Comparison of Zernike and Fourier wavefront reconstruction algorithms in representing the corneal aberration of normal and abnormal eyes

Purpose: The goal of this study was to investigate with what accuracy the Zernike and Fourier reconstruction algorithms can describe the corneal aberration in normal and abnormal eyes. Methods: Corneal topography (Orbscan IIz) was collected on 87 normal, 27 keratoconus, 9 penetrating keratoplasty (PKP), and 20 post-LASIK symptomatic eyes. Raw topography images were converted into elevation maps, which were then re-sampled at resolutions of 100, 300, and 500 μm. Differences in elevation between adjacent pixels were used to generate simulated wavefront slope data. Both conventional Zernike and iterative Fourier algorithms were used to reconstruct the elevation map from the same slope information. The difference between the reconstructed and original maps was used to evaluate reconstruction performance, quantified by the residual root mean square (RMS) error. Results: Residual RMS error decreased logarithmically as the number of Zernike modes used in the reconstruction increased. Both algorithms had the least error with normal eyes and the greatest error with PKP eyes. Using a large number of Zernike modes when sampling resolution was low lead to inaccurate reconstruction. The Fourier method had better reconstruction reliability centrally than peripherally. Only 5 order Zernike modes were required to produce less residual RMS error than that produced by the Fourier method. Conclusions: For all conditions tested, the Zernike method out-performed the Fourier method when representing the corneal aberration in topography maps. Even 5 order Zernike polynomials were enough to out-perform the Fourier method in all populations. Up to 9 order Zernike modes may be required to accurately describe the corneal aberration in some abnormal eyes. Introduction: For nearly 30 years, the eye’s wavefront aberration has been objectively measured using a number of different aberrometry techniques. 1-4 Since then, it has become necessary to express the eye’s aberration mathematically to facilitate application of the technology clinically. The Zernike polynomials, an orthogonal base function used to describe optical systems with circular pupils, have thus far fulfilled this need. 5 In 2001, Porter showed through a principal components analysis that the Zernike polynomials efficiently described the eye’s wavefront aberration. 6 Although they did demonstrate that the Zernike fitting was not perfect and that it resulted in residual root mean square (RMS) error, the amount was undefined and uncorrelated to the number of modes used. Despite previous acceptance of the Zernike polynomials, there has been recent scrutiny concerning the accuracy with which they can represent the total ocular wave aberration, especially in abnormal eyes such as those having keratoconus, penetrating keratoplasty, or severe trauma. 7-10 In 2003 Smolek et. al. showed that the corneal fit error of the Zernike polynomials strongly correlated with visual acuity. 9 They concluded from this that the polynomials did not fully characterize the surface features that affect vision. They also observed an unpredictable increase in wavefront error when using a larger expansion series (10 order) to represent the corneal aberration. In 2004 Klyce et. al. reiterated their concern with respect to using Zernike polynomials in surgical or pathological eyes. They found that using Zernike polynomials in normal eyes was acceptable, but that moderate to severe amounts of higher-order aberration caused significant fit error and an underestimation of the total higher-order corneal aberration present. Other methods including zonal reconstruction using bi-cubic splines and use of a Fourier series have been suggested as alternatives for representation of the wave aberration in eyes. 12 Of these, the Fourier series has become of particular interest recently. Like the Zernike polynomials, the Fourier series is an infinite expansion that can be used to represent any complex shape (including a wave aberration) by breaking it into its frequency components. Previous studies by Smolek and Klyce that used a Fourier-based algorithm worked by directly fitting the surface of the cornea. However, most commercially available ophthalmic wavefront sensors use a different method, namely wavefront slope fitting. Therefore, their results may not be directly applicable to most scientific and clinical studies involving wavefront sensors. More recently, Dai published the first study comparing Fourier and Zernike reconstruction algorithms using the wavefront slope fitting method. He concluded that the Fourier algorithm outperformed the Zernike one on highly aberrated wavefront shapes, but the study made several assumptions that may compromise objectivity and clinical applicability. For example, his choice of randomly regenerated Zernike coefficients up to the 15 order as a gold standard makes it difficult to correlate performance of these algorithms to wavefront shapes similar to those found in normal and abnormal eyes. Perhaps more importantly, Dai started with a 10mm pupil but based reconstruction performance on only the central 6mm. Since it is impractical to measure wavefront slope over a 10mm pupil clinically, true reconstruction performance should be based on a smaller reference pupil size. These shortcomings are outlined in greater detail in the discussion. The purpose of this study was to investigate the relative accuracy with which the Zernike polynomials and Fourier series can represent the corneal surface aberration in both normal and abnormal eyes.