Abstract

Recently Sicam et al. [J. Opt. Soc. Am. A 21, 1300 (2004) ] presented a new corneal reconstruction algorithm for estimating corneal sag by Zernike polynomials. An equivalent but simpler derivation of the model equations is presented. The algorithm is tested on a sphere, a conic, and a toric. These tests reveal significant height errors that accrue with distance from the corneal apex. Additional postprocessing steps are introduced to circumvent these errors. A consistent and significant reduction in height errors is observed across the test surfaces. Finally, Sicam et al. used the conic p-value p as a measure of algorithm efficacy. Further investigation shows that the finite Zernike representation affected the reported results. The p-value should therefore be used with caution as an efficacy measure.

© 2007 Optical Society of America

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References

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  1. S. E. Wilson, "Computerized corneal topography and its importance to wavefront technology," Cornea 20, 441-454 (2001).
    [CrossRef] [PubMed]
  2. R. A. Applegate, "Noninvasive measurement of corneal topography," IEEE Eng. Med. Biol. Mag. 14, 30-42 (1995).
    [CrossRef]
  3. S. A. Klein, "Axial curvature and the skew ray error in corneal topography," Optom. Vision Sci. 74, 931-944 (1997).
    [CrossRef]
  4. 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]
  5. M. A. Halstead, B. A. Barsky, S. A. Klein, and 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]
  6. F. M. Vos, H. J. W. Spoelder, D. M. Germans, R. Hofman, and H. Bal, "Real-time, adaptive measurement of corneal shapes," Comput. Sci. Eng. 4, 66-76 (2002).
    [CrossRef]
  7. F. M. Vos, R. G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, and F. C. A. Groen, "A new instrument to measure the shape of the cornea based on pseudorandom color coding," IEEE Trans. Instrum. Meas. 46, 794-797 (1997).
    [CrossRef]
  8. V. A. Sicam, J. Coppens, T. J. T. P. van der Berg, and R. G. L. van der Heijde, "Corneal surface reconstruction algorithm that uses Zernike polynomial representation," J. Opt. Soc. Am. A 21, 1300-1306 (2004).
    [CrossRef]
  9. M. A. Halstead, B. A. Barsky, S. A. Klein, and R. B. Mandell, "Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals," in Proceedings of the Twenty-Third Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 335-342.
  10. ANSI Z80.28-2004 Methods for reporting optical aberrations of the eye.
  11. ISO 19980:2005 Ophthalmic instruments--corneal topographers.
  12. J. Schwiegerling, J. E. Greivenkamp, and J. M. Miller, "Representation of videokeratoscopic height data with Zernike polynomials," J. Opt. Soc. Am. A 12, 2105-2113 (1995).
    [CrossRef]
  13. R. H. Rand, H. C. Howland, and 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]
  14. J. Turuwhenua and J. Henderson, "A novel low-order method for recovery of the corneal shape," Optom. Vision Sci. 81, 863-871 (2004).
    [CrossRef]

2004

2002

F. M. Vos, H. J. W. Spoelder, D. M. Germans, R. Hofman, and H. Bal, "Real-time, adaptive measurement of corneal shapes," Comput. Sci. Eng. 4, 66-76 (2002).
[CrossRef]

2001

S. E. Wilson, "Computerized corneal topography and its importance to wavefront technology," Cornea 20, 441-454 (2001).
[CrossRef] [PubMed]

1997

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]

F. M. Vos, R. G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, and F. C. A. Groen, "A new instrument to measure the shape of the cornea based on pseudorandom color coding," IEEE Trans. Instrum. Meas. 46, 794-797 (1997).
[CrossRef]

R. H. Rand, H. C. Howland, and 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]

1995

J. Schwiegerling, J. E. Greivenkamp, and J. M. Miller, "Representation of videokeratoscopic height data with Zernike polynomials," J. Opt. Soc. Am. A 12, 2105-2113 (1995).
[CrossRef]

M. A. Halstead, B. A. Barsky, S. A. Klein, and 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. A. Applegate, "Noninvasive measurement of corneal topography," IEEE Eng. Med. Biol. Mag. 14, 30-42 (1995).
[CrossRef]

Applegate, R. A.

R. H. Rand, H. C. Howland, and 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]

R. A. Applegate, "Noninvasive measurement of corneal topography," IEEE Eng. Med. Biol. Mag. 14, 30-42 (1995).
[CrossRef]

Bal, H.

F. M. Vos, H. J. W. Spoelder, D. M. Germans, R. Hofman, and H. Bal, "Real-time, adaptive measurement of corneal shapes," Comput. Sci. Eng. 4, 66-76 (2002).
[CrossRef]

Barsky, B. A.

M. A. Halstead, B. A. Barsky, S. A. Klein, and 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]

M. A. Halstead, B. A. Barsky, S. A. Klein, and R. B. Mandell, "Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals," in Proceedings of the Twenty-Third Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 335-342.

Coppens, J.

Germans, D. M.

F. M. Vos, H. J. W. Spoelder, D. M. Germans, R. Hofman, and H. Bal, "Real-time, adaptive measurement of corneal shapes," Comput. Sci. Eng. 4, 66-76 (2002).
[CrossRef]

Greivenkamp, J. E.

Groen, F. C. A.

F. M. Vos, R. G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, and F. C. A. Groen, "A new instrument to measure the shape of the cornea based on pseudorandom color coding," IEEE Trans. Instrum. Meas. 46, 794-797 (1997).
[CrossRef]

Halstead, M. A.

M. A. Halstead, B. A. Barsky, S. A. Klein, and 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]

M. A. Halstead, B. A. Barsky, S. A. Klein, and R. B. Mandell, "Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals," in Proceedings of the Twenty-Third Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 335-342.

Henderson, J.

J. Turuwhenua and J. Henderson, "A novel low-order method for recovery of the corneal shape," Optom. Vision Sci. 81, 863-871 (2004).
[CrossRef]

Hofman, R.

F. M. Vos, H. J. W. Spoelder, D. M. Germans, R. Hofman, and H. Bal, "Real-time, adaptive measurement of corneal shapes," Comput. Sci. Eng. 4, 66-76 (2002).
[CrossRef]

Howland, H. C.

R. H. Rand, H. C. Howland, and 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, "Corneal topography reconstruction algorithm that avoids the skew-ray ambiguity and the skew-ray error," Optom. Vision Sci. 74, 945-962 (1997).
[CrossRef]

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

M. A. Halstead, B. A. Barsky, S. A. Klein, and 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]

M. A. Halstead, B. A. Barsky, S. A. Klein, and R. B. Mandell, "Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals," in Proceedings of the Twenty-Third Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 335-342.

Mandell, R. B.

M. A. Halstead, B. A. Barsky, S. A. Klein, and 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]

M. A. Halstead, B. A. Barsky, S. A. Klein, and R. B. Mandell, "Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals," in Proceedings of the Twenty-Third Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 335-342.

Miller, J. M.

Rand, R. H.

R. H. Rand, H. C. Howland, and 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.

Sicam, V. A.

Spoelder, H. J. W.

F. M. Vos, H. J. W. Spoelder, D. M. Germans, R. Hofman, and H. Bal, "Real-time, adaptive measurement of corneal shapes," Comput. Sci. Eng. 4, 66-76 (2002).
[CrossRef]

F. M. Vos, R. G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, and F. C. A. Groen, "A new instrument to measure the shape of the cornea based on pseudorandom color coding," IEEE Trans. Instrum. Meas. 46, 794-797 (1997).
[CrossRef]

Turuwhenua, J.

J. Turuwhenua and J. Henderson, "A novel low-order method for recovery of the corneal shape," Optom. Vision Sci. 81, 863-871 (2004).
[CrossRef]

van der Berg, T. J. T. P.

van der Heijde, R. G. L.

V. A. Sicam, J. Coppens, T. J. T. P. van der Berg, and R. G. L. van der Heijde, "Corneal surface reconstruction algorithm that uses Zernike polynomial representation," J. Opt. Soc. Am. A 21, 1300-1306 (2004).
[CrossRef]

F. M. Vos, R. G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, and F. C. A. Groen, "A new instrument to measure the shape of the cornea based on pseudorandom color coding," IEEE Trans. Instrum. Meas. 46, 794-797 (1997).
[CrossRef]

van Stokkum, I. H. M.

F. M. Vos, R. G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, and F. C. A. Groen, "A new instrument to measure the shape of the cornea based on pseudorandom color coding," IEEE Trans. Instrum. Meas. 46, 794-797 (1997).
[CrossRef]

Vos, F. M.

F. M. Vos, H. J. W. Spoelder, D. M. Germans, R. Hofman, and H. Bal, "Real-time, adaptive measurement of corneal shapes," Comput. Sci. Eng. 4, 66-76 (2002).
[CrossRef]

F. M. Vos, R. G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, and F. C. A. Groen, "A new instrument to measure the shape of the cornea based on pseudorandom color coding," IEEE Trans. Instrum. Meas. 46, 794-797 (1997).
[CrossRef]

Wilson, S. E.

S. E. Wilson, "Computerized corneal topography and its importance to wavefront technology," Cornea 20, 441-454 (2001).
[CrossRef] [PubMed]

Comput. Sci. Eng.

F. M. Vos, H. J. W. Spoelder, D. M. Germans, R. Hofman, and H. Bal, "Real-time, adaptive measurement of corneal shapes," Comput. Sci. Eng. 4, 66-76 (2002).
[CrossRef]

Cornea

S. E. Wilson, "Computerized corneal topography and its importance to wavefront technology," Cornea 20, 441-454 (2001).
[CrossRef] [PubMed]

IEEE Eng. Med. Biol. Mag.

R. A. Applegate, "Noninvasive measurement of corneal topography," IEEE Eng. Med. Biol. Mag. 14, 30-42 (1995).
[CrossRef]

IEEE Trans. Instrum. Meas.

F. M. Vos, R. G. L. van der Heijde, H. J. W. Spoelder, I. H. M. van Stokkum, and F. C. A. Groen, "A new instrument to measure the shape of the cornea based on pseudorandom color coding," IEEE Trans. Instrum. Meas. 46, 794-797 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Optom. Vision Sci.

R. H. Rand, H. C. Howland, and 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]

J. Turuwhenua and J. Henderson, "A novel low-order method for recovery of the corneal shape," Optom. Vision Sci. 81, 863-871 (2004).
[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]

M. A. Halstead, B. A. Barsky, S. A. Klein, and 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]

Other

M. A. Halstead, B. A. Barsky, S. A. Klein, and R. B. Mandell, "Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals," in Proceedings of the Twenty-Third Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 335-342.

ANSI Z80.28-2004 Methods for reporting optical aberrations of the eye.

ISO 19980:2005 Ophthalmic instruments--corneal topographers.

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Figures (7)

Fig. 1
Fig. 1

Overall setup used to recover corneal shape. A ray from a source point S reflects at the corneal surface P, passes through the nodal point O to the CCD plane. The correspondence between a measured point C on the CCD plane and a source point S allows a reconstruction of corneal shape.

Fig. 2
Fig. 2

Closer view of the region where the ray strikes the corneal surface. The distance z = D F is the corneal sag. This point is to be determined from the nearby ray that passes through D and strikes the surface at intersection point P.

Fig. 3
Fig. 3

Reconstruction results for the Sicam (RC) method applied to (a) a sphere ( R = 7 mm ) and (b) a conic ( R = 7.87 , p = 0.82 ). The discrete points show error profiles at each pupil setting ( 7 to 11 mm ) . The continuous lines show predicted deviation, that is, the predicted error incurred by the Sicam (RC) method.

Fig. 4
Fig. 4

Reconstruction results for the Sicam ( RC + M ) method applied to (a) a sphere ( R = 7 mm ) and (b) a conic ( R = 7.87 , p = 0.82 ). The discrete points show error profiles at each pupil setting ( 7 to 11 mm ) . Note the large reductions in error after the improved method has been applied.

Fig. 5
Fig. 5

Reconstruction results for a toric surface along the vertical meridian applied (a) using the Sicam (RC) method and (b) the Sicam ( RC + M ) method. The predicted deviation has been determined for the vertical meridian and is plotted as the continuous line in (a). Note the large reductions in error after the improved method has been applied.

Fig. 6
Fig. 6

Reconstruction results over the entire toric surface (a) using the Sicam (RC) method and (b) the Sicam ( RC + M ) method.

Fig. 7
Fig. 7

Reconstruction results for the RK example showing errors as a function of meridian. Errors are shown at the outer edge ( 2 mm ) of the band of corrugations for angles up to 90°. The resulting curves are consistently submicrometer for Zernike orders greater than or equal to 16. However, increasing the order beyond that limit gives relatively small reductions, within the order of magnitude ( 0.1 μ m ) . For order 20, the results are similar to those presented in Ref. [8].

Tables (3)

Tables Icon

Table 1 Estimates of R and p for the Sphere (Radial Order 10) a

Tables Icon

Table 2 Estimates of R and p for the Conic (Radial Order 10) a

Tables Icon

Table 3 Estimates of R and p for the Sphere (Radial Order 14) a

Equations (47)

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

x p = u ( u O A ) z p ,
y p = v ( v O A ) z p ,
z p = z ( u , v ) ( 1 + u 2 + v 2 ( O A ) ( A R ) ) ,
z ( u , v ) = c n m ( Z n m ( u , v ) Z n m ( 0 , 0 ) ) ,
z p z + z ( ρ ) Δ ρ .
ρ 2 2 R z + p z 2 = 0 ,
z = ρ 2 2 R ,
z ( ρ ) = ρ R
Δ ρ = z ρ O A .
Δ ρ Δ ρ .
n = i ̂ + r ̂ ( i ̂ + r ̂ ) z ,
i ̂ = S P S P ,
r ̂ = C O C O ,
n P u = 0 ,
n P v = 0 .
n x = m , n c n m U n m ,
n y = m , n c n m V n m ,
U n m = ( 1 + n y v + n x u O A ) γ n m u n x O A γ n m ,
V n m = ( 1 + n y v + n x u O A ) γ n m v n y O A γ n m ,
γ n m = ( Z n m ( u , v ) Z n m ( 0 , 0 ) ) ( 1 + u 2 + v 2 ( O A ) ( A R ) ) .
z p = n , m c n m ( Z n m ( u , v ) Z n m ( 0 , 0 ) ) ( 1 + u 2 + v 2 ( O A ) ( A R ) ) = n , m c n m γ n m .
y = ( n x ( u 1 , v 1 ) n y ( u 1 , v 1 ) n x ( u N , v N ) n y ( u N , v N ) ) ,
M = [ U 0 ( u 1 , v 1 ) U 1 ( u 1 , v 1 ) U J 1 ( u 1 , v 1 ) V 0 ( u 1 , v 1 ) V 1 ( u 1 , v 1 ) V J 1 ( u 1 , v 1 ) U 0 ( u N , v N ) U 1 ( u N , v N ) U J 1 ( u N , v N ) V 0 ( u N , v N ) V 1 ( u N , v N ) V J 1 ( u N , v N ) ] ,
C = ( c 0 c 1 c J 1 ) .
C = [ M T M ] 1 [ M T y ] ,
y = ( z p ( u 1 , v 1 ) z p ( u N , v N ) ) ,
M = [ Z 0 ( x p , 1 , y p , 1 ) Z J 1 ( x p , 1 , y p , 1 ) Z 0 ( x p , N , y p , N ) Z J 1 ( x p , N + y p , N ) ] , C = ( c ̂ 0 c ̂ 1 c ̂ J 1 ) ,
R = a 2 2 ( 2 3 c 2 0 6 5 c 4 0 + 12 7 c 6 0 60 c 8 0 + 30 11 c 10 0 ) ,
p = ( 8 R 3 a 4 ) ( 6 5 c 4 0 30 7 c 6 0 + 270 c 8 0 210 11 c 10 0 ) ,
c n m = 1 π 0 2 π 0 1 z a ( a ρ , θ ) Z n m ( ρ , θ ) ρ d ρ d θ
z = ρ 2 2 R + g ( ρ , θ ) ,
g ( ρ , θ ) = { ε sin ( 8 θ ) , ρ 2 mm 2 ( ρ 1.5 ) ε sin ( 8 θ ) , 1.5 < ρ < 2 mm 0 , ρ 1.5 mm }
P u = u ( x p , y p , z p ) ,
n x x p u + n y y p u z p u = 0 ,
x p u = 1 ( z p O A ) ( u O A ) z p u ,
y p u = ( v O A ) z p u .
n x ( 1 + z p O A ) + ( 1 + n y v + n x u O A ) z p u = 0 ,
z p = n , m c n m γ n m ,
γ n m = ( Z n m ( u , v ) Z n m ( 0 , 0 ) ) ( 1 + u 2 + v 2 ( O A ) ( A R ) )
z p u = c n m γ n m u .
n x = n , m c n m U n m ,
U n m = ( 1 + n y v + n x u O A ) γ n m u n x O A γ n m ,
γ n m u = ( Z n m ( u , v ) u ) ( 1 + u 2 + v 2 ( O A ) ( A R ) ) + ( Z n m ( u , v ) Z n m ( 0 , 0 ) ) ( 2 u ( OA ) ( A R ) ) .
P v = v ( x p , y p , z p )
n y = c n m V n m ,
V n m = ( 1 + n y v + n x u O A ) γ n m v n y O A γ n m ,
γ n m v = ( Z n m ( u , v ) v ) ( 1 + u 2 + v 2 ( O A ) ( A R ) ) + ( Z n m ( u , v ) Z n m ( 0 , 0 ) ) ( 2 v ( O A ) ( A R ) ) .

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