## Forward ray tracing for image projection prediction and surface reconstruction in the evaluation of corneal topography systems |

Optics Express, Vol. 18, Issue 18, pp. 19324-19338 (2010)

http://dx.doi.org/10.1364/OE.18.019324

Acrobat PDF (1253 KB)

### Abstract

A forward ray tracing (FRT) model is presented to determine the exact image projection in a general corneal topography system. Consequently, the skew ray error in Placido-based topography is demonstrated. A quantitative analysis comparing FRT-based algorithms and Placido-based algorithms in reconstructing the front surface of the cornea shows that arc step algorithms are more sensitive to noise (imprecise). Furthermore, they are less accurate in determining corneal aberrations particularly the quadrafoil aberration. On the other hand, FRT-based algorithms are more accurate and more precise showing that point to point corneal topography is superior compared to its Placido-based counterpart.

© 2010 OSA

## 1. Introduction

1. J. Amanatides, “Ray tracing with cones,” Comput. Graph. **18**(3), 129–135 (1984). [CrossRef]

2. C. Bauer, “Direct illuminance caching: a way to enhance the performance of RADIANCE,” Lighting Res. Tech. **34**(4), 333–345 (2002). [CrossRef]

2. C. Bauer, “Direct illuminance caching: a way to enhance the performance of RADIANCE,” Lighting Res. Tech. **34**(4), 333–345 (2002). [CrossRef]

3. V. A. Sicam, J. J. Snellenburg, R. G. van der Heijde, and I. H. van Stokkum, “Pseudo forward ray-tracing: a new method for surface validation in cornea topography,” Optom. Vis. Sci. **84**(9), 915–923 (2007). [CrossRef] [PubMed]

4. J. D. Doss, R. L. Hutson, J. J. Rowsey, and D. R. Brown, “Method for calculation of corneal profile and power distribution,” Arch. Ophthalmol. **99**(7), 1261–1265 (1981). [PubMed]

5. J. Y. Wang, D. A. Rice, and S. D. Klyce, “A new reconstruction algorithm for improvement of corneal topographical analysis,” Refract. Corneal Surg. **5**(6), 379–387 (1989). [PubMed]

6. S. A. Klein, “A corneal topography algorithm that produces continuous curvature,” Optom. Vis. Sci. **69**(11), 829–834 (1992). [CrossRef] [PubMed]

8. J. Turuwhenua, “An improved low order method for corneal reconstruction,” Optom. Vis. Sci. **85**(3), 211–217 (2008). [CrossRef] [PubMed]

8. J. Turuwhenua, “An improved low order method for corneal reconstruction,” Optom. Vis. Sci. **85**(3), 211–217 (2008). [CrossRef] [PubMed]

10. S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. **74**(11), 931–944 (1997). [CrossRef] [PubMed]

9. N. K. Tripoli, K. L. Cohen, P. Obla, J. M. Coggins, and D. E. Holmgren, “Height measurement of astigmatic test surfaces by a keratoscope that uses plane geometry surface reconstruction,” Am. J. Ophthalmol. **121**(6), 668–676 (1996). [PubMed]

10. S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. **74**(11), 931–944 (1997). [CrossRef] [PubMed]

12. 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**(4), 794–797 (1997). [CrossRef]

11. S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. **74**(11), 945–962 (1997). [CrossRef] [PubMed]

13. T. Swartz, L. Marten, and M. Wang, “Measuring the cornea: the latest developments in corneal topography,” Curr. Opin. Ophthalmol. **18**(4), 325–333 (2007). [CrossRef] [PubMed]

12. 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**(4), 794–797 (1997). [CrossRef]

15. Y. Mejía and J. C. Galeano, “Corneal topographer based on the Hartmann test,” Optom. Vis. Sci. **86**(4), 370–381 (2009). [CrossRef] [PubMed]

16. 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. Vis. Sci. **72**(11), 821–827 (1995). [CrossRef] [PubMed]

18. J. Turuwhenua, “Corneal surface reconstruction algorithm using Zernike polynomial representation: improvements,” J. Opt. Soc. Am. A **24**(6), 1551–1561 (2007). [CrossRef]

19. V. A. Sicam and R. G. VAN der Heijde, “Topographer reconstruction of the nonrotation-symmetric anterior corneal surface features,” Optom. Vis. Sci. **83**(12), 910–918 (2006). [CrossRef] [PubMed]

20. V. Sokurenko and V. Molebny, “Damped least-squares approach for point-source corneal topography,” Ophthalmic Physiol. Opt. **29**(3), 330–337 (2009). [CrossRef] [PubMed]

3. V. A. Sicam, J. J. Snellenburg, R. G. van der Heijde, and I. H. van Stokkum, “Pseudo forward ray-tracing: a new method for surface validation in cornea topography,” Optom. Vis. Sci. **84**(9), 915–923 (2007). [CrossRef] [PubMed]

19. V. A. Sicam and R. G. VAN der Heijde, “Topographer reconstruction of the nonrotation-symmetric anterior corneal surface features,” Optom. Vis. Sci. **83**(12), 910–918 (2006). [CrossRef] [PubMed]

## 2. The FRT model for nodal point CT systems

_{s}, y

_{s}, z

_{s}). These are traced towards the intersection points on the corneal surface (x

_{c}, y

_{c}, z

_{c}) and proceed with the reflected ray towards the image points (x

_{i}, y

_{i}, z

_{i}) captured by a camera. Because only the chief ray is considered the reflected ray is constrained to pass through the nodal point (0, 0, 0) of the camera lens system. These definitions are summarized in Table 1 . A telecentric system can also be employed instead of the nodal point system described here, by adequately modifying the nodal-point constraint. It is known that surface reconstruction for both systems are quite similar and are described well in literature [8

8. J. Turuwhenua, “An improved low order method for corneal reconstruction,” Optom. Vis. Sci. **85**(3), 211–217 (2008). [CrossRef] [PubMed]

*f*(x

_{c},y

_{c}) in Eq. (1) represents a function composed of Zernike polynomials parameterized by Zernike polynomial coefficients evaluated over an 6 mm corneal zone. A Zernike polynomials expansion is an appropriate way of expressing the corneal surface [17

17. V. A. Sicam, J. Coppens, T. J. van den Berg, and R. G. van der Heijde, “Corneal surface reconstruction algorithm that uses Zernike polynomial representation,” J. Opt. Soc. Am. A **21**(7), 1300–1306 (2004). [CrossRef]

18. J. Turuwhenua, “Corneal surface reconstruction algorithm using Zernike polynomial representation: improvements,” J. Opt. Soc. Am. A **24**(6), 1551–1561 (2007). [CrossRef]

3. V. A. Sicam, J. J. Snellenburg, R. G. van der Heijde, and I. H. van Stokkum, “Pseudo forward ray-tracing: a new method for surface validation in cornea topography,” Optom. Vis. Sci. **84**(9), 915–923 (2007). [CrossRef] [PubMed]

21. L. A. Carvalho, “Accuracy of Zernike polynomials in characterizing optical aberrations and the corneal surface of the eye,” Invest. Ophthalmol. Vis. Sci. **46**(6), 1915–1926 (2005). [CrossRef] [PubMed]

*N*on the corneal surface. The normal vector

*N*is derived by taking the surface gradient using the partial derivatives of Eq. (1) and is given by Eq. (6). The relation the corneal normal describes with the incident unit vector

*I*and the reflected vector

*R*is captured by Eqs. (4) and 5. Equation (4) constrains the direction of the surface normal to be in the same direction of

*x*, where

_{c}, y_{c}, z_{c}, I_{x}, I_{y}, I_{z}, R_{x}, R_{y}, R_{z}, l_{I}, l_{R}*I*and

_{x}, I_{y}, I_{z}, R_{x}, R_{y}, R_{z}*l*are the directional cosines and the lengths of the incident and reflected vectors respectively. Note that Eqs. (2), 3 and 5 are vector equations which can be decomposed into three scalar equations each, however the dot product equation (Eq. (4) results into only one scalar equation. This balanced set of equations and unknowns makes it possible to determine the solution of the corneal intersection points x

_{I}, l_{R}_{c}, y

_{c}and z

_{c}by standard mathematical software (e.g. using the

*fsolve*routine in Matlab [22]). Consequently, the location of the image points captured by the camera can be determined by direct ray tracing from the corneal intersection points to the nodal point of the lens and further to the camera plane.

## 3. Surface reconstruction method derived from FRT

**84**(9), 915–923 (2007). [CrossRef] [PubMed]

17. V. A. Sicam, J. Coppens, T. J. van den Berg, and R. G. van der Heijde, “Corneal surface reconstruction algorithm that uses Zernike polynomial representation,” J. Opt. Soc. Am. A **21**(7), 1300–1306 (2004). [CrossRef]

**STEP 1**: Create matrix equations derived from the equations in Table 1 and Eq. (7):This is the surface equation expressed in Zernike convention:

*r*is the pupil radius used as a reference for a unit circle,

_{p}*M*is the matrix representation of the Zernike polynomials and

**C**represents the Zernike coefficients. Finally, we have two matrix equations from the cross product (Eq. (5):

**STEP 2**: Use zero initial values for the Zernike coefficients to represent a flat surface in the corneal apex plane. Via ray tracing, image points can be traced to the apex plane and form the points (x

_{c}, y

_{c}, z

_{c}).

**STEP 3**: Use (x

_{c}, y

_{c}, z

_{c}) as information to calculate a new set of Zernike coefficients

**C**by exploiting the matrix Eqs. (8-10). These matrix equations can be cast into the form:The solution for Eq. (11) is given by:Which can be solved numerically using the Moore-Penrose pseudo inverse algorithm.

**STEP 4**: Go back to STEP 2 to build a new surface with the new Zernike coefficients. Proceed to STEP 3 to determine more accurate Zernike coefficients. Repeat this iteration until a desired accuracy has been reached.

## 4. Simulation of corneal topography systems

### 4.1 Qualitative evaluation

9. N. K. Tripoli, K. L. Cohen, P. Obla, J. M. Coggins, and D. E. Holmgren, “Height measurement of astigmatic test surfaces by a keratoscope that uses plane geometry surface reconstruction,” Am. J. Ophthalmol. **121**(6), 668–676 (1996). [PubMed]

10. S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. **74**(11), 931–944 (1997). [CrossRef] [PubMed]

19. V. A. Sicam and R. G. VAN der Heijde, “Topographer reconstruction of the nonrotation-symmetric anterior corneal surface features,” Optom. Vis. Sci. **83**(12), 910–918 (2006). [CrossRef] [PubMed]

23. 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. Vis. Sci. **74**(11), 926–930 (1997). [CrossRef] [PubMed]

23. 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. Vis. Sci. **74**(11), 926–930 (1997). [CrossRef] [PubMed]

**83**(12), 910–918 (2006). [CrossRef] [PubMed]

9. N. K. Tripoli, K. L. Cohen, P. Obla, J. M. Coggins, and D. E. Holmgren, “Height measurement of astigmatic test surfaces by a keratoscope that uses plane geometry surface reconstruction,” Am. J. Ophthalmol. **121**(6), 668–676 (1996). [PubMed]

24. S. Marcos, P. Rosales, L. Llorente, and I. Jiménez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract Refract. Surg. **33**(2), 217–226 (2007). [CrossRef] [PubMed]

### 4.2 Quantitative evaluation

- 1) FRT-based surface reconstruction algorithm (FRT algorithm) described in section 3.
- 2) FRT-based surface reconstruction algorithm with meridional constraint (MRT algorithm). This means that the tracing of rays from the image points to the source points is constrained to lie on the same meridian. This algorithm will show the effect of neglecting skew ray errors entirely.
- 3) Basic arc step algorithm (Basic AS algorithm) described in ref [6]. The arc step algorithm does calculations on a per meridian basis. The algorithm tries to converge to calculated corneal height values by assuming that the corneal surface is spherical on a local scope and therefore considers a set of intersections of light rays on the cornea. Adjacent points within one meridian should form an arc and the algorithm makes sure that the connections between consecutive arcs are continuous, in other words the derivatives at the arc boundaries match.
6. S. A. Klein, “A corneal topography algorithm that produces continuous curvature,” Optom. Vis. Sci.

**69**(11), 829–834 (1992). [CrossRef] [PubMed] - 4) Arc step algorithm with skew ray error correction (SKEC AS algorithm) described in ref [8
**85**(3), 211–217 (2008). [CrossRef] [PubMed]

- - Identical stimulator pattern for all algorithms; the typical square one-to-one stimulator design was modified to match a Placido disk based design. However, where the one-to-one correspondence algorithm had full information on the location of the detected points, for the arc step algorithms only the information on the meridians was made available.
- - An equal number of data points available to both algorithms, equal to the number of rings times the number of meridians.
- - Equal noise amplitude applied; however due to the nature of the difference between Placido disk systems and one-to-one systems the effect of the applied noise varies slightly. The implementation of the noise is described more precisely in the later part of this section.

- 1) A toric surface with 8.0 mm and 7.5 mm maximum and minimum meridional radius of curvature; both meridional axes are perpendicular to each other.
- 2) A spherical surface of 8mm radius of curvature with trefoil aberration.
- 3) A spherical surface of 8mm radius of curvature with quadrafoil aberration.
- 4) A spherical surface of 8mm radius of curvature with octafoil aberration.
- 5) A typical corneal surface without abnormalities.

23. 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. Vis. Sci. **74**(11), 926–930 (1997). [CrossRef] [PubMed]

*r*is the surface base radius of curvature set at 8 mm,

*n*= 3, 4, 8 for trefoil, quadrafoil and octafoil respectively.

25. O. Muftuoglu, P. Prasher, R. W. Bowman, J. P. McCulley, and V. V. Mootha, “Corneal higher-order aberrations after Descemet’s stripping automated endothelial keratoplasty,” Ophthalmology **117**(5), 878–884, e6 (2010). [CrossRef] [PubMed]

^{th}radial order. The stimulator point pattern chosen for the simulation is a dotted ring system (10 rings, 180 points per ring within a 6 mm corneal zone). This allows for a clear comparison between a one-to-one correspondence topography system and a Placido-based topography system. The source points are arbitrarily chosen to be located at a plane 50 mm away from the corneal apex. These points were generated using back ray tracing from the camera plane, where the image pattern forms rings that have equidistant spacing from each other, to an 8 mm radius of curvature spherical surface and to the plane assigned for the source points. The distance between the nodal point of the lens and the corneal apex is chosen to be 105 mm for this simulation. These dimensions roughly correspond with the dimensions of the experimental corneal topography machine described in refs [3

**84**(9), 915–923 (2007). [CrossRef] [PubMed]

**83**(12), 910–918 (2006). [CrossRef] [PubMed]

^{−15}μm). The MRT algorithm produced the biggest error, under-estimating the true value by an order of 1 μm. The arc step algorithms produce larger errors compared to the FRT algorithm but the amount of error is clinically not relevant, noting that 1 μm RMS error is equal to 0.77 equivalent Diopter error [26

26. L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A **19**(12), 2329–2348 (2002). [CrossRef]

^{−15}μm) in reconstructing the corneal aberrations. The results are summarized in Table 4 . The MRT algorithm shows larger errors in the order of a small fraction of a micrometer.

### 4.3 Effect of noise

11. S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. **74**(11), 945–962 (1997). [CrossRef] [PubMed]

16. 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. Vis. Sci. **72**(11), 821–827 (1995). [CrossRef] [PubMed]

20. V. Sokurenko and V. Molebny, “Damped least-squares approach for point-source corneal topography,” Ophthalmic Physiol. Opt. **29**(3), 330–337 (2009). [CrossRef] [PubMed]

**83**(12), 910–918 (2006). [CrossRef] [PubMed]

20. V. Sokurenko and V. Molebny, “Damped least-squares approach for point-source corneal topography,” Ophthalmic Physiol. Opt. **29**(3), 330–337 (2009). [CrossRef] [PubMed]

27. L. A. Carvalho, M. Stefani, A. C. Romão, L. Carvalho, J. C. de Castro, S. Tonissi, P. Schor, and W. Chamon, “Videokeratoscopes for dioptric power measurement during surgery,” J. Cataract Refract. Surg. **28**(11), 2006–2016 (2002). [CrossRef] [PubMed]

11. S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. **74**(11), 945–962 (1997). [CrossRef] [PubMed]

**83**(12), 910–918 (2006). [CrossRef] [PubMed]

## 5. Discussion

29. T. O. Salmon and L. N. Thibos, “Videokeratoscope-line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations,” J. Opt. Soc. Am. A **19**(4), 657–669 (2002). [CrossRef]

### 5.1 Simulations without noise

**85**(3), 211–217 (2008). [CrossRef] [PubMed]

**74**(11), 931–944 (1997). [CrossRef] [PubMed]

**29**(3), 330–337 (2009). [CrossRef] [PubMed]

### 5.2 Simulations with noise

30. R. A. Applegate, J. D. Marsack, and L. N. Thibos, “Metrics of retinal image quality predict visual performance in eyes with 20/17 or better visual acuity,” Optom. Vis. Sci. **83**(9), 635–640 (2006). [CrossRef] [PubMed]

31. B. Braaf, M. Dubbelman, R. G. van der Heijde, and V. A. Sicam, “Performance in specular reflection and slit-imaging corneal topography,” Optom. Vis. Sci. **86**(5), 467–475 (2009). [CrossRef] [PubMed]

**85**(3), 211–217 (2008). [CrossRef] [PubMed]

**74**(11), 945–962 (1997). [CrossRef] [PubMed]

**69**(11), 829–834 (1992). [CrossRef] [PubMed]

### 5.3 Clinical implications

**84**(9), 915–923 (2007). [CrossRef] [PubMed]

## 6. Conclusion

## Acknowledgements

## References and links

1. | J. Amanatides, “Ray tracing with cones,” Comput. Graph. |

2. | C. Bauer, “Direct illuminance caching: a way to enhance the performance of RADIANCE,” Lighting Res. Tech. |

3. | V. A. Sicam, J. J. Snellenburg, R. G. van der Heijde, and I. H. van Stokkum, “Pseudo forward ray-tracing: a new method for surface validation in cornea topography,” Optom. Vis. Sci. |

4. | J. D. Doss, R. L. Hutson, J. J. Rowsey, and D. R. Brown, “Method for calculation of corneal profile and power distribution,” Arch. Ophthalmol. |

5. | J. Y. Wang, D. A. Rice, and S. D. Klyce, “A new reconstruction algorithm for improvement of corneal topographical analysis,” Refract. Corneal Surg. |

6. | S. A. Klein, “A corneal topography algorithm that produces continuous curvature,” Optom. Vis. Sci. |

7. | R. Mattioli and N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vis. Sci. |

8. | J. Turuwhenua, “An improved low order method for corneal reconstruction,” Optom. Vis. Sci. |

9. | N. K. Tripoli, K. L. Cohen, P. Obla, J. M. Coggins, and D. E. Holmgren, “Height measurement of astigmatic test surfaces by a keratoscope that uses plane geometry surface reconstruction,” Am. J. Ophthalmol. |

10. | S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. |

11. | S. A. Klein, “Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error,” Optom. Vis. Sci. |

12. | 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. |

13. | T. Swartz, L. Marten, and M. Wang, “Measuring the cornea: the latest developments in corneal topography,” Curr. Opin. Ophthalmol. |

14. | J. H. Massig, E. Lingelbach, and B. Lingelbach, “Videokeratoscope for accurate and detailed measurement of the cornea surface,” Appl. Opt. |

15. | Y. Mejía and J. C. Galeano, “Corneal topographer based on the Hartmann test,” Optom. Vis. Sci. |

16. | 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. Vis. Sci. |

17. | V. A. Sicam, J. Coppens, T. J. van den Berg, and R. G. van der Heijde, “Corneal surface reconstruction algorithm that uses Zernike polynomial representation,” J. Opt. Soc. Am. A |

18. | J. Turuwhenua, “Corneal surface reconstruction algorithm using Zernike polynomial representation: improvements,” J. Opt. Soc. Am. A |

19. | V. A. Sicam and R. G. VAN der Heijde, “Topographer reconstruction of the nonrotation-symmetric anterior corneal surface features,” Optom. Vis. Sci. |

20. | V. Sokurenko and V. Molebny, “Damped least-squares approach for point-source corneal topography,” Ophthalmic Physiol. Opt. |

21. | L. A. Carvalho, “Accuracy of Zernike polynomials in characterizing optical aberrations and the corneal surface of the eye,” Invest. Ophthalmol. Vis. Sci. |

22. | Matlab, The Mathworks, Massachusetts, USA. |

23. | 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. Vis. Sci. |

24. | S. Marcos, P. Rosales, L. Llorente, and I. Jiménez-Alfaro, “Change in corneal aberrations after cataract surgery with 2 types of aspherical intraocular lenses,” J. Cataract Refract. Surg. |

25. | O. Muftuoglu, P. Prasher, R. W. Bowman, J. P. McCulley, and V. V. Mootha, “Corneal higher-order aberrations after Descemet’s stripping automated endothelial keratoplasty,” Ophthalmology |

26. | L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A |

27. | L. A. Carvalho, M. Stefani, A. C. Romão, L. Carvalho, J. C. de Castro, S. Tonissi, P. Schor, and W. Chamon, “Videokeratoscopes for dioptric power measurement during surgery,” J. Cataract Refract. Surg. |

28. | F. Lu, J. X. Wu, and J. Qu., “Association between offset of the pupil center from the corneal vertex and wavefront aberration,” J. Opt. |

29. | T. O. Salmon and L. N. Thibos, “Videokeratoscope-line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations,” J. Opt. Soc. Am. A |

30. | R. A. Applegate, J. D. Marsack, and L. N. Thibos, “Metrics of retinal image quality predict visual performance in eyes with 20/17 or better visual acuity,” Optom. Vis. Sci. |

31. | B. Braaf, M. Dubbelman, R. G. van der Heijde, and V. A. Sicam, “Performance in specular reflection and slit-imaging corneal topography,” Optom. Vis. Sci. |

**ToC Category:**

Vision, Color, and Visual Optics

**History**

Original Manuscript: July 6, 2010

Revised Manuscript: July 23, 2010

Manuscript Accepted: July 28, 2010

Published: August 26, 2010

**Virtual Issues**

Vol. 5, Iss. 13 *Virtual Journal for Biomedical Optics*

**Citation**

Joris J. Snellenburg, Boy Braaf, Erik A. Hermans, Rob G. L. van der Heijde, and Victor Arni D. P. Sicam, "Forward ray tracing for image projection prediction and surface reconstruction in the evaluation of corneal topography systems," Opt. Express **18**, 19324-19338 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19324

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### References

- J. Amanatides, “Ray tracing with cones,” Comput. Graph. 18(3), 129–135 (1984). [CrossRef]
- C. Bauer, “Direct illuminance caching: a way to enhance the performance of RADIANCE,” Lighting Res. Tech. 34(4), 333–345 (2002). [CrossRef]
- V. A. Sicam, J. J. Snellenburg, R. G. van der Heijde, and I. H. van Stokkum, “Pseudo forward ray-tracing: a new method for surface validation in cornea topography,” Optom. Vis. Sci. 84(9), 915–923 (2007). [CrossRef] [PubMed]
- J. D. Doss, R. L. Hutson, J. J. Rowsey, and D. R. Brown, “Method for calculation of corneal profile and power distribution,” Arch. Ophthalmol. 99(7), 1261–1265 (1981). [PubMed]
- J. Y. Wang, D. A. Rice, and S. D. Klyce, “A new reconstruction algorithm for improvement of corneal topographical analysis,” Refract. Corneal Surg. 5(6), 379–387 (1989). [PubMed]
- S. A. Klein, “A corneal topography algorithm that produces continuous curvature,” Optom. Vis. Sci. 69(11), 829–834 (1992). [CrossRef] [PubMed]
- R. Mattioli and N. K. Tripoli, “Corneal geometry reconstruction with the Keratron videokeratographer,” Optom. Vis. Sci. 74(11), 881–894 (1997). [CrossRef] [PubMed]
- J. Turuwhenua, “An improved low order method for corneal reconstruction,” Optom. Vis. Sci. 85(3), 211–217 (2008). [CrossRef] [PubMed]
- N. K. Tripoli, K. L. Cohen, P. Obla, J. M. Coggins, and D. E. Holmgren, “Height measurement of astigmatic test surfaces by a keratoscope that uses plane geometry surface reconstruction,” Am. J. Ophthalmol. 121(6), 668–676 (1996). [PubMed]
- S. A. Klein, “Axial curvature and the skew ray error in corneal topography,” Optom. Vis. Sci. 74(11), 931–944 (1997). [CrossRef] [PubMed]
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