Fourier Wavefront Analysis

Fourier analysis was recently introduced as an alternative to Zernike polynomials to describe an optical wavefront and provide the basis for designing a customized corneal ablation profile. This article provides a conversational understanding of the principles of Fourier analysis that will aid clinicians in describing customized corneal ablation to their patients as well as provide background information relevant to laser refractive surgery.

WAVEFRONT RECONSTRUCTION
The fundamental steps of customized laser vision correction include data acquisition, wavefront reconstruction, ablation-profile calculation, and the delivery of laser energy to the level of the cornea. In wavefront reconstruction, Hartmann-Shack information (known as centroids) is the basic raw data. To generate a centroid, a thin beam of monochromatic light is projected down the visual axis, and the reflected light from the retina is captured at individual points. The disparity between the anticipated and actual location of the reflected light is calculated, and a mathematical algorithm is used to derive a wavefront map (Figure 1).

Zernike or Fourier algorithms are two primary methods by which surgeons process Hartmann-Shack data to produce a wavefront map. They are different mathematical ways of achieving the same goal of reconstructing the wavefront of the visual system.

WHY WERE ZERNIKE'S CHOSEN ORIGINALLY?
The familiarity of the lower-order Zernike terms in clinical optics make Zernike algorithms a natural choice for the diagnostic applications of wavefront analysis. Defocus, astigmatism, and spherical aberration are quantified by Zernike terms. Zernike polynomials provide a common language for the description of the various components of aberrations that cumulatively account for blur. However, there are significant limitations inherent in the use of Zernike polynomials that are a consequence of using a polar coordinate-based system.

With higher-order Zernike polynomials (such as trefoil and tetrafoil), my colleagues and I describe mathematical shapes of aberrations that do not have an analogous visual component, as do lower-order aberrations, such as astigmatism, defocus, or spherical aberration. Zernike polynomials are more the building blocks inherent in mathematical models as opposed to relevant shapes we are trying to extract. Zernike polynomials can reasonably approximate the wavefront error in an eye, but the fitting of complex patterns induces averaging and smoothing, which are a consequence of the mathematical model.

Therefore, an increasing number of terms are necessary to provide resolution of complex wavefront measurements. Limiting the number of Zernike terms used to describe a shape will result in the attenuation of detail (Figure 2). Clinical correlation of this effect has been demonstrated: when Zernikes are used to fit a complex surface, such as a keratoconic cornea, the fitting error is substantial.1

However, there is a practical limit to how many orders of polynomial expressions we can calculate based on the fidelity of the raw data that we enter. The data captured lose fidelity beyond nine orders of Zernikes, and more noise than information is introduced into the solution beyond 10 orders. Zernike polynomials are also mathematically unable to describe a straight line. This limitation has implications in terms of analyzing visual aberrations that result from striae, cap amputations, or any number of other aberration defects—either natural or induced—that have a linear quality.

WHAT IS FOURIER ANALYSIS?
The Fourier method involves the use of a series of chosen sine waves to reconstruct a complex pattern. The capability of Fourier analysis is simply and graphically illustrated by its ability to describe a square wave. It is possible to take an appropriate set of sine waves and add them together to describe a given shape (Figure 3). That more complex patterns can similarly be described is the basis of the Fourier method in optical wavefront analysis.

Fourier transformation analysis uses a mathematic process of harmonic analysis that does not split the visual system into individual terms but rather creates a description of the whole wavefront. The building of a complex wave with chosen sine waves is called Fourier synthesis. The breaking apart of complex waves into components of sine waves is called Fourier analysis. For the process known as wavefront analysis, a complex pattern is decomposed into sine waves.

Currently, the Fourier method is a widely used mathematical model for problem solving in science and engineering. The algorithm is used in linear systems analysis, antenna studies, optics, probability theory, quantum physics, random process modeling, and boundary-value problems. The Fourier method also is employed commonly in the refinement of astronomic data from multiple lens systems. For example, the Keck telescope in Hawaii uses Fourier transformation to provide ultra high resolution for its ground-based observatory.2

ANALYZING WAVEFRONT DATA
The mathematics of Zernike polynomial interpretation requires a cluster of points to be analyzed and totaled. There must be adjacent data to each individual data point to allow interpretation. A consequence of using Zernike polynomials is that centroids at the pupillary margin are not included in the wavefront analysis. This method also eliminates data points in which the adjacent centroids are noninterpretable Out of the approximately 240 Hartmann-Shack centroids in a 7-mm pupil, an average of 26 data points are generated for use in fitting the Zernike polynomials. The polar-coordinate nature of the Zernike system necessitates that all data are obtained with respect to their correlation to a central point. Therefore, Zernicke polynomials define a circular shape that works well for round pupils but will ignore some data in those that are oval or irregularly shaped (Figure 4).

In Fourier analysis, all data points are weighed equally and incorporated into the derived wavefront shape, the result of which depends on the quantity of data used to determine the shape. Fourier analysis incorporates each unique centroid and does not require the inclusion of adjacent data points. When using the Fourier method, data points from the margin of the pupil and those outside a round pupil are incorporated in the analysis.

TRANSIENT NOISE
Is it possible that the Fourier algorithm could provide too much resolution? Could it be measuring at a level that introduces variability attributable to transient noise, caused by measurement error or tear film breakup or other characteristics of the human optical system? Does a Fourier-based procedure permanently etch transient visual aberrations into the eye?

Theoretically, in a continuous optical system, one of the benefits of Zernike polynomials is that smoothing occurs between data points. However, the smoothing may, in fact, be a disadvantage in eyes that are aberrated from previous surgery or other causes. This smoothing can be graphically represented by a test pattern that is analyzed by both mathematical methods (Figure 5). The Zernike solution demonstrates a resemblance to the test pattern shape but has a noticeable loss of detail. The Fourier transformation provides an increased level of detail and shows a greater resemblance to the initial image.

It is still not definitively determined if it is more desirable to have smoothing with Zernike or to have a more detailed transformation with Fourier. These ideas are worthy of consideration, and examining actual wavefront data helps illustrate the issue. Because these algorithms both capture information using Hartmann-Shack data points, the same information can be collected on a single patient, analyzed in two different ways, and then compared.

For example, three wavefront examinations were taken for the left eye of a patient (Figure 6). The examinations were captured several minutes apart, and each was processed using both Zernike polynomials up to the sixth order and Fourier reconstruction. The resultant wavefront maps were then translated into ablation target maps. They show that the eye contains repeatable, correctable aberrations that cannot be measured using the Zernike notation. The size, orientation, and shape of the aberrations are very consistent from one examination to the next, a finding indicating that the features cannot be caused by measurement noise or fluctuations in the tear film. Furthermore, each of the Zernike-based target maps shows the smoothing effect of Zernike reconstruction. The wavefront aberrations are minimized and distorted using Zernike polynomials.

CONCLUSION
The use of Fourier transformation to analyze a wavefront represents an alternative approach to Zernike polynomials. Despite potential advantages to either method, the increased resolution of complex shapes with Fourier transformation suggests that it may prove to be a useful modality in treating patients.

Both techniques represent well-validated mathematical models of describing complex wavefronts, and it is likely they will be available for clinical use in the future. Continued use will ultimately authenticate which method provides superior results in the long run.

John A. Vukich, MD, is Assistant Clinical Professor at the University of Wisconsin, Madison. He is a consultant for Visx, Inc.
Dr. Vukich may be reached at (608) 282-2002; javukich@facstaff.wisc.edu.
1. Smolek MK, Klyce SD. Zernike polynomials are inadequate to represent higher order aberrations in the eye. Invest Ophthalmol Vis Sci. 2003;44:4676-4681.
2. Ghez AM, Morris M, Becklin EE, et al. The accelerations of stars orbiting the Milky Way's central black hole. Nature. 2000;407:349-351.