Accuracy and reliability of orthogonal polynomials in representing corneal topography

Abstract

Fitting of corneal topography data to analytical surfaces has been necessary in many clinical and experimental applications, yet absolute superiority of fitting methods was still unclear, and their overfitting risks were not well studied. This study aimed to evaluate the accuracy and reliability of orthogonal polynomials as fitting routines to represent corneal topography. Four orthogonal polynomials, namely, Zernike polynomials (ZPs), pseudo-Zernike polynomials (PZPs), Gaussian-Hermite polynomials (GHPs) and Orthogonal Fourier-Mellin polynomials (OFMPs), were employed to fit anterior and posterior corneal topographies collected from 200 healthy and 174 keratoconic eyes using Pentacam topographer. The fitting performance of these polynomials were compared, and the potential overfitting risks were assessed through a prediction exercise. The results showed that, except for low orders, the fitting performance differed little among polynomials with orders ≥10 regarding surface reconstruction (RMSEs ∼0.3 ​μm). Anterior surfaces of normal corneas were fitted more efficiently, followed by those of keratoconic corneas, then posterior corneal surfaces. The results, however, revealed an alarming fact that all polynomials tended to overfit the data beyond certain orders. GHPs, closely followed by ZPs, were the most robust in predicting unmeasured surface locations; while PZPs and especially OFMPs overfitted the surfaces drastically. Order 10 appeared to be optimum for corneal surfaces with 10-mm diameter, ensuring accurate reconstruction and avoiding overfitting. The optimum order however varied with topography diameters and data resolutions. The study concluded that continuing to use ZPs as fitting routine for most topography maps, or using GHPs instead, remains a good choice. Choosing polynomial orders close to the topography diameters (millimeters) is generally suggested to ensure both reconstruction accuracy and prediction reliability and avoid overfitting for both normal and complex (e.g., keratoconic) corneal surfaces.