Wavefront sensing with critical sampling

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.

Recursive formula to compute Zernike radial polynomials

In optics, Zernike polynomials are widely used in testing, wavefront sensing, and aberration theory. This unique set of radial polynomials is orthogonal over the unit circle and finite on its boundary. This Letter presents a recursive formula to compute Zernike radial polynomials using a relationship between radial polynomials and Chebyshev polynomials of the second kind. Unlike the previous algorithms, the derived recurrence relation depends neither on the degree nor on the azimuthal order of the radial polynomials. This leads to a reduction in the computational complexity.