title: The recovery of posterior cornea and anterior lens radii by a novel ray-tracing method.
abstract: BACKGROUND
  Methods for estimating ocular surface radii are typically based on paraxial vergence calculations, need to account for finite source positions, and require refocusing of the camera. This article describes (1) a telecentric ray-finding method; and (2) its application to the problem of determining posterior cornea (R2) and anterior lens (R3) radii by a regression procedure that addresses these issues.
  
  
  METHODS
  The ray-finding algorithm simulates Purkinje image heights Pj(h) (for j = 2, 3; for the Le Grand eye) for the expected range of Rj. A two-step cubic regression procedure fits this image data globally and then over a refined interval to estimate Rj locally. The goodness of fit is measured by the R statistic. A standard method is compared with the new method in simulation. Mean absolute errors and SD's are recorded for 10 randomly chosen R2 and R3. The effect of errors caused by (1) axial shift of the posterior cornea and anterior lens (axially up to +/-0.1 mm); and (2) camera digitization (pixel sizes of 20 microm) are simulated. The method can make use of general surface height information; therefore, it is tested on an eye with nonspherical cornea shape (a toroidal surface).
  
  
  RESULTS
  The time to generate Rj is no more than 45 s (R3), with R > 0.9999. The errors for the unmodified eye are (1.3 +/- 2.1) x 10 mm (R2) and (2.4 +/- 1.6) x 10 mm (R3) (new) vs. (5.6 +/- 0.9) x 10 mm (R2) and (1.1 +/- 1.0) x 10 mm (R3) (standard). Digitization increases errors to 0.32 +/- 0.16 mm (R2) and 0.10 +/- 0.10 (R3) (new) vs. 0.45 +/- 0.33 mm (R2) and 0.48 +/- 0.13 mm (R3) (standard). Axial shift error for R2 is similar between methods, with a tendency toward lower error for the new method given digitization error. This trend is found for R3, although the new method now does consistently better with digitization error. Shift contributes the smaller proportion of total error (approximately 10 mm) compared with digitization (approximately 10 mm). The errors for the toroidal surface are (5.7 +/- 7.6) x 10 mm (R2) and (1.1 +/- 0.7) x 10 mm (R3) (new) compared with errors of 0.18 +/- 0.01 mm (R2) and 0.44 +/- 0.07 mm (R3) (standard). The new method produces better results in this case.
  
  
  CONCLUSIONS
  A telecentric image-computing algorithm produces accurate image positions. A two-step cubic regression produces accurate estimates of R2 and R3 (errors approximately 10 mm). This error increases with axial shift (approximately 10 mm) and digitization (approximately 10 mm). The new method does better than a standard method ignoring all the errors and tends to handle digitization error better. The new method works well on a toroidal anterior cornea. Testing on model/real eyes is required. Efforts are continuing to refine methods for videophakometry.
authors.affiliation: Department of Physics and Electronic Engineering, University of Waikato, Hamilton, New Zealand. j.turuwhenua@auckland.ac.nz
authors.forename: Jason
authors.initials: J
authors.lastname: Turuwhenua
authors.forename: Jon
authors.initials: J
authors.lastname: Henderson
mesh_terms: D003198:Computer Simulation; D003315:Cornea / Q000033:anatomy & histology*; D019781:Corneal Topography; D006801:Humans; D007091:Image Processing, Computer-Assisted; D007908:Lens, Crystalline / Q000033:anatomy & histology*; D008962:Models, Theoretical; D055095:Optics and Photonics; D012044:Regression Analysis; D013690:Television