Abstract

We analyze the accuracy of a laser keratopographer in the evaluation of corneal topography for nonsmooth corneal surfaces and when some noise is introduced into measured data. Through some numerical simulations, cosinesoidal deformations with different amplitudes and spatial periods are introduced on theoretical surfaces. Gaussian noise is introduced on the simulated x and y position coordinates for the measured position of the reflected beam in order to simulate detection errors that are due to vibrations or electric and other noise on the position-sensing detector. We found that the topography of the surface could be obtained with reliable accuracy if the height-to-width ratio of the deformations of the surface is smaller than 0.02 and the error in the detection of position at the position-sensing detector is under 0.5 mm.

© 2002 Optical Society of America

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  1. R. A. Applegate, H. C. Howland, “Noninvasive measurement of corneal topography,” IEEE Eng. Med. Biol. 14, 30–42 (1995).
    [CrossRef]
  2. C. Roberts, “Characterization of the inherent error in a spherical-biased corneal topography system in mapping a radially aspheric surface,” Refract. Corneal Surg. 10, 103–116 (1994).
  3. D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vis. Sci. 74, 895–898 (1997).
    [CrossRef] [PubMed]
  4. G. O. Waring, S. B. Hannush, S. J. Bogan, R. K. Maloney, “Classification of corneal topography with videokeratopography,” in Corneal Topography: Measuring and Modifying the CorneaD. T. Schanzlin, J. B. Robin, eds. (Springer-Verlag, New York, 1992), pp. 47–73.
    [CrossRef]
  5. M. Jeandervin, J. Barr, “Comparison of repeat videokeratography: repeatability and accuracy,” Optom. Vis. Sci. 75, 663–669 (1998).
    [CrossRef] [PubMed]
  6. S. A. Klein, “Axial curvature and skew ray error in corneal topography,” Optom. Vis. Sci. 74, 931–944 (1997).
    [CrossRef] [PubMed]
  7. J. R. Díaz-Uribe, P. Anaszkiewicz, R. Suárez-Sánchez, “Laser deflectometry keratopography,” in Optics in Medicine, Biology, and Environmental Research: Selected contributions to the First Conference on Optics within Life Sciences (OWLS I), G. von Bally, S. Khanna, eds. (Elsevier Science, Amsterdam, 1993), pp. 236–239.
  8. J. R. Díaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vis. Sci. 76, 40–49 (1999).
    [CrossRef] [PubMed]
  9. It is worth pointing out that currently the measuring time is less than 1/70 s; see G. Ascanio, A. Caballero, L. Ruiz, M. González, R. Díaz, “Scanning laser system to determine the corneal shape,” Applied Mechanics in the Americas (American Academy of Mechanics and the Brazilian Society of Mechanical Sciences, Rio de Janeiro, Brazil, 1998), Vol. 6, pp. 109–112.
  10. R. Lindsay, G. Smith, D. Atchison, “Descriptors of corneal shape,” Optom. Vis. Sci. 75, 156–158 (1998).
    [CrossRef] [PubMed]
  11. P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, New York, 1992), pp. 84–87.
  12. N. Barkakati, TURBO C++: Bible, 1st ed. (The Waite Group, Carmel, Ind., 1990).

1999 (1)

J. R. Díaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vis. Sci. 76, 40–49 (1999).
[CrossRef] [PubMed]

1998 (2)

R. Lindsay, G. Smith, D. Atchison, “Descriptors of corneal shape,” Optom. Vis. Sci. 75, 156–158 (1998).
[CrossRef] [PubMed]

M. Jeandervin, J. Barr, “Comparison of repeat videokeratography: repeatability and accuracy,” Optom. Vis. Sci. 75, 663–669 (1998).
[CrossRef] [PubMed]

1997 (2)

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

D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vis. Sci. 74, 895–898 (1997).
[CrossRef] [PubMed]

1995 (1)

R. A. Applegate, H. C. Howland, “Noninvasive measurement of corneal topography,” IEEE Eng. Med. Biol. 14, 30–42 (1995).
[CrossRef]

1994 (1)

C. Roberts, “Characterization of the inherent error in a spherical-biased corneal topography system in mapping a radially aspheric surface,” Refract. Corneal Surg. 10, 103–116 (1994).

Anaszkiewicz, P.

J. R. Díaz-Uribe, P. Anaszkiewicz, R. Suárez-Sánchez, “Laser deflectometry keratopography,” in Optics in Medicine, Biology, and Environmental Research: Selected contributions to the First Conference on Optics within Life Sciences (OWLS I), G. von Bally, S. Khanna, eds. (Elsevier Science, Amsterdam, 1993), pp. 236–239.

Applegate, R. A.

R. A. Applegate, H. C. Howland, “Noninvasive measurement of corneal topography,” IEEE Eng. Med. Biol. 14, 30–42 (1995).
[CrossRef]

Ascanio, G.

It is worth pointing out that currently the measuring time is less than 1/70 s; see G. Ascanio, A. Caballero, L. Ruiz, M. González, R. Díaz, “Scanning laser system to determine the corneal shape,” Applied Mechanics in the Americas (American Academy of Mechanics and the Brazilian Society of Mechanical Sciences, Rio de Janeiro, Brazil, 1998), Vol. 6, pp. 109–112.

Atchison, D.

R. Lindsay, G. Smith, D. Atchison, “Descriptors of corneal shape,” Optom. Vis. Sci. 75, 156–158 (1998).
[CrossRef] [PubMed]

Barkakati, N.

N. Barkakati, TURBO C++: Bible, 1st ed. (The Waite Group, Carmel, Ind., 1990).

Barr, J.

M. Jeandervin, J. Barr, “Comparison of repeat videokeratography: repeatability and accuracy,” Optom. Vis. Sci. 75, 663–669 (1998).
[CrossRef] [PubMed]

Bevington, P. R.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, New York, 1992), pp. 84–87.

Bogan, S. J.

G. O. Waring, S. B. Hannush, S. J. Bogan, R. K. Maloney, “Classification of corneal topography with videokeratopography,” in Corneal Topography: Measuring and Modifying the CorneaD. T. Schanzlin, J. B. Robin, eds. (Springer-Verlag, New York, 1992), pp. 47–73.
[CrossRef]

Brenner, D.

D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vis. Sci. 74, 895–898 (1997).
[CrossRef] [PubMed]

Caballero, A.

It is worth pointing out that currently the measuring time is less than 1/70 s; see G. Ascanio, A. Caballero, L. Ruiz, M. González, R. Díaz, “Scanning laser system to determine the corneal shape,” Applied Mechanics in the Americas (American Academy of Mechanics and the Brazilian Society of Mechanical Sciences, Rio de Janeiro, Brazil, 1998), Vol. 6, pp. 109–112.

Díaz, R.

It is worth pointing out that currently the measuring time is less than 1/70 s; see G. Ascanio, A. Caballero, L. Ruiz, M. González, R. Díaz, “Scanning laser system to determine the corneal shape,” Applied Mechanics in the Americas (American Academy of Mechanics and the Brazilian Society of Mechanical Sciences, Rio de Janeiro, Brazil, 1998), Vol. 6, pp. 109–112.

Díaz-Uribe, J. R.

J. R. Díaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vis. Sci. 76, 40–49 (1999).
[CrossRef] [PubMed]

J. R. Díaz-Uribe, P. Anaszkiewicz, R. Suárez-Sánchez, “Laser deflectometry keratopography,” in Optics in Medicine, Biology, and Environmental Research: Selected contributions to the First Conference on Optics within Life Sciences (OWLS I), G. von Bally, S. Khanna, eds. (Elsevier Science, Amsterdam, 1993), pp. 236–239.

González, M.

It is worth pointing out that currently the measuring time is less than 1/70 s; see G. Ascanio, A. Caballero, L. Ruiz, M. González, R. Díaz, “Scanning laser system to determine the corneal shape,” Applied Mechanics in the Americas (American Academy of Mechanics and the Brazilian Society of Mechanical Sciences, Rio de Janeiro, Brazil, 1998), Vol. 6, pp. 109–112.

Granados-Agustin, F.

J. R. Díaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vis. Sci. 76, 40–49 (1999).
[CrossRef] [PubMed]

Hannush, S. B.

G. O. Waring, S. B. Hannush, S. J. Bogan, R. K. Maloney, “Classification of corneal topography with videokeratopography,” in Corneal Topography: Measuring and Modifying the CorneaD. T. Schanzlin, J. B. Robin, eds. (Springer-Verlag, New York, 1992), pp. 47–73.
[CrossRef]

Howland, H. C.

R. A. Applegate, H. C. Howland, “Noninvasive measurement of corneal topography,” IEEE Eng. Med. Biol. 14, 30–42 (1995).
[CrossRef]

Jeandervin, M.

M. Jeandervin, J. Barr, “Comparison of repeat videokeratography: repeatability and accuracy,” Optom. Vis. Sci. 75, 663–669 (1998).
[CrossRef] [PubMed]

Klein, S. A.

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

Lindsay, R.

R. Lindsay, G. Smith, D. Atchison, “Descriptors of corneal shape,” Optom. Vis. Sci. 75, 156–158 (1998).
[CrossRef] [PubMed]

Maloney, R. K.

G. O. Waring, S. B. Hannush, S. J. Bogan, R. K. Maloney, “Classification of corneal topography with videokeratopography,” in Corneal Topography: Measuring and Modifying the CorneaD. T. Schanzlin, J. B. Robin, eds. (Springer-Verlag, New York, 1992), pp. 47–73.
[CrossRef]

Roberts, C.

C. Roberts, “Characterization of the inherent error in a spherical-biased corneal topography system in mapping a radially aspheric surface,” Refract. Corneal Surg. 10, 103–116 (1994).

Robinson, D. K.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, New York, 1992), pp. 84–87.

Ruiz, L.

It is worth pointing out that currently the measuring time is less than 1/70 s; see G. Ascanio, A. Caballero, L. Ruiz, M. González, R. Díaz, “Scanning laser system to determine the corneal shape,” Applied Mechanics in the Americas (American Academy of Mechanics and the Brazilian Society of Mechanical Sciences, Rio de Janeiro, Brazil, 1998), Vol. 6, pp. 109–112.

Smith, G.

R. Lindsay, G. Smith, D. Atchison, “Descriptors of corneal shape,” Optom. Vis. Sci. 75, 156–158 (1998).
[CrossRef] [PubMed]

Suárez-Sánchez, R.

J. R. Díaz-Uribe, P. Anaszkiewicz, R. Suárez-Sánchez, “Laser deflectometry keratopography,” in Optics in Medicine, Biology, and Environmental Research: Selected contributions to the First Conference on Optics within Life Sciences (OWLS I), G. von Bally, S. Khanna, eds. (Elsevier Science, Amsterdam, 1993), pp. 236–239.

Waring, G. O.

G. O. Waring, S. B. Hannush, S. J. Bogan, R. K. Maloney, “Classification of corneal topography with videokeratopography,” in Corneal Topography: Measuring and Modifying the CorneaD. T. Schanzlin, J. B. Robin, eds. (Springer-Verlag, New York, 1992), pp. 47–73.
[CrossRef]

IEEE Eng. Med. Biol. (1)

R. A. Applegate, H. C. Howland, “Noninvasive measurement of corneal topography,” IEEE Eng. Med. Biol. 14, 30–42 (1995).
[CrossRef]

Optom. Vis. Sci. (5)

D. Brenner, “Modeling the cornea with the topographic modeling system videokeratoscope,” Optom. Vis. Sci. 74, 895–898 (1997).
[CrossRef] [PubMed]

M. Jeandervin, J. Barr, “Comparison of repeat videokeratography: repeatability and accuracy,” Optom. Vis. Sci. 75, 663–669 (1998).
[CrossRef] [PubMed]

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

J. R. Díaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vis. Sci. 76, 40–49 (1999).
[CrossRef] [PubMed]

R. Lindsay, G. Smith, D. Atchison, “Descriptors of corneal shape,” Optom. Vis. Sci. 75, 156–158 (1998).
[CrossRef] [PubMed]

Refract. Corneal Surg. (1)

C. Roberts, “Characterization of the inherent error in a spherical-biased corneal topography system in mapping a radially aspheric surface,” Refract. Corneal Surg. 10, 103–116 (1994).

Other (5)

G. O. Waring, S. B. Hannush, S. J. Bogan, R. K. Maloney, “Classification of corneal topography with videokeratopography,” in Corneal Topography: Measuring and Modifying the CorneaD. T. Schanzlin, J. B. Robin, eds. (Springer-Verlag, New York, 1992), pp. 47–73.
[CrossRef]

It is worth pointing out that currently the measuring time is less than 1/70 s; see G. Ascanio, A. Caballero, L. Ruiz, M. González, R. Díaz, “Scanning laser system to determine the corneal shape,” Applied Mechanics in the Americas (American Academy of Mechanics and the Brazilian Society of Mechanical Sciences, Rio de Janeiro, Brazil, 1998), Vol. 6, pp. 109–112.

J. R. Díaz-Uribe, P. Anaszkiewicz, R. Suárez-Sánchez, “Laser deflectometry keratopography,” in Optics in Medicine, Biology, and Environmental Research: Selected contributions to the First Conference on Optics within Life Sciences (OWLS I), G. von Bally, S. Khanna, eds. (Elsevier Science, Amsterdam, 1993), pp. 236–239.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, New York, 1992), pp. 84–87.

N. Barkakati, TURBO C++: Bible, 1st ed. (The Waite Group, Carmel, Ind., 1990).

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Figures (11)

Fig. 1
Fig. 1

Setup of the LK: PSD, position-sensing detector; PBS, polarizing beam splitter.

Fig. 2
Fig. 2

Surface with oscillations (a) A = 0.3 mm, Λ = 10 mm; (b) A = 0.3 mm, Λ = 2.5 mm.

Fig. 3
Fig. 3

Deformation degree: (sph) reference sphere; (a) A = 0.3 mm, Λ a = 2.5 mm, M a = 0.24; (b) A = 0.6 mm, Λ b = 2.5 mm, M b = 0.48; (c) A = 0.6 mm, Λ c = 5 mm, M c = 0.24.

Fig. 4
Fig. 4

Simulated detection points obtained from the simulated corneal surface of parabolic shape: (a) R = 10 mm, N = 1000; (b) R = 8 mm, N = 933.

Fig. 5
Fig. 5

Plots of the differences between actual and calculated radial coordinates Δr for the parabolic surface case: (a) R = 10 mm, N = 1000; (b) R = 8 mm, N = 933.

Fig. 6
Fig. 6

Simulated detection points from the surface with oscillations: (a) M = 0.0005, A = 0.01 mm, Λ = 4 mm, N = 1000; (b) M = 0.015, A = 0.05 mm, Λ = 6.67 mm, N = 894.

Fig. 7
Fig. 7

Differences between actual and calculated radial coordinates for the surface with oscillations (a) M = 0.0005, A = 0.01 mm, Λ = 4 mm, N = 1000; (b) M = 0.015, A = 0.05 mm, Λ = 6.67 mm, N = 894.

Fig. 8
Fig. 8

Deformation degree M versus differences between the radial coordinates Δr for the surface with oscillations.

Fig. 9
Fig. 9

Noisy simulated detection points for (a) η = 0.1 mm, R = 10.0 mm, N = 1000; (b) η = 1.0 mm, R = 10.0 mm, N = 1000.

Fig. 10
Fig. 10

Differences between actual and calculated radial coordinate Δr for noisy data (parabolic surface case: η = 0.1 mm, R = 10.0 mm, N = 1000).

Fig. 11
Fig. 11

Noise parameter η versus the maximum values of the differences between the radial coordinates Δr max: (a) parabolic surface, (b) surface with oscillations.

Tables (4)

Tables Icon

Table 1 Δr max and Δr rms Values Obtained for the Parabolic Surface

Tables Icon

Table 2 Δr max and Δr rms Values Obtained for the Surface with Oscillations

Tables Icon

Table 3 Δr max and Δr rms Values Obtained for the Parabolic Surface with Noisy Data

Tables Icon

Table 4 Δr max and Δr rms Values Obtained for the Surface with Oscillations with Noisy Data

Equations (9)

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

r θ, ϕ=r θ0, ϕ0exp-nθnrdθ+nϕnrsin θdϕ,
Z=-12RX2+Y2+d,
X2+Y2+Z2= d+A cos2πYΛ2;
dZdY=- YZ+2AΛπdZsin2πYΛ.
slopetotal-slopespheremax= 2AΛdZsin2πYΛmax= 2AΛdZ,
M=2AΛ
x, yx+δx, y+δy.
δx=η2 χ1r1, r2, δy=η2 χ2r1, r2,
χ1r1, r2= -2ln r1cos2πr2, χ2r1, r2= -2ln r1sin2πr2,

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