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Inadequacy of Zernike Polynomial Fitting: Case Not Made
Author Response: Inadequacy of Zernike Polynomial Fitting: Case Not Made
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Charles E. Campbell
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ceccec{at}pacbell.net Charles E. Campbell
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I have read with interest the recent paper by Michael K. Smolek and Stephen D. Klyce entitled "Zernike Polynomial Fitting Fails to Represent All Visually Significant Corneal Aberrations" and have the following comments. Representation of a wavefront error by a set of coefficients generated by Zernike polynomial decomposition is a compact and convenient way of transmitting this information to others, and, if correctly used, can give insight into resulting degradation of vision due to the error. But in this paper Zernike decomposition has been used in a way quite different from the way it is customarily used. Customarily Zernike decompostion is used to analyze the difference between the actual wavefront and a "perfect wavefront." Here, however, Zernike decomposition is also used as fitting method to fit elevation data collected by a corneal topography system, and this fit has then been compared against the original data, characterized by a root mean square error (RMS) value. This is certainly a good way to render corneal topography data "smooth," thereby removing noise, and it is a method I and others have successfully used. But it is a method that must be used with care, and certainly truncating the decomposition at 4th order is insufficient for almost all real cases. So it seems to me that the authors' claim that Zernike polynomial decomposition cannot represent the aberrations of the eye induced by corneal irregularities using 4th order Zernike decomposition mistakes the expressing of the wavefront error itself to this order to that of fitting the corneal surface itself to this order. They are quite different cases. I can agree that in cases of high corneal irregularity, such as those considered in this paper, 4th order Zernike decomposition is insufficient, but this is no reason to claim that Zernike decomposition is inadequate in general. The authors have used a correlation between Best Spectacle Corrected Visual Acuity (BSCVA) and wavefront RMS error as a measure of the quality of the fit using Zernike polynomials. The measurement of BSCVA is really a measure of the medium to high spatial frequency band of the eye's modulation transfer function (MTF), because the letter details are small (high frequency), and people use edge contrast and similar clues to judge which letter they are seeing which also are composed of high spatial frequencies. In the presence of aberrations, especially those of higher-order, this MTF band becomes complex, and a simple measure like RMS wavefront error is inadequate. RMS wavefront error is best used to establish a threshold over which vision starts to degrade. But once this threshold has been passed, RMS wavefront error no longer is a good measure of the degradation. For this reason I feel the author's use of RMS error as a correlate to BSCVA is inappropriate. It would have been much better to calculate the MTF from the measured wavefront error and correlate perhaps the mean value in an appropriate band to the BSCVA. The program CTView was used by the authors to calculate the wavefront error, and in so doing this program subtracts a reference sphere from the actual corneal elevation data to create an "error lens." I and others have noted that a sphere is not the appropriate reference surface, because a spherical surface will not refract an incoming plane wavefront to create a spherical wavefront, which will then create a diffraction limited point spread. The correct reference is a Cartesian oval, which in this case is an ellipsoid of revolution. However, in fairness to the authors, they had to use what was available to them, and the error induced by so doing would primarily have been in indicate a difference amount of spherical aberration than would truly have been induced by the cornea. What would have been of more interest than analyses presented would have been an analysis of effect on the MTF of changing from a total wavefront error based on the difference between the corneal elevation data as received from the corneal topographer and a reference surface to the wavefront error based on the difference between the corneal elevation reconstructed from the Zernike fitting and a reference surface. This would tell if the fitting error is significant or not. I do not think that the authors have made their case that "Zernike Polynomial Fitting Fails to Represent All Visually Significant Corneal Aberrations" because they have not shown that the difference between using elevation data directly from a corneal topographer and using elevation data from a surface fitted to this data using Zernike polynomial fitting is significant from the point of view of image formation. The fact that there is a correlation between a decrease in BSCVA and an increase in fitting error by no means shows that the fitted surface is missing important visual information. The result the authors present is better interpreted as showing that as the corneal surface becomes more irregular and hence becomes a poorer refracting element, not only does this make vision poorer as one might expect, but it also means that a more complex surface than is represented by the fitted surface is needed to fit the actual surface well. The question of whether or not one could, by using the fitted surface only, predict the decrease in BSCVA reasonably is completely left open, and that is the question that must be answered if one is to judge if using Zernike fitting methods is appropriate. Finally, there is an interesting piece of information that one can glean from one of the maps in Figure 7. Examination of the lower right hand map showing the difference between the corneal topographer elevation values and those from the reconstructed surface using 10th order Zernike fitting reveals a finely spaced radial spoke pattern. This pattern has at least two components. One that varies in a meridional direction about every 36 degrees appears to be a residual artifact of the symmetry found in the Z(10,10) and Z(10,-10) terms as they varying at this rate meridionally. The second component varies meridionally at a higher, but not as constant, a rate. It probably reveals discontinuities of the corneal topographer's reconstruction of the surface from its videokeratoscope data. These can occur due to the fact that corneal topographers typically reconstruct along meridians so that surface continuity is forced along meridians, but not across them. I have also seen these fine radial spoke patterns in data from a different corneal topographer than the one the authors used and commented on them in a presentation I gave this past summer, which may be viewed online at http://research.opt.indiana.edu/default.html by selecting "Meetings," then the date, August 5, 2003, and then selecting my presentation. The relevant images are presented in my Slide #10. These fine radial patterns serve as a measure of an error in the corneal topographer's representation of the corneal surface created by its reconstruction algorithm. High frequency radial variations are unlikely in the cornea itself, and, of more importance, cannot be supported the tear film due to surface tension effects in a thin fluid film. Charles Campbell |
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Michael K. Smolek , Stephen D. Klyce
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msmole{at}lsuhsc.edu Michael K. Smolek, et al.
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We thank Mr. Charles Campbell for his thoughtful remarks on our recent paper: "Zernike polynomial fitting fails to represent all visually significant corneal aberrations." Mr. Campbell is a recognized expert in the field of physiologic optics, has chaired the committee that formulated the ANSI Standards for Corneal Topography, and has published on the use of Zernike polynomials in wavefront analysis. However, with all due respect, we take issue with his comments. Historically, Zernike polynomials have been used to measure and correct aberrations within optical instruments, as well as for computing the imperfections within object space (such as turbulence in the atmosphere). Unlike ophthalmic applications, it should be kept in mind that some specialized applications of Zernike fitting, such as astronomical imaging, can involve multiple sub-aperture analysis and temporal integration of images, which will lead to significantly improved resolution. Such methods are unavailable in ophthalmology, and a direct comparison of performance across different applications is grossly unfair and uninformative. However, we believe that the Zernike polynomial is still a very powerful ophthalmic tool, allowing the extraction and measurement of aberrations from wavefronts obtained in the living eye. Such data allows individualized corneal refractive corrections to be made with laser systems that go beyond the second order spherocylinder corrections of spectacles and contact lenses. The question is, how far beyond the second order and at what level of accuracy? It is interesting to note that the visual outcome of individualized laser surgery may be worse than expected, which suggests that perhaps some aberrations, notably the higher order aberrations, are unaccounted for by the Zernike fit now commonly used. In order to judge the accuracy of any fitting routine, including Zernike polynomials, we need to determine the correlation between patient visual acuity and the error accounted for by the fitted surface, as well as the correlation between visual acuity and the residual error not accounted for by the fitting routine. Our results indicate that the Zernike polynomial tends to work well in accounting for errors due to the lower order terms, but it fails to fully capture all of the visually significant information present in the higher order aberrations, such as those due to disease, trauma, or induced by surgery. This should not be surprising given that even the best fitting routines are seldom perfect and may fail in capturing the highly detailed information of a surface. In addition, we must realize that error is distributed piecemeal among the higher order Zernike terms. The more higher order terms you add, the better able in theory you can account for surface details, but in return you must give up on processing speed. In many instances, this is not practical. Furthermore, the higher the radial order and angular frequency of a basis function, the less likely it is to contribute to visual performance. In these respects, we can say that Zernike fitting becomes increasingly less efficient for higher order aberrations, and that suggests that more efficient algorithms are likely to drive lasers in the near future. Bear in mind that when it comes to driving a laser beam, there is no particular need to reference traditional aberration shapes such as coma and spherical aberration. The main concerns will be accuracy and speed of the algorithm. Mr. Campbell implied that we are not using Zernike polynomials as they should customarily be used, which is to examine the difference between an "actual wavefront and a perfect wavefront." Instead, we measured the error between the true corneal elevation (as defined by topographic data) and the Zernike representation of the elevation surface, in order to assess the goodness-of-fit under specific test conditions such as pupil size and numbers of terms. There is nothing unorthodox about using a fitting routine in this manner, and in fact it is an important and popular feature of the CTView (now called Vol-CT) software application. Mr. Campbell indicated that fitting the cornea to a 4th order Zernike polynomial will make it overly smooth, and in this we completely agree. Apparently the disagreement is that we conclude that a 4th order fit (or any fit that uses insufficient numbers of terms) to a wavefront surface is also overly smooth. If an eye is normal and with few higher order aberrations, then the 4th order fit is perhaps marginally acceptable, but for any abnormal eye (including retreated eyes), the Zernike fit will be unacceptably worse than a normal eye. Because laser systems use at most only several Zernike orders, any procedure performed on an abnormal eye is likely not to be fully corrected. Even a 10th order fit may be less than ideal when the eye is highly aberrated. Mr. Campbell further argues that we used highly irregular surfaces in our study and that the Zernike polynomial performs better with more normal surfaces. He apparently fails to see the dichotomy of his argument. If all surgical candidates had normal wavefronts, then perhaps there would be less of a need to fit to higher orders, but not all eyes are close to being normal and there is no reason to believe that normal eyes do not have higher order aberrations naturally. Based on our data we concluded that the Zernike polynomial is generally acceptable for normal, presurgical, virgin eyes, but we show compelling evidence that the fit tends to be unacceptable for abnormal eyes (including laser retreatment eyes), because the polynomial cannot adequately account for all of the error that correlates to vision. Our study is not simply an exercise in future limitations of the Zernike approach. The results are clinically significant today because: 1) patients seeking visual corrections often fall within the abnormal category; 2) surgeons want to satisfy their patients and not turn them away; and 3) refractive surgeons will continue to push the treatment envelope to include more irregular wavefronts from abnormal eyes. Consequently, it is clear to us that we need a reliable fitting routine today that can account for the full amount of wavefront error. The Zernike polynomial is apparently limited in fulfilling our current and future needs. In addition, we now better appreciate that surgeons need to know the accuracy of the Zernike fit given the number of terms being used and the irregularity of the wavefront error. To our knowledge, no aberrometry system currently provides even a rudimentary estimate of the goodness-of-fit of the polynomial. Consequently, surgeons have no clue as to whether the chosen Zernike fit is anywhere close to being accurate enough to correct the wavefront error. We next disagree with Mr. Campbell's assertion that BSCVA cannot be used in a study of optical performance with higher order aberrations. We found a strong and statistically significant correlation between BSCVA and wavefront error as well as the elevation fit error. Clearly the Zernike polynomial fitting routine does account for a large portion of the aberrations that affect visual acuity, but no fitting routine is perfect, and the Zernike fit does not account for all visually significant aberrations. Mr. Campbell tells us that RMS error is not the correct metric to use when evaluating visual performance. We certainly agree that there are metrics with even better correlations to vision than RMS error as the research of Applegate, Thibos, Williams, and others shows, but these metrics are not available for clinical use at this time. The fact that the RMS of the fit error has a statistically significant correlation with visual acuity and the fact that it is a measure that is available to researchers underscores its utility in studies such as ours. Mr. Campbell writes that it would be more appropriate to use the modulation transfer function (MTF) for correlation to the wavefront error, but we must warn the reader that if the MTF is obtained from the Zernike fit of the wavefront error, then it will be a potentially flawed MTF based on a poor surface fit. In summary and with respect to Mr. Campbell's assertion that we have not made our case, we simply say that the main conclusion of our study speaks for itself and should not be ignored. There is a significant correlation between BSCVA and the corneal surface shape error that remains after accounting for the contribution by the Zernike fitted surface at both 4th and 10th orders. If the Zernike polynomial actually fit surfaces well, there would be no correlation because the residual error would be insignificant. The limits of curve fitting need to be better understood and appreciated by those who perform wavefront analysis. We would also point out that after submitting our paper to IOVS, we learned that Douglas Koch, MD and others on his research team, including VISX researchers, have been working independently on the same issue of Zernike fitting accuracy. They have come to the same conclusion we have concerning the limitation of the Zernike polynomial to account for all visually significant and correctable aberrations, and are now working on a proprietary fitting routine that appears to be an improvement over the Zernike polynomial. Michael K. Smolek and Stephen D. Klyce Department of Ophthalmology, LSU Eye Center, New Orleans, LA |
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