Eigencorneas: application of principal component analysis to corneal topography

Technical variability of cornea parameters derived from anterior segment OCT fitted with Fringe Zernike polynomials

BACKGROUND

This study uses bootstrapping to evaluate the technical variability (in terms of model parameter variation) of Zernike corneal surface fit parameters based on Casia2 biometric data.

METHODS

Using a dataset containing N = 6953 Casia2 biometric measurements from a cataractous population, a Fringe Zernike polynomial surface of radial degree 10 (36 components) was fitted to the height data. The fit error (height - reconstruction) was bootstrapped 100 times after normalisation. After reversal of normalisation, the bootstrapped fit errors were added to the reconstructed height, and characteristic surface parameters (flat/steep axis, radii, and asphericities in both axes) extracted. The median parameters refer to a robust surface representation for later estimates of elevation, whereas the SD of the 100 bootstraps refers to the variability of the surface fit.

RESULTS

Bootstrapping gave median radius and asphericity values of 7.74/7.68 mm and -0.20/-0.24 for the corneal front surface in the flat/steep meridian and 6.52/6.37 mm and -0.22/-0.31 for the corneal back surface. The respective SD values for the 100 bootstraps were 0.0032/0.0028 mm and 0.0093/0.0082 for the front and 0.0126/0.0115 mm and 0.0366/0.0312 for the back surface. The uncertainties for the back surface are systematically larger as compared to the uncertainties of the front surface.

CONCLUSION

As measured with the Casia2 tomographer, the fit parameters for the corneal back surface exhibit a larger degree of variability compared with those for the front surface. Further studies are needed to show whether these uncertainties are representative for the situation where actual repeat measurements are possible.

Bootstrapping of Corneal Optical Coherence Tomography Data to Investigate Conic Fit Robustness

BACKGROUND

Fitting of parametric model surfaces to corneal tomographic measurement data is required in order to extract characteristic surface parameters. The purpose of this study was to develop a method for evaluating the uncertainties in characteristic surface parameters using bootstrap techniques.

METHODS

We included 1684 measurements from a cataractous population performed with the tomographer Casia2. Both conoid and biconic surface models were fitted to the height data. The normalised fit error (height-reconstruction) was bootstrapped 100 times and added to the reconstructed height, extracting characteristic surface parameters (radii and asphericity for both cardinal meridians and axis of the flat meridian) for each bootstrap. The width of the 90% confidence interval of the 100 bootstraps was taken as uncertainty and quoted as a measure of the robustness of the surface fit.

RESULTS

As derived from bootstrapping, the mean uncertainty for the radii of curvature was 3 µm/7 µm for the conoid and 2.5 µm/3 µm for the biconic model for the corneal front/back surface, respectively. The corresponding uncertainties for the asphericity were 0.008/0.014 for the conoid and 0.001/0.001 for the biconic. The respective mean root mean squared fit error was systematically lower for the corneal front surface as compared to the back surface (1.4 µm/2.4 µm for the conoid and 1.4 µm/2.6 µm for the biconic).

CONCLUSION

Bootstrapping techniques can be applied to extract uncertainties of characteristic model parameters and yield an estimate for robustness as an alternative to evaluating repeat measurements. Further studies are required to investigate whether bootstrap uncertainties accurately reproduce those from repeat measurement analysis.

Image quality eigenfunctions for the human eye

This work presents a compact statistical model of the retinal image quality in a large population of human eyes following two objectives. The first was to develop a general modal representation of the optical transfer function (OTF) in terms of orthogonal functions and construct a basis composed of cross-correlations between pairs of complex Zernike polynomials. That basis was not orthogonal and highly redundant, requiring the application of singular value decomposition (SVD) to obtain an orthogonal basis with a significantly lower dimensionality. The first mode is the OTF of the perfect system, and hence the modal representation, is highly compact for well-corrected optical systems, and vice-versa. The second objective is to apply this modal representation to the OTFs of a large population of human eyes for a pupil diameter of 5 mm. This permits an initial strong data compression. Next, principal component analysis (PCA) is applied to obtain further data compression, leading to a compact statistical model of the initial population. In this model each OTF is approximated by the sum of the population mean plus a linear combination of orthogonal eigenfunctions (eigen-OTF) accounting for a selected percentage (90%) of the population variance. This type of models can be useful for Monte Carlo simulations among other applications.

Average biometry of the cornea in a large population of Iranian school children

This work establishes the average Scheimpflug corneal tomography for a population of 4953 healthy Iranian primary school children. These data were transformed to determine the corneal position and orientation in three-dimensional space, followed by a model fit that combines a biconic with a Zernike expansion. Girls were found to have slightly steeper corneas than boys. Both corneal surfaces show negative conic constants and significant higher-order aspheric Zernike terms. The corneal surfaces are decentered and misaligned with respect to each other and to the line of sight. Consequently, the average corneal surfaces may be considered as decentered and misaligned higher-order aspheres.

Bigaussian Wavefront Model for Normal and Keratoconic Eyes

PURPOSE

To report bigaussian multivariate wavefront models capable of stochastically generating an unlimited amount of plausible wavefront data for either normal or keratoconic eyes.

METHODS

The models use centroid wavefront data measured previously with an iTrace in 330 healthy right eyes and 122 keratoconic right eyes. These centroids were fitted to an 11th-order Zernike series, followed by principal component analysis to reduce dimensionality and remove correlations. The remaining parameters were then fitted to a sum of two multivariate Gaussian distributions. This fit then forms the core of the stochastic model, which may be used to generate synthetic data. Finally, the agreement between the original and synthetic data was tested using two one-sided t tests.

RESULTS

For normal eyes, the first eigenvectors mostly represent pure Zernike polynomials, with a decreasing degree of purity with increasing order. For keratoconic eyes, eigenvector purity was considerably lower than for normal eyes. Depending on the data set, series of 22 to 29 eigenvectors were found sufficient for accurate wavefront reconstruction (i.e., root-mean-square errors below 0.05 μm). These eigenvectors were then used as a base for the stochastic models. In all models and all Zernike coefficients, the mean of the synthetic data was significantly equal to that of the original data (two one-sided t test, P > .05/75), but the variability of the synthetic data is often significantly lower (F test, P < .05/75).

CONCLUSIONS

This synthetic wavefront model may be safely used in calculations as an alternative to actual measurements should such data not be available.

SyntEyes KTC: higher order statistical eye model for developing keratoconus

PURPOSE

To present and validate a stochastic eye model for developing keratoconus to e.g. improve optical corrective strategies. This could be particularly useful for researchers that do not have access to original keratoconic data.

METHODS

The Scheimpflug tomography, ocular biometry and wavefront of 145 keratoconic right eyes were collected. These data were processed using principal component analysis for parameter reduction, followed by a multivariate Gaussian fit that produces a stochastic model for keratoconus (SyntEyes KTC). The output of this model is filtered to remove the occasional incorrect topography patterns by either an automatic or manual procedure. Finally, the output of this keratoconus model is matched to that of the original model for normal eyes using the non-corneal biometry to obtain a description of keratoconus development.

RESULTS

The synthetic data generated by the model were found to be significantly equal to the original data (non-parametric Mann-Whitney equivalence test; 145/154 passed). The variability of the synthetic data, however, was often significantly less than that of the original data, especially for the higher order Zernike terms of corneal elevation (non-parametric Levene test; p < 0.05/154). These results remained generally the same after applying either filter procedure to remove the synthetic eyes with incorrect topographies. Interpolation between matched pairs of normal and keratoconic SyntEyes appears to provide an adequate model for keratoconus progression.

CONCLUSION

The synthetic data provided by the proposed keratoconus model closely resembles actual clinical data and may be used for a range of research applications when (sufficient) real data is not available.

Smoothing Splines on Unit Ball Domains with Application to Corneal Topography

Optical coherence tomography (OCT) is a non-invasive imaging technique used to study and understand internal structures of biological tissues such as the anterior chamber of the human eye. An interesting problem is the reconstruction of the shape of the biological tissue from OCT images, that is not only a good fit of the data but also respects the smoothness properties observed in the images. A similar problem arises in Magnetic Resonance Imaging (MRI). We cast the problem as a penalized weighted least squares regression with a penalty on the magnitude of the second derivative (Laplacian) of the surface. We present a novel algorithm to construct the Kimeldorf-Wahba solution for unit ball domains. Our method unifies the ad-hoc approaches currently in the literature. Application of the theory to data from an anterior segment optical coherence tomographer is presented. A detailed comparison of the reconstructed surface using different approaches is presented.