Comparison of parametric methods for modeling corneal surfaces

Corneal topography is a medical imaging technique to get the 3D shape of the cornea as a set of 3D points of its anterior and posterior surfaces. From these data, topographic maps can be derived to assist the ophthalmologist in the diagnosis of disorders. In this paper, we compare three different mathematical parametric representations of the corneal surfaces leastsquares fitted to the data provided by corneal topography. The parameters obtained from these models reduce the dimensionality of the data from several thousand 3D points to only a few parameters and could eventually be useful for diagnosis, biometry, implant design etc. The first representation is based on Zernike polynomials that are commonly used in optics. A variant of these polynomials, named Bhatia-Wolf will also be investigated. These two sets of polynomials are defined over a circular domain which is convenient to model the elevation (height) of the corneal surface. The third representation uses Spherical Harmonics that are particularly well suited for nearly-spherical object modeling, which is the case for cornea. We compared the three methods using the following three criteria: the root-mean-square error (RMSE), the number of parameters and the visual accuracy of the reconstructed topographic maps. A large dataset of more than 2000 corneal topographies was used. Our results showed that Spherical Harmonics were superior with a RMSE mean lower than 2.5 microns with 36 coefficients (order 5) for normal corneas and lower than 5 microns for two diseases affecting the corneal shapes: keratoconus and Fuchs’ dystrophy

Comparative analysis of some modeal reconstruction methods of the cornea from corneal elevation data

Purpose. A comparative study of the ability of some modal schemes to reproduce corneal shapes of varying complexity is performed, using both standard radial polynomials and the radial basis functions (RBF). Our claim is that the correct approach in the case of highly irregular corneas should combine several bases. Methods. Standard approaches of reconstruction by Zernike and other types of radial polynomials are compared with the discrete least squares fit (LSF) by the RBF in three theoretical surfaces, synthetically generated by computer algorithms in the lack of measurement noise. For the reconstruction by polynomials the maximal radial order 6 was chosen, which corresponds to the first 28 Zernike polynomials or the first 49 Bhatia-Wolf polynomials. The fit with the RBF has been carried out using a regular grid of centers. Results. The quality of fit was assessed by computing for each surface the mean square errors (MSE) of the reconstruction by LSF, measured at the same nodes where the heights were collected. Another criterion of the fitting quality used was the accuracy in recovery of the Zernike coefficients, especially in the case of incomplete data. Conclusions. The Zernike (and especially, the Bhatia-Wolf) polynomials constitute a reliable reconstruction method of a non-severely aberrated surface with a small surface regularity index (SRI). However, they fail to capture small deformations of the anterior surface of a synthetic cornea. The most promising is a combined approach that balances the robustness of the Zernike fit with the localization of the RBF

An adaptive algorithm for the cornea modeling from keratometric data

In this paper we describe an adaptive and multi-scale algorithm for the parsimonious t of the corneal surface data that allows to adapt the number of functions used in the reconstruction to the conditions of each cornea. The method implements also a dynamical selection of the parameters and the management of noise. It can be used for the real-time reconstruction of both altimetric data and corneal power maps from the data collected by keratoscopes, such as the Placido rings based topographers, decisive for an early detection of corneal diseases such as keratoconus. Numerical experiments show that the algorithm exhibits a steady exponential error decay, independently of the level of aberration of the cornea. The complexity of each anisotropic gaussian basis functions in the functional representation is the same, but their parameters vary to fit the current scale. This scale is determined only by the residual errors and not by the number of the iteration. Finally, the position and clustering of their centers, as well as the size of the shape parameters, provides an additional spatial information about the regions of higher irregularity. These results are compared with the standard approximation procedures based on the Zernike polynomials expansions

Comparative Analysis of Some Modal Reconstruction Methods of the Shape of the Cornea from Corneal Elevation Data

Purpose: A comparative study of the ability of some modal schemes to reproduce corneal shapes of varying complexity was performed, by using both standard radial polynomials and radial basis functions (RBFs). The hypothesis was that the correct approach in the case of highly irregular corneas should combine several bases. Methods: Standard approaches of reconstruction by Zernike and other types of radial polynomials were compared with the discrete least-squares fit (LSF) by the RBF in three theoretical surfaces, synthetically generated by computer algorithms in the absence of measurement noise. For the reconstruction by polynomials, the maximal radial order 6 was chosen, which corresponds to the first 28 Zernike polynomials or the first 49 Bhatia-Wolf polynomials. The fit with the RBF was performed by using a regular grid of centers. Results: The quality of fit was assessed by computing for each surface the mean square errors (MSEs) of the reconstruction by LSF, measured at the same nodes where the heights were collected. Another criterion of the fit quality used was the accuracy in recovery of the Zernike coefficients, especially in the case of incomplete data. Conclusions: The Zernike (and especially, the Bhatia-Wolf) polynomials constitute a reliable reconstruction method of a nonseverely aberrated surface with a small surface regularity index (SRI). However, they fail to capture small deformations of the anterior surface of a synthetic cornea. The most promising approach is a combined one that balances the robustness of the Zernike fit with the localization of the RBF

Procedimiento de reconstrucción de la topografía corneal a partir de datos altímetros o de curvatura

Número de publicación: ES2392619 A1 (12.12.2012) También publicado como: ES2392619 B1 (22.10.2013) Número de Solicitud: Consulta de Expedientes OEPM (C.E.O.) P201000842(08.06.2010)Procedimiento de reconstrucción de la topografía corneal a partir de datos altimétricos o de curvatura. La invención consiste en un método de reconstrucción de la superficie de la cara anterior de la córnea, a partir de los datos medidos en un conjunto discreto de puntos por medio de un topógrafo corneal o equipo equivalente. Se trata de un procedimiento que obtiene una expresión analítica de la superficie, combinando un ajuste por polinomios de Zernike o con esfera de mejor ajuste, con una reconstrucción por funciones de base radial gaussianas. Se logra obtener una descripción detallada de la superficie corneal, permitiendo un diagnóstico más fiable de patologías, o la implementación de tratamientos customizados. Este procedimiento es fácilmente implementable en cualquier topógrafo corneal, tomógrafo de coherencia óptica, equipos de lámpara de hendidura y equivalentes, de los existentes en el mercado, como sustituto del método estándar basado en polinomios de Zernike.Universidad de Almerí

Astigmatism and Pseudoaccommodation in Pseudophakic Eyes

noAdvanced IOLs with circumferential zones of different power provide pseudoaccommodation. We investigated the potential for power variation with meridian, namely astigmatism, to provide pseudo-accommodation. With appropriate power and axis orientations, acceptable pseudo-accommodation can be achieved

Investigating Keratoconus Using Optical Eye Modeling

Keratoconus (KC) is a corneal atrophy which causes a conical extrusion (cone). It is often misdiagnosed as optical defocus with astigmatism. In this thesis, computer KC eye models for various conditions are constructed. Using the KC eye model the influence on visual performance including the consequent refractive errors and the higher-order aberrations of KC eyes was investigated. The affects of cone shape, dimension (volume), and location on visual performance are also discussed. The modeled KC eyes are additionally used to evaluate the performance of a common eye vision-screening instrument. The simple photorefraction (PR) technique uses only one small light source with a camera to photograph ocular reflection patterns. Computer optical ray tracing was performed to simulate the PR images of both KC eyes and astigmatic eyes. The simulation results shown indicate the ability to detect and differentiate KC from normal refractive errors

Forward ray tracing for image projection prediction and surface reconstruction in the evaluation of corneal topography systems

A forward ray tracing (FRT) model is presented to determine the exact image projection in a general corneal topography system. Consequently, the skew ray error in Placido-based topography is demonstrated. A quantitative analysis comparing FRT-based algorithms and Placido-based algorithms in reconstructing the front surface of the cornea shows that arc step algorithms are more sensitive to noise (imprecise). Furthermore, they are less accurate in determining corneal aberrations particularly the quadrafoil aberration. On the other hand, FRT-based algorithms are more accurate and more precise showing that point to point corneal topography is superior compared to its Placido-based counterpart