 2 3 10 C 4 2 10 3 14 C 6 2 r max 2 .
W r ( ρ r max , θ ) = 1 r max k C k Z k ρ ( ρ , θ ) , = 1 r max k C k [ l A k l Y l ( ρ , θ ) ] .
W r ( ρ r max , θ ) = 1 r max l ( k A k l C k ) Y l ( ρ , θ ) = 1 r max l ( k A l k T C k ) Y l ( ρ , θ ) .
{ B l } = S { C k } , where S = A T .
{ C k } = T { B l } , where T = S 1 .
S 2 = [ 2 0 0 0 0 0 2 0 0 0 0 0 6 0 0 0 0 0 2 6 0 0 0 0 0 6 ] .
S 3 = [ 0 5 0 0 0 0 5 0 S 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 2 3 ] .
S 4 = [ 0 0 0 0 0 0 0 0 0 0 0 7 10 3 0 0 0 0 0 2 10 0 0 S 3 0 0 0 7 10 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 5 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 5 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 5 ] .
W ( r , θ ) = Z 2 2 + 3 Z 2 0 + 2 Z 2 2 ,
= 6 ρ 2 sin 2 θ + 3 3 ( 2 ρ 2 1 ) + 2 6 ρ 2 cos 2 θ , ρ = r r max .
W ( r , θ ) r = r V ( r , θ ) = r [ M + J 0 cos 2 θ + J 45 sin 2 θ ] .
W ( r , θ ) = r 2 2 [ M + J 0 cos 2 θ + J 45 sin 2 θ ] + C ,
W e ( r , θ ) = r 2 2 [ M + J 0 cos 2 θ + J 45 sin 2 θ ] .
W ( r , θ ) = 10 ρ 4 + 2 ρ 2 cos 2 θ + 3 ρ 2 , ρ = r r max .
M = 0.8889 , J 0 = 0.2222 , J 45 = 0 .
M ( Method 1 ) = 0 , M ( Method 2 ) = 2.2222 ,
M ( Method 3 ) = 1.4815 , in diopters .

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