Fitting behaviors of Fourier transform and Zernike polynomials

PURPOSE: To evaluate the residual fit errors for wavefront measurements and corneal surfaces in virgin and post-surgery eyes using the Fourier transform and Zernike 6th- and 10th-order expansions. METHODS: Using MatLab (The MathWorks, Inc.) and VOLCT program (Sarver and Associates, Inc.), wavefront gradient fields derived from the Hartmann-Shack lenslet array of the CustomVue System (Visx, Inc.) and the corneal surface obtained from the Humphrey Atlas topographer (Carl Zeiss, Inc.) were fitted with the Fourier transform and Zernike 6th- and 10th-order expansions. The wavefront gradient fields and corneal surfaces reconstructed by the 3 methods were compared with original ones, and the residual fit errors were analyzed (6.0 mm pupil). Ten eyes in each of the 4 groups (virgin eyes, post-laser in situ keratomileusis (LASIK) for myopia, post-LASIK for hyperopia, and post-radial keratotomy) were included. RESULTS: Wavefront gradient fields reconstructed with Fourier transform produced significantly smaller residual fit errors than Zernike 6th-order in all groups and Zernike 10th-order in all eyes except post-myopic-LASIK eyes. In all groups, wavefront gradient fields reconstructed with Zernike 10th order yielded significantly smaller residual errors than Zernike 6th order (all P<.05). Higher residual errors were produced by these 3 methods in more highly aberrated eyes. When corneal surfaces for all groups were reconstructed, the Fourier transform produced significantly lower residual fit errors than Zernike 6th order and 10th order and the Zernike 10th order yielded significantly lower residual errors than Zernike 6th order (all P<.05). As corneal higher-order aberrations increased, higher residual surface fit errors were produced by Zernike 6th-order and 10th-order expansions but not by Fourier transform. CONCLUSIONS: Fourier transform reconstructed ocular wavefront and corneal topographic maps more accurately than Zernike polynomials up to the 10th order. Clinical implications require further study.     

Goodness-of-prediction of Zernike polynomial fitting to corneal surfaces

PURPOSE: To determine the goodness-of-prediction of the fitting routine by measuring the difference between topographic corneal surfaces and their Zernike reconstructions as a function of polynomial order and optical zone size for various corneal conditions. SETTING: Corneal research laboratory in a university eye center. METHODS: Corneal topography maps (N = 253) were obtained from the Louisiana State University Eye Center. A variety of corneal conditions were used: normals; astigmatism; laser in situ keratomileusis, photorefractive keratoplasty (PRK), and radial keratotomy (RK) postoperative cases (myopic spherical corrections); keratoconus suspect; mild, moderate, and severe keratoconus; pellucid marginal degeneration; contact lens-induced corneal warpage; and penetrating keratoplasty. The root-mean-square (RMS) error of the goodness-of-prediction of the Zernike representation of corneal surface elevation was extracted for 4, 6, and 10 mm optical zones, whereas Zernike radial orders were varied from 3 to 14 in 1-order steps. The mean +/- SEM of the RMS error was plotted as a function of Zernike order and compared with criteria for normal surface fits. RESULTS: Fitting accuracy improved as more Zernike terms were included, but some conditions showed significant errors (when compared with normal surfaces), even with many added terms. For a 6 mm optical zone, the normal cornea group had the lowest RMS error and did not require terms above the 4th order to achieve <0.25 microm RMS error. Astigmatism met the 0.25 microm threshold at the 5th order, whereas keratoconus suspect required 7 orders. Laser in situ keratomileusis and PRK met the 0.25 microm threshold at the 8th order, whereas RK required 10 orders. Contact lens-induced corneal warpage and mild keratoconus needed 12 orders to meet the 0.25 microm threshold, whereas pellucid marginal degeneration, moderate and severe keratoconus, and keratoplasty categories were not well fitted even at 14 orders. CONCLUSIONS: A 4th-order Zernike polynomial appeared reliable for modeling the normal cornea, but using a 4th-order fitting routine with an abnormal corneal surface caused a loss of fine-detail shape information. As more Zernike terms were added, the accuracy of the fit improved, and the result approached the minimum error found with normal corneas. Unless sufficient higher-order Zernike terms are included when analyzing irregular surfaces, some diagnostic applications of Zernike coefficients may not be rigorous. This conclusion also suggests that wavefront shape analysis is similarly dependent on the number of orders used. Current surgical corrections based on normal-eye wavefronts may fail to capture all visually relevant aberrations in abnormal eyes, such as those having laser retreatments or experiencing corneal warpage from contact lens wear. A clinical goodness-of-fit or goodness-of-prediction index would indicate whether the number of terms in use has fully accounted for all of the visually significant aberrations present in the eye.     

Accuracy of Zernike polynomials in characterizing optical aberrations and the corneal surface of the eye

PURPOSE: Zernike polynomials have been successfully used for approximately 70 years in many different fields of optics. Nevertheless, there are some recent discussions regarding the precision and accuracy of these polynomials when applied to surfaces such as the human cornea. The main objective of this work was to investigate the absolute accuracy of Zernike polynomials of different orders when fitting several types of theoretical corneal and wave-front surface data. METHODS: A set of synthetic surfaces resembling several common corneal anomalies was sampled by using cylindrical coordinates to simulate the height output files of commercial videokeratography systems. The same surfaces were used to compute the optical path difference (wave-front [WF] error), by using a simple ray-tracing procedure. Corneal surface and WF error was fit by using a least-squares algorithm and Zernike polynomials of different orders, varying from 1 to 36 OSA-VSIA convention terms. RESULTS: The root mean square error (RMSE) ranged-from the most symmetric corneal surface (spherical shape) through the most complex shape (after radial keratotomy [RK]) for both the optical path difference and the surface elevation for 1 through 36 Zernike terms-from 421.4 to 0.8 microm and 421.4 to 8.2 microm, respectively. The mean RMSE for the maximum Zernike terms for both surfaces was 4.5 microm. CONCLUSIONS: These results suggest that, for surfaces such as that present after RK, in keratoconus, or after keratoplasty, even more than 36 terms may be necessary to obtain minimum accuracy requirements. The author suggests that the number of Zernike polynomials should not be a global fixed conventional or generally accepted value but rather a number based on specific surface properties and desired accuracy.     

A computer model for the evaluation of the effect of corneal topography on optical performance

We developed a method that models the effect of irregular corneal surface topography on corneal optical performance. A computer program mimics the function of an optical bench. The method generates a variety of objects (single point, standard Snellen letters, low contrast Snellen letters, arbitrarily complex objects) in object space. The lens is the corneal surface evaluated by a corneal topography analysis system. The objects are refracted by the cornea by using raytracing analysis to produce an image, which is displayed on a video monitor. Optically degraded images are generated by raytracing analysis of selected irregular corneal surfaces, such as those from patients with keratoconus and those from patients having undergone epikeratophakia for aphakia.     

Correlation between the dioptric power, astigmatism and surface shape of the anterior and posterior corneal surfaces

A knowledge of the shape of the cornea is of major importance for the planning and monitoring of surgery, and for the correct diagnosis of corneal diseases. Many authors have studied the geometry of the second corneal surface in the central region and it has been stated that there is a high correlation between the central radii of curvature and asphericities of the two corneal surfaces. In this work we extend this study to a larger, central, 6 mm diameter of the cornea. Surface height data, obtained with an Oculus Pentacam from 42 eyes of 21 subjects, were analysed to yield surface power vectors. Corneal heights of both surfaces were also decomposed into low-order Zernike polynomials and the correlations between each of the power vectors and low-order Zernike coefficients for the two surfaces were studied. There was not only a strong correlation between spherical powers and Zernike defocus coefficients, but also between the astigmatic components. The correspondence between the astigmatism in both surfaces found here can be of the utmost importance in planning optical surgery, since perfect spherical ablation of the first surface does not assure total correction of corneal astigmatism.     

Comparison of topographic corneal irregularity after LASIK and intrastromal corneal ring segments in the same patients

PURPOSE: To retrospectively compare the irregularity of the corneal surfaces of 14 patients after LASIK in 1 eye and placement of intrastromal corneal ring segments (ICRS) in the other eye. METHODS: In a within-patient comparison, Orbscan corneal topography was used to retrospectively compare the corneal surface irregularity of LASIK-treated and ICRS-treated eyes at an outpatient tertiary-care ophthalmology clinic in Sao Paulo, Brazil. For the anterior corneal surface, irregularity measurements were compared for both the central and peripheral areas of the cornea. The differences between each group were analyzed for statistical significance. RESULTS: The corneal surfaces of eyes treated with ICRS were found to be more irregular than the corneal surfaces of eyes treated with LASIK, the mean irregularity being 1.91 for LASIK-treated eyes and 3.12 for ICRS-treated eyes in the anterior corneal surface and 0.51 for LASIK-treated eyes and 0.87 for ICRS-treated eyes in the posterior corneal surface. A statistically significant difference was noted only in the posterior surfaces. CONCLUSIONS: When measured with Orbscan topography, ICRS-treated eyes show more corneal surface irregularity than LASIK-treated eyes. The difference in outcome for the two types of treatment may be due to the mechanical effect of the ICRS on the shape of the cornea.     

近视眼角膜高阶像差的特征及影响因素

目的 分析角膜高阶像差的分布特点,角膜前表面、后表面与整体角膜高阶像差三者之间的关系,以及相关因素对各项高阶像差的影响.方法 横断面研究.随机选择在天津市眼科医院屈光手术中心欲行近视眼准分子激光手术的患者108例,其中男55例,女53例,年龄18～38岁,平均(23.3±4.1)岁;等效球镜度数为-1.38～-12.50 D,平均(-5.81±2.19)D;应用Pentacam眼前节分析系统采集右眼角膜像差及角膜最薄点厚度(TCT).采用Mann-Whitney秩和检验和Spearman相关分析.结果 近视眼角膜各高阶像差均方根(RMS)值随阶数增加大致呈递减趋势,以三阶和四阶RMS值的数值较大.角膜后表面的波前像差值远远小于前表面,整体角膜RMS值介于角膜前、后表面之间.角膜前表面三阶RMS值、角膜前表面六阶RMS值、角膜后表面五阶RMS值、角膜后表面六阶RMS值、角膜三阶RMS值、角膜六阶RMS值与眼压呈正相关(r=0.217、0.248、0.269、0.210、0.222、0.282,P＜0.05).高阶像差与TCT之间无相关性.男性与女性各高阶像差之间的差异无明显统计学意义.左右眼高阶像差差异无统计学意义.结论 近视眼角膜各高阶像差RMS值随阶数增加大致呈递减趋势.角膜后表面的高阶像差RMS值远远小于角膜前表面,角膜前、后表面像差相互补偿.性别和眼别对角膜高阶像差无明显影响.     

Zernike representation of corneal topography height data after nonmechanical penetrating keratoplasty

PURPOSE: To demonstrate a mathematical method for decomposition of discrete corneal topography height data into a set of Zernike polynomials and to demonstrate the clinical applicability of these computations in the postkeratoplasty cornea. METHODS: Fifty consecutive patients with either Fuchs' dystrophy (n = 20) or keratoconus (n = 30) were seen at 3 months, 6 months, and 1 year (before suture removal) and again after suture removal following nonmechanical trephination with the excimer laser. Patients were assessed using regular keratometry, corneal topography (TMS-1, simulated keratometry [SimK]), subjective refraction, and best-corrected visual acuity (VA) at each interval. A set of Zernike coefficients with radial degree 8 was calculated to fit two model surfaces: a complete representation (TOTAL) and a representation with parabolic terms only to define an approximate spherocylindrical surface (PARABOLIC). The root mean square error (RMS) was calculated comparing the corneal raw height data with TOTAL (TOTALRMS) and PARABOLIC (PARABOLICRMS). The cylinder of subjective refraction was correlated with the keratometric readings, the SimK, and the respective Zernike parameter. Visual acuity was correlated with the tilt components of the Zernike expansion. RESULTS: The measured corneal surface could be approximated by the composed surface 1 with TOTALRMS < or = 1.93 microm and by surface 2 with PARABOLICRMS < or = 3.66 microm. Mean keratometric reading after suture removal was 2.8+/-0.6 D. At all follow-up examinations, the SimK yielded higher values, whereas the keratometric reading and the refractive cylinder yielded lower values than the respective Zernike parameter. The correlation of the Zernike representation and the refractive cylinder (P = 0.02 at 3 months, P = 0.05 at 6 months and at 1 year, and P = 0.01 after suture removal) was much better than the correlation of the SimK and refractive cylinder (P = 0.3 at 3 months, P = 0.4 at 6 months, P = 0.2 at 1 year, and P = 0.1 after suture removal). Visual acuity increased from 0.23+/-0.10 at the 3-month evaluation to 0.54+/-0.19 after suture removal. After suture removal, there was a statistically significant inverse correlation between VA and tilt (P = 0.02 in patients with keratoconus and P = 0.05 in those with Fuchs' dystrophy). CONCLUSIONS: Zernike representation of corneal topography height data renders a reconstruction of clinically relevant corneal topography parameters with a marked reduction of redundance and a small error. Correlation of amount/axis of refractive cylinder with respective Zernike parameters is more accurate than with keratometry or respective SimK values of corneal topography analysis.     

An improved low order method for corneal reconstruction.

PURPOSE: An iterative and cubic arc-step method for corneal reconstruction developed previously, was retested including improved skew-ray compensation. The method was compared with a similar method described by Klein, and also with a least squares based approach using Zernike polynomials. METHODS: Images of an asphere, a cornea after photorefractive keratectomy (PRK), an ellipsoid and radially keratotomized (RK) cornea were generated. Three reconstruction methods were applied: (i) the cubic method (without skew-ray compensation; with old and improved compensation), (ii) the method due to Klein and (iii) a least squares approach based on Zernike polynomials (order taken up to 25). Errors were recorded for all conditions tested. RESULTS: The root mean square errors for the improved method were well below micron level, and consistently lower than the method of Klein. The Zernike-based method produced lowest errors for aspheric and ellipsoidal surfaces, when order was > or =(10, 8) respectively. However, the improved method produced the lowest errors for the PRK and RK examples. In this case, the Zernike-based method produced submicron errors for orders > or =14 (both surfaces), but errors comparable with the arc-step methods could not be achieved for polynomial orders < or =25. The improved method completed in three to four iterations (abs. height error <1 x 10(-7) mm) in all cases. CONCLUSIONS: Compensation for skew-rays was incorporated in a straightforward way, yielding an efficient and effective low-order method. The Zernike method produced the lowest root mean square errors for asphere and ellipsoid, provided order was > or =(10, 8). However, the arc-step methods (the cubic and Klein methods) were more accurate for RK and PRK surfaces, at least for Zernike orders < or =25. The results suggest that low order methods provide a good solution to the reconstruction of corneal shape, at least for Placido disk videokeratography applications.     

The topography of the central and peripheral cornea

PURPOSE: To investigate the topography of the central and peripheral cornea in a group of young adult subjects with a range of normal refractive errors. METHODS: Corneal topography data were acquired for 100 young adult subjects by a method that allows central and peripheral maps to be combined to produce one large, extended corneal topography map. This computer-based method involves matching the common topographical features in the overlapping maps. Corneal height, axial radius of curvature, and axial power data were analyzed. The corneal height data were also fit with Zernike polynomials. RESULTS: Conic fitting to the corneal height data revealed the average apical radius (Ro) was 7.77 +/- 0.2-mm and asphericity (Q) was -0.19 +/- 0.1 for a 6-mm corneal diameter. The conic fit parameters were both found to change significantly for increasing corneal diameters. For a 10-mm corneal diameter, Ro was 7.72 +/- 0.2 mm and Q was -0.36 +/- 0.1. A slight but significant meridional variation was found in Q, with the steepest principal corneal meridian found to flatten at a slightly greater rate than the flattest meridian. The RMS fit error for the conic section was found to increase markedly for larger corneal diameters. Higher-order polynomial fits were needed to fit the peripheral corneal data adequately. Analysis of the axial power data revealed highly significant changes occurring in the corneal best-fit spherocylinder with increasing distance from the corneal center. The peripheral cornea was found to become significantly flatter and to decrease slightly in its toricity. Individual subjects exhibited a range of different patterns of central and peripheral corneal topography. Several of the higher order corneal surface Zernike coefficients were found to change significantly with increasing corneal diameter. CONCLUSIONS: Highly significant changes occur in the shape of the cornea in the periphery. On average, the peripheral cornea becomes significantly flatter and slightly less astigmatic than the central cornea. A conic section is a poor estimator of the peripheral cornea.     

Pellucid marginal corneal degeneration: evaluation of the corneal surface and contact lens fitting

AIM: To quantify corneal irregularities, to describe the fitting with contact lenses, and to answer the question whether or not contact lenses with a special back surface design could improve visual acuity in patients with pellucid marginal corneal degeneration (PMCD). METHODS: 13 eyes were fitted with contact lenses with a special back surface. Videokeratographic data were assessed. The patients were followed up for an average period of 22.2 months. Lens tolerance and corrected visual acuity were evaluated. RESULTS: The mean eccentricity did not exceed 0.7 in all patients. Either the superior or the inferior eccentricity, or both, were negative in all patients. Using Fourier analysis all PMCD subjects showed an increased irregular astigmatism of the anterior cornea. Using Zernike coefficients seven eyes (53.8%) had a higher order aberration root mean square error (HOA RMS error) out of the normal range. The visual acuity with contact lenses improved in all eyes with an average increase of 2.7 lines (maximum eight lines). No serious complications were observed. CONCLUSIONS: Quantitative evaluation of videokeratographic data may help to diagnose PMCD and to distinguish PMCD from other ectatic corneal diseases. Contact lenses with a special back surface design can improve visual acuity and lens tolerance.     

From scattering to wavefronts--what's in between?

PURPOSE: Some optical errors are too localized and random to be detected by commercial wavefront devices and Zernike polynomial expression. We looked beyond aberrations defined by Zernike expression to discuss implications of fine irregularities associated with highly aberrated corneal surfaces and complex surface roughness that can lead to light scattering. METHODS: Most fine irregularities are related to postoperative surface roughness, complexities of corneal ablation, and the laser in situ keratomileusis (LASIK) flap. These can be characterized mathematically by a random function that includes local surface tilts, the correlation radius of irregularities (Ic), surface roughness, and other terms. The Kirchoff method of scatter analysis characterizes fine surface irregularities by replacing each point on the surface with a tangential plane, allowing it to be governed by Snellen and Fresnel laws. RESULTS: The joint action of the continuum of microbeams defines a complex point spread function that can be expressed by the Strehl ratio. Small, highly irregular steep central islands and flap striae may not be adequately detected by Zernike expression and may have a surface irregularity diameter of 0.1 to 2.0 mm and height of 10 to 20 microm that results in a reduced Strehl ratio below 0.8. Laser ablation inhomogeneities may have dimensions of 1 to 10 microm, resulting in a root mean square tilt value approaching 1.0 and a Strehl ratio below 0.5. CONCLUSION: Corneal surface irregularities after laser vision correction may induce significant optical aberrations and distortions apart from classical wavefront or scattering errors. As these may not be detected by commercial wavefront devices, and yet contribute to the degradation of optical performance, alternate techniques should be evaluated to detect and describe these surface irregularities.     

Zernike多项式分析全角膜前后表面和厚度三维不规则分布在圆锥角膜诊断中的价值

目的评价利用Zernike多项式分析全角膜前后表面和厚度三维不规则分布在圆锥角膜中的诊断作用。方法病例对照研究。选取圆锥角膜患者20例（30眼）作为圆锥角膜组和同期健康志愿者30例（30眼）作为正常对照组。应用Pentacam 系统测量所有受试者的眼前节形态,记录角膜前后表面高度以及全角膜厚度数据。利用Zernike多项式分析得出2~7阶和高阶系数的均方根（RMS）值。采用独立样本t检验对参数进行统计比较,通过受试者工作特征曲线（ROC）对测量参数的诊断性能进行评估。结果经Zernike多项式分析处理后的2~7阶和高阶系数的RMS值在圆锥角膜组和正常对照组中的差异均有统计学意义（P＜0.001）。角膜前表面高度参数中,以3阶、5阶和高阶系数的RMS值对圆锥角膜的诊断性能较好,曲线下面积（AUC）分别为0.999、0.982、0.994：在角膜后表面高度参数中,以3阶、5阶、7阶和高阶系数的RMS值对圆锥角膜的诊断性能较好,AUC分别为0.997、0.991、0.993、0.981;在全角膜厚度参数中,以3阶、5阶和高阶系数的RMS值对圆锥角膜的诊断性能较好,AUC分别为0.993、0.989、0.992。其中3阶系数的RMS值诊断圆锥角膜的性能最好,在角膜前后表面高度和全角膜厚度中,其AUC均接近1.000。结论基于Pentacam系统的Zernike多项式分析全角膜前后表面高度和厚度三维不规则分布在圆锥角膜的诊断中有一定的临床指导价值。以3阶系数的RMS值诊断性能最高。     

Videokeratoscopes for dioptric power measurement during surgery

PURPOSE: To evaluate 2 versions of a computerized surgical videokeratoscope that measures a central region of the cornea of approximately 4.0 to 6.0 mm in diameter to provide information on dioptric power and astigmatism. SETTING: Grupo de Optica Oftalmica Universidade de S?o Paulo, and Departamento de Oftalmologia da Escola Paulista de Medicina, S?o Paulo, Brazil. METHODS: The first videokeratoscope system with 10 Placido rings is based on a 15 000 fiber-optic illuminated disk attached to the objective lens of a surgical microscope. With this system, the light intensity can be adjusted during surgery for better contrast of the Placido image. The second system with 14 Placido rings is based on a neon light source behind the Placido disk. With both systems, a 480 x 640 pixel resolution charge-coupled device camera and an IBM-compatible personal computer frame grabber were used. Image processing was used for boundary detection of the rings. An axial curvature algorithm based on the spherical surfaces was used to calculate the dioptric power for each examination. Measurements of 4 real spherical surfaces and 4 simulated aspheric surfaces (ellipsotoric surfaces with different apical radii and different shape factors) were performed. RESULTS: Twenty corneas of 10 healthy volunteers were measured on both videokeratoscope systems and an EyeSys System 2000 corneal topographer. The root- mean-square error for the spherical and simulated aspheric surfaces was, respectively, less than 0.20 diopter (D) and 1.30 D for the 10-disk videokeratoscope system and less than 0.16 D and 1.40 D for the 14-disk system. The mean deviation in corneal measurements with both systems was 0.09 mm for the radius of curvature, 0.51 D for dioptric power, and 5 degrees for cylinder. CONCLUSIONS: The results indicate that the 2 surgical videokeratoscopes are sufficiently precise to aid the anterior segment surgeon in reducing residual astigmatism in cataract surgery and keratoplasty.     

Evaluation of apparent ectasia of the posterior surface of the cornea after keratorefractive surgery

PURPOSE: To calculate the apparent posterior corneal changes after keratorefractive surgery and reevaluate corneal ectasia displayed by Orbscan (Orbtek). SETTING: Department of Ophthalmology, Nara Medical University, Nara, Japan. METHODS: Postoperative:preoperative magnification ratio of the posterior surface of the cornea was calculated in a theoretical eye model. RESULTS: Assuming the preoperative corneal thickness is 600.00 microm, the preoperative refractive power of the anterior corneal surface is 48.0 diopters (D), the refractive power of the cornea is 1.376, the ablation diameter is 6.0 mm, the postoperative corneal thickness is 480.00 microm, the postoperative refractive power of the anterior corneal surface is 38.0 D, and the posterior surface of the cornea does not change postoperatively, the apparent image of the posterior surface of the cornea becomes 0.778% smaller postoperatively. If the posterior radius of curvature of the cornea is 6.2 mm, it becomes smaller by 48.24 microm. If this change directly affects the difference map, the posterior surface of the cornea moves forward by 48.24 microm. CONCLUSION: The results correspond to the amount of ectasia in previous reports. This artifact may explain the apparent ectasia detected by Orbscan.     

Residual bed thickness and corneal forward shift after laser in situ keratomileusis

PURPOSE: To prospectively assess the forward shift of the cornea after laser in situ keratomileusis (LASIK) in relation to the residual corneal bed thickness. SETTING: Miyata Eye Hospital, Miyazaki, Japan. METHODS: Laser in situ keratomileusis was performed in 164 eyes of 85 patients with a mean myopic refractive error of -5.6 diopters (D) +/- 2.8 (SD) (range -1.25 to -14.5 D). Corneal topography of the posterior corneal surface was obtained using a scanning-slit topography system before and 1 month after surgery. Similar measurements were performed in 20 eyes of 10 normal subjects at an interval of 1 month. The amount of anteroposterior movement of the posterior corneal surface was determined. Multiple regression analysis was used to assess the factors that affected the forward shift of the corneal back surface. RESULTS: The mean residual corneal bed thickness after laser ablation was 388.0 +/- 35.9 microm (range 308 to 489 microm). After surgery, the posterior corneal surface showed a mean forward shift of 46.4 +/- 27.9 microm, which was significantly larger than the absolute difference of 2 measurements obtained in normal subjects, 2.6 +/- 5.7 microm (P<.0001, Student t test). Variables relevant to the forward shift of the corneal posterior surface were, in order of magnitude of influence, the amount of laser ablation (partial regression coefficient B = 0.736, P<.0001) and the preoperative corneal thickness (B = -0.198, P<.0001). The residual corneal bed thickness was not relevant to the forward shift of the cornea. CONCLUSIONS: Even if a residual corneal bed of 300 microm or thicker is preserved, anterior bulging of the cornea after LASIK can occur. Eyes with thin corneas and high myopia requiring greater laser ablation are more predisposed to an anterior shift of the cornea.     

Measuring corneal thickness with a rotary scanning system

PURPOSE: Recently the authors presented a technique that allows measurement of the corneal thickness along a vertical meridian from optical sections obtained using a slit-lamp microscope. We present a new system that allows rotary scanning of the cornea and consequent measurement of the corneal thickness along any meridian. METHOD: The corneal thickness along two perpendicular meridians was measured from optical sections obtained with a new rotary scanning system. To obtain the optical sections, the cornea is illuminated with a light beam that is previously expanded in a fan by a small cylindrical lens rotated on a plane-containing axis to explore the cornea. The light diffused by both corneal surfaces is collected by two video cameras placed at an angle with the light beam. The axes of the two cameras define, with the visual axis, two planes normal to each other. After image acquisition, a processing algorithm is applied in order to compute the corneal thickness along the two meridians. RESULTS: Some preliminary results for the vertical and the horizontal meridians of one eye are shown, as well as all the intermediate computations. CONCLUSIONS: With this system, we expect to be able to measure corneal topography and thickness along an arbitrary meridian. The rotary scanning of the cornea is mechanically simple and will eventually allow automated scanning.     

Absolute accuracy of Placido-based videokeratographs to measure the optical aberrations of the cornea.

Advances in excimer laser spot size and ablation pattern capabilities, together with the development of Hartmann-Shack sensors for high-resolution eye aberration measurements, have taken refractive surgery to a new stage. Nevertheless, studies have shown that the lens contributes differently to total aberration and depends on accommodation and age factors. Therefore, the choice of which aberrations to correct is also related to the internal optics of the eye, although only the cornea undergoes surface changes. A simple model is to subtract the total aberrations of the eye from the aberrations of only the cornea, which are calculated based on its three-dimensional profile. This has been done successfully, in practice, for certain commercial videokeratographs (VKS), but it is important to analyze the consequences of general flaws associated with VKS systems when used to estimate the corneal aberrations. A computer simulation of Placido-based VKS has been developed and applied to several theoretical surfaces resembling corneal profiles. The resulting Placido images were used to retrieve the corneal profiles. Corneal surface data were input into an algorithm that calculates the wave-front aberration as the difference in optical path length of the principal ray of an on-axis object point and several marginal rays. Wave aberration was then fit by a set of Zernike polynomials using the least-square method. From the Zernike coefficients, the correlation of different corneal profiles to their optical aberrations could be estimated. Misalignment (decentralization), improper focusing, and tilt were also implemented to the VKS simulation, and a quantitative analysis of the consequences on precision of wave aberration estimation was undertaken.     

Corneal model

PURPOSE: To describe the optical region of the cornea with as few parameters as possible and to compare this approach to commonly used mathematical models for the cornea. SETTING: University eye hospital, Mainz, Germany. METHODS: Corneal surface is approximated by a simple model (SM) that is defined by 2 perpendicular vertex radii, their angle to the horizontal, and a unique numerical eccentricity. These parameters, together with a parameter quantifying the decentration of the recording, are obtained in a consistent fit of corneal topographic data. The SM is compared to Zernike polynomial approximations of the 4th (Z4 model) and 8th (Z8 model) radial orders. Residual refraction errors for these approximations are calculated by numerical ray tracing, allowing a comparison of the different approaches. The statistical evaluation was carried out in 100 healthy eyes. RESULTS: The model approximation accuracy for the SM was at least as high as the reproducibility of the topographic measurements. For small optical zones up to 4.0 mm in diameter, the SM was on average more accurate than the Z4 model. CONCLUSIONS: The parameters of the SM, which are closely related to conventional parameters of the cornea, provided a highly accurate basis for following refractive interventions (customized corneal or cataract surgery). Zernike polynomials tend to improve peripheral optical quality at the expense of the central quality. Except in cases of technical optics, this is an unwanted effect in the human eye.