Abstract

A method of ray tracing for free-form optical surfaces has been developed. The ray tracing through such surfaces is based on Delaunay triangulation of the discrete data of the surface and is related to finite-element modeling. Some numerical examples of applications to analytical, noisy, and experimental free-form surfaces (in particular, a corneal topography map) are presented. Ray-tracing results (i.e., spot diagram root-mean-square error) with the new method are in agreement with those obtained using a modal fitting of the surface, for sampling densities higher than 40×40 elements. The method competes in flexibility, simplicity, and computing times with standard methods for surface fitting and ray tracing.

© 2009 Optical Society of America

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  1. D. T. Moore, “Ray tracing in tilted, decentered, displaced gradient-index optical systems,” J. Opt. Soc. Am. 66, 789-795 (1976).
    [CrossRef]
  2. A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984-987 (1982).
    [CrossRef] [PubMed]
  3. E. W. Marchand, “Rapid ray tracing in radial gradients,” Appl. Opt. 27, 465-467 (1988).
    [CrossRef] [PubMed]
  4. R. A. Egorchenkov and Y. A. Kravtsov, “Complex ray-tracing algorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650-656 (2001).
    [CrossRef]
  5. O. N. Stavroudis, “Ray-tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187-191 (1962).
    [CrossRef]
  6. Q. T. Liang and X. D. Zheng, “Ray-tracing calculations for uniaxial optical components with curved surfaces,” Appl. Opt. 30, 4521-4525 (1991).
    [CrossRef] [PubMed]
  7. G. Beyerle and I. S. McDermid, “Ray-tracing formulas for refraction and internal reflection in uniaxial crystals,” Appl. Opt. 37, 7947-7953 (1998).
    [CrossRef]
  8. M. Izdebski, “Ray and wave tracing in uniaxial crystals perturbed by an external field,” Appl. Opt. 47, 2729-2738 (2008).
    [CrossRef] [PubMed]
  9. X. Tian, G. Lai, and T. Yatagai, “Characterization of asymmetric optical waveguides by ray tracing,” J. Opt. Soc. Am. A 6, 1538-1543 (1989).
    [CrossRef]
  10. J. S. Witkowski and A. Grobelny, “Ray tracing method in a 3D analysis of fiber-optic elements,” Opt. Appl. 38, 281-294(2008).
  11. W. A. Allen and J. R. Snyder, “Ray tracing through uncentered and aspheric surfaces,” J. Opt. Soc. Am. 42, 243-249 (1952).
    [CrossRef]
  12. M. Herzberger, “Automatic ray tracing,” J. Opt. Soc. Am. 47, 736-739 (1957).
    [CrossRef]
  13. G. H. Spencer and M. V. R. K. Murty, “General ray-tracing procedure,” J. Opt. Soc. Am. 52, 672-678 (1962).
    [CrossRef]
  14. A. Sharma and A. K. Ghatak, “Ray tracing in gradient-index lenses: computation of ray-surface intersection,” Appl. Opt. 25, 3409-3412 (1986).
    [CrossRef] [PubMed]
  15. R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, Jr., R. P. Sharp, and 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, 1994), pp. 58-61.
  16. H. C. Howland, J. Buettner, and 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, 1994), pp. 54-57.
  17. P. Artal and A. Guirao, “Contributions of the cornea and the lens to the aberrations of the human eye,” Opt. Lett. 23, 1713-1715 (1998).
    [CrossRef]
  18. A. Guirao and P. Artal, “Corneal wave aberration from videokeratography: accuracy and limitations of the procedure,” J. Opt. Soc. Am. A 17, 955-965 (2000).
    [CrossRef]
  19. A. Guirao, M. Redondo, and P. Artal, “Optical aberrations of the human cornea as a function of age,” J. Opt. Soc. Am. A 17, 1697-1702 (2000).
    [CrossRef]
  20. S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28, 1594-1600 (2002).
    [CrossRef] [PubMed]
  21. L. A. Carvalho, “Computer algorithm for simulation of the human optical system and contribution of the cornea to the optical imperfections of the eye,” Revista de Física Aplicada e Instrumentação 16, 7-17 (2003).
  22. J. C. He, J. Gwiazda, F. Thorn, and R. Held, “Wave-front aberrations in the anterior corneal surface and the whole eye,” J. Opt. Soc. Am. A 20, 1155-1163 (2003).
    [CrossRef]
  23. R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83, 371-381 (2006).
    [CrossRef] [PubMed]
  24. P. Rosales and S. Marcos, “Customized computer models of eyes with intraocular lenses,” Opt. Express 15, 2204-2218 (2007).
    [CrossRef] [PubMed]
  25. B. Delaunay, “Sur la sphère vide,” Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793-800 (1934).
  26. L. Guibas, D. Knuth, and M. Sharir, “Randomized incremental construction of Delaunay and Voronoi diagrams,” Algorithmica 7, 381-413 (1992).
    [CrossRef]
  27. C. Touma and C. Gotsman, “Triangle mesh compression,” in Proceedings of Graphics Interface '98 26-34 (Canadian Information Processing Society, 1998).
  28. J. Kohout, J. I. Kolingerová, and J. Žára, “Parallel Delaunay triangulation in E2 and E3 for computers with shared memory,” Parallel Comput. 31, 491-522 (2005).
    [CrossRef]
  29. D. S. Kang, Y. J. Kim, and B. S. Shin, “Efficient large-scale terrain rendering method for real-world game simulation,” in Technologies for E-Learning and Digital Entertainment: First International Conference, Edutainment 2006 (Springer, 2006), pp. 597-605.
    [CrossRef] [PubMed]
  30. Zemax Optical Design Program User's Manual (Zemax Development Corporation, 2007).
  31. S. Ortiz, D. Siedlecki, L. Remon, and S. Marcos, “Three-dimensional optical distortion correction for quantitative anterior segment OCT,” Invest. Ophthalmol. Vis. Sci. 50, 5796(2009), abstract.
  32. A. Podoleanu, I. Charalambous, L. Plesea, A. Dogariu, and R. Rosen, “Correction of distortions in optical coherence tomography imaging of the eye,” Phys. Med. Biol. 49, 1277-1294(2004).
    [CrossRef] [PubMed]
  33. V. Westphal, A. M. Rollins, S. Radhakrishnan, and J. A. Izatt, “Correction of geometric and refractive image distortions in optical coherence tomography applying Fermat's principle,” Opt. Express 10, 397-404 (2002).
    [PubMed]
  34. G. Strang and G. Fix, An Analysis of the Finite Element Method (Prentice-Hall, 1973).
  35. P. L. George and H. Borouchaki, Delaunay Triangulation and Meshing (Editions Hermes, 1998).
  36. R. Vanselow, “About Delaunay triangulations and discrete maximum principles for the linear conforming FEM applied to the Poisson equation,” Appl. Math. 46, 13-28 (2001).
    [CrossRef]
  37. N. Calvo, S. R. Idelsohn, and E. Onate, “The extended Delaunay tessellation,” Eng. Computat. 20, 583-600 (2003).
    [CrossRef]
  38. I. Babuska, U. Banerjee, and J. E. Osborn, “Generalized finite element methods: main ideas, results, and perspective,” Int. J. Computat. Methods 1, 67-103 (2004).
    [CrossRef]
  39. H. S. M. Coxeter, “Barycentric Coordinates,” in Introduction to Geometry, H. S. M. Coxeter, ed. (Wiley, 1969), pp. 216-221.
  40. It may happen that the point P is located on the edge of the triangle (when either u or v from Eq. is equal to 0, or u+v=1), or even in the vertex (when both u and v are equal to 0 or either u or v is equal to 1), which means that it is common to two or more of the triangles that have a common edge (vertex), but the vectors normal to these triangles are different. In this situation, it is convenient to consider the mean value of the normal vector coordinates for further ray tracing calculations.
  41. H. Wendland, Scattered Data Approximation (Cambridge U. Press, 2004).
    [CrossRef]
  42. F. L. Bookstein, “Principal warps: thin plate splines and the decomposition of deformations,” IEEE Trans. Pattern Anal. Machine Intell. 11, 567-585 (1989).
    [CrossRef]
  43. J. Allison, “Multiquadric radial basis functions for representing multidimensional high energy physics data,” Comput. Phys. Commun. 77, 377-395 (1993).
    [CrossRef]
  44. G. R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method (CRC Press, 2002).
    [CrossRef]
  45. It should be noted that in the original Sharma formula in , a square term of υ is missing. This has been corrected on Eq. .
  46. 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]
  47. S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the calculation of corneal aberrations from videokeratography: a test on keratoconus eyes,” J. Refract. Surg. 18, 263-270 (2002).
    [PubMed]
  48. D. Malacara, Optical Shop Testing (Wiley, 1978).
  49. L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, “Random rough surface simulations,” in Scattering of Electromagnetic Waves: Numerical Simulations, L. Tsang, J. A. Kong, K. -H. Ding, and C. O. Ao, eds. (Wiley, 2001), pp. 124-132.
  50. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1967), pp. 55-69.
  51. W. N. Charman, “The eye in focus: accommodation and presbyopia,” Clin. Exp. Optom. 91, 207-225 (2008).
    [CrossRef] [PubMed]
  52. D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin. Exp. Optom. 91, 251-264 (2008).
    [CrossRef] [PubMed]
  53. M. K. Smolek and S. D. Klyce, “Is keratoconus a true ectasia? An evaluation of corneal surface area,” Arch. Ophthalmol. 118, 1179-1186 (2000).
    [PubMed]
  54. W. A. Douthwaite, “Application of linear regression to videokeratoscope data for tilted surfaces,” Ophthal. Physiol. Opt. 22, 46-54 (2002).
    [CrossRef]
  55. V. A. D. P. Sicam, J. Coppens, T. J. T. P. van den 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]
  56. J. Turuwhenua, “Corneal surface reconstruction algorithm using Zernike polynomial representation: improvements,” J. Opt. Soc. Am. A 24, 1551-1561 (2007).
    [CrossRef]
  57. D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87-95 (2001).
    [CrossRef] [PubMed]
  58. L. Llorente, S. Marcos, C. Dorronsoro, and S. A. Burns, “Effect of sampling on real ocular aberration measurements,” J. Opt. Soc. Am. A 24, 2783-2796 (2007).
    [CrossRef]

2009 (1)

S. Ortiz, D. Siedlecki, L. Remon, and S. Marcos, “Three-dimensional optical distortion correction for quantitative anterior segment OCT,” Invest. Ophthalmol. Vis. Sci. 50, 5796(2009), abstract.

2008 (4)

J. S. Witkowski and A. Grobelny, “Ray tracing method in a 3D analysis of fiber-optic elements,” Opt. Appl. 38, 281-294(2008).

W. N. Charman, “The eye in focus: accommodation and presbyopia,” Clin. Exp. Optom. 91, 207-225 (2008).
[CrossRef] [PubMed]

D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin. Exp. Optom. 91, 251-264 (2008).
[CrossRef] [PubMed]

M. Izdebski, “Ray and wave tracing in uniaxial crystals perturbed by an external field,” Appl. Opt. 47, 2729-2738 (2008).
[CrossRef] [PubMed]

2007 (3)

2006 (1)

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83, 371-381 (2006).
[CrossRef] [PubMed]

2005 (1)

J. Kohout, J. I. Kolingerová, and J. Žára, “Parallel Delaunay triangulation in E2 and E3 for computers with shared memory,” Parallel Comput. 31, 491-522 (2005).
[CrossRef]

2004 (3)

A. Podoleanu, I. Charalambous, L. Plesea, A. Dogariu, and R. Rosen, “Correction of distortions in optical coherence tomography imaging of the eye,” Phys. Med. Biol. 49, 1277-1294(2004).
[CrossRef] [PubMed]

I. Babuska, U. Banerjee, and J. E. Osborn, “Generalized finite element methods: main ideas, results, and perspective,” Int. J. Computat. Methods 1, 67-103 (2004).
[CrossRef]

V. A. D. P. Sicam, J. Coppens, T. J. T. P. van den 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]

2003 (3)

N. Calvo, S. R. Idelsohn, and E. Onate, “The extended Delaunay tessellation,” Eng. Computat. 20, 583-600 (2003).
[CrossRef]

L. A. Carvalho, “Computer algorithm for simulation of the human optical system and contribution of the cornea to the optical imperfections of the eye,” Revista de Física Aplicada e Instrumentação 16, 7-17 (2003).

J. C. He, J. Gwiazda, F. Thorn, and R. Held, “Wave-front aberrations in the anterior corneal surface and the whole eye,” J. Opt. Soc. Am. A 20, 1155-1163 (2003).
[CrossRef]

2002 (4)

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28, 1594-1600 (2002).
[CrossRef] [PubMed]

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the calculation of corneal aberrations from videokeratography: a test on keratoconus eyes,” J. Refract. Surg. 18, 263-270 (2002).
[PubMed]

W. A. Douthwaite, “Application of linear regression to videokeratoscope data for tilted surfaces,” Ophthal. Physiol. Opt. 22, 46-54 (2002).
[CrossRef]

V. Westphal, A. M. Rollins, S. Radhakrishnan, and J. A. Izatt, “Correction of geometric and refractive image distortions in optical coherence tomography applying Fermat's principle,” Opt. Express 10, 397-404 (2002).
[PubMed]

2001 (3)

R. A. Egorchenkov and Y. A. Kravtsov, “Complex ray-tracing algorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650-656 (2001).
[CrossRef]

R. Vanselow, “About Delaunay triangulations and discrete maximum principles for the linear conforming FEM applied to the Poisson equation,” Appl. Math. 46, 13-28 (2001).
[CrossRef]

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87-95 (2001).
[CrossRef] [PubMed]

2000 (3)

1998 (2)

1995 (1)

1993 (1)

J. Allison, “Multiquadric radial basis functions for representing multidimensional high energy physics data,” Comput. Phys. Commun. 77, 377-395 (1993).
[CrossRef]

1992 (1)

L. Guibas, D. Knuth, and M. Sharir, “Randomized incremental construction of Delaunay and Voronoi diagrams,” Algorithmica 7, 381-413 (1992).
[CrossRef]

1991 (1)

1989 (2)

X. Tian, G. Lai, and T. Yatagai, “Characterization of asymmetric optical waveguides by ray tracing,” J. Opt. Soc. Am. A 6, 1538-1543 (1989).
[CrossRef]

F. L. Bookstein, “Principal warps: thin plate splines and the decomposition of deformations,” IEEE Trans. Pattern Anal. Machine Intell. 11, 567-585 (1989).
[CrossRef]

1988 (1)

1986 (1)

1982 (1)

1976 (1)

1962 (2)

1957 (1)

1952 (1)

1934 (1)

B. Delaunay, “Sur la sphère vide,” Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793-800 (1934).

Allen, W. A.

Allison, J.

J. Allison, “Multiquadric radial basis functions for representing multidimensional high energy physics data,” Comput. Phys. Commun. 77, 377-395 (1993).
[CrossRef]

Ao, C. O.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, “Random rough surface simulations,” in Scattering of Electromagnetic Waves: Numerical Simulations, L. Tsang, J. A. Kong, K. -H. Ding, and C. O. Ao, eds. (Wiley, 2001), pp. 124-132.

Applegate, R. A.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, Jr., R. P. Sharp, and 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, 1994), pp. 58-61.

H. C. Howland, J. Buettner, and 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, 1994), pp. 54-57.

Artal, P.

Babuska, I.

I. Babuska, U. Banerjee, and J. E. Osborn, “Generalized finite element methods: main ideas, results, and perspective,” Int. J. Computat. Methods 1, 67-103 (2004).
[CrossRef]

Banerjee, U.

I. Babuska, U. Banerjee, and J. E. Osborn, “Generalized finite element methods: main ideas, results, and perspective,” Int. J. Computat. Methods 1, 67-103 (2004).
[CrossRef]

Barbero, S.

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the calculation of corneal aberrations from videokeratography: a test on keratoconus eyes,” J. Refract. Surg. 18, 263-270 (2002).
[PubMed]

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28, 1594-1600 (2002).
[CrossRef] [PubMed]

Beckmann, P.

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1967), pp. 55-69.

Beyerle, G.

Bookstein, F. L.

F. L. Bookstein, “Principal warps: thin plate splines and the decomposition of deformations,” IEEE Trans. Pattern Anal. Machine Intell. 11, 567-585 (1989).
[CrossRef]

Borouchaki, H.

P. L. George and H. Borouchaki, Delaunay Triangulation and Meshing (Editions Hermes, 1998).

Buettner, J.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, Jr., R. P. Sharp, and 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, 1994), pp. 58-61.

H. C. Howland, J. Buettner, and 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, 1994), pp. 54-57.

Burns, S. A.

Calvo, N.

N. Calvo, S. R. Idelsohn, and E. Onate, “The extended Delaunay tessellation,” Eng. Computat. 20, 583-600 (2003).
[CrossRef]

Carvalho, L. A.

L. A. Carvalho, “Computer algorithm for simulation of the human optical system and contribution of the cornea to the optical imperfections of the eye,” Revista de Física Aplicada e Instrumentação 16, 7-17 (2003).

Charalambous, I.

A. Podoleanu, I. Charalambous, L. Plesea, A. Dogariu, and R. Rosen, “Correction of distortions in optical coherence tomography imaging of the eye,” Phys. Med. Biol. 49, 1277-1294(2004).
[CrossRef] [PubMed]

Charman, W. N.

W. N. Charman, “The eye in focus: accommodation and presbyopia,” Clin. Exp. Optom. 91, 207-225 (2008).
[CrossRef] [PubMed]

Collins, M. J.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87-95 (2001).
[CrossRef] [PubMed]

Coppens, J.

Cottinghan, A. J.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, Jr., R. P. Sharp, and 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, 1994), pp. 58-61.

Coxeter, H. S. M.

H. S. M. Coxeter, “Barycentric Coordinates,” in Introduction to Geometry, H. S. M. Coxeter, ed. (Wiley, 1969), pp. 216-221.

Davis, B.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87-95 (2001).
[CrossRef] [PubMed]

Delaunay, B.

B. Delaunay, “Sur la sphère vide,” Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793-800 (1934).

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, “Random rough surface simulations,” in Scattering of Electromagnetic Waves: Numerical Simulations, L. Tsang, J. A. Kong, K. -H. Ding, and C. O. Ao, eds. (Wiley, 2001), pp. 124-132.

Dogariu, A.

A. Podoleanu, I. Charalambous, L. Plesea, A. Dogariu, and R. Rosen, “Correction of distortions in optical coherence tomography imaging of the eye,” Phys. Med. Biol. 49, 1277-1294(2004).
[CrossRef] [PubMed]

Dorronsoro, C.

Douthwaite, W. A.

W. A. Douthwaite, “Application of linear regression to videokeratoscope data for tilted surfaces,” Ophthal. Physiol. Opt. 22, 46-54 (2002).
[CrossRef]

Egorchenkov, R. A.

Fisher, S. W.

D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin. Exp. Optom. 91, 251-264 (2008).
[CrossRef] [PubMed]

Fix, G.

G. Strang and G. Fix, An Analysis of the Finite Element Method (Prentice-Hall, 1973).

George, P. L.

P. L. George and H. Borouchaki, Delaunay Triangulation and Meshing (Editions Hermes, 1998).

Ghatak, A. K.

González, L.

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83, 371-381 (2006).
[CrossRef] [PubMed]

Gotsman, C.

C. Touma and C. Gotsman, “Triangle mesh compression,” in Proceedings of Graphics Interface '98 26-34 (Canadian Information Processing Society, 1998).

Greivenkamp, J. E.

Grobelny, A.

J. S. Witkowski and A. Grobelny, “Ray tracing method in a 3D analysis of fiber-optic elements,” Opt. Appl. 38, 281-294(2008).

Guibas, L.

L. Guibas, D. Knuth, and M. Sharir, “Randomized incremental construction of Delaunay and Voronoi diagrams,” Algorithmica 7, 381-413 (1992).
[CrossRef]

Guirao, A.

Gwiazda, J.

He, J. C.

Held, R.

Hernández-Matamoros, J. L.

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83, 371-381 (2006).
[CrossRef] [PubMed]

Herzberger, M.

Howland, H. C.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, Jr., R. P. Sharp, and 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, 1994), pp. 58-61.

H. C. Howland, J. Buettner, and 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, 1994), pp. 54-57.

Idelsohn, S. R.

N. Calvo, S. R. Idelsohn, and E. Onate, “The extended Delaunay tessellation,” Eng. Computat. 20, 583-600 (2003).
[CrossRef]

Iskander, D. R.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87-95 (2001).
[CrossRef] [PubMed]

Izatt, J. A.

Izdebski, M.

Kang, D. S.

D. S. Kang, Y. J. Kim, and B. S. Shin, “Efficient large-scale terrain rendering method for real-world game simulation,” in Technologies for E-Learning and Digital Entertainment: First International Conference, Edutainment 2006 (Springer, 2006), pp. 597-605.
[CrossRef] [PubMed]

Kim, Y. J.

D. S. Kang, Y. J. Kim, and B. S. Shin, “Efficient large-scale terrain rendering method for real-world game simulation,” in Technologies for E-Learning and Digital Entertainment: First International Conference, Edutainment 2006 (Springer, 2006), pp. 597-605.
[CrossRef] [PubMed]

Klyce, S. D.

M. K. Smolek and S. D. Klyce, “Is keratoconus a true ectasia? An evaluation of corneal surface area,” Arch. Ophthalmol. 118, 1179-1186 (2000).
[PubMed]

Knuth, D.

L. Guibas, D. Knuth, and M. Sharir, “Randomized incremental construction of Delaunay and Voronoi diagrams,” Algorithmica 7, 381-413 (1992).
[CrossRef]

Kohout, J.

J. Kohout, J. I. Kolingerová, and J. Žára, “Parallel Delaunay triangulation in E2 and E3 for computers with shared memory,” Parallel Comput. 31, 491-522 (2005).
[CrossRef]

Kolingerová, J. I.

J. Kohout, J. I. Kolingerová, and J. Žára, “Parallel Delaunay triangulation in E2 and E3 for computers with shared memory,” Parallel Comput. 31, 491-522 (2005).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, “Random rough surface simulations,” in Scattering of Electromagnetic Waves: Numerical Simulations, L. Tsang, J. A. Kong, K. -H. Ding, and C. O. Ao, eds. (Wiley, 2001), pp. 124-132.

Kravtsov, Y. A.

Kumar, D. V.

Lai, G.

Liang, Q. T.

Liu, G. R.

G. R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method (CRC Press, 2002).
[CrossRef]

Llorente, L.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1978).

Marchand, E. W.

Marcos, S.

S. Ortiz, D. Siedlecki, L. Remon, and S. Marcos, “Three-dimensional optical distortion correction for quantitative anterior segment OCT,” Invest. Ophthalmol. Vis. Sci. 50, 5796(2009), abstract.

L. Llorente, S. Marcos, C. Dorronsoro, and S. A. Burns, “Effect of sampling on real ocular aberration measurements,” J. Opt. Soc. Am. A 24, 2783-2796 (2007).
[CrossRef]

P. Rosales and S. Marcos, “Customized computer models of eyes with intraocular lenses,” Opt. Express 15, 2204-2218 (2007).
[CrossRef] [PubMed]

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28, 1594-1600 (2002).
[CrossRef] [PubMed]

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the calculation of corneal aberrations from videokeratography: a test on keratoconus eyes,” J. Refract. Surg. 18, 263-270 (2002).
[PubMed]

McDermid, I. S.

Meister, D. J.

D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin. Exp. Optom. 91, 251-264 (2008).
[CrossRef] [PubMed]

Merayo-Lloves, J.

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the calculation of corneal aberrations from videokeratography: a test on keratoconus eyes,” J. Refract. Surg. 18, 263-270 (2002).
[PubMed]

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28, 1594-1600 (2002).
[CrossRef] [PubMed]

Miller, J. M.

Moore, D. T.

Moreno-Barriuso, E.

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the calculation of corneal aberrations from videokeratography: a test on keratoconus eyes,” J. Refract. Surg. 18, 263-270 (2002).
[PubMed]

Murty, M. V. R. K.

Navarro, R.

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83, 371-381 (2006).
[CrossRef] [PubMed]

Onate, E.

N. Calvo, S. R. Idelsohn, and E. Onate, “The extended Delaunay tessellation,” Eng. Computat. 20, 583-600 (2003).
[CrossRef]

Ortiz, S.

S. Ortiz, D. Siedlecki, L. Remon, and S. Marcos, “Three-dimensional optical distortion correction for quantitative anterior segment OCT,” Invest. Ophthalmol. Vis. Sci. 50, 5796(2009), abstract.

Osborn, J. E.

I. Babuska, U. Banerjee, and J. E. Osborn, “Generalized finite element methods: main ideas, results, and perspective,” Int. J. Computat. Methods 1, 67-103 (2004).
[CrossRef]

Plesea, L.

A. Podoleanu, I. Charalambous, L. Plesea, A. Dogariu, and R. Rosen, “Correction of distortions in optical coherence tomography imaging of the eye,” Phys. Med. Biol. 49, 1277-1294(2004).
[CrossRef] [PubMed]

Podoleanu, A.

A. Podoleanu, I. Charalambous, L. Plesea, A. Dogariu, and R. Rosen, “Correction of distortions in optical coherence tomography imaging of the eye,” Phys. Med. Biol. 49, 1277-1294(2004).
[CrossRef] [PubMed]

Radhakrishnan, S.

Redondo, M.

Remon, L.

S. Ortiz, D. Siedlecki, L. Remon, and S. Marcos, “Three-dimensional optical distortion correction for quantitative anterior segment OCT,” Invest. Ophthalmol. Vis. Sci. 50, 5796(2009), abstract.

Rollins, A. M.

Rosales, P.

Rosen, R.

A. Podoleanu, I. Charalambous, L. Plesea, A. Dogariu, and R. Rosen, “Correction of distortions in optical coherence tomography imaging of the eye,” Phys. Med. Biol. 49, 1277-1294(2004).
[CrossRef] [PubMed]

Schwiegerling, J.

Sharir, M.

L. Guibas, D. Knuth, and M. Sharir, “Randomized incremental construction of Delaunay and Voronoi diagrams,” Algorithmica 7, 381-413 (1992).
[CrossRef]

Sharma, A.

Sharp, R. P.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, Jr., R. P. Sharp, and 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, 1994), pp. 58-61.

Shin, B. S.

D. S. Kang, Y. J. Kim, and B. S. Shin, “Efficient large-scale terrain rendering method for real-world game simulation,” in Technologies for E-Learning and Digital Entertainment: First International Conference, Edutainment 2006 (Springer, 2006), pp. 597-605.
[CrossRef] [PubMed]

Sicam, V. A. D. P.

Siedlecki, D.

S. Ortiz, D. Siedlecki, L. Remon, and S. Marcos, “Three-dimensional optical distortion correction for quantitative anterior segment OCT,” Invest. Ophthalmol. Vis. Sci. 50, 5796(2009), abstract.

Smolek, M. K.

M. K. Smolek and S. D. Klyce, “Is keratoconus a true ectasia? An evaluation of corneal surface area,” Arch. Ophthalmol. 118, 1179-1186 (2000).
[PubMed]

Snyder, J. R.

Spencer, G. H.

Stavroudis, O. N.

Strang, G.

G. Strang and G. Fix, An Analysis of the Finite Element Method (Prentice-Hall, 1973).

Thorn, F.

Tian, X.

Touma, C.

C. Touma and C. Gotsman, “Triangle mesh compression,” in Proceedings of Graphics Interface '98 26-34 (Canadian Information Processing Society, 1998).

Tsang, L.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, “Random rough surface simulations,” in Scattering of Electromagnetic Waves: Numerical Simulations, L. Tsang, J. A. Kong, K. -H. Ding, and C. O. Ao, eds. (Wiley, 2001), pp. 124-132.

Turuwhenua, J.

van den Berg, T. J. T. P.

van der Heijde, R. G. L.

Vanselow, R.

R. Vanselow, “About Delaunay triangulations and discrete maximum principles for the linear conforming FEM applied to the Poisson equation,” Appl. Math. 46, 13-28 (2001).
[CrossRef]

Wendland, H.

H. Wendland, Scattered Data Approximation (Cambridge U. Press, 2004).
[CrossRef]

Westphal, V.

Witkowski, J. S.

J. S. Witkowski and A. Grobelny, “Ray tracing method in a 3D analysis of fiber-optic elements,” Opt. Appl. 38, 281-294(2008).

Yatagai, T.

Yee, R. W.

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, Jr., R. P. Sharp, and 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, 1994), pp. 58-61.

Žára, J.

J. Kohout, J. I. Kolingerová, and J. Žára, “Parallel Delaunay triangulation in E2 and E3 for computers with shared memory,” Parallel Comput. 31, 491-522 (2005).
[CrossRef]

Zheng, X. D.

Algorithmica (1)

L. Guibas, D. Knuth, and M. Sharir, “Randomized incremental construction of Delaunay and Voronoi diagrams,” Algorithmica 7, 381-413 (1992).
[CrossRef]

Appl. Math. (1)

R. Vanselow, “About Delaunay triangulations and discrete maximum principles for the linear conforming FEM applied to the Poisson equation,” Appl. Math. 46, 13-28 (2001).
[CrossRef]

Appl. Opt. (6)

Arch. Ophthalmol. (1)

M. K. Smolek and S. D. Klyce, “Is keratoconus a true ectasia? An evaluation of corneal surface area,” Arch. Ophthalmol. 118, 1179-1186 (2000).
[PubMed]

Clin. Exp. Optom. (2)

W. N. Charman, “The eye in focus: accommodation and presbyopia,” Clin. Exp. Optom. 91, 207-225 (2008).
[CrossRef] [PubMed]

D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin. Exp. Optom. 91, 251-264 (2008).
[CrossRef] [PubMed]

Comput. Phys. Commun. (1)

J. Allison, “Multiquadric radial basis functions for representing multidimensional high energy physics data,” Comput. Phys. Commun. 77, 377-395 (1993).
[CrossRef]

Eng. Computat. (1)

N. Calvo, S. R. Idelsohn, and E. Onate, “The extended Delaunay tessellation,” Eng. Computat. 20, 583-600 (2003).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48, 87-95 (2001).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Machine Intell. (1)

F. L. Bookstein, “Principal warps: thin plate splines and the decomposition of deformations,” IEEE Trans. Pattern Anal. Machine Intell. 11, 567-585 (1989).
[CrossRef]

Int. J. Computat. Methods (1)

I. Babuska, U. Banerjee, and J. E. Osborn, “Generalized finite element methods: main ideas, results, and perspective,” Int. J. Computat. Methods 1, 67-103 (2004).
[CrossRef]

Invest. Ophthalmol. Vis. Sci. (1)

S. Ortiz, D. Siedlecki, L. Remon, and S. Marcos, “Three-dimensional optical distortion correction for quantitative anterior segment OCT,” Invest. Ophthalmol. Vis. Sci. 50, 5796(2009), abstract.

Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk (1)

B. Delaunay, “Sur la sphère vide,” Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793-800 (1934).

J. Cataract Refract. Surg. (1)

S. Barbero, S. Marcos, and J. Merayo-Lloves, “Corneal and total optical aberrations in a unilateral aphakic patient,” J. Cataract Refract. Surg. 28, 1594-1600 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (9)

V. A. D. P. Sicam, J. Coppens, T. J. T. P. van den 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]

J. C. He, J. Gwiazda, F. Thorn, and R. Held, “Wave-front aberrations in the anterior corneal surface and the whole eye,” J. Opt. Soc. Am. A 20, 1155-1163 (2003).
[CrossRef]

J. Turuwhenua, “Corneal surface reconstruction algorithm using Zernike polynomial representation: improvements,” J. Opt. Soc. Am. A 24, 1551-1561 (2007).
[CrossRef]

L. Llorente, S. Marcos, C. Dorronsoro, and S. A. Burns, “Effect of sampling on real ocular aberration measurements,” J. Opt. Soc. Am. A 24, 2783-2796 (2007).
[CrossRef]

A. Guirao and P. Artal, “Corneal wave aberration from videokeratography: accuracy and limitations of the procedure,” J. Opt. Soc. Am. A 17, 955-965 (2000).
[CrossRef]

X. Tian, G. Lai, and T. Yatagai, “Characterization of asymmetric optical waveguides by ray tracing,” J. Opt. Soc. Am. A 6, 1538-1543 (1989).
[CrossRef]

R. A. Egorchenkov and Y. A. Kravtsov, “Complex ray-tracing algorithms with application to optical problems,” J. Opt. Soc. Am. A 18, 650-656 (2001).
[CrossRef]

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]

A. Guirao, M. Redondo, and P. Artal, “Optical aberrations of the human cornea as a function of age,” J. Opt. Soc. Am. A 17, 1697-1702 (2000).
[CrossRef]

J. Refract. Surg. (1)

S. Barbero, S. Marcos, J. Merayo-Lloves, and E. Moreno-Barriuso, “Validation of the calculation of corneal aberrations from videokeratography: a test on keratoconus eyes,” J. Refract. Surg. 18, 263-270 (2002).
[PubMed]

Ophthal. Physiol. Opt. (1)

W. A. Douthwaite, “Application of linear regression to videokeratoscope data for tilted surfaces,” Ophthal. Physiol. Opt. 22, 46-54 (2002).
[CrossRef]

Opt. Appl. (1)

J. S. Witkowski and A. Grobelny, “Ray tracing method in a 3D analysis of fiber-optic elements,” Opt. Appl. 38, 281-294(2008).

Opt. Express (2)

Opt. Lett. (1)

Optom. Vis. Sci. (1)

R. Navarro, L. González, and J. L. Hernández-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom. Vis. Sci. 83, 371-381 (2006).
[CrossRef] [PubMed]

Parallel Comput. (1)

J. Kohout, J. I. Kolingerová, and J. Žára, “Parallel Delaunay triangulation in E2 and E3 for computers with shared memory,” Parallel Comput. 31, 491-522 (2005).
[CrossRef]

Phys. Med. Biol. (1)

A. Podoleanu, I. Charalambous, L. Plesea, A. Dogariu, and R. Rosen, “Correction of distortions in optical coherence tomography imaging of the eye,” Phys. Med. Biol. 49, 1277-1294(2004).
[CrossRef] [PubMed]

Revista de Física Aplicada e Instrumentação (1)

L. A. Carvalho, “Computer algorithm for simulation of the human optical system and contribution of the cornea to the optical imperfections of the eye,” Revista de Física Aplicada e Instrumentação 16, 7-17 (2003).

Other (15)

D. Malacara, Optical Shop Testing (Wiley, 1978).

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, “Random rough surface simulations,” in Scattering of Electromagnetic Waves: Numerical Simulations, L. Tsang, J. A. Kong, K. -H. Ding, and C. O. Ao, eds. (Wiley, 2001), pp. 124-132.

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1967), pp. 55-69.

G. R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method (CRC Press, 2002).
[CrossRef]

It should be noted that in the original Sharma formula in , a square term of υ is missing. This has been corrected on Eq. .

G. Strang and G. Fix, An Analysis of the Finite Element Method (Prentice-Hall, 1973).

P. L. George and H. Borouchaki, Delaunay Triangulation and Meshing (Editions Hermes, 1998).

R. A. Applegate, H. C. Howland, J. Buettner, A. J. Cottinghan, Jr., R. P. Sharp, and 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, 1994), pp. 58-61.

H. C. Howland, J. Buettner, and 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, 1994), pp. 54-57.

C. Touma and C. Gotsman, “Triangle mesh compression,” in Proceedings of Graphics Interface '98 26-34 (Canadian Information Processing Society, 1998).

D. S. Kang, Y. J. Kim, and B. S. Shin, “Efficient large-scale terrain rendering method for real-world game simulation,” in Technologies for E-Learning and Digital Entertainment: First International Conference, Edutainment 2006 (Springer, 2006), pp. 597-605.
[CrossRef] [PubMed]

Zemax Optical Design Program User's Manual (Zemax Development Corporation, 2007).

H. S. M. Coxeter, “Barycentric Coordinates,” in Introduction to Geometry, H. S. M. Coxeter, ed. (Wiley, 1969), pp. 216-221.

It may happen that the point P is located on the edge of the triangle (when either u or v from Eq. is equal to 0, or u+v=1), or even in the vertex (when both u and v are equal to 0 or either u or v is equal to 1), which means that it is common to two or more of the triangles that have a common edge (vertex), but the vectors normal to these triangles are different. In this situation, it is convenient to consider the mean value of the normal vector coordinates for further ray tracing calculations.

H. Wendland, Scattered Data Approximation (Cambridge U. Press, 2004).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Function z ( x , y ) = x 2 - y 2 represented as a set of triangles to illustrate the Delaunay triangulation. The discrete data of the function are situated in vertices of triangles. The axes are given in arbitrary units.

Fig. 2
Fig. 2

Graphical illustration of a point P m that is a common vertex of six neighboring triangles and of the barycentric technique.

Fig. 3
Fig. 3

Example of a random grid on a plane. The dashed lines form triangles surrounding point P m .

Fig. 4
Fig. 4

Noise generated for simulations of realistic noisy surfaces: (a)  rms = 855 / 8 nm and correlation length equal to 1 μm and (b)  rms = 855 / 8 nm and correlation length equal to 1000 μm .

Fig. 5
Fig. 5

Example of a free-form surface: (a) raw (discrete) data from a corneal topography obtained from Placido ring corneal videokeratography, (b) the elevation of the anterior corneal surface, and (c) ray-tracing visualization through the anterior surface of the cornea.

Fig. 6
Fig. 6

Results of the seventh-order Zernike polynomial expansion of the corneal elevation map of Fig. 5. (a) Zernike coefficients for a 3 mm normalization radius. The inset in (a) shows the values of the zeroth and fourth Zernike terms as they are beyond the original scale. (b) Difference between raw data (interpolated with cubic splines) and Zernike polynomial fit. The rms of the difference is 1.4 μm . The surfaces were set to have the same Z value in the center.

Fig. 7
Fig. 7

(a) RMS of the spot diagram taken at the distance 27.72 mm behind the cornea and (b) computational time, as a function of sampling density for different methods of interpolation in the finite-element ray-tracing algorithm. The position “RawData” in the horizontal axis denotes results from the ray tracing on a direct Delaunay reconstruction from the raw elevation data (with no sampling or Zernike smoothing). For all other cases, the raw data were first approximated by a seventh-order Zernike polynomial and then sampled with different densities. The results for semianalytical ray tracing on the Zernike surface are shown for comparison, as they do not depend on the sampling density (number of facets).

Fig. 8
Fig. 8

Time consumption as a function of the number of traced rays for finite-element ray-tracing algorithm with various methods of data interpolation in comparison to the semianalytical procedure used with Zernike polynomial expansion.

Tables (1)

Tables Icon

Table 1 Accuracy of the Algorithm for Different Sampling Grid Densities (Noisy Surface) a

Equations (10)

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

t = 2 m 2 l .
R 0 = ( x 0 y 0 z 0 ) , K 0 ¯ = [ k x 0 k y 0 k z 0 ] ,
R = R 0 + t K 0 ¯ ,
d i = | w i ¯ ( w i ¯ · K 0 ¯ ) K 0 ¯ | ,
P = P m + u · P m B ¯ + v · P m C ¯ ,
u = ( P m B ¯ · P m B ¯ ) ( P m P ¯ · P m C ¯ ) ( P m B ¯ · P m C ¯ ) ( P m P ¯ · P m B ¯ ) ( P m C ¯ · P m C ¯ ) ( P m B ¯ · P m B ¯ ) ( P m C ¯ · P m B ¯ ) ( P m B ¯ · P m C ¯ ) , v = ( P m C ¯ · P m C ¯ ) ( P m P ¯ · P m B ¯ ) ( P m C ¯ · P m B ¯ ) ( P m P ¯ · P m C ¯ ) ( P m C ¯ · P m C ¯ ) ( P m B ¯ · P m B ¯ ) ( P m C ¯ · P m B ¯ ) ( P m B ¯ · P m C ¯ ) ,
f ( x , y ) = c 0 + c 1 x + c 2 y + i = 1 n c i + 2 φ ( d i ) ,
N ^ = P m B ¯ × P m C ¯ | P m B ¯ × P m C ¯ | or N ^ = P m C ¯ × P m B ¯ | P m C ¯ × P m B ¯ | ,
n K 1 ¯ = n K 0 ¯ + w N ^ ,
w = n 2 n 2 + υ 2 υ ,

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