Abstract

Videokeratoscopic data are generally displayed as a color-coded map of corneal refractive power, corneal curvature, or surface height. Although the merits of the refractive power and curvature methods have been extensively debated, the display of corneal surface height demands further investigation. A significant drawback to viewing corneal surface height is that the spherical and cylindrical components of the cornea obscure small variations in the surface. To overcome this drawback, a methodology for decomposing corneal height data into a unique set of Zernike polynomials is presented. Repeatedly removing the low-order Zernike terms reveals the hidden height variations. Examples of the decomposition-and-display technique are shown for cases of astigmatism, keratoconus, and radial keratotomy.

© 1995 Optical Society of America

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References

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  1. C. Roberts, “The accuracy of ‘power’ maps to display curvature data in corneal topography systems,” Invest. Ophthalmol. Vis. Sci. 35, 3525–3532 (1994).
    [PubMed]
  2. R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, R. P. Sharp, R. W. Yee, “Corneal aberrations before and after radial keratotomy (RK) calculated from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 58–61.
  3. H. C. Howland, J. Buettner, R. A. Applegate, “Computation of the shapes of normal corneas and their monochromatic aberrations from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 54–57.
  4. J. P. Carroll, “A method to describe corneal topography,” Optom. Vis. Sci. 71, 259–264 (1994).
    [CrossRef] [PubMed]
  5. S. R. Lange, E. H. Thall, “Interoperative corneal topographic measurement using phase-shifted projected fringe contouring,” in Ophthalmic and Visual Optics, Vol. 3 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 28–31.
  6. R. H. Webb, “Zernike polynomial description of ophthalmic surfaces,” in Ophthalmic and Visual Optics, Vol. 3 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 38–41.
  7. C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. Wyant, eds. (Academic, New York, 1992), Vol. 10, pp. 193–221.
  8. D. Malacara, “Wavefront fitting with discrete orthogonal polynomials in a units radius circle,” Opt. Eng. 29, 672–675 (1990).
    [CrossRef]
  9. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).Note: Eq. (34) should read aj=(bj-∑k=j+1Nαkjak)/αjj.
    [CrossRef] [PubMed]
  10. E. H. Thall, S. R. Lange, “Preliminary results of a new intraoperative corneal topography technique,” J. Cataract Refract. Surg. 19, 193–197 (1993).
    [CrossRef] [PubMed]
  11. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1968).
  12. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 99–107.
  13. Robert Parks, NIST, Gaithersburg, Md. 20899 (personal communication, March1995).
  14. J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Videokeratoscopic measurement of toric surfaces: accuracy analysis of the Computed Anatomy TMS-1, the EyeSys Laboratories Corneal Measurement System, and the Visioptic EH-270,” submitted to J. Refract. Corneal Surg.

1994 (2)

C. Roberts, “The accuracy of ‘power’ maps to display curvature data in corneal topography systems,” Invest. Ophthalmol. Vis. Sci. 35, 3525–3532 (1994).
[PubMed]

J. P. Carroll, “A method to describe corneal topography,” Optom. Vis. Sci. 71, 259–264 (1994).
[CrossRef] [PubMed]

1993 (1)

E. H. Thall, S. R. Lange, “Preliminary results of a new intraoperative corneal topography technique,” J. Cataract Refract. Surg. 19, 193–197 (1993).
[CrossRef] [PubMed]

1990 (1)

D. Malacara, “Wavefront fitting with discrete orthogonal polynomials in a units radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

1980 (1)

Applegate, R. A.

H. C. Howland, J. Buettner, R. A. Applegate, “Computation of the shapes of normal corneas and their monochromatic aberrations from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 54–57.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, R. P. Sharp, R. W. Yee, “Corneal aberrations before and after radial keratotomy (RK) calculated from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 58–61.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1968).

Buettner, J.

H. C. Howland, J. Buettner, R. A. Applegate, “Computation of the shapes of normal corneas and their monochromatic aberrations from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 54–57.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, R. P. Sharp, R. W. Yee, “Corneal aberrations before and after radial keratotomy (RK) calculated from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 58–61.

Carroll, J. P.

J. P. Carroll, “A method to describe corneal topography,” Optom. Vis. Sci. 71, 259–264 (1994).
[CrossRef] [PubMed]

Cottinghan, A. J.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, R. P. Sharp, R. W. Yee, “Corneal aberrations before and after radial keratotomy (RK) calculated from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 58–61.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 99–107.

Greivenkamp, J. E.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Videokeratoscopic measurement of toric surfaces: accuracy analysis of the Computed Anatomy TMS-1, the EyeSys Laboratories Corneal Measurement System, and the Visioptic EH-270,” submitted to J. Refract. Corneal Surg.

Howland, H. C.

H. C. Howland, J. Buettner, R. A. Applegate, “Computation of the shapes of normal corneas and their monochromatic aberrations from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 54–57.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, R. P. Sharp, R. W. Yee, “Corneal aberrations before and after radial keratotomy (RK) calculated from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 58–61.

Kim, C.-J.

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. Wyant, eds. (Academic, New York, 1992), Vol. 10, pp. 193–221.

Lange, S. R.

E. H. Thall, S. R. Lange, “Preliminary results of a new intraoperative corneal topography technique,” J. Cataract Refract. Surg. 19, 193–197 (1993).
[CrossRef] [PubMed]

S. R. Lange, E. H. Thall, “Interoperative corneal topographic measurement using phase-shifted projected fringe contouring,” in Ophthalmic and Visual Optics, Vol. 3 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 28–31.

Lowman, A. E.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Videokeratoscopic measurement of toric surfaces: accuracy analysis of the Computed Anatomy TMS-1, the EyeSys Laboratories Corneal Measurement System, and the Visioptic EH-270,” submitted to J. Refract. Corneal Surg.

Malacara, D.

D. Malacara, “Wavefront fitting with discrete orthogonal polynomials in a units radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Mellinger, M. D.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Videokeratoscopic measurement of toric surfaces: accuracy analysis of the Computed Anatomy TMS-1, the EyeSys Laboratories Corneal Measurement System, and the Visioptic EH-270,” submitted to J. Refract. Corneal Surg.

Miller, J. M.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Videokeratoscopic measurement of toric surfaces: accuracy analysis of the Computed Anatomy TMS-1, the EyeSys Laboratories Corneal Measurement System, and the Visioptic EH-270,” submitted to J. Refract. Corneal Surg.

Parks, Robert

Robert Parks, NIST, Gaithersburg, Md. 20899 (personal communication, March1995).

Roberts, C.

C. Roberts, “The accuracy of ‘power’ maps to display curvature data in corneal topography systems,” Invest. Ophthalmol. Vis. Sci. 35, 3525–3532 (1994).
[PubMed]

Schwiegerling, J. T.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Videokeratoscopic measurement of toric surfaces: accuracy analysis of the Computed Anatomy TMS-1, the EyeSys Laboratories Corneal Measurement System, and the Visioptic EH-270,” submitted to J. Refract. Corneal Surg.

Shannon, R. R.

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. Wyant, eds. (Academic, New York, 1992), Vol. 10, pp. 193–221.

Sharp, R. P.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, R. P. Sharp, R. W. Yee, “Corneal aberrations before and after radial keratotomy (RK) calculated from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 58–61.

Silva, D. E.

Snyder, R. W.

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Videokeratoscopic measurement of toric surfaces: accuracy analysis of the Computed Anatomy TMS-1, the EyeSys Laboratories Corneal Measurement System, and the Visioptic EH-270,” submitted to J. Refract. Corneal Surg.

Thall, E. H.

E. H. Thall, S. R. Lange, “Preliminary results of a new intraoperative corneal topography technique,” J. Cataract Refract. Surg. 19, 193–197 (1993).
[CrossRef] [PubMed]

S. R. Lange, E. H. Thall, “Interoperative corneal topographic measurement using phase-shifted projected fringe contouring,” in Ophthalmic and Visual Optics, Vol. 3 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 28–31.

Wang, J. Y.

Webb, R. H.

R. H. Webb, “Zernike polynomial description of ophthalmic surfaces,” in Ophthalmic and Visual Optics, Vol. 3 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 38–41.

Yee, R. W.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, R. P. Sharp, R. W. Yee, “Corneal aberrations before and after radial keratotomy (RK) calculated from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 58–61.

Appl. Opt. (1)

Invest. Ophthalmol. Vis. Sci. (1)

C. Roberts, “The accuracy of ‘power’ maps to display curvature data in corneal topography systems,” Invest. Ophthalmol. Vis. Sci. 35, 3525–3532 (1994).
[PubMed]

J. Cataract Refract. Surg. (1)

E. H. Thall, S. R. Lange, “Preliminary results of a new intraoperative corneal topography technique,” J. Cataract Refract. Surg. 19, 193–197 (1993).
[CrossRef] [PubMed]

Opt. Eng. (1)

D. Malacara, “Wavefront fitting with discrete orthogonal polynomials in a units radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Optom. Vis. Sci. (1)

J. P. Carroll, “A method to describe corneal topography,” Optom. Vis. Sci. 71, 259–264 (1994).
[CrossRef] [PubMed]

Other (9)

S. R. Lange, E. H. Thall, “Interoperative corneal topographic measurement using phase-shifted projected fringe contouring,” in Ophthalmic and Visual Optics, Vol. 3 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 28–31.

R. H. Webb, “Zernike polynomial description of ophthalmic surfaces,” in Ophthalmic and Visual Optics, Vol. 3 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 38–41.

C.-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Applied Optics and Optical Engineering, R. R. Shannon, J. Wyant, eds. (Academic, New York, 1992), Vol. 10, pp. 193–221.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1968).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 99–107.

Robert Parks, NIST, Gaithersburg, Md. 20899 (personal communication, March1995).

J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiegerling, A. E. Lowman, J. M. Miller, “Videokeratoscopic measurement of toric surfaces: accuracy analysis of the Computed Anatomy TMS-1, the EyeSys Laboratories Corneal Measurement System, and the Visioptic EH-270,” submitted to J. Refract. Corneal Surg.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, R. P. Sharp, R. W. Yee, “Corneal aberrations before and after radial keratotomy (RK) calculated from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 58–61.

H. C. Howland, J. Buettner, R. A. Applegate, “Computation of the shapes of normal corneas and their monochromatic aberrations from videokeratometric measurements,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 54–57.

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Figures (5)

Fig. 1
Fig. 1

Zernike polynomials: (a) Z00, (b) Z11 (Z1−1 is the same as Z11 but rotated 90°), (c) Z20, (d) Z2−2 (Z22 is the same as Z2−2 but rotated 45°).

Fig. 2
Fig. 2

Height maps of corneal astigmatism: (a) raw height data, (b) raw height data minus the parabolic term, (c) height data of (b) minus the cylindrical term. White represents a high point on the cornea, and black represents a low point. Diameter 7.0 mm.

Fig. 3
Fig. 3

Height maps of advanced keratoconus: (a) raw height data, (b) raw height data minus the parabolic term, (c) height data of (b) minus the cylindrical term. White represents a high point on the cornea, and black represents a low point. Diameter 5.5 mm.

Fig. 4
Fig. 4

Height maps of six-incision radial keratotomy: (a) raw height data, (b) raw height data minus the parabolic term, (c) height data of (b) minus the Zernike terms, with n ≤ 6 and m < 6. White represents a high point on the cornea, and black represents a low point. Diameter 8.0 mm.

Fig. 5
Fig. 5

Height maps of eight-incision radial keratotomy: (a) raw height data, (b) raw height data minus the parabolic term, (c) height data of (b) minus the Zernike terms, with n ≤ 8 and m < 8. White represents a high point on the cornea, and black represents a low point. Diameter 7.6 mm.

Tables (1)

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Table 1 Actual and Calculated Radii of Curvature in Terms of R × R for Several Toric Surfaces

Equations (34)

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f ( x , y ) j = 1 a j g j ( x , y ) ,
f ( x , y ) = a 1 + a 2 x + a 3 y + a 4 x 2 + a 5 y 2 + a 6 x y + ,
A g j ( x , y ) g k ( x , y ) d x d y = { 0 for j k 1 for j = k .
a j = A g j ( x , y ) f ( x , y ) d x d y .
Z n ± m = { 2 ( n + 1 ) R n m ( ρ ) cos m θ for + m 2 ( n + 1 ) R n m ( ρ ) sin m θ for - m ( n + 1 ) R n m ( ρ ) for m = 0 ,
R n m ( ρ ) = s = 0 ( n - m ) / 2 ( - 1 ) s ( n - s ) ! s ! [ ( n + m ) 2 - s ] ! [ ( n + m ) 2 - s ] ! ρ n - 2 s ,
Z 0 0 ( ρ , θ ) = 1 ,
Z 1 1 ( ρ , θ ) = 2 ρ cos θ ,
Z 1 - 1 ( ρ , θ ) = 2 ρ sin θ ,
Z 2 0 ( ρ , θ ) = 3 ( 2 ρ 2 - 1 ) ,
Z 2 2 ( ρ , θ ) = 6 ρ 2 cos 2 θ ,
Z 2 - 2 ( ρ , θ ) = 6 ρ 2 sin 2 θ .
i Z n ± m ( ρ i , θ i ) Z n ± m ( ρ i , θ i ) δ n n δ m m ,
U n ± m ( ρ i , θ i ) = n , ± m n , ± m b n , ± m , n , ± m Z n ± m ( ρ i , θ i ) .
i U n ± m ( ρ i , θ i ) U n ± m ( ρ i , θ i ) = δ n n δ m m .
f ( ρ i , θ i ) = n , ± m a n , ± m Z n ± m ( ρ i , θ i ) ,
6 a 2 , - 2 ρ 2 sin ( 2 θ ) + 6 a 2 , 2 ρ 2 cos ( 2 θ ) .
a 2 , - 2 cos ( 2 θ ) - a 2 , 2 sin ( 2 θ ) = 0.
θ 0 = 1 2 tan - 1 ( a 2 , - 2 a 2 , 2 ) ,
θ a = { θ 0 for a 2 , - 2 sin 2 θ 0 + a 2 , 2 cos 2 θ 0 < 0 θ 0 + 90 ° for a 2 , - 2 sin 2 θ 0 + a 2 , 2 cos 2 θ 0 > 0 .
Z 2 0 ( ρ , θ ) = 3 ( 2 ρ 2 - 1 ) ,
Z 2 2 ( ρ , θ ) = 6 ρ 2 cos 2 θ ,
Z 2 - 2 ( ρ , θ ) = 6 ρ 2 sin 2 θ ,
Z 4 0 ( ρ , θ ) = 5 ( 6 ρ 4 - 6 ρ 2 + 1 ) ,
Z 4 2 ( ρ , θ ) = 10 ( 4 ρ 2 - 3 ) ρ 2 cos 2 θ ,
Z 4 - 2 ( ρ , θ ) = 10 ( 4 ρ 2 - 3 ) ρ 2 sin 2 θ .
sag = r 2 2 R 0 = ρ 2 r max 2 2 R 0 ,
ρ 2 r max 2 2 R = 2 3 a 2 , 0 ρ 2 + 6 a 2 , 2 ρ 2 cos 2 θ 0 + 6 a 2 , - 2 ρ 2   sin 2 θ 0 - 6 5 a 4 , 0 ρ 2 - 3 10 a 4 , 2 ρ 2   cos 2 θ 0 - 3 10 a 4 , - 2 ρ 2 sin 2 θ 0 + .
R = r max 2 2 [ 2 3 a 2 , 0 - 6 5 a 4 , 0 + 6 ( a 2 , 2 cos 2 θ 0 + a 2 , - 2 sin 2 θ 0 ) - 3 10 ( a 4 , 2 cos 2 θ 0 + a 4 , - 2 sin 2 θ 0 ) ] .
Φ = 1000 n - 1 R ,
R = r max 2 2 [ 2 3 a 2 , 0 - 6 5 a 4 , 0 - 6 ( a 2 , 2 cos 2 θ 0 + a 2 , - 2 sin 2 θ 0 ) + 3 10 ( a 4 , 2 cos 2 θ 0 + a 4 , - 2 sin 2 θ 0 ) ] ,
Φ = 1000 n - 1 R .
Φ a = Φ - Φ .
aj=(bj-k=j+1Nαkjak)/αjj.

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