March 2007
Volume 48, Issue 3
Free
Cornea  |   March 2007
Construction of a 3-D Atlas of Corneal Shape
Author Affiliations
  • Jean-François Laliberté
    From the Maisonneuve-Rosemont Hospital, Montreal, QC, Canada; the
  • Jean Meunier
    Departments of Computer Science and Operations Research,
    Ophthalmology, and
  • Miguel Chagnon
    Mathematics and Statistic, University of Montreal, Montreal, Québec, Canada; and the
  • Jean-Claude Kieffer
    INRS-EMT (Institut National de la Recherche Scientifique-Énergie, Matériaux et Télécommunications), Université du Québec, Varennes, Québec, Canada.
  • Isabelle Brunette
    From the Maisonneuve-Rosemont Hospital, Montreal, QC, Canada; the
    Ophthalmology, and
Investigative Ophthalmology & Visual Science March 2007, Vol.48, 1072-1078. doi:10.1167/iovs.06-0681
  • Views
  • PDF
  • Share
  • Tools
    • Alerts
      ×
      This feature is available to authenticated users only.
      Sign In or Create an Account ×
    • Get Citation

      Jean-François Laliberté, Jean Meunier, Miguel Chagnon, Jean-Claude Kieffer, Isabelle Brunette; Construction of a 3-D Atlas of Corneal Shape. Invest. Ophthalmol. Vis. Sci. 2007;48(3):1072-1078. doi: 10.1167/iovs.06-0681.

      Download citation file:


      © ARVO (1962-2015); The Authors (2016-present)

      ×
  • Supplements
Abstract

purpose. A methodology is proposed to build population-based average three dimensional (3-D) atlases or standards of the human cornea based on topographic data, along with variation maps. Also, methodologies for comparing populations or screening populations, based on these atlases are proposed.

methods. Topographies (Orbscan II; Bausch & Lomb, Rochester, NY) of 516 normal subjects were used. Methodology for the construction of a corneal atlas consisted of (1) data acquisition from both anterior and posterior corneal surfaces in the format of a 101 × 101 grid of z elevations evenly spaced (every 0.1 mm) along the x and y axes; (2) spatial normalization of the topographies on a unique average best-fit sphere to reduce the large variability in size and spatial location between corneas; (3) generation of the average 3-D model; and (4) statistics maps including average, median, and SD for each point of the grid.

results. To demonstrate the informative potential of this methodology, examples of atlases were generated. Numerical corneal atlases allow (1) characterization of a population, (2) comparison of two or more populations, (3) comparison of an individual with a reference population, and (4) screening of a population for the detection of specific corneal shape abnormalities, such as keratoconus or previous refractive surgery.

conclusions. The concept of a 3-D corneal atlas was developed. The proposed technique was meant to be simple, accurate, reliable, and robust and can be extended easily to any type of topographer capable of providing tridimensional corneal maps.

Over the past 20 years, corneal shape analysis has evolved from the subjective analysis of a Placido disc corneal reflection (photokeratoscopy) to the sophisticated computer assisted analyses offered by current videokeratographs. Corneal videokeratography provides detailed information regarding the shape of the entire cornea that is useful to assess morphologic changes caused by aging, diseases, trauma, or surgery. 
Videokeratographs are based on three acquisition principles: Most systems are based on the reflection of a Placido disc on the anterior corneal surface (EyeSys; EyeSys Vision, Houston, TX; Topographic Modeling System (TMS); Tomey Technology and Vision, Nagoya, Japan). The most recent generation of videokeratographs is based on slit scanning of the cornea, either alone (Pentacam; Oculus, Wetzlar, Germany) or in combination with a Placido disc (Orbscan; Bausch & Lomb, Rochester, NY). A video camera captures the slit of light projected on and through the cornea to calculate the shape of its anterior and posterior surfaces. Finally, in the fluorescein profilometry technique (Corneal Topography System; PAR Vision Systems Corp., New Hartford, NY), fluorescein is instilled to delineate the anterior corneal surface, and a stereo triangulation algorithm is used to calculate the shape of this surface. 
Despite significant progress in the concept of corneal shape analysis, these systems are only able to study one topography at the time. At the most, two topographies from the same subject can be compared by using a difference map. Some groups have attempted to characterize sets of topographies. In 1994, Hayashi et al. 1 2 generated average, standard deviation (SD) and count maps from the TMS-1 topographies of a group of 104 subjects. They also provided difference maps between average maps. Their goal was to compare superolateral versus superior phacoemulsification incisions. Buehren et al. 3 facilitated the comparison between two populations by providing a Student’s t-test map. Fam et al. 4 also generated SD maps to evaluate the repeatability of the Orbscan pachymetry maps that they used to generate an average corneal pachymetry model. The repeatability was greatest in the center and decreased toward the periphery. The notion of alignment of the topographies before their averaging was raised by Buehren et al. 5 who were concerned about eye movement between successive topographies. In their technique, a regression plane was used to correct for the tilt between two topographies, the best-fit sphere (BFS) apex was used to adjust for x, y, z shifts, and a best-fit spherocylinder alignment was used to rectify the cyclodeviation error. In 2005, Grzybowski et al. 6 addressed the usefulness of three different fitting zones (apex, global, and peripheral fit) for the alignment of pre- and post-LASIK topographies. They concluded that the choice of the fitting zone could influence the appearance of the central posterior elevation after LASIK. Dealing with right and left eyes also necessitates data conversion to account for the natural symmetry between both eyes with respect to the sagittal plane (enantiomorphism). The temporal cornea being steeper than the nasal cornea, 7 simple averaging (without alignment of the corneas) results in an increased variability. Topuz et al. 8 and Smolek and Klyce (Smolek MK et al. IOVS 2001;42:ARVO Abstract 2839) used the mirrored images of left eyes to convert them in right eyes before merging the data from both eyes. 
These studies represent the first steps toward a population-based average corneal model. Average representations of topographies from more than one individual were given by Hayashi et al., 1 Topuz et al., 8 Fam et al., 4 and Smolek and Klyce groups (Smolek MK et al. IOVS 2001;42:ARVO Abstract 2839). However, alignment of the topographies before averaging was only used for topographies belonging to the same subject, and in none of the cases was the methodology for the development and optimization of these average models reported. 
The present study was a logical extension of past studies. The goal was to describe and assess a methodology for the construction of a population-based 3-D atlas of the cornea and to provide useful tools to characterize this population. The particularities of this work include the 3-D combination of both anterior and posterior surfaces data, the detailed description of the alignment technique used before averaging, and the definition of the indications and contraindications for such alignment. The methodology described herein was meant to be simple, accurate, reliable, and robust, and it was conceived for easy and universal implementation with any type of topographer capable of providing tridimensional corneal surface maps. 
Materials and Methods
Population
The topographies used to construct atlases were selected from a pool of 5448 topographies taken at the Department of Ophthalmology of the Maisonneuve-Rosemont Hospital and the LASIK MD Clinic (Montreal, QC, Canada). Inclusion criteria were a refractive spherical equivalent within ± 3.00 D from emmetropia, a refractive cylinder of <1.00 D, and the absence of corneal diseases or previous ocular surgery. Contact lens wearers had removed their lenses 72 hours before topography if they wore soft contact lenses or at least 4 weeks before if they wore rigid contact lenses. All subjects were aged between 18 and 70 years. The research protocol adhered to the tenets of the Declaration of Helsinki and was approved by the Maisonneuve-Rosemont Hospital human experimentation committee. A signed informed consent was obtained from the subjects. 
Definitions
We propose a method for the construction of an atlas from any set of corneal topographies, using any topographer available on the market. The quality of the atlas will depend on the quality of individual topographies. The greater the precision and accuracy of individual topographies and the greater the number of points available for computation, the greater the value of the resultant atlas. 
In the present study, the topographies were obtained with the Orbscan II (Bausch & Lomb, Rochester, NY). This apparatus acquires approximately 10,000 data points (40 slits × 240 points/slit) from the anterior and posterior corneal surfaces within 1.5 seconds. The Orbscan software includes an optional recording utility that saves the data into text files with numerical arrays of 101 × 101 points uniformly spaced and centered on the visual axis. The horizontal and vertical distance between two points is 0.1 mm. 
We used as raw data the 3-D coordinates of the anterior and posterior surfaces of the cornea provided by Orbscan. For each point (x, y) of the 101 × 101 grid, the z coordinate indicates the distance from the corneal surface to a reference plane perpendicular to the line of sight. Since the reference plane may vary from one topography to another, we made use of the anterior (or posterior) best-fit sphere (BFS) (i.e., the sphere that best adjusts to the anterior, or posterior, surface of the cornea in the sense of the least mean squares in a central adjustment zone of 10.0 mm in diameter). The BFS is characterized by its radius and center. Once the BFS was calculated from the elevation with respect to the reference plane, we used the elevation with respect to the BFS itself. We will explain in the next section how to compare two topographies using the BFS. The BFS also facilitates visualization of the corneal shape—namely, the highlighting of small elevation changes. Note that Orbscan provides its own BFS, but we preferred to compute the BFS ourselves, based on the raw elevation maps. 
Orbscan computes corneal thickness as the distance between a point on the anterior surface and its corresponding (interpolated) point on the posterior surface along the anterior BFS radius. The manufacturer has added a correction factor (called the acoustic factor) to these calculations so that Orbscan and ultrasound central pachymetry values match more closely—ultrasound still being recognized as the gold standard for pachymetry. 9 10 This acoustic factor reduces all pachymetry values by 8% (Orbscan default acoustic factor of 0.92). In this study, we computed pachymetry according to the same method, using the acoustic correction factor. 
To facilitate interpretation, and unless otherwise specified, the color scales used for the atlases were the same as those used by Orbscan (5-μm color steps for the elevation maps, green representing a point on the BFS, and 20-μm color steps for pachymetry maps). Color steps of 1 μm were used for the SD maps. 
Spatial Normalization
Spatial normalization of the surfaces before their averaging is a fundamental step for the construction of a 3-D corneal atlas. It consists of an overall resizing of the corneas to a standard (average) size and a translation to a common reference point to minimize variation between corneas. The scaling process is isotropic for the entire corneal surface, which means that local variations and individual features remain. 
For a given sample set of anterior surfaces, the reference BFS is generated by first identifying the BFS radius and center of each anterior surface in the sample set. The average radius rant/avg of all radii obtained and the average coordinates (x ant/ avg, y ant/ avg, z ant/ avg) of all corresponding centers are then computed. 
Next, each anterior surface is normalized on the average BFSant/avg, as follows: 
Step 1.
Translation of the anterior surface, to align its BFS center to coordinate (0, 0, 0). This step is essential to get an equal scaling on each axis. 
Step 2.
Isotropic scaling (equal scaling on x, y, and z axes) of the surface points, to normalize the BFS radius to rant/avg
Step 3.
Translation of the BFS center from (0, 0, 0) to the average BFS center (x ant/ avg, y ant/ avg, z ant/ avg). 
Step 4.
Resampling of the new (transformed) surface points on the original 101 × 101 discrete grid with a cubic spline interpolation algorithm. 11 This technique calculates a smooth surface passing by all realigned points and retrieves the surface elevation for the predefined grid points’ position. 
The adjustment of the posterior surface follows these same steps. For the calculation of the pachymetry maps, the same transformation (translations and scaling) is applied to both anterior and posterior surfaces, to preserve the relative position between the two. The pachymetry is then computed along the radius emerging from the reference center (x ant/ avg, y ant/ avg, z ant/ avg) and the acoustic factor is taken into account, as just discussed. 
Construction of the Numerical Atlas
After spatial normalization of the surfaces of the topography set, we are ready to build an atlas for the population under investigation. For each point (x, y) of the 101 × 101 grid, the average elevation and SD are calculated. As some of the points can be missing on the topographies, usually in the periphery (lashes, lid borders, and surface irregularities are usually responsible for this loss of information), statistics are computed only for the points for which at least 25% of the data are available. 
We have decided to generate an average model paired with an SD map, because average and SD are commonly used in statistics and easily understood by the scientific community. Average and SD are interesting choices if the data are not (or are slightly) corrupted by artifacts. However, one could choose to generate a median model paired with percentile maps. For small data sets, median maps are less affected by outliers or artifacts caused by the limitations of the acquisition system. 12 Large tear meniscus, surface irregularities, poor fixation by the subject, and/or incorrect alignment by the technician during acquisition are several possible causes of artifacts. 
In the next sections, we mention some of the applications of corneal atlases. 
Characterization of a Population
Corneal atlases allow population-based qualitative and quantitative description of the 3-D corneal shape. This could not be done with traditional single surface topography analysis. The methodology that we have described can be applied to all kinds of normal, pathologic, or postoperative eye populations. 
Comparison of Two Populations
Atlases also represent a powerful tool for the numerical comparison between two or more populations. The typical process for comparing two atlases consists of the following:
  •  
    Construction of an atlas for each population, as described herein.
  •  
    Computation of the elevation difference between the two atlases at each point (x, y) of the 101 × 101 grid.
  •  
    Computation of the appropriate P statistics: For the comparison of two atlases, the Student’s t-test or the midrank Wilcoxon test 13 probability maps can be used, combined with the technique of Benjamini and Hochberg 14 to control the false-discovery rate. For the comparison of more than two atlases, this typical process can be generalized, and the computation of the probability can be replaced by an ANOVA or by an analysis of covariance, with the adjustment of the atlases values with one (or more) covariate(s) (age or time, for instance).
Comparison of an Individual with a Reference Population
A third significant advantage of atlases is the objective numerical comparison between one individual and a reference population, another process that was not possible with individual topographies. The typical methodology to test a topography against a standard consists of the following:
  •  
    Normalization of the tested topographic surface to fit the atlas average BFS.
  •  
    Computation of the point-based difference between the normalized tested topography and the atlas.
  •  
    Illustration with appropriate difference maps. Three difference illustration techniques are proposed: (1) The simple difference map is the computation of the point-based difference between the normalized map of the subject and the atlas average map. (2) The SD difference map is the computation of the point-based difference between the normalized map of the subject and the atlas average map, but where all differences that are less then three SD are set equal to zero. Thus, all points where the difference is greater than three SD from the average value will stand out. (3) The percentile map illustrates the subject corneal shape position with respect to the topographies composing the atlas. The percentile analysis requires, for each point of the 101 × 101 grid, the sorting of all z elevation values of the topographies that were used in building the atlas. Formula 1 presents the percentile value perc as a function of the elevation z, whereas formula 2 gives the z elevation as a function of the percentile value:
 
\[\mathrm{perc}(x,y,z)\ {=}\ (\ \frac{100}{N_{x,y}})\ {\times}\ \mathrm{rank\ of}\ z\ \mathrm{in\ the}\begin{array}{l}\mathrm{ordered\ list}\ {[}z_{x,y}{]}\\_{\mathrm{all\ valid\ point\ at\ (x,y)}}\end{array}\]
 
\[z_{\mathrm{percentile}}(x,y,p)\ {=}\ (\ \frac{\mathrm{perc}}{100}\ {\times}\ N_{x,y})^{\mathrm{th}}\mathrm{element\ of\ the}\begin{array}{l}\mathrm{ordered\ list}\ {[}z_{x,y}{]}\\_{\mathrm{all\ valid\ point\ at\ (x,y)}}\end{array}\]
where N x,y is the number of topographic maps covering the point (x, y). The parameter perc is the percentile between 1 and 100. A perc of 1 means that the elevation of the tested cornea, at that point, is at the level of the lowest z elevation observed among the topographies used to build the atlas, whereas a perc of 100 means that, at that point, the elevation of the tested cornea is at the level of the highest z elevation.
Screening and Classification
Screening of a population for a specific disease requires a clear definition of the absence of the disease, which can be given by a normal cornea reference atlas. The methodology used is then the same as that described for testing a topography against a reference atlas. One could also use several atlases from different populations for classification purposes. Advanced classification algorithms could be used, including neural networks 15 based on shape indices derived from differences between the reference atlas and the tested subject, to automate the screening process. 
Results
Characterization of a Population: Example of a Normal Population
The methodology described herein was used to build a corneal atlas using 516 topographies from 516 right eyes of healthy subjects aged between 19.3 and 69.3 years (mean ± SD = 40.5 ± 11.3). The mean (±SD) refractive spherical equivalent was −1.08 ± 1.87 D, with a mean astigmatism of 0.35 ± 0.32 D. Table 1summarizes the amplitude of the isotropic scaling and translation applied to corneas for the normalization of the surfaces to the average BFS. We also calculated the RMS difference between the original elevation values of the 101 × 101 discrete grid points and the elevation values obtained after normalization and resampling. The RMS difference was calculated for isotropic scaling alone (step 2), translation alone (step 3), and both, for the anterior surface, posterior surface, and pachymetry maps (Table 2)
Figure 1Agives a color representation of the anterior elevation, posterior elevation, and pachymetry maps of this atlas. The anterior surface average BFS had a radius rant/avg of 7.94 mm and center coordinates (x ant/ avg, y ant/ avg, z ant/ avg) of (0.01, 0.01, −5.10) mm. The central corneal elevation was slightly positive (yellow-orange), with a mean elevation of 7.8 μm above the anterior BFS in the 2.0-mm diameter central zone). The SD was lower in the central region and increased toward the periphery (changing from cold to warm colors). For the posterior elevation map average BFS, we obtained rpost/avg =6.59 mm and (x post/ avg, y post/ avg, z post/ avg) = (−0.02, −0.02, −4.37) mm. The posterior surface showed a more negative coefficient Q (increased prolateness), which resulted in an increased positive elevation centrally (23.7 μm above the posterior BFS in the 2.0-mm-diameter central zone). The SD map showed a slightly higher variability than for the anterior surface and an increased variability in the periphery (which could be attributed in part to a greater intersubject variability and a lower accuracy of the Orbscan for posterior surface and peripheral measurements. 16 17 The average pachymetry map (Fig. 1A)showed a thinner central region. The thinnest point measured 547 μm and was slightly displaced temporally and inferiorly with respect to the topographic center (−0.2, −0.1). The variability was lower in the central region than in the periphery, a finding also reported by others. 4  
Meridional profile representations of the corneal atlas can also be displayed. Figure 2shows horizontal and vertical profile representations of the atlas, with a 3-SD interval. For better appreciation of the changes from the center to the periphery in that figure, the distance between the surface and its BFS was amplified by a factor of 5, as well as the 3-SD interval width. 
Comparison of Two Populations Based on Age Group
To demonstrate the usefulness of corneal atlases, preliminary results showing changes in corneal shape with age are illustrated in Figure 3 . The eyes of two groups of young (20–25 years, n = 89 OD) and older (55–60 years, n = 136 OD) subjects were studied. Overall, taken independently, both atlases looked quite similar, whether anterior elevation, posterior elevation, or pachymetry were considered. However, by subtracting the 20- to 25-year-old atlas from the 55- to 60-year-old atlas, the following observations were made. The anterior elevation difference map (Fig. 3 , top, third from left) revealed that the superior and inferior periphery was higher between 55 and 60 years than between 20 and 25 years, whereas the reverse was true of the nasal and temporal periphery. This corroborates the well-documented shift in astigmatism observed with age, from with the rule to against the rule. 18 The posterior elevation difference map (Fig. 3 , center, third from left) revealed a slight temporal shift of the central bulge with age. Finally, the pachymetry difference map displayed a slight increase in the central pachymetry with age (Fig. 3 , bottom, 3rd from left). 
Comparison of an Individual with a Reference Population: Three Examples
To illustrate how a single topography can be tested against a population reference model, three typical anterior elevation topographies from a normal eye, a postmyopic LASIK eye, and a keratoconus, were compared with a reference atlas. The right eye atlas described in Figure 1Awas used as the reference. 
The normal eye topography (Fig. 4A ; left column) was very similar to that of the reference population. The elevation difference map showed no difference greater than 5 μm in the central zone and the entirely green ±3-SD map confirmed that all elevation points were within the ±3-SD range. The normal range of the percentile map illustrates that the tested eye is within the normal range, since the map appears completely green. 
The second case had undergone LASIK to correct mild (−1.75 D) myopia. This case is illustrated in Figure 4B(middle). The patient’s anterior elevation map revealed the typical central flattening resulting from a myopic LASIK. The difference between the reference atlas and the surgically treated cornea clearly exhibits the central ablation region (in blue) in the difference and percentile maps. We can see that it lies outside the normal range indicated in green. 
The third case represents a clinically diagnosed keratoconus (Fig. 4C , right column). The inferotemporal protrusion is easily delineated in all difference and percentile maps as a reddish region. The annular area surrounding the cone appears blue rather than green, due in part to the forward shift of the BFS resulting from the keratoconus protrusion. 
Discussion
The results demonstrate the informative potential of population-based 3-D corneal atlases constructed with the proposed methodology. In addition to a better knowledge of the shape of the human cornea, atlases provide powerful tools for comparisons between populations, or between individual topographies and a reference population. 
The methodology described in this article is not limited to the Orbscan topography system and could be used with any other type of instrument capable of providing corneal 3-D surface points (e.g., Pentacam, Oculus, Wetzlar, Germany; Artemis 2, Ultralink, St. Petersburg, FL; Visante OCT, Carl Zeiss Meditec, Dublin, CA). The choice of the parameter used to build the average model is also optional (e.g., elevation with respect to the BFS, true elevation, axial curvature, tangential curvature). We chose to use the raw elevation because it basically localizes the corneal shape in a specified volume, while several definitions have been proposed for corneal curvature depending on the method of calculation (axial, tangential, or Gaussian curvature), the index of refraction (corneal values or keratometric equivalents), and the definition of the corneal center. The choice of the statistic is also determined according to the needs (e.g., average and SD maps, median and percentile maps, difference map, Student’s t-test map, or mid-rank Wilcoxon test with the percentile information, with or without Benjamini’s correction, ANOVA). 
Spatial normalization of the surfaces before their averaging (described herein as steps 1–4) represents an important part of the average corneal model construction. Spatial normalization is a well-recognized technique in the field of medical imaging. 19 When subjects are compared, it has been shown that the major problem is excessive shape variability and that without alignment, the usefulness of the atlas is limited. 19 20 21 Spatial normalization (e.g., translation, rotation, and scaling) was used and validated in brain imaging for shape analysis of brain tissues 19 20 22 23 and for the construction of population-based lung shape average models. 24  
The four-step spatial normalization proposed herein allows analysis of the corneal shape independently of corneal size and location. As it implies that all corneas have the same size and location, this model cannot be used for the analysis of corneal size or location. The model is used for the analysis of proportions and relative variations, not for the analysis of absolute (real) dimensions. If pachymetry or keratometry values are to be studied, isotropic scaling before averaging should be omitted. Similarly, when corneal topography is used to calculate optical aberrations, 25 26 spatial normalization should be avoided, as the corneal refractive power would be altered proportionally to the stretching factor (scaling). 
In the literature, spatial normalization is often underestimated, and more naive methodologies are usually adopted. For instance, the easiest way to compute a topographic average is to use the elevation raw data directly, without prior normalization, as illustrated on Figure 1B(left): Although the average elevation map remains adequate, the SD map shows much higher variability than that obtained with the 4-step normalization technique described herein (Fig. 1A , left). Such a variability makes it difficult to detect statistically significant differences between populations. The topographies could also be aligned with respect to a specified region, such as: the apex (e.g., central 1.0-mm-diameter zone; Fig. 1B , center), the entire surface (e.g., central 10.0-mm-diameter region; Fig. 1B , right), 6 or the periphery (e.g., 8.8- to 9.0-mm-diameter annulus). Again, this yields SD maps with low variations near the registration locus but much higher variations anywhere else. 
We used the BFS as a reference surface because it is routinely used with Orbscan topographies and because spheres are easy to calculate and align. Elevation maps, however, could also be treated without the BFS concept. In fact, it could be interesting to replace the BFS by the best-fit average surface of a normal reference population. The deviation maps generated from this more natural nonspherical standard could be more informative than the current Orbscan maps derived from a sphere, because they are more sensitive to smaller deviations from the normal shape. 
Conclusion
A new concept of a population-based 3-D numerical atlas is proposed for the characterization of corneal shape, using topographies from both anterior and posterior corneal surfaces. This methodology will allow broad application, including comparisons between populations or between a subject and a reference population, as well as large-scale topographical screening. The technique is meant to be simple, accurate, reliable, and robust, and it has been conceived for easy and universal implementation with any type of topographer capable of providing tridimensional corneal surface maps. 
 
Table 1.
 
Amount of Isotropic Scaling and Translation Applied to Corneas for the Normalization of the Surfaces to the Average BFS
Table 1.
 
Amount of Isotropic Scaling and Translation Applied to Corneas for the Normalization of the Surfaces to the Average BFS
Isotropic Scaling (%) X,Y Translation (μm)
Avg. ± SD (Min; Median; Max) Avg. ± SD (Min; Median; Max)
Anterior elevation 0.00 ± 3.16 (−11.06; −0.20; 12.63) 18.37 ± 10.50 (0.38; 16.27; 77.46)
Posterior elevation 0.00 ± 3.95 (−11.25; 0.04; 14.83) 48.84 ± 23.10 (3.60; 46.93; 135.11)
Table 2.
 
RMS Difference between the Original 101 × 101 Discrete Grid Elevation Points and Those Obtained with Isotropic Scaling Alone (Step 2), Translation Alone (Step 3), and Combined Isotropic Scaling and Translation (Steps 2 and 3), for the Anterior Surface, Posterior Surface, and Pachymetry Maps
Table 2.
 
RMS Difference between the Original 101 × 101 Discrete Grid Elevation Points and Those Obtained with Isotropic Scaling Alone (Step 2), Translation Alone (Step 3), and Combined Isotropic Scaling and Translation (Steps 2 and 3), for the Anterior Surface, Posterior Surface, and Pachymetry Maps
Isotropic Scaling (μm) Avg. ± SD (max) X,Y Translation (μm) Avg. ± SD (max) Scaling and Translation (μm) Avg. ± SD (max)
Anterior elevation 0.18 ± 0.01 (0.93) 0.07 ± 0.01 (0.31) 0.20 ± 0.01 (0.96)
Posterior elevation 0.83 ± 0.03 (2.39) 0.20 ± 0.01 (0.60) 0.85 ± 0.03 (2.50)
Pachymetry 3.87 ± 0.17 (24.67) 0.11 ± 0.01 (0.51) 2.26 ± 0.09 (12.53)
Figure 1.
 
Example of a normal population atlas based on corneal elevation topography. This atlas was built with 516 normal right eyes, by using four different normalization techniques. (A) Four-step spatial normalization described in the present paper: Average anterior elevation, posterior elevation, and pachymetry maps are illustrated (first row), paired with their corresponding SD maps (second row). (B) Anterior elevation and the corresponding SD map for three other normalization techniques: (left) no alignment; (center) alignment with respect to the entire surface (10.0-mm diameter); (right) alignment with respect to the apex (1.0-mm diameter). The SD maps obtained in (A) show less variability than those illustrated in (B).
Figure 1.
 
Example of a normal population atlas based on corneal elevation topography. This atlas was built with 516 normal right eyes, by using four different normalization techniques. (A) Four-step spatial normalization described in the present paper: Average anterior elevation, posterior elevation, and pachymetry maps are illustrated (first row), paired with their corresponding SD maps (second row). (B) Anterior elevation and the corresponding SD map for three other normalization techniques: (left) no alignment; (center) alignment with respect to the entire surface (10.0-mm diameter); (right) alignment with respect to the apex (1.0-mm diameter). The SD maps obtained in (A) show less variability than those illustrated in (B).
Figure 2.
 
Examples of profile cut along the (A) horizontal and (B) vertical meridians of the atlas described in Figure 1A . The distance between the surface and its BFS was amplified by a factor of 5 to illustrate the variation better, as well as the 3-SD interval width. Dotted curves: the anterior and posterior atlas BFS.
Figure 2.
 
Examples of profile cut along the (A) horizontal and (B) vertical meridians of the atlas described in Figure 1A . The distance between the surface and its BFS was amplified by a factor of 5 to illustrate the variation better, as well as the 3-SD interval width. Dotted curves: the anterior and posterior atlas BFS.
Figure 3.
 
Example of a comparison between two populations: an atlas from 136 normal subjects aged between 55 and 60 years is compared to that from 89 younger individuals (20–25 years). These elevation maps were built with right eyes only.
Figure 3.
 
Example of a comparison between two populations: an atlas from 136 normal subjects aged between 55 and 60 years is compared to that from 89 younger individuals (20–25 years). These elevation maps were built with right eyes only.
Figure 4.
 
Examples of comparisons between individual topographies and a reference atlas. In this case, the atlas described in Figure 1Awas used as the reference. Three topographies were tested: (A) A normal cornea, (B) a cornea that had undergone LASIK to correct myopia (amount of treatment: −1.75 D), and (C) a keratoconus cornea. Four types of maps are proposed to illustrate the comparisons (1) The difference map gives the difference between the tested topography and the reference atlas; (2) the ±3 SD difference map represents the region within 3 SD of the mean in green; (3) the percentile map gives the surface percentile value with respect to the atlas topographies; (4) the second percentile map represents in green the region of the cornea that is inside the percentile range of 2.5% and 97.5%.
Figure 4.
 
Examples of comparisons between individual topographies and a reference atlas. In this case, the atlas described in Figure 1Awas used as the reference. Three topographies were tested: (A) A normal cornea, (B) a cornea that had undergone LASIK to correct myopia (amount of treatment: −1.75 D), and (C) a keratoconus cornea. Four types of maps are proposed to illustrate the comparisons (1) The difference map gives the difference between the tested topography and the reference atlas; (2) the ±3 SD difference map represents the region within 3 SD of the mean in green; (3) the percentile map gives the surface percentile value with respect to the atlas topographies; (4) the second percentile map represents in green the region of the cornea that is inside the percentile range of 2.5% and 97.5%.
The authors thank François Destrempes, PhD, for technical assistance and the LASIK MD Clinic (Montreal, QC, Canada) for recruitment of the patients. 
HayashiK, NakaoF, HayashiF. Corneal topographic analysis of superolateral incision cataract surgery. J Cataract Refract Surg. 1994;20:392–399. [CrossRef] [PubMed]
HayashiK, HayashiH, HayashiF. Topographic analysis of the changes in corneal shape due to aging. Cornea. 1995;14:527–532. [PubMed]
BuehrenT, CollinsMJ, IskanderDR, DavisB, LingelbachB. The stability of corneal topography in the post-blink interval. Cornea. 2001;20:826–833. [CrossRef] [PubMed]
FamHB, LimKL, ReinsteinDZ. Orbscan global pachymetry: analysis of repeated measures. Optom Vis Sci. 2005;82:1047–1053. [CrossRef] [PubMed]
BuehrenT, LeeBJ, CollinsMJ, IskanderDR. Ocular microfluctuations and videokeratoscopy. Cornea. 2002;21:346–351. [CrossRef] [PubMed]
GrzybowskiDM, RobertsCJ, MahmoudAM, ChangJS, Jr. Model for nonectatic increase in posterior corneal elevation after ablative procedures. J Cataract Refract Surg. 2005;31:72–81. [CrossRef] [PubMed]
DingeldeinSA, KlyceSD. The topography of normal corneas. Arch Ophthalmol. 1989;107:512–518. [CrossRef] [PubMed]
TopuzH, OzdemirM, CinalA, GumusalanY. Age-related differences in normal corneal topography. Ophthalmic Surg Lasers Imaging. 2004;35:298–303. [PubMed]
FakhryMA, ArtolaA, BeldaJI, AyalaMJ, AlioJL. Comparison of corneal pachymetry using ultrasound and Orbscan II. J Cataract Refract Surg. 2002;28:248–252. [CrossRef] [PubMed]
SalletG. Comparison of optical and ultrasound central corneal pachymetry. Bull Soc Belge Ophtalmol. 2001;281:35–38. [PubMed]
UeberhuberCW. Numerical Computation. 1997;1Springer New York.
PressWH, FlanneryBP, TeukolskySA, VettelingWT. Robust Estimation. Numerical Recipes in C: The Art of Scientific Computing. 1988;15-7:699–707.Cambridge University Press Cambridge.
BickelPJ, DoksumKA. Mathematical Statistics: Basic Ideas and Selected Topics. 1977;Holden-Day San Francisco.
BenjaminiJ, HochbergY. Controlling the false discovery rate: a practical and powerful approach to multiple testing. J Roy Stat Soc. 1995;57:289–300.
BishopCM. Neural Networks for Pattern Recognition. 1995;Oxford University Press New York.
CairnsG, McGheeCN, CollinsMJ, OwensH, GambleGD. Accuracy of Orbscan II slit-scanning elevation topography. J Cataract Refract Surg. 2002;28:2181–2187. [CrossRef] [PubMed]
CairnsG, McGheeCN. Orbscan computerized topography: attributes, applications, and limitations. J Cataract Refract Surg. 2005;31:205–220. [CrossRef] [PubMed]
BaldwinWR, MillsD. A longitudinal study of corneal astigmatism and total astigmatism. Am J Optom Physiol Opt. 1981;58:206–211. [CrossRef] [PubMed]
TogaAW, ThompsonP. An introduction to brain warping.TogaAW eds. Brain Mapping. 1999;1–26.Academic Press San Diego, CA.
DavatzikosC. Spatial normalization of 3D brain images using deformable models. J Comput Assist Tomogr. 1996;20:656–665. [CrossRef] [PubMed]
MazziottaJC, TogaAW, EvansA, FoxP, LancasterJ. A probabilistic atlas of the human brain: theory and rationale for its development. The International Consortium for Brain Mapping (ICBM). Neuroimage. 1995;2:89–101. [CrossRef] [PubMed]
FristonKJ, AshburnerJ, FrithCD, PolineJ-B, HeatherJD, FrackowiakRSJ. Spatial registration and normalization of images. Hum Brain Mapp. 1995;3:165–189. [CrossRef]
LaliberteJF, MeunierJ, MignotteM, SoucyJP. Detection of diffuse abnormal perfusion in SPECT using a normal brain atlas. Neuroimage. 2004;23:561–568. [CrossRef] [PubMed]
DanilouchkineMG, WestenbergJJ, van AssenHC, van ReiberJH, LelieveldtBP. 3D model-based approach to lung registration and prediction of respiratory cardiac motion. Med Image Comput Comput Assist Interv Int Conf Med Image Comput Comput Assist Interv. 2005;8:951–959.
ArtalP, GuiraoA, BerrioE, WilliamsDR. Compensation of corneal aberrations by the internal optics in the human eye. J Vision. 2001;1:1–8.
SalmonTO, ThibosLN. Videokeratoscope-line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations. J Opt Soc Am A Opt Image Sci Vis. 2002;19:657–669. [CrossRef] [PubMed]
Figure 1.
 
Example of a normal population atlas based on corneal elevation topography. This atlas was built with 516 normal right eyes, by using four different normalization techniques. (A) Four-step spatial normalization described in the present paper: Average anterior elevation, posterior elevation, and pachymetry maps are illustrated (first row), paired with their corresponding SD maps (second row). (B) Anterior elevation and the corresponding SD map for three other normalization techniques: (left) no alignment; (center) alignment with respect to the entire surface (10.0-mm diameter); (right) alignment with respect to the apex (1.0-mm diameter). The SD maps obtained in (A) show less variability than those illustrated in (B).
Figure 1.
 
Example of a normal population atlas based on corneal elevation topography. This atlas was built with 516 normal right eyes, by using four different normalization techniques. (A) Four-step spatial normalization described in the present paper: Average anterior elevation, posterior elevation, and pachymetry maps are illustrated (first row), paired with their corresponding SD maps (second row). (B) Anterior elevation and the corresponding SD map for three other normalization techniques: (left) no alignment; (center) alignment with respect to the entire surface (10.0-mm diameter); (right) alignment with respect to the apex (1.0-mm diameter). The SD maps obtained in (A) show less variability than those illustrated in (B).
Figure 2.
 
Examples of profile cut along the (A) horizontal and (B) vertical meridians of the atlas described in Figure 1A . The distance between the surface and its BFS was amplified by a factor of 5 to illustrate the variation better, as well as the 3-SD interval width. Dotted curves: the anterior and posterior atlas BFS.
Figure 2.
 
Examples of profile cut along the (A) horizontal and (B) vertical meridians of the atlas described in Figure 1A . The distance between the surface and its BFS was amplified by a factor of 5 to illustrate the variation better, as well as the 3-SD interval width. Dotted curves: the anterior and posterior atlas BFS.
Figure 3.
 
Example of a comparison between two populations: an atlas from 136 normal subjects aged between 55 and 60 years is compared to that from 89 younger individuals (20–25 years). These elevation maps were built with right eyes only.
Figure 3.
 
Example of a comparison between two populations: an atlas from 136 normal subjects aged between 55 and 60 years is compared to that from 89 younger individuals (20–25 years). These elevation maps were built with right eyes only.
Figure 4.
 
Examples of comparisons between individual topographies and a reference atlas. In this case, the atlas described in Figure 1Awas used as the reference. Three topographies were tested: (A) A normal cornea, (B) a cornea that had undergone LASIK to correct myopia (amount of treatment: −1.75 D), and (C) a keratoconus cornea. Four types of maps are proposed to illustrate the comparisons (1) The difference map gives the difference between the tested topography and the reference atlas; (2) the ±3 SD difference map represents the region within 3 SD of the mean in green; (3) the percentile map gives the surface percentile value with respect to the atlas topographies; (4) the second percentile map represents in green the region of the cornea that is inside the percentile range of 2.5% and 97.5%.
Figure 4.
 
Examples of comparisons between individual topographies and a reference atlas. In this case, the atlas described in Figure 1Awas used as the reference. Three topographies were tested: (A) A normal cornea, (B) a cornea that had undergone LASIK to correct myopia (amount of treatment: −1.75 D), and (C) a keratoconus cornea. Four types of maps are proposed to illustrate the comparisons (1) The difference map gives the difference between the tested topography and the reference atlas; (2) the ±3 SD difference map represents the region within 3 SD of the mean in green; (3) the percentile map gives the surface percentile value with respect to the atlas topographies; (4) the second percentile map represents in green the region of the cornea that is inside the percentile range of 2.5% and 97.5%.
Table 1.
 
Amount of Isotropic Scaling and Translation Applied to Corneas for the Normalization of the Surfaces to the Average BFS
Table 1.
 
Amount of Isotropic Scaling and Translation Applied to Corneas for the Normalization of the Surfaces to the Average BFS
Isotropic Scaling (%) X,Y Translation (μm)
Avg. ± SD (Min; Median; Max) Avg. ± SD (Min; Median; Max)
Anterior elevation 0.00 ± 3.16 (−11.06; −0.20; 12.63) 18.37 ± 10.50 (0.38; 16.27; 77.46)
Posterior elevation 0.00 ± 3.95 (−11.25; 0.04; 14.83) 48.84 ± 23.10 (3.60; 46.93; 135.11)
Table 2.
 
RMS Difference between the Original 101 × 101 Discrete Grid Elevation Points and Those Obtained with Isotropic Scaling Alone (Step 2), Translation Alone (Step 3), and Combined Isotropic Scaling and Translation (Steps 2 and 3), for the Anterior Surface, Posterior Surface, and Pachymetry Maps
Table 2.
 
RMS Difference between the Original 101 × 101 Discrete Grid Elevation Points and Those Obtained with Isotropic Scaling Alone (Step 2), Translation Alone (Step 3), and Combined Isotropic Scaling and Translation (Steps 2 and 3), for the Anterior Surface, Posterior Surface, and Pachymetry Maps
Isotropic Scaling (μm) Avg. ± SD (max) X,Y Translation (μm) Avg. ± SD (max) Scaling and Translation (μm) Avg. ± SD (max)
Anterior elevation 0.18 ± 0.01 (0.93) 0.07 ± 0.01 (0.31) 0.20 ± 0.01 (0.96)
Posterior elevation 0.83 ± 0.03 (2.39) 0.20 ± 0.01 (0.60) 0.85 ± 0.03 (2.50)
Pachymetry 3.87 ± 0.17 (24.67) 0.11 ± 0.01 (0.51) 2.26 ± 0.09 (12.53)
×
×

This PDF is available to Subscribers Only

Sign in or purchase a subscription to access this content. ×

You must be signed into an individual account to use this feature.

×