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A. Alchagirov, S.D. Klyce, M.K. Smolek, M.D. Karon; Comparison of Accuracy of Zernike Polynomials and Fourier Series in Corneal Surface Representation . Invest. Ophthalmol. Vis. Sci. 2005;46(13):851.
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© ARVO (1962-2015); The Authors (2016-present)
There is a controversy about the advantages of Fast Fourier Transforms (FFT) versus Zernike series for the fitting of corneal data. Unlike the Zernike method, FFT is not an approximation, but an exact transformation between spatial and frequency domains. FFT error is introduced during data preparation (mapping onto an evenly spaced grid), and therefore is not comparable with the fitting error of the Zernike series. We propose a novel method that allows direct comparison of the accuracy of both approaches.
Reconstructed elevation data was analyzed for 5 Tomey corneal topography exams for each of 4 categories: mild keratoconus (KC1), penetrating keratoplasty (PKP), myopic refractive surgery (MRS), and normal (NRM). Best fit reference spheres were used to obtain residual elevation data. For the Fourier approach, data was translated into the first quadrant of the XY plane to allow the use of the even extension of the Fourier series. Incomplete 3D Fourier series and Zernike polynomials of various orders were fitted to the corneal elevation data using the least–squares and the RMS errors of the fits were compared. Mean RMS errors and standard deviations were calculated.
Both Fourier and Zernike methods exhibit good fitting accuracy; however, at > 19 Zernike orders the design matrix of the fit becomes singular and additional terms result in an increase and instability of RMS error (see figure). The Fourier fit was stable and non–singular at > 35 orders. It's best RMS error was ∼3 times smaller than the Zernike's for all cornea categories, as shown in the figure. The limiting number of fitting parameters for the Fourier series was of the same order as the Nyquist limit on the number of frequencies for the FFT algorithm. This indicates that the Fourier series in this limit is an adequate substitute for the FFT.
The proposed method allows direct comparison of fitting accuracy of the Zernike polynomial series and the Fourier cosine series. The Zernike polynomials were shown to become unstable after a critical number of terms, which limited its accuracy. The Fourier series was shown to be significantly more accurate at higher orders.
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