Goodness-of-prediction of Zernike polynomial fitting to corneal surfaces

To determine the goodness-of-prediction of the fitting routine by measuring the difference between topographic corneal surfaces and their Zernike reconstructions as a function of polynomial order and optical zone size for various corneal conditions.

Corneal research laboratory in a university eye center.

Corneal topography maps (N = 253) were obtained from the Louisiana State University Eye Center. A variety of corneal conditions were used: normals; astigmatism; laser in situ keratomileusis, photorefractive keratoplasty (PRK), and radial keratotomy (RK) postoperative cases (myopic spherical corrections); keratoconus suspect; mild, moderate, and severe keratoconus; pellucid marginal degeneration; contact lens-induced corneal warpage; and penetrating keratoplasty. The root-mean-square (RMS) error of the goodness-of-prediction of the Zernike representation of corneal surface elevation was extracted for 4, 6, and 10 mm optical zones, whereas Zernike radial orders were varied from 3 to 14 in 1-order steps. The mean +/- SEM of the RMS error was plotted as a function of Zernike order and compared with criteria for normal surface fits.

Fitting accuracy improved as more Zernike terms were included, but some conditions showed significant errors (when compared with normal surfaces), even with many added terms. For a 6 mm optical zone, the normal cornea group had the lowest RMS error and did not require terms above the 4th order to achieve <0.25 microm RMS error. Astigmatism met the 0.25 microm threshold at the 5th order, whereas keratoconus suspect required 7 orders. Laser in situ keratomileusis and PRK met the 0.25 microm threshold at the 8th order, whereas RK required 10 orders. Contact lens-induced corneal warpage and mild keratoconus needed 12 orders to meet the 0.25 microm threshold, whereas pellucid marginal degeneration, moderate and severe keratoconus, and keratoplasty categories were not well fitted even at 14 orders.

A 4th-order Zernike polynomial appeared reliable for modeling the normal cornea, but using a 4th-order fitting routine with an abnormal corneal surface caused a loss of fine-detail shape information. As more Zernike terms were added, the accuracy of the fit improved, and the result approached the minimum error found with normal corneas. Unless sufficient higher-order Zernike terms are included when analyzing irregular surfaces, some diagnostic applications of Zernike coefficients may not be rigorous. This conclusion also suggests that wavefront shape analysis is similarly dependent on the number of orders used. Current surgical corrections based on normal-eye wavefronts may fail to capture all visually relevant aberrations in abnormal eyes, such as those having laser retreatments or experiencing corneal warpage from contact lens wear. A clinical goodness-of-fit or goodness-of-prediction index would indicate whether the number of terms in use has fully accounted for all of the visually significant aberrations present in the eye.