Abstract

We developed an algorithm that directly determines Zernike coefficients for the corneal anterior surface derived from the reflection image of a stimulus with pseudorandom encoding. This algorithm does not need to include calculation of corneal height maps. The numerical performance of the algorithm is good. It has the potential of determining corneal shape with submicrometer accuracy in obtaining Zernike coefficients. When applied to real eye measurements the accuracy of the procedure will be limited by the topographer that is used.

© 2004 Optical Society of America

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References

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  1. P. Artal, A. Guirao, “Contributions of the cornea and the lens to the aberrations of the human eye,” Opt. Lett. 23, 1713–1715 (1998).
    [CrossRef]
  2. A. Guirao, P. Artal, “Corneal wave aberration from videokeratography: accuracy and limitations of the procedure,” J. Opt. Soc. Am. A 17, 955–965 (2000).
    [CrossRef]
  3. F. M. Vos, G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, F. C. A. Groen, “A new PRBA-based instrument to measure the shape of the cornea,” IEEE Trans. Instrum. Meas. 46, 794–797 (1997).
    [CrossRef]
  4. M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of corneal topography from a videokeratographic reflection pattern,” Optom. Vision Sci. 72, 821–827 (1995).
    [CrossRef]
  5. R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997).
    [CrossRef]
  6. S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vision Sci. 74, 931–944 (1997).
    [CrossRef]
  7. S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vision Sci. 74, 945–962 (1997).
    [CrossRef]
  8. A. W. Greynolds, “Superconic and subconic surfaces in optical design,” presented at the 2002 International Lens Design Conference, Tucson, Arizona, June 3–7, 2002, Postconference Digest, p. 2.
  9. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, 652–660 (2002).
  10. M. Dubbelman, H. A. Weeber, G. L. van der Heijde, H. J. Völker-Dieben, “Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,” Acta Ophthalmol. Scand. 80, 379–383 (2002).
    [CrossRef] [PubMed]
  11. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).
  12. R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vision Sci. 74, 926–930 (1997).
    [CrossRef]

2002

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, 652–660 (2002).

M. Dubbelman, H. A. Weeber, G. L. van der Heijde, H. J. Völker-Dieben, “Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,” Acta Ophthalmol. Scand. 80, 379–383 (2002).
[CrossRef] [PubMed]

2000

1998

1997

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vision Sci. 74, 926–930 (1997).
[CrossRef]

F. M. Vos, G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, F. C. A. Groen, “A new PRBA-based instrument to measure the shape of the cornea,” IEEE Trans. Instrum. Meas. 46, 794–797 (1997).
[CrossRef]

R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997).
[CrossRef]

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vision Sci. 74, 931–944 (1997).
[CrossRef]

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vision Sci. 74, 945–962 (1997).
[CrossRef]

1995

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of corneal topography from a videokeratographic reflection pattern,” Optom. Vision Sci. 72, 821–827 (1995).
[CrossRef]

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, 652–660 (2002).

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vision Sci. 74, 926–930 (1997).
[CrossRef]

Artal, P.

Barsky, B. A.

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of corneal topography from a videokeratographic reflection pattern,” Optom. Vision Sci. 72, 821–827 (1995).
[CrossRef]

Dubbelman, M.

M. Dubbelman, H. A. Weeber, G. L. van der Heijde, H. J. Völker-Dieben, “Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,” Acta Ophthalmol. Scand. 80, 379–383 (2002).
[CrossRef] [PubMed]

Greynolds, A. W.

A. W. Greynolds, “Superconic and subconic surfaces in optical design,” presented at the 2002 International Lens Design Conference, Tucson, Arizona, June 3–7, 2002, Postconference Digest, p. 2.

Groen, F. C. A.

F. M. Vos, G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, F. C. A. Groen, “A new PRBA-based instrument to measure the shape of the cornea,” IEEE Trans. Instrum. Meas. 46, 794–797 (1997).
[CrossRef]

Guirao, A.

Halstead, M. A.

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of corneal topography from a videokeratographic reflection pattern,” Optom. Vision Sci. 72, 821–827 (1995).
[CrossRef]

Howland, H. C.

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vision Sci. 74, 926–930 (1997).
[CrossRef]

Klein, S. A.

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vision Sci. 74, 931–944 (1997).
[CrossRef]

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vision Sci. 74, 945–962 (1997).
[CrossRef]

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of corneal topography from a videokeratographic reflection pattern,” Optom. Vision Sci. 72, 821–827 (1995).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

Mandell, R. B.

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of corneal topography from a videokeratographic reflection pattern,” Optom. Vision Sci. 72, 821–827 (1995).
[CrossRef]

Mattioli, R.

R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997).
[CrossRef]

Rand, R. H.

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vision Sci. 74, 926–930 (1997).
[CrossRef]

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, 652–660 (2002).

Spoelder, H. J. W.

F. M. Vos, G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, F. C. A. Groen, “A new PRBA-based instrument to measure the shape of the cornea,” IEEE Trans. Instrum. Meas. 46, 794–797 (1997).
[CrossRef]

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, 652–660 (2002).

Tripoli, N. K.

R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997).
[CrossRef]

van der Heijde, G. L.

M. Dubbelman, H. A. Weeber, G. L. van der Heijde, H. J. Völker-Dieben, “Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,” Acta Ophthalmol. Scand. 80, 379–383 (2002).
[CrossRef] [PubMed]

F. M. Vos, G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, F. C. A. Groen, “A new PRBA-based instrument to measure the shape of the cornea,” IEEE Trans. Instrum. Meas. 46, 794–797 (1997).
[CrossRef]

van Stokkum, I. H. M.

F. M. Vos, G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, F. C. A. Groen, “A new PRBA-based instrument to measure the shape of the cornea,” IEEE Trans. Instrum. Meas. 46, 794–797 (1997).
[CrossRef]

Völker-Dieben, H. J.

M. Dubbelman, H. A. Weeber, G. L. van der Heijde, H. J. Völker-Dieben, “Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,” Acta Ophthalmol. Scand. 80, 379–383 (2002).
[CrossRef] [PubMed]

Vos, F. M.

F. M. Vos, G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, F. C. A. Groen, “A new PRBA-based instrument to measure the shape of the cornea,” IEEE Trans. Instrum. Meas. 46, 794–797 (1997).
[CrossRef]

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, 652–660 (2002).

Weeber, H. A.

M. Dubbelman, H. A. Weeber, G. L. van der Heijde, H. J. Völker-Dieben, “Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,” Acta Ophthalmol. Scand. 80, 379–383 (2002).
[CrossRef] [PubMed]

Acta Ophthalmol. Scand.

M. Dubbelman, H. A. Weeber, G. L. van der Heijde, H. J. Völker-Dieben, “Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,” Acta Ophthalmol. Scand. 80, 379–383 (2002).
[CrossRef] [PubMed]

IEEE Trans. Instrum. Meas.

F. M. Vos, G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, F. C. A. Groen, “A new PRBA-based instrument to measure the shape of the cornea,” IEEE Trans. Instrum. Meas. 46, 794–797 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Refract. Surg.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, 652–660 (2002).

Opt. Lett.

Optom. Vision Sci.

M. A. Halstead, B. A. Barsky, S. A. Klein, R. B. Mandell, “A spline surface algorithm for reconstruction of corneal topography from a videokeratographic reflection pattern,” Optom. Vision Sci. 72, 821–827 (1995).
[CrossRef]

R. Mattioli, N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vision Sci. 74, 881–894 (1997).
[CrossRef]

S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vision Sci. 74, 931–944 (1997).
[CrossRef]

S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vision Sci. 74, 945–962 (1997).
[CrossRef]

R. H. Rand, H. C. Howland, R. A. Applegate, “Mathematical model of a Placido disk keratometer and its implications for recovery of corneal topography,” Optom. Vision Sci. 74, 926–930 (1997).
[CrossRef]

Other

A. W. Greynolds, “Superconic and subconic surfaces in optical design,” presented at the 2002 International Lens Design Conference, Tucson, Arizona, June 3–7, 2002, Postconference Digest, p. 2.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

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Figures (10)

Fig. 1
Fig. 1

Diagram showing reference axis used in describing a conic surface.

Fig. 2
Fig. 2

Schematic diagram of the Vrije University topographer.

Fig. 3
Fig. 3

Schematic diagram showing backward-ray-tracing principle used in the surface reconstruction algorithm.

Fig. 4
Fig. 4

Diagram used to derive Eq. (10).

Fig. 5
Fig. 5

Blowup of a section in Fig. 4.

Fig. 6
Fig. 6

Theoretical shape function.

Fig. 7
Fig. 7

Grayscale contour map (in millimeters) of (a) the theoretical shape function and its reconstruction for varying radial orders of (b) 8, (c) 12, (d) 16, (e) 20, (f) 24.

Fig. 8
Fig. 8

Signal-to-noise ratio versus radial order.

Fig. 9
Fig. 9

Effect of alignment on Zernike tilt and power term.

Fig. 10
Fig. 10

Effect of z position of surface on Zernike power term.

Tables (5)

Tables Icon

Table 1 Numerical Performance of the Algorithm on a Small Sphere a

Tables Icon

Table 2 Numerical Performance of the Algorithm on a Typical Cornea a

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Table 3 Numerical Performance of the Algorithm Applied to an Astigmatic Surface a

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Table 4 Numerical Performance of the Algorithm Applied to a Surface Described by Higher-Order Zernike Polynomial Z8-8 a

Tables Icon

Table 5 Measurements on Spherical Balls with Varying Radii

Equations (35)

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

s2-2rz+kz2=0.
s2=x2+y2.
z=s22r+ks48r3+k2s616r5+5k3s8128r7+ .
z=CnmZnm(ρ, θ),
s=rpρ,
r=rp2(23C20-65C40+127C60),
k=8r3rp3 (65C40-307C60).
xp=-u-(u/OA)zp,
yp=-v-(v/OA)zp,
zp=iKCi[Zi(-u, -v)-Zi(0, 0)]*1+u2+v2OA*AR,
tan α=u2+v2/OA,
tan β=u2+v2/AR,
AR=r/rp.
FPGβ.
tan βFGPG.
tan βFGEF.
tan α=EFDF.
FGDF tan α tan β.
OP=xp, yb, zp-0, 0, 0,
PS=xs, ys, zs-xp, yp, zp.
n=PSPS-OPOP.
tu=xpu xˆ+ypu yˆ+zpu zˆ,
tv=xpv xˆ+ypv yˆ+zpv zˆ.
tu=-1+u-uzpOAxˆ+u-vzpOAyˆ+zpu zˆ,
tv=v-uzpOAxˆ+-1-v-vzpOAyˆ+zpv zˆ.
ntu=-nx+nxu-uzpOA+nyu-vzpOA+nzzpu,
ntv=-ny+nxv-uzpOA+nyv-vzpOA+nzzpv.
ntu=-nx+iKMi(u, v)Ci,
ntv=-ny+iKNi(u, v)Ci,
y=nx1nxNny1nyN,B=M11M1KMN1MNKN11N1KNN1NNK,C=C1C2CK.
C=[BTB]-1[BTy].
z=s22r+qs2sin 2θ,
z=s22r+qs8sin 8θ.
z=s22r+g(s, θ),
g(s, θ)= sin θfors2mm2(s-1.5) sin θfor1.5<s<2mm0fors1.5mm.

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