Schwiegerling et al. Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2105

Representation of videokeratoscopicheight data with Zernike polynomials

Jim Schwiegerling

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

John E. Greivenkamp and Joseph M. Miller

Optical Sciences Center and Department of Ophthalmology, University of Arizona, Tucson, Arizona 85721

Received November 23, 1994; revised manuscript received April 17, 1995; accepted May 15, 1995

Videokeratoscopic data are generally displayed as a color-coded map of corneal refractive power, cornealcurvature, or surface height. Although the merits of the refractive power and curvature methods havebeen extensively debated, the display of corneal surface height demands further investigation. A significantdrawback to viewing corneal surface height is that the spherical and cylindrical components of the corneaobscure small variations in the surface. To overcome this drawback, a methodology for decomposing cornealheight data into a unique set of Zernike polynomials is presented. Repeatedly removing the low-order Zerniketerms reveals the hidden height variations. Examples of the decomposition-and-display technique are shownfor cases of astigmatism, keratoconus, and radial keratotomy. 1995 Optical Society of America

1. INTRODUCTION

Use of commercially available videokeratoscopes by eye-care providers and vision scientists has become wide-spread. The majority of these devices are based uponPlacido-disk technology, in which a series of concentricilluminated rings (mires) reflect off the cornea of an ob-server and are imaged, along with the cornea, by a videocamera. From knowledge of the geometry of the kerato-scope and the variations in ring spacing in the image,the slope of the corneal surface can be determined inthe meridional direction. For display and analysis of thedata obtained from videokeratoscopes, the corneal slopedata are usually converted to a more intuitive form. Thederivative of the acquired slope data gives corneal curva-ture and is related to the optical power of the surface.There are some ambiguities associated with this tech-nique that stem mainly from alternative definitions ofcurvature and power. The accuracy and validity of thesedefinitions have been extensively debated, and a thoroughanalysis is given by Roberts.1 The corneal slope data canalso be integrated for determining the height or sag ofthe corneal surface. This method has the advantage thatthe true topography of the cornea is given. However,higher-order height variations tend to be concealed by thelower-order components of the cornea. Examples of in-terpreting videokeratographic height data are given byseveral authors.2 – 6

The most common method for displaying videokerato-scopic data is a color-coded map that gives some mea-sure of the dioptric power distribution of the cornea. Forthese displays, the corneal slope data are used to calcu-late the axial or instantaneous curvature at a given pointon the cornea. By assuming an effective index of thecornea, one can calculate a local dioptric power from thecurvature data. These maps use a color scale on whicheach color corresponds to a range of power, so that the

0740-3232/95/102105-09$06.00

power map is a contour plot of corneal dioptric power.Recently, however, cases for using curvature maps andheight maps as alternatives or complements to powermaps have been made. Eye-care providers and visionscientists use videokeratographic data for a variety of ap-plications, and a given display method may prove morebeneficial depending on the application. The corneal cur-vature map may be advantageous to the optometrist fit-ting contacts or spectacles. The refractive surgeon maywish to examine changes in dioptric power resulting fromsurgery. The optical engineer may prefer a height mapof the cornea in order to determine the optical quality ofthe surface and model the effects of various corneal condi-tions. Regardless of the application, a thorough under-standing of the benefits and drawbacks of each method isnecessary to enable one to choose the appropriate analysisand display method for the appropriate application.

In this paper a methodology for analyzing videoker-atographic height data is presented. The technique in-volves decomposing the corneal height data in terms ofthe orthonormal set of Zernike polynomials. The use ofZernike polynomials to represent the corneal surface hasbeen suggested by several authors.2 – 6 This is a directanalogy to the widespread use of Zernike polynomialsin the optical fabrication and testing area.7 – 9 We havealso been told that this feature is an unpublished ca-pability of a prototype corneal topographer.10 Once theZernike polynomial decomposition is complete, the fun-damental components of the corneal surface are relatedto more-familiar quantities such as spherical and cylin-drical curvature and power. A drawback to viewing thevideokeratographic height data is that the fine varia-tions in corneal height are obscured by the spherical andcylindrical components of the cornea. In order to visu-alize these residual higher-order height variations of thecornea, we subtract the lower-order components from theoriginal height data. In this paper this decomposition-

1995 Optical Society of America

2106 J. Opt. Soc. Am. A/Vol. 12, No. 10 /October 1995 Schwiegerling et al.

and-display technique is applied to topographies of aneye with regular corneal astigmatism, an eye diagnosedwith advanced keratoconus, and two eyes that under-went radial keratotomy. Advantages of this techniqueover previous efforts and its accuracy and limitations arediscussed.

2. DECOMPOSITION OF SURFACESThe shape of a surface is represented mathematically as afunction of two variables. In a Cartesian coordinate sys-tem this function has the form z ­ f sx, yd, where z is theheight or sag of the surface at a given point sx, yd. Whenthe functional form of f sx, yd is complicated, analysisof the surface is sometimes simplified by representationof the surface as a linear combination of simpler surfaces.These simpler surfaces are described mathematically bythe set of functions hgj sx, ydj and are combined such that

f sx, yd >P̀

j­1aj gj sx, yd , (1)

where aj is a constant describing the weighting on eachfunction gj sx, yd.11,12

The set of functions hgj sx, ydj is said to be complete ifany arbitrary square-integrable function f sx, yd is exactlyrepresented by the linear combination in relation (1).The most commonly used set of complete functions ishxmynj, where m and n are positive integers. Relation (1)in this case becomes

f sx, yd ­ a1 1 a2x 1 a3y 1 a4x2 1 a5y2 1 a6xy 1 . . . ,

(2)

or the Taylor series expansion of f sx, yd in two dimen-sions. The set of coefficients haj j in relation (1) is usuallyfound by the method of least squares. This techniqueminimizes the square of the difference between f sx, ydand the series expansion, and this difference approacheszero as the number of terms in the series approachesinfinity.

A set of complete functions hgj sx, ydj is said to be or-thonormal over a region A if they satisfyZ

Agj sx, ydgksx, yddxdy ­

(0 for j fi k1 for j ­ k

. (3)

The Fourier series, a commonly used expansion for peri-odic functions, uses the complete orthonormal set of si-nusoidal functions. The polynomials in the Taylor seriesexpansion of Eq. (2), however, do not satisfy the orthogo-nality conditions of Eq. (3).

A simple method for determining the set of expansioncoefficients haj j exists when the functions hgj sx, ydj areorthonormal over a region A. If both sides of relation (1)are multiplied by gksx, yd and integrated over the regionA, then utilizing Eq. (3) yields

aj ­Z

Agj sx, ydf sx, yddxdy . (4)

Equation (4) illustrates a significant advantage of or-thonormal decomposition of surfaces over the least-squares method with use of nonorthonormal functions.Each of the coefficients aj in Eq. (4) depends only on itscorresponding function gj sx, yd and the original function.

This relationship indicates that the set of coefficients haj jare independent of one another and independent of thenumber of terms taken in the series expansion. In otherwords, the coefficients do not need to be recalculated whena more exact fit of f sx, yd is desired. If, however, the setof functions hgj sx, ydj are not orthonormal, then the setof coefficients haj j are interrelated. Each coefficient aj

depends on the entire set hgj sx, ydj and changes everytime terms are added to the series expansion.

3. ZERNIKE POLYNOMIALSThe Zernike polynomials are a set of functions hZn

6m

sr, udj that are orthonormal over the continuous unitcircle.6 – 9 They have been used extensively for phase-contrast microscopy, optical aberration theory, and inter-ferometric testing to fit wave-front data. These functionsare characterized by a polynomial variation in the radialdirection r (for 0 # r # 1) and a sinusoidal variation inthe azimuthal direction u. The polynomials are definedmathematically by

Zn6m ­

8>><>>:p

2sn 1 1d Rnmsrdcos mu for 1 mp

2sn 1 1d Rnmsrdsin mu for 2 mp

sn 1 1d Rnmsrd for m ­ 0

,

(5)

where

Rnmsrd ­

sn2md/2Xs­0

s21dssn 2 sd!

s!∑

sn1md2 2 s

∏!∑

sn2md2 2 s

∏!

rn22s,

(6)

n is the order of the polynomial in the radial directionr, and m is the frequency in the azimuthal directionu. Several numbering systems are used for the Zerniketerms; we have chosen the numbering system that is inthe proposed ISO standards for optical components (ISO-10110).13 The first six Zernike polynomials are given by

Z00sr, ud ­ 1 , (7)

Z11sr, ud ­ 2r cos u , (8)

Z121sr, ud ­ 2r sin u , (9)

Z20sr, ud ­

p3 s2r2 2 1d , (10)

Z22sr, ud ­

p6 r2 cos 2u , (11)

Z222sr, ud ­

p6 r2 sin 2u . (12)

Figure 1 shows several of these functions. The func-tion Z0

0 in Fig. 1(a) describes a surface of constant height.When a function is decomposed into Zernike polynomi-als, the coefficient of Z0

0 is the mean height of the sur-face. When higher-order polynomials are added to theexpansion, each term must have zero mean in order tosatisfy the orthonormality condition of Eq. (3) and toleave the mean height of the fitted surface unchanged.Figures 1(b)–(d) show several of the zero-mean higher-order terms.

Another interesting property of the Zernike expansionis the redundant nature of the nonrotationally symmetricfunctions such as Z1

1 and Z121 or Z2

2 and Z222. For

Schwiegerling et al. Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2107

Fig. 1. Zernike polynomials: (a) Z00, (b) Z1

1 (Z121 is the same as Z1

1 but rotated 90±), (c) Z20, (d) Z2

22 (Z22 is the same as

Z222 but rotated 45±).

instance, the function Z11 describes a plane tilted about

the y axis, and Z121 similarly describes a plane tilted

about the x axis (or Z11 rotated 90±). If a plane tilted

about an axis making an angle u0 relative to the y axisis expanded into Zernike polynomials, the coefficients Z1

1

and Z121 are appropriately weighted such that the ratio of

their coefficients equals tan u0. This relationship, whereone Zernike polynomial describes a surface and anotherdescribes a rotated version of the same surface, recursthroughout the set of Zernike polynomials. This featureallows the Zernike expansion to match a surface orientedat any angle.

Several of the low-order Zernike terms represent famil-iar corneal shapes. The Z2

0 term in Fig. 1(c) is a parabo-loid and represents an average curvature of the cornea.The functions Z2

2 and Z222 in Fig. 1(d), which describe

corneal astigmatism, are two saddle-shaped surfaces ro-tated 45± with respect to each other. When the saddleshape of the Z2

2 and Z222 terms is added to the paraboloid

described by Z20, the radius of curvature of the paraboloid

is shortened along one axis and lengthened along a per-pendicular axis. The longer radius of curvature definesthe base sphere of the cornea, and the difference betweenthe radii of curvature defines the cylindrical component ofthe cornea (in plus-cylinder form). The astigmatic axis is

given by the orientation of the saddle surface formed bythe sum of the Z2

2 and Z222 terms.

Since the Zernike polynomials are orthogonal over thecontinuous unit circle and the lower-order terms repre-sent familiar corneal shapes such as sphere and cylin-der, the Zernike polynomials appear to be an ideal setof functions for decomposing and analyzing corneal sur-face height. Videokeratoscopes, however, measure thecorneal height only at a discrete number of points, andunfortunately the Zernike polynomials are not orthogo-nal over a discrete set of points. A technique knownas Gram–Schmidt orthogonalization, however, allows thediscrete set of corneal height data to be expanded in termsof the Zernike polynomials and still keep the advantagesof an orthogonal expansion.

The procedure for Gram–Schmidt orthogonalization isoutlined by Wang and Silva9 (with one minor correctionnoted after the Ref. 9 entry in the References) and is notrepeated here. The basic premise of the orthogonaliza-tion routine is that the Zernike polynomials are not or-thogonal over a discrete set of points, orP

iZn

6msri, uidZn06m0

sri, uid fi dnn0dmm0 , (13)

for all discrete points sri, uid. By taking various linear

2108 J. Opt. Soc. Am. A/Vol. 12, No. 10 /October 1995 Schwiegerling et al.

combinations of the Zernike functions, however, one canconstruct a new set of functions Un

6msri, uid such that

Un6msri, uid ­

n,6mPn0,6m0

bn,6m,n0,6m0 Zn06m0

sri, uid . (14)

Through the appropriate choice of the coefficientsbn,6m,n0,6m0 , the functions Un

6msri, uid can be made or-thogonal, orP

iUn

6msri, uidUn06m0

sri, uid ­ dnn0dmm0 . (15)

The corneal height data are decomposed into a linear com-bination of the functions Un

6msri, uid. The expansion co-efficients for these new functions are given by Eq. (4);however, the integral collapses to a sum because of thediscrete number of points. Once the expansion is com-plete, the orthogonal functions are converted back intoterms of the Zernike polynomials. The result is a uniqueset of Zernike coefficients.

4. DECOMPOSITION AND DISPLAYOF REAL TOPOGRAPHY DATAThe decomposition techniques outlined above were used toanalyze real corneal height data taken from a ComputedAnatomy TMS-1 videokeratoscope (New York, N.Y.).The corneal height data f sri, uid and a set of radial co-ordinates ri are provided as direct output of the TMS-1.Since the TMS-1 uses continuous mires, no informationabout azimuthal coordinates is obtained. The azimuthalcoordinates ui are therefore assumed to be 256 uniformlyspaced sectors on a polar grid. The effects of this ap-proximation are discussed in more detail in Section 5.Since the Zernike polynomials are orthonormal only overthe unit circle, the radial coordinates ri need to be nor-malized by the maximum radial extent of the data rmax

such that ri ­ riyrmax.The discrete set of data points are expanded into

Zernike polynomials by use of the Gram–Schmidt or-thogonalization procedure outlined above, such that

f sri, uid ­P

n,6man,6mZn

6msri, uid , (16)

for all points sri, uid. From the set of coefficients han,6mj,values for the base and the astigmatic radii of curva-ture and power, as well as for the cylinder axis, can becalculated.

For determining the spherical and cylindrical compo-nents of the series expansion, the cylindrical axis needs tobe found. This axis is defined by the lowest-order astig-matic terms of the expansion. These astigmatic termsare given by

p6 a2,22r2 sins2ud 1

p6 a2,2r2 coss2ud . (17)

By taking the derivative of this expression with respectto u and finding the extremum,

a2,22 coss2ud 2 a2,2 sins2ud ­ 0 . (18)

If the axis u0 is defined as

u0 ­12

tan21

√a2,22

a2,2

!, (19)

then two relevant solutions exist for Eq. (18). These so-lutions are u ­ u0 and u ­ u0 1 90±. The astigmatic axisis given by

ua ­

(u0 for a2,22 sin 2u0 1 a2,2 cos 2u0 , 0u0 1 90± for a2,22 sin 2u0 1 a2,2 cos 2u0 . 0

.

(20)If ua is negative then add 180±, so that ua always liesin the range 0 # ua , 180±. For the base spherical andcylindrical powers to be determined, the radii of curva-ture along u0 and the axis perpendicular to it need to bedetermined.

The parabolic terms of the expansion oriented alongu0 can be used as an approximation to the spherical andcylindrical components of f sri, uid. The Zernike polyno-mials with even radial order n $ 2 and azimuthal fre-quencies m ­ 0 or m ­ 62 all contain a parabolic term.The first six of these polynomials are

Z20sr, ud ­

p3 s2r2 2 1d , (21)

Z22sr, ud ­

p6 r2 cos 2u , (22)

Z222sr, ud ­

p6 r2 sin 2u , (23)

Z40sr, ud ­

p5 s6r4 2 6r2 1 1d , (24)

Z42sr, ud ­

p10 s4r2 2 3dr2 cos 2u , (25)

Z422sr, ud ­

p10 s4r2 2 3dr2 sin 2u . (26)

For obtaining the spherical and cylindrical components ofthe corneal height data, the parabolic terms of the Zernikeexpansion are compared with a paraboloid of the form

sag ­r2

2R0­

r2rmax2

2R0

, (27)

where r is the radial coordinate and R0 is the radius ofcurvature of the paraboloid. This value R0 is used as anapproximation of the radius of curvature of the cornealsurface. The radius of curvature in general will differalong the axes of the cornea as a result of astigmatism.Therefore define R0 ­ R' for u ­ u0, and define R0 ­ Rfor u ­ u0 1 90±. Equating the Zernike expansion termscontaining r2 oriented along u0 with Eq. (27) yields

r2rmax2

2R'

­ 2p

3 a2,0r2 1p

6 a2,2r2 cos 2u0

1p

6 a2,22r2 sin 2u0 2 6p

5 a4,0r2

2 3p

10 a4,2r2 cos 2u0

2 3p

10 a4,22r2 sin 2u0 1 . . . . (28)

Solving for R' and truncating higher-order terms, weobtain

R' ­rmax

2

2f2p

3 a2,0 2 6p

5 a4,0 1p

6 sa2,2 cos 2u0 1 a2,22 sin 2u0d 2 3p

10sa4,2 cos 2u0 1 a4,22 sin 2u0dg. s29d

If R' is in millimeters, the power F' along u ­ u0 in

Schwiegerling et al. Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2109

diopters is given by

F' ­ 1000n 2 1

R'

, (30)

where n is usually 1.3375.Similarly, for u ­ u0 1 90±,

R ­rmax

2

2f2p

3 a2,0 2 6p

5 a4,0 2p

6 sa2,2 cos 2u0 1 a2,22 sin 2u0d 1 3p

10sa4,2 cos 2u0 1 a4,22 sin 2u0dg, s31d

and the power F along u ­ u0 1 90± in diopters is given by

F ­ 1000n 2 1

R, (32)

where n is usually 1.3375.The cylindrical power Fa is diopters is therefore given

by

Fa ­ F' 2 F . (33)

These results can be expressed in the familiar plus-cylinder form sF 1 Fa 3 uad.

Equations (29)–(33) display one of the drawbacks tousing Zernike polynomials. For determining the radiiof curvature along the principal meridians, an infinitenumber of Zernike coefficients are needed. The radii-of-curvature equations above have been truncated so thatthey contain only the first six Zernike terms with para-bolic dependence. However, the values for the radii ofcurvature converge quickly, so that the truncation errorbecomes small. The accuracy of these equations is dis-cussed in greater detail in Section 5. A second drawbackis that the value of rmax can change from data set to dataset. The values of the expansion coefficients cannot bedirectly compared between two decompositions unless thevalues of rmax are equal.

When one is viewing corneal height data, fine heightvariations in the corneal surface are obscured by the basespherical and cylindrical components. The decomposi-tion technique allows a simple method for displaying someof these higher-order variations. Repeatedly removinglower-order expansion terms causes the higher-order vari-ations to become increasingly apparent. For example, topermit astigmatism to be seen more clearly, the parabolicterm is subtracted from the original height data. Forhigher-order height variations to be seen, the astigmaticcomponent of the corneal surface is subtracted. In somecases, as will be seen below, removing additional termswill reveal additional interesting corneal height artifacts.

The corneal height maps take on characteristics differ-ent from those of the more-familiar power map distribu-tions. These differences may initially prove confusing tothe viewer who is used to seeing corneal power displays.The novice viewer must remember that the height mapdisplays the sag of the corneal surface as opposed to sur-face curvature or power. The uniform power map dis-tribution associated with spherical surfaces (under onedefinition of power) is seen as a central peak falling offevenly in all radial directions in a height map. The fa-miliar “bow-tie” pattern seen in an astigmatic power mapis seen as a saddle shape in the height map. Interpret-ing height maps requires only a modest amount of reedu-cation on the part of the new viewer. The technique of

removing low-order terms from the original corneal heightdata can be thought of as a generalization of fluores-cein maps. For fluorescein maps, a spherical term is re-moved from the corneal height data by means of a contactlens, resulting in a display of the residual height vari-ation. For the Zernike decomposition technique, more-

complex shapes are removed to reveal the residual cornealheight data.

Figures 2 through 5 show several examples of thedecomposition-and-display technique. Each of the fig-ures displays a gray-scale height map of the cornea andnotes the range of heights. The patient whose data aredisplayed in Fig. 2 has corneal astigmatism. Figure 2(a)shows the raw height data obtained from the TMS-1.In Fig. 2(b) the parabolic component Z2

0 of the originalheight data has been removed. The astigmatism (saddleshape) in the resultant display is apparent. In Fig. 2(c)the cylindrical terms Z2

22 and Z22 have been removed to

reveal the residual higher-order variations in the cornea.The prescription for spherical power and astigmatismpredicted by Eqs. (16)–(22) is 46.9 D 1 2.1 D 3 1± com-pared with a Sim K value of 48.9 D 3 86±y46.9 D 3 176±

and a Min K value of 46.8 D 3 1± determined by theTMS-1. The Sim K value finds the maximum poweralong a meridian in the paraxial region of the corneaand gives this power, its orientation, and the cornealpower in a perpendicular meridian. The Min K simi-larly finds the minimum power along a meridian in theparaxial region and gives its magnitude and orientation.The data displayed in Fig. 3 are for the patient who hasbeen diagnosed with advanced keratoconus. Figure 3(a)once again shows the raw topography height data. InFig. 3(b) the base curvature has been subtracted. InFig. 3(c) the cylindrical terms Z2

22 and Z22 have been

removed to reveal the cone.The patients whose data are displayed in Figs. 4 and

5 have undergone six- and eight-incision radial kera-totomy, respectively. Figures 4(a) and 5(a) once againshow the raw topography height data. In Figs. 4(b) and5(b) the base curvature has been subtracted. In Fig. 4(c)terms of the Zernike expansion with radial order n # 6and azimuthal frequency m , 6 have been subtractedfrom the original height data, revealing the height vari-ations resulting from the six RK incisions. Only theseZernike terms are removed from the six-incision RK, be-cause some of the higher terms contain a sixfold symmetrythat matches the cuts. Similarly, in Fig. 5(c) terms of theZernike expansion with radial order n # 8 and azimuthalfrequency m , 8 have been subtracted from the origi-nal height data, revealing the height variations resultingfrom the eight RK incisions. Higher-order Zernike termscontain an eightfold symmetry that matches the cuts.The terms below the anticipated symmetry of the RK areassociated with the refractive power, residual astigma-tism, and spherical aberration of the postoperative cornea.Once these terms have been removed, only the artifactresulting from the refractive surgery remains. The six-incision RK artifact shows a peak-to-valley height vari-ation of , 18 mm. The eight-incision RK artifact shows

2110 J. Opt. Soc. Am. A/Vol. 12, No. 10 /October 1995 Schwiegerling et al.

Fig. 2. Height maps of corneal astigmatism: (a) raw heightdata, (b) raw height data minus the parabolic term, (c) heightdata of (b) minus the cylindrical term. White represents a highpoint on the cornea, and black represents a low point. Diameter7.0 mm.

a peak-to-valley height variation of ,20 mm. Expansionterms of higher order than the anticipated symmetry ofthe artifacts can be used to fit these RK artifacts.

5. NUMERICAL ACCURACYVideokeratoscopes measure corneal surface slope in themeridional direction. Some inaccuracies will result in

converting this slope data to height data. Greivenkampet al.14 have measured the surface height of a vari-ety of toric surfaces on several commercially availablevideokeratoscopes. The calculated surface heights werecompared with the actual heights of the surfaces to de-termine the accuracy of the videokeratoscope. The errorin calculated surface height results from several sources:the lack of azimuthal information about the corneal slope

Fig. 3. Height maps of advanced keratoconus: (a) raw heightdata, (b) raw height data minus the parabolic term, (c) heightdata of (b) minus the cylindrical term. White represents a highpoint on the cornea, and black represents a low point. Diameter5.5 mm.

Schwiegerling et al. Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2111

Fig. 4. Height maps of six-incision radial keratotomy: (a) rawheight data, (b) raw height data minus the parabolic term,(c) height data of (b) minus the Zernike terms, with n # 6 andm , 6. White represents a high point on the cornea, and blackrepresents a low point. Diameter 8.0 mm.

data, tilt between the optical axis of the cornea andthe optical axis of the keratoscope head, and defocus ofthe corneal image. Under ideal circumstances the rmssurface-height errors ranged from 0.7 mm for a 0 D toricup to 4.2 mm for a 7 D toric. In order to determine theaccuracy of the Gram–Schmidt orthogonalization routine,we fitted simulated perfect keratographic data sets of 0,1, 3, 5, and 7 D torics, using the Zernike expansion, and

calculated the rms error of the fit. The rms fit error forthese five ideal surfaces is less than 0.02 mm for an expan-sion with radial orders n # 8 and azimuthal frequenciesm , 8. This rms fit error is well below the measure-ment accuracy of the videokeratoscope. In other words,the numerical accuracy of the decomposition methodologyoutlined here is limited by the measurement device andnot by the orthogonalization routine.

Fig. 5. Height maps of eight-incision radial keratotomy:(a) raw height data, (b) raw height data minus the parabolicterm, (c) height data of (b) minus the Zernike terms, with n # 8and m , 8. White represents a high point on the cornea, andblack represents a low point. Diameter 7.6 mm.

2112 J. Opt. Soc. Am. A/Vol. 12, No. 10 /October 1995 Schwiegerling et al.

Table 1. Actual and Calculated Radii of Curvature in Terms of R 3 R'R 3 R'R 3 R' for Several Toric Surfaces

Astigmatism of Toric Surface(D) Actual Calculated

0 7.80 mm 3 7.80 mm 7.84 mm 3 7.81 mm1 7.80 mm 3 7.62 mm 7.89 mm 3 7.69 mm3 7.80 mm 3 7.29 mm 7.80 mm 3 7.30 mm5 7.80 mm 3 6.99 mm 7.85 mm 3 7.03 mm7 7.80 mm 3 6.71 mm 7.85 mm 3 6.76 mm

The accuracy of Eqs. (29)–(33) depends on both theaccuracy of the videokeratoscope height data and theZernike expansion coefficients. We have truncatedthe expressions for spherical and cylindrical radii of cur-vature and power to keep the equations simple. In deriv-ing these formulas, we find expansion terms containingr2 ­ sryrmaxd2 oriented along and perpendicular to u0

(i.e., spherical or cylindrical terms) and compare themwith Eq. (27). Equations (29)–(33) contain only the firsttwo spherical terms (a2,0 and a4,0) and the first two setsof astigmatic terms (a2,2, a2,22 and a4,2, a4,22). As theorder of the expansion is increased, additional terms con-taining a r2 dependence arise (the next logical terms area6,0 and a6,2, a6,22 for the spherical and the cylindricalcomponents, respectively). The accuracy of the radii-of-curvature and the power calculations can be increasedby inclusion of these and additional higher-order terms;however, the expressions become cumbersome.

For most corneas the lower-order Zernike polynomialterms carry most of the significant information. Thehigher-order terms therefore tend to become less reliableas the measurement noise approaches the amount of sur-face variation represented by these terms. However, forsurfaces with significant variations that can be describedonly by higher-order terms, such as the incision patterndue to an RK procedure, the coefficients for these high-order terms are found to be stable and reliable. As a test,we measured the cornea of an RK patient four times, de-composed the results, and compared the coefficients. Allof the corneal height data have the same rmax. For all ofthe coefficients examined (45 terms), the ratio of the meanvalue to the standard deviation of the coefficient measure-ments exceeds unity. For polynomial terms representingsignificant surface variations (such as the eightfold sym-metry described above), this signal-to-noise measure ofthe coefficient quality is on the order of 10:1.

The toric surfaces mentioned above were used to de-termine the accuracy of the spherical and the astig-matic radii-of-curvature and power calculations. Thekeratographs of the torics obtained with the TMS-1 weredecomposed into a set of Zernike polynomials and val-ues for the radii of curvature calculated with Eqs. (29)and (31). Table 1 shows the actual and the calculatedprescriptions of the five toric surfaces in terms of theradius of curvature R along one principal meridian of thetoric and the radius of curvature R' along the perpen-dicular axis. The rms radius error is , 50 mm for thefive surfaces, which corresponds to a rms power error ofless than 0.3 D. The errors produced by Eqs. (29)–(33)are therefore approaching the accuracy level claimed byvideokeratoscope manufacturers. If additional accuracyis needed, higher-order terms can be included.

6. CONCLUSION

A technique for analyzing and displaying corneal heightdata was presented. The height data were decomposedinto an orthonormal set of polynomials and related tothe Zernike polynomials. The Zernike polynomials pro-vide a numerically stable expansion of corneal height dataand contain terms representative of fundamental cornealshapes such as sphere and cylinder. Some previous ef-forts at decomposing corneal height data use a nonorthog-onal expansion such as the Taylor series. The Zernikepolynomials have an advantage over the nonorthogonalexpansions when high-order fits such as those found inthe radial keratotomy examples above are performed.Whereas the Zernike polynomials remain numericallystable, the Taylor series expansion quickly becomes ill-conditioned.9

A significant drawback to viewing corneal height dataconcerns the display of pertinent height data. Small,high-order variations in corneal height are hidden by thespherical and cylindrical components of the cornea. Toeliminate this drawback, one can display the height datain stages in order to examine the different componentsthat make up the shape of the cornea. The spherical,cylindrical, and higher-order terms can be individuallyremoved. This decomposition technique has several ob-vious applications. One application is as a complementto corneal power maps. Surface height provides an easymethod of determining contact lens fitting, since the spac-ing between the base curve of the lens and the cornea canbe calculated directly. This process is a generalization ofthe fluorescein maps routinely used in contact lens evalu-ation. The height maps also provide information aboutcone location in keratoconus. Finally, the height mapscan be used to evaluate irregularities and asymmetriesin radial keratotomy incisions. Height maps can also beapplied to optical modeling. The height variations mea-sured on the cornea can be applied to a schematic eyemodel and the optical effects determined with use of op-tical ray traces. The height variation in the cornea inthe RK examples is ,20 mm peak to valley and may havesignificant optical effects. The RK artifacts generated bythis technique also demonstrate that the post-RK corneahas high spots or bumps corresponding to the location ofthe incisions.

We examined the numerical accuracy of the decompo-sition and analysis techniques in order to validate themethodology. Commercially available videokeratoscopeshave intrinsic error in measuring the surface height ofthe cornea. These errors arise from necessary assump-tions about the alignment of the keratoscope and aboutthe azimuthal slope of the cornea. The fit errors result-

Schwiegerling et al. Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2113

ing from the orthogonalization routine are significantlysmaller than the errors produced by the videokeratoscope.Therefore, as future generations of videokeratoscopes im-prove their height-measuring abilities and as new cornealmeasurement technologies arise, the methodology out-lined here will still be a viable technique for analyzingcorneal height data.

We presented several examples of the decomposition-and-display technique in order to introduce and educatethe reader to residual height maps. These maps appearvastly different from more-familiar power map displays.With a fundamental understanding of the residual heightmap displays and a little practice, analyzing theseresidual height maps will become second nature. Theexamples of the technique included artifacts such asastigmatism, keratoconus, and radial keratotomy. Theseanalysis techniques can obviously be extended to otherprocedures such as penetrating keratoplasty and photore-fractive keratectomy. With the popularity of refractivesurgery ever increasing, eye-care providers and visionscientists need additional tools in order to provide high-quality care for these patients. The Zernike decomposi-tion technique and display methods presented here pro-vide a sophisticated analysis tool to complement currenttechniques of evaluating the corneal surface.

ACKNOWLEDGMENTThe authors thank Raymond Applegate and his associatesfor providing some of the vidoekeratographic data pre-sented in this paper.

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aj ­

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NPk­j11

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