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An alternative mathematical algorithm for the photo- and videokeratoscope. (English) Zbl 1112.34320
Summary: Due to the resolution of current laser technology, the accuracy of corneal topography as measured by the videokeratoscope is no longer adequate to provide precise enough data for refractive surgery or for the fitting of customized contact lenses. We present an algorithm for recovering corneal topography that makes use of modern differential geometric techniques and numerical descent in Sobolev spaces. We believe this algorithm may be used with the photo- and videokeratoscope to increase the accuracy of the recovered corneal topography.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
92C55 Biomedical imaging and signal processing
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[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0314.46030
[2] Alberto, L.; Carvalho, V.; Castro, J.C.; Antonio, L.; Carvalho, V., Measuring higher order optical aberrations of the human eye: techniques and applications, Braz. J. med. biol. res., 35, 11, 1395-1406, (2002)
[3] Cauchy, P.L., Méthode générale pour la résolution des systemes d’équations simultanées, C. R. acad. sci. Paris, 25, (1847)
[4] Dix, J.G.; McCabe, T.W., On finding equilibria for isotropic hyperelastic materials, Nonlinear anal., 15, 437-444, (1990) · Zbl 0721.73003
[5] Doss, J.D.; Hutson, R.L.; Rowsey, J.J.; Brown, R., Method for calculation of corneal profile and power distribution, Arch. ophthalmol., 99, 1261-1265, (1981)
[6] J. Garza, Using steepest descent to find energy-minimizing maps satisfying nonlinear constraints, Dissertation, University of North Texas, 1994.
[7] Hannush, S.B.; Crawford, S.L.; Waring III, G.O.; Gemmill, M.C.; Lynn, M.J.; Nizam, A., Accuracy and precision of keratometry, photokeratoscopy, and corneal modeling on calibrated steel balls, Arch. ophthalmol., 107, 1235-1239, (1989)
[8] Hannush, S.B.; Crawford, S.L.; Waring III, G.O.; Gemmill, M.C.; Lynn, M.J.; Nizam, A., Reproducibility of normal corneal power measurements with a keratometer, photokeratoscope, and video imaging system, Arch. ophthalmol., 108, 539-544, (1990)
[9] Hemenger, R.P.; Tomlinson, A.; Oliver, K., Corneal optics from videokeratographs, Ophthalmic. physiol. optom., 15, 63-68, (1995)
[10] K. Kim, Steepest descent for partial differential equations of mixed type, Dissertation, University of North Texas, 1992.
[11] Klyce, S.D., Corneal topography and the new wave, Cornea, 19, 5, 723-729, (2000)
[12] Knoll, A.H., Corneal contours in the general population as revealated by the photokeratoscope, Am. J. optom., 38, 389-397, (1961)
[13] Kufner, A., Weighted Sobolev spaces, (1985), Wiley New York · Zbl 0567.46009
[14] Luenberger, D.G., Linear and nonlinear programming, (1989), Addison-Wesley Reading, MA
[15] Mahavier, W.T., A convergence result for discrete steepest descent in weighted Sobolev spaces, J. appl. ABS. anal., 2, 1-2, 67-72, (1997) · Zbl 0941.65051
[16] Mahavier, W.T., A numerical method utilizing weighted Sobolev descent to solve singular differential equations, Nonlinear world, 4, 4, 435-456, (1997) · Zbl 0908.65060
[17] Mahavier, W.T., Solving boundary value problems numerically using steepest descent in Sobolev spaces, Missouri J. math. sci., 11, 1, 19-32, (1999) · Zbl 1097.65535
[18] Maloney, R.K.; Bogan, S.J.; Waring III, G.O., Determination of corneal image-forming properties from corneal topography, Am. J. ophthalmol., 115, 31-41, (1993)
[19] Mejía-Barbosa, Y.; Malacara-Hernández, D., A review of methods for measuring corneal topography, Optometry vision sci., 78, 4, 240-253, (2001)
[20] Neuberger, J.W., Steepest descent and differential equations, J. math. soc. Japan, 37, 187-195, (1985) · Zbl 0576.65053
[21] J.W. Neuberger, Sobolev Gradients and Differential Equations, Springer-Verlag Lecture Notes in Mathematics, vol. 67, Springer, Berlin, 1997. · Zbl 0935.35002
[22] Newman, E.; Oliker, V.I., Determining the intensities produced by reflected and refracted wave fronts in geometrical optics, J. opt. soc. am., 12, 4, 784-793, (1995)
[23] Oliker, V.I., Near radially symmetric solutions of an inverse problem in geometric optics, 3, 743-756, (1987) · Zbl 0658.35094
[24] Oliker, V.I., On reconstructing a reflecting surface from the scattering data in the geometric optics approximation, 5, 1, 51-65, (1989) · Zbl 0702.35242
[25] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[26] Plácido, A., Novo instrumento de exploracão da córnea, Periodico d’oftalmologia pratica, 5, 27-30, (1880)
[27] Renka, R.J.; Neuberger, J.W., Minimal surfaces and Sobolev gradients, SIAM J. sci. comput., 16, 6, 1412-1427, (1995) · Zbl 0857.35004
[28] Richardson, W.B., Sobolev preconditioning for the poisson – boltzman equation, Comput. meth. appl. mech. eng., 181, 4, 425-436, (2000) · Zbl 0960.82035
[29] Salmon, T.O.; Horner, D.G., Comparison of elevation, curvature, and power descriptors for corneal topographic mapping, Optom. vis. sci., 72, 800-808, (1995)
[30] Schuchman, V., On behavior of nonlinear differential equations in Hilbert space, Int. J. math. sci., 11, 143-165, (1988) · Zbl 0659.34009
[31] Schwiegerling, J.; Greivenkamp, J.E.; Miller, J.M.; Snyder, R.W.; Palmer, M.L., Optical modeling of radial keratotomy incision patterns, Am. J. ophthalmol., 122, 808-817, (1996)
[32] Tripoli, N.K.; Cohen, K.L.; Obla, P.; Coggins, J.M.; Holmgren, D.E., Height measurement of astigmatic test surfaces by a keratoscope that uses plane geometry surface reconstruction, Am. J. ophthalmol., 121, 668-676, (1996)
[33] Turuwhenua, J.; Henderson, J., A novel low-order method for recovery of the corneal shape, Optometry vision sci., 81, 11, 863-871, (2004)
[34] van Saarloos, P.P.; Constable, I.J., Improved method for calculation of corneal topolgraphy for any photokeratoscope geometry, Optom. vis. sci., 68, 960-965, (1991)
[35] Wang, J.; Rice, D.A.; Klyce, S.D., A new reconstruction algorithm for improvement of corneal topographical analysis, Refractive corneal surg., 6, 379-387, (1989)
[36] Younes, M.; Boltz, R.; Leach, N.E.; Bedell, H., Short- and long-term repeatability of visioptic alcon eyemap (visioptic EH-270) corneal topographer on normal human corneas, Optom. vis. sci., 72, 838-844, (1995)
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