Smoothing Splines on Unit Ball Domains with Application to Corneal Topography

Simultaneous Estimation of Corneal Topography, Pachymetry, and Curvature

Identification of objective criteria to correctly diagnose ectatic diseases of the cornea or to detect early stages of corneal ectasia is of great interest in ophthalmology and optometry. Metrics for diagnosis typically employed are curvature maps (axial/sagittal, tangential); elevation map of the anterior surface of the cornea with respect to a reference sphere; and pachymetry (thickness) map of the cornea. We present evidence that currently used curvature maps do not represent the actual curvatures (principal or mean) in a human cornea. A novel contribution of this paper is the computation of the true mean curvature over every point of a central region of the cornea. We show that the true mean curvature can accurately identify the location of the ectasia. We present a quartic smoothing spline algorithm for the simultaneous computation of elevation maps for anterior and posterior corneal surfaces, pachymetry, and true mean curvature. The input to the algorithm is data from a single measurement from imaging devices such as an anterior segment optical coherence tomographer or a Scheimpflug imager. We show that a different combination of metrics is useful for the diagnosis of existing ectasia (true mean curvature and anterior elevation map) as opposed to subclinical ectasia (pachymetry and posterior elevation map). We compare our results with existing algorithms, and present applications to a normal cornea, a forme fruste keratoconic cornea, and an advanced keratoconic cornea.