title: Corneal topography reconstruction algorithm that avoids the skew ray ambiguity and the skew ray error.
abstract: BACKGROUND
  Because corneal shape is typically measured with Placido ring targets, there is an ambiguity in determining which point on the ring corresponds to which point on the image. This ambiguity is expected to lead to errors (which we call the skew ray error) in reconstructing the corneal shape from Placido ring videokeratoscope (VKS) images.
  
  
  METHODS
  A coordinate system based on cylindrical coordinates is developed for handling the case when the corneal normal does not lie in the meridional plane. A simple solution is found for the forward problem of locating the Placido ring object given the VKS image and the corneal slope.
  
  
  RESULTS
  Algorithms for solving two inverse problems are developed: (1) the inverse problem of determining the Placido image given the Placido ring object and the corneal shape, and (2) the inverse problem of reconstructing the corneal shape based on knowledge of the Placido ring object and image. The algorithm compensates for the skew ray error and does not assume the cornea to be small. Both inverse algorithms are applied to a corneal shape with eight-fold corrugations similar to what might occur in radial keratotomy (RK). The reconstruction algorithm assumes sufficient smoothness for an arc-step algorithm to be possible. We find that the reconstruction algorithm does an excellent job of obtaining the correct corneal shape without the skew ray error.
  
  
  CONCLUSIONS
  The concrete reconstruction algorithm demonstrates that one can recover the correct corneal shape just from the reflection of the Placido rings. A dart board pattern is not necessary for removing the skew ray ambiguity, as long as the cornea has sufficient smoothness in the radial direction, an assumption common to all reconstruction algorithms.
authors.affiliation: School of Optometry, University of California at Berkeley, USA.
authors.forename: S A
authors.initials: SA
authors.lastname: Klein
mesh_terms: D000465:Algorithms; D003315:Cornea / Q000033:anatomy & histology*; D006801:Humans; D007091:Image Processing, Computer-Assisted; D008953:Models, Anatomic