This document is an electronic version of a paper published and copyrighted by the Optical Society of America. It is provided in electronic form for the convenience of those who wish to use the methods described. This electronic version contains additional appendices containing worked examples not contained in the published version.

Any work which quotes these methods should refer to the original publication. The full reference is:

Thibos, L. N., Wheeler, W. & Horner, D. (1994). A vector method for the analysis of astigmatic refractive errors. Vision Science and Its Applications, (Optical Society of America, Washington, DC), 2, 14-17.

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A Vector Method for the Analysis of Astigmatic Refractive Errors

Larry N. Thibos, William Wheeler*, and Douglas Horner, School of Optometry and Department of Mathematics*, Indiana University, Bloomington, IN 47405

Introduction

Refractive errors of the eye are specified by the sphero-cylinder lens required to render the eye emmetropic. Unfortunately, the standard ophthalmic conventions used to describe correcting lenses is not well suited for mathematical manipulation or statistical analysis. There are two reasons for this state of affairs. The first is a pragmatic complaint. The astigmatic component of refractive error is specified in polar form (i.e. power at some axis) which is not convenient for computing the sum or difference of one or more lenses. Consequently, it is difficult to perform even the simplest statistical computation (e.g. mean, variance) of refractive error data, or to compute the total power of several lenses placed in close aposition (the dreaded "crossed cylinder" problem).

The second difficulty with current practice is more conceptual in nature. Standard notation for refractive errors naturally leads to the view that a sphero-cylinder lens is the sum of a spherical lens and a cylindrical lens. Unfortunately, these two components are not independent of each other since a cylindrical lens contains some spherical power, called the "mean spherical equivalent" (MSE).

To deal with these difficulties, two methods have been suggested. Gartner [1] proposed an operational method in which the two parameters used to specify astigmatic lenses (power, axis) be represented graphically as a 2-dimensional vector. Gartner and others have pointed out that if such a vector is expressed in rectangular (rather than polar) form then optometer design [2] and analysis of refractive data [3] would be greatly simplified.

Taking a different approach, Keating [4,5,6] and Harris [7,8,9] have developed statistical methods for analyzing refractive errors based on Long's representation of a sphero-cylinder lens as a dioptric power matrix [10]. In Long's scheme, the power in the principal meridia and the two oblique meridia are arranged as a 2x2 matrix which then plays the same role for sphero-cylinder lenses as the dioptric power plays for spherical lenses.

Purpose and Summary of Results

The present study takes a fresh look at the problem from the point of view of Fourier analysis. Starting from an analysis of the curvature of arbitrary surfaces, we show why optical power at any given point varies sinusoidally with meridian. Consequently this function, called a power profile, is describable by a simple Fourier series with a single harmonic component. Such a series contains just three Fourier coefficients, which we show correspond to the three natural parameters of a sphero-cylinder lens. This observation allows for the economical and intuitively satisfying representation of the complete power profile of a sphero-cylinder lens by a single point, or power vector, in a three dimensional dioptric space. These results are a natural extension of Gartner's and Harris' earlier operational methods, and thus provide a sound theoretical foundation for vector methods of analyzing refractive error data.

Power Profiles

The optical power of a refracting surface is directly proportional to its curvature [[kappa]] and to the difference of refractive index

. (1)

Curvature of an arbitrary surface depends only on first and second derivatives at the point in question. Consequently, Euler's theorem states that [[kappa]] varies with meridian [[theta]] as follows

(2)

where [[kappa]]x and [[kappa]]y are the surface curvatures in the principle sections (i.e. the sections which give greatest and least curvature, respectively) which are inclined at the angle [[alpha]]. Combining eqns. (1) and (2) leads to the textbook formula for the variation of lens power P with meridian [[theta]] for an arbitrary refracting surface

(3)

where

(4)

We refer to the function P([[theta]] ) as a power profile. In its present form eqn. (3) disguises the fact that power varies as a simple sinusoidal function of a double angle. To see this, apply the identity to get

. (5)

Fourier Analysis

The point of departure for our study was the interpretation of eqn. (5). In conventional ophthalmic practice, the constant term C/2 is grouped with the cosine term so that together they represent a cylindrical lens. The remaining term corresponds to a spherical lens of power S and thus together these two lenses make up a sphero-cylinder lens. However, we also recognize eqn. (5) as a formal Fourier series which contains a constant term (S+C/2) plus one harmonic term (C/2 cos(2([[theta]]-[[alpha]])). An important benefit of this Fourier interpretation, which keeps the constant terms together, is that the constant term is mathematically independent of the cosine term. This independence follows from the fact that the basis functions of a Fourier series are mutually orthogonal .

At first glance the Fourier interpretation of the power profile of a sphero-cylinder lens may seem more abstract than the conventional ophthalmic interpretation for which there is a direct correspondence between mathematical terms in eqn. (5) and physical lenses. However, it turns out that the terms of the Fourier series also correspond naturally to physical lenses. The constant term is, of course, a spherical lens whereas the pure cosine term describes the power profile of a Jackson cross-cylinder (JCC) lens. Although Jackson cross-cylinders are generally regarded as specialty lenses in clinical practice, they take on a central role in the new method of analysis we propose here.

To conveniently specify a JCC lens requires that we first establish a suitable convention. To do so we note that a power profile of the form

(6)

corresponds to a JCC lens made from the combination of an ordinary cylinder of positive power J at axis [[alpha]]+90 and a negative cylinder of the same power at axis [[alpha]]. We will use the notation J[[alpha]] for such a lens, where J denotes the power and the subscript [[alpha]] denotes the axis. By this convention, the cosine term in eqn. (5) corresponds to a JCC lens of power J=C/2 with its axis inclined at the angle [[alpha]]. An example is illustrated in Fig. 1, which demonstrates that a JCC lens (+1D, x45deg.) is equal to the combination of two conventional cylindrical lenses.

Fig. 1. The power profile of a Jackson cross-cylinder (JCC) lens.

In summary, the Fourier approach decomposes an arbitrary sphero-cylinder lens S +C x ([[alpha]] +90). into the sum of a spherical lens, of power M=S+C/2, and a JCC lens, of power J=C/2 with its axis inclined at the angle [[alpha]]. This allows us to re-write eqn. (5) in the proposed Fourier convention as

. (7)

Equation (7) describes the power profile in polar form. For computational purposes it is usually more useful to write the same equation in rectangular form by making use of the identity to re-write eqn. (7) as

(8)

where is the power of a JCC lens at axis 0deg. and is the power of a JCC lens at axis 45deg.. Expressed in words, equation (8) says that an arbitrary sphero-cylinder lens is equivalent to the sum of a spherical lens of power M and two JCCs, one at axis 0deg. with power J0 and the other at axis 45deg. with power J45.

Power Vectors

The foregoing analysis shows that the power profile of an arbitrary sphero-cylinder lens is completely specified by just three parameters: M, J0 and J45. Since each of these parameters has the units of diopters, it makes sense to plot them as a single point in a 3-dimensional dioptric space in a manner similar to that proposed by others [8,11]. In other words, let x=M, y=J0 , and z=J45 and then draw a vector from the origin to the (x,y,z) coordinates as illustrated in Fig. 2. By analogy with Long's power matrix concept, we call the result a power vector.

Fig. 2 A 3-dimensional dioptric space for representing refractive errors.

Discussion

Numerous benefits accrue from the power vector representation of lenses. First, it is a very economical scheme since an entire curve (the power profile) is replaced by a single point (the tip of the vector) in three-dimensional space. Second, the axes of the space are easily understood in optical terms since they represent the three components of an arbitrary sphero-cylinder lens: a sphere, and two JCC lenses. Third, the geometrical optical rules for lens combinations are equivalent to the ordinary rules of vector addition. This is true because the three coordinate axes correspond to the three Fourier coefficients of the power profile. Since Fourier components are mutually orthogonal, linearity applies in the sense that if two power profiles are combined linearly (i.e. added or subtracted) , then so are the corresponding Fourier coefficients. Thus the summation of power profiles (as in Fig. 1) is equivalent to the addition of their corresponding power vectors.

This direct correspondence between geometrical optics and vector algebra suggests that the dioptric space of Fig. 2 is a useful domain for formulating optical lens problems. For example, determining the sum of obliquely crossed sphero-cylinders is often described as a difficult problem, but when lenses are represented as vectors the combination of two or more sphero-cylinder lenses is easily computed by the ordinary rules of vector addition (i.e. separate summation of x,y,z components). Another useful feature of the dioptric space of Fig. 2 is that the magnitude of the difference between two lenses is just the Euclidean distance between two points, in units of diopters.

The dioptric space of Fig. 2 is also useful for visualizing the time course of changes in refraction. If the correcting lens is described by the tip of the power vector, then changes in refraction over time will be described by the trajectory of the tip of this moving vector. Such a trajectory could provide insight into a variety of clinical phenomena, such as the changing refractive state of the eye following refractive surgery or ortho-keratology, during normal accommodation, or during the course of development or of a disease which may affect the dioptric apparatus of the eye.

(Supported by NEI grant EY05l09 to LNT.)

References

1. W. F. Gartner, "Astigmatism and optometric vectors," Am. J. Optom. & Arch. Am. Acad. Optom. 459-463 (1965).

2. W. E. Humphrey, "Automatic retinoscopy: the Humphrey vision analyser," The Optician 173, 17-27 (1977).

3. A. G. Bennett and R. B. Rabbetts, Clinical Visual Optics, (Butterworths, London, 1984).

4. M. P. Keating, "An easier method to obtain the sphere, cylinder, and axis from an off-axis dioptric power matrix," Am. J. Optom. Physiol. Opt. 57, 734-737 (1980).

5. M. P. Keating, "On the use of matrices for the mean value of refractive error," Ophthalmic. Physiol. Opt. 3, 201-203 (1983).

6. M. P. Keating, Geometric, Physical, and Visual Optics, (Butterworths, Boston, 1988).

7. W. F. Harris, "Statistical inference on mean dioptric power: hypothesis testing and confidence regions," Ophthalmic. Physiol. Opt. 10, 363-372 (1990).

8. W. F. Harris, "Representation of dioptric power in Euclidean 3-space," Ophthalmic. Physiol. Opt. 11, 130-136 (1991).

9. W. F. Harris, "Testing hypotheses on dioptirc power," Optom. Vis. Sci. 69, 835-845 (1992).

10. W. F. Long, "A matrix formalism for decentration problems," Am. J. Optom. Physiol. Opt. 53, 27-33 (1976).

11. F. C. J. Deal and J. Toop, "Recommended coordinate systems for thin spherocylind-rical lenses," Optom. Vis. Sci. 70, 409-413 (1993).

Appendix: Worked examples

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Example 1: A brute-force example of transforming published refraction data (which happen to be in Ophthalmological +cyl form) into an intermediate stage of Polar Fourier form, then into Rectangular Fourier form. The latter is required to do such things as finding the mean of a set of refractions. You can easily skip the Polar Fourier form and go straight to Rectangular Fourier form if you like (see other examples). The analysis carried out is simply the calculation of the refractive difference between a pre-op and a post-operation condition. Data source: Holladay, Cravy, and Koch, (1992). "Calculating the surgically induced refractive change following ocular surgery", J. Cataract Refract. Surg., 18: 429-443 .

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Example 2: longitudinal study of the changes in refractive error following refractive surgery. Conventional refractions are converted to PowerVector notation in a single step. Changes relative to the pre-operative condition are computed as simple, linear differences between corresponding vector components and may be visualized as a trajectory in 3-dimensional (x,y,z) space. Data courtesy of Dr. Ray Applegate, U. Texas.