Direct Matrix Reconstruction Method for a Hartmann-Based Corneal Topographer Using Local Polynomial Fitting

Abstract

Purpose: : One of the interesting features of spot based corneal topography^1,2 is that the source locations can be arranged so that after reflection of the cornea, the spots form a nearly uniform grid. However, due to corneal irregularities and various corneal shapes, a perfectly uniform grid is not guaranteed and in practice never achieved; leading to the problem of working with arbitrary point clouds of data. While modal reconstructors (for example based on Zernike Polynomials) can handle arbitrary point locations, for surfaces containing high spatial frequency content they do not have enough fidelity unless they are extended to include higher orders. A direct solution zonal reconstruction method for a Hartmann based corneal topographer is detailed that does not rely on a uniform grid of points for reconstruction unlike other common wavefront reconstruction methods based on iterative methods in uniform space.³

Methods: : The first step in the reconstruction process is to associate each spot found in the camera image with a source spot on the topographer cone and assign gradients based on the problem geometry. Then, local neighborhoods of points are formed by computing the Delaunay triangulation⁴, a common calculation step used in computational geometry^5,6, enabling localized polynomials to describe the surface of the triangles. The algorithm marches through the resulting triangulation, computing coefficients of the local modal fits, connecting nearby triangles to each other, and generating a larger system matrix in order to minimize the overall error of the model to the measurement.

Results: : When the reconstruction algorithms were validated using simulations that emulate the corneal topographer hardware, the results demonstrate sub-micron accuracy. Measurements made using model eyes and test spheres show that the technique is capable of sub-micron Elevation Surface accuracy on test surfaces as well. Finally, several interesting measurement examples will be presented and compared to Zernike modal reconstruction results.

Conclusions: : Spot-based or full gradient corneal topography enables the measurement of orthogonal (x and y) slopes that when used with an appropriate reconstructor, also enables accurate reconstruction of surfaces containing high spatial frequency content.

Keywords: topography • aberrations • shape and contour