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Abstract
In this paper, the use of four flat-null-screens forming a quadrangular prism setup is employed as a target for an experimental corneal topographer. Topographic maps of three reflective calibration spheres with different radii of curvature: 7.70 mm, 9.42 mm, and 6.20 mm, and the corneal surface evaluation of a volunteer subject are presented. The experimental setup is based on the null-screen method. Zonal reconstruction of the surface shape of the human cornea is obtained by numerical integration of the normal vectors. The corresponding elevation map was found fitting an spherical model to the recovered data. The differences obtained in radius of curvature were 0.02% or less for the calibration spheres, and the cornea under test showed a difference of 1.39% (0.6 diopters) compared to the value obtained with ORBSCAN topographer.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The anterior corneal surface provides about three-quarters-parts of the total dioptric power of the human eye [1]; also, refractive corrections, monitoring of corneal ectasias, and keratoconus play a fundamental role in the assessment of patients. Thus, quantifying corneal parameters such as: radius of curvature, refractive power, and elevation maps, is a very important task for opticians, optometrists, and ophthalmologists. All these parameters are measured by corneal topography. A complete knowledge of the shape and optical power distribution of the human cornea is essential for the monitoring and diagnostics of degenerative diseases.
It is well known that studies about corneal topography have typically been made using Placido disk or Scheimpflug corneal topography [1]. However, the main drawbacks in the use of the Placido system, was the data ambiguity in azimuthal direction, as it is reported by Schwiegerling, et al [2]; also, S.A. Klein pointed out the skew ray error in corneal topography [3,4]. Consequently, new generations of topographers were developed making use of different principles; for instance, in 2001 Mejia et al, proposed a target with an ovoid shape for flattening the virtual image formed by the cornea and a set of strategically distributed point illuminators to get almost a perfect square array of bright spots on the image plane, for applying a modified Hartmann test to measure corneal topography [5]. In the same year, Munson et al, published a Shack-Hartmann aberrometer to assess the optical outcome of corneal transplantation in a keratoconic eye employing a set of sources produced by an individual array of lenses [6]. More recently, in 2010, Snellenburg et al, published a forward ray tracing for image projection prediction a surface reconstruction in the evaluation of corneal topography system [7]. All these advances have been tested successfully in corneal topography modifying the geometrical pattern can overcome the skew ray error [3,4]. It is well worth to recognize that recently Gómez, et. al., propose an iterative method for evaluating optical surfaces from circular rings without the skew ray error [8].
On the other hand, the null-screen test has been used effectively to test different types of optical surfaces as: aspherical fast convex surfaces [9–14], aspherical surfaces with deformation coefficients [15,16], off-axis conical surfaces [17], and in the last years it has been a good choice for testing freeform surfaces [18–20]; also, this idea has been useful to test the corneal surface successfully [21–30]. Furthermore, in 1997 Funes-Maderey et al. [31], demonstrated, using the parabasal theory, that the object surface producing a flat virtual image by reflection on a convex spherical surface is an ellipsoid of revolution; later, in 2001 Mejia et al., using the Coddington Equations, demonstrated that a better approximation is a target with an ovoidal shape [5]. In 2005 R. Colin-Flores et al., found the cylinder that better approximates the ellipsoidal target for having an almost flat virtual image [32]. In 2015, a target built with three LCD´s was proposed to apply the DyPoS (Dynamic Point Shifting) method [24], three flat null screens displayed on LCD´s were used in a triangular prism setup as a corneal topographer; that configuration was used to approximate a cylinder as a target, in trying to have an almost flat virtual image. One problem with such a configuration is that for producing a square array of dots on the image, each of the three null screen design must be different.
In the present work we propose to use a quadrangular prism geometry setup, because this is more symmetric and simpler for the evaluation of the corneal surface; at the same time a target composed of individual spots helps to avoid the skew ray problem, In addition, it is mentioned that, In the quadrangular prism, all the null-screens are identical, which will facilitate their calculation, it is an advantage of avoiding the design of a different null-screen for each section of the quadrangular prism, as in the case of the triangular arrangement. In addition, in this work it is shown the evaluation of three calibration spheres of different radius of curvature, as well as the evaluation of a human cornea. The paper is organized in the next form. In Section 2 the proposed method is completely described and in Section 3 the null screen design procedure for this setup is described. The specific experimental setup and methodology are fully explained in Section 4. Later, in Section 5, the experimental results obtained in the measurement of three calibration spheres and the reconstruction of the cornea of a volunteer are presented. Additional comments are presented in Section 6 and, the conclusions are listed in Section 7.
2. Proposed setup and method
The proposed setup is shown in Fig. 1. The quadrangular acrylic prism with the null-screen target is placed in front of the surface to be measured (a calibration surface or a cornea). The specular reflection on the convex surface yields a virtual image that is captured by a CMOS camera. The optical target and its image reflected by the human cornea will provide an almost square pattern, similar to the modified Hartmann test [5].
Fig. 1. Layout of the test configuration for a quadrangular corneal topographer based in null-screen method.
The null-screen test has the advantage that this technique does not need any additional optical element with a specific design to correct the aberrations of the system under test, and the spots of the screen can be easily computed [24,27]. The essential idea consists of designing a target composed by null-screens with a set of spots in such a way that if the test surface is perfect and there is no decentering, the image reflected by the optical surface under test gives a perfectly ordered arrangement type Hartmann test [5]. For that, is very important to consider the procedure described below.
2.1 Design dots in the image plane
Figure 2(a) shows a transverse view of the quadrangular acrylic prism. It is easy to see that the layout of the test configuration is divided into 4 specific sectors and each sector is bisected at 45° respectively. It is important to mention that each region corresponds to one flat-null-screen. In the same Fig. 2(a), X3 and Y3 are the corresponding distances from the center of the optical surface to the flat-null-screen. This is the beginning of the null-screen designed for the quadrangular acrylic prism. On the other hand, Fig. 2(b) schematically represents the square array of dots at the image plane on the CCD sensor.
Fig. 2. Front of view quadrangular acrylic prism and optical surface under test: (a) Transverse of layout of testing configuration, (b) Schematic ideal drop shaped spots on the spherical surface in the CCD image plane.
Figure 2(b) shows the design of the regular array of dots proposed, which is described by Eqs. (1) and (2) respectively.
where, xo and yo represent the vector position of each dot in the image plane; ll and cc are integers that count the dots along the x and y directions respectively, for the square array of (2N + 1)2 centered dots, and N is the number of points proposed along each of the x and y positive directions and d represents the shortest side of the CCD sensor (see Table 1). In addition, in Fig. 2(b) we can see the only part of the image plane that is covered by the image of the dots represented by the region contained within a yellow circle and the central yellow square. The yellow square is the image of the far edges of the same quadrangular acrylic prism.
Table 1. Elements of the layout of testing configuration.
Considering an imaginary circle of radius r1 (red one), that does not contain the central surface information, the limits to the radial coordinates of the evaluation area on the CCD image plane can be expressed as:
The image of the acrylic prism edges, which are parallel to the z-axis, are straight lines at: 45°, 135°, 225° and 315°, which are shown with green straight lines in Fig. 2(a) and (b). Each flat-null-screen covers a 90° sector on the image plane. Thus, the real number of dots is reduced accordingly to the dots surrounded by the yellow circle and the red circle with radius r1 and r2 respectively. The total number of dots observed in the image can be determined by
where, m1 = tan(θ1), and m2 = tan(θ2), are the slope of the straight line to each sector with; θ1 = 45°, and θ2 = 135° respectively, then, x1 and y1 are the final coordinates to the square sectors on the CCD image plane, the detailed definition of the sectors is shown in Fig. 3.3. Parameters for designing the flat-null-screen
To determine the points on the null-screen belonging to a square array of dots on the CCD image plane, we performed a reverse exact ray-tracing calculation, similarly as they have been obtained in Ref. [24]; we use the same expression to get the coordinates for each flat-null-screen, the calculation only differs in the use of four flat-null-screens parallel to the optical axis [27,28]. The variables involved in the design of the flat-null-screen are shown in Fig. 4.
Fig. 3. (a) Ideal dots for sectors: 1, 2, 3, and 4, (b) Ideal dots considered in the CCD image plane.
Fig. 4. Variables involved in the calculation of flat null-screens (a0 and b0 are for designing the null-screen)*.
The design of the null-screen requires finding the coordinates of the points P3 = (x3, y3, z3), on the corresponding flat-null-screen, which, produces a perfect square array of dots on the CCD image plane; for this, we start at the CCD image plane. A ray is traced back with the direction defined by the image principal point P, as it is shown with blue rays in Fig. 4. The point P1 = (x1, y1, -a0-b0), represents the Cartesian coordinates of a dot on the CCD image plane, a0 represents the distance from the nodal point P to the CCD image plane, and b0 is the distance from the object principal point P’ to the vertex of the optical surface under test as it is shown in Fig. 4. Indeed, for the thin lens approximation, P and P’ are so close that can be represented for only one principal point P.
To design the flat-null-screen the parameters ao and bo are obtained with the Gauss lens approximation and the definition of the transversal magnification given by
where, hi is the height of the experimental image and ho is the clear diameter of the spherical surface of the plano-convex lens used as reference; in our case hi = 1.33 mm, and ho = 12.7 mm, therefore; MT =0.1047.On the other hand,
f is focal length of the lens, f = 25 mm. In this case Sag<<b0, thus Sag→0. Consequently, and,Finally, to the design of the null-screens, the parameters ao and bo take the numerical values: ao = 27.6181 mm and bo = 263.7218 mm respectively. Nevertheless, later, these values will be recalculated using the algorithm described in Ref. [16], for a better calibration of the system.
According to Fig. 4, a ray is traced back from P1 to P2 inversely as it is explained in Ref. [24,27], this ray has a direction defined by the vector I.
The coordinates P2 = (x2, y2, z2) of the incidence points on the optical surface are obtained by intersecting the incident ray with a spherical reference surface (see Fig. 4). Then, to obtain the coordinates at the point P3 at the corresponding flat-null-screen, it is important obtain the unit reflected vector $\hat{{\boldsymbol R}}$, which is obtained from the vector form of the Reflection Law.
where $\hat{I}$ and $\widehat {N\; }$ are the unit incident and normal vectors at the incidence point. The normal vector is obtained from the reference surface f(x, y, z) = 0, by its normalized gradient operator evaluated at the point of incidenceSo that, the reflected ray is defined by
The ray hits the corresponding flat-null-screen at P3 = (x3, y3, z3) (see Fig. 4). The coordinates of the flat-null-screen parallel to the optical axis are defined by
where β is a real number, then, intersecting each ray with its corresponding flat-null-screen, the proper value of the parameter β for each point on the screen is found; in this case, Rx, Ry, and Rz represent the Cartesian components of the reflected rays. The details for the calculation of the null screen are fully explained in previously published papers [24,25,27].4. Experimental setup
To demonstrate the feasibility of the proposed device and method, an experimental test is next described. The numerical value of the components employed for this layout is shown in Table 1.
4.1 Methodology
To build this practical system, we used four identical acrylic pieces. Each section was joined each side in order to obtain a quadrangular prism as shown in Fig. 5. To ensure a better alignment on each cover the quadrangular profile of the prism was cut with a laser and each designed null-screen was placed on the corresponding side in the acrylic prism.
The individual spots on the screen can be easily computed according to the Eqs. (14)–(17). The flat-null-screen for the quadrangular acrylic prism is designed as an image and printed on a piece of paper. We can see the corresponding flat-null-screens in Fig. 6. These null screens have been designed in such a way that the observed image is a square array of circular spots when a 7.8 radius of curvature spherical surface is located a distance b0 from the point P’ as is shown in Fig. 4. In Fig. 6 is clearly observed that the spots are not circular in shape, they are not located along straight lines, nor the distance to their neighbors is constant. These facts can be understood as the distortion introduced by the spherical surface used for the design; the rays traced are far from being paraxial rays.
In each side of the rectangular prism the flat-null-screen is the same, however, in order to classify the null-screens, we have painted strategic colors marks: red, green, blue, and yellow, (R, G, B, and Y), following the next basic rules: a) each flat-null-screen has different basic color; R, G, B, and Y respectively, this identifies clearly, which screen produces each part of the experimental image. b) Eight square markings are positioned along the x-axis direction; also, the colored square mark, is the first spot along the z-axis of the null-screen, just to be redundant, in the same Fig. 6, it is easy to see six; red, green, blue and yellow square marks along the z-axis. In addition, the same reference color marks designed (R, G, B, and Y) helps to find the correct correspondence during quantitative evaluation. The set of colored spots allows to relate “x” and “y” directions on the screens even if one-or-two spots are missing due to sticking of the null-screens on the edges of the prism.
Figure 6 shows the actual design to the four null-screens; the layout of the null-screens is done by reverse ray tracing. In Section 3, it is explained in detail how the coordinates of each null-screen have been obtained, and the result is the image shown in Fig. 6. The fact that they are not round spots is due to the reflection suffered on the surface under test, this gives as a result a geometry with shape of ovals; i.e., deformed ellipses. This is because is a null test. The image obtained by the reflection of the null-screen on a spherical surface 7.7 mm of radius of curvature, will be a quadrangular array of dots, as we can see in the right of the Fig. 7.
4.2 Experimental setup
Figure 7 shows an experimental setup for testing the three calibration spheres. Each calibration sphere is placed in front of the quadrangular target; at the right side the image captured for the sphere ball1 is shown. The lines around the four flat surfaces are due to a stripe of white LEDs used to increase the contrast in the image and be able to better find the centroids on the image.
For the corneal surface, in Fig. 8(a), the setup and the subject are shown; whereas in Fig. 8(b), the corresponding image is shown. The image reflected on the corneal surface and captured by the CCD sensor shows a set of dots slightly deformed from a square array; the deformations are due to the fact that the cornea is not an spherical surface. This pattern provides data allowing accurate reconstruction of the corneal surface.
Fig. 8. (a) Experimental setup for corneal topographer testing the anterior human cornea, (b) Detail in the evaluations of the reflected pattern on the corneal surface (left emmetropic eye).
On the other hand, it is important to mention that; in this proposal, there is no alignment system such as in commercial topographers. The experimental arrangement does not have a chin support, for this reason, it is necessary to recalculate the parameters a0 and b0. Following the idea published by D. Aguirre-Aguirre, et al. [16], the parameters a, b (now called a0 and b0) used for designing the null-screen are randomly varied within some defined intervals a0 +Δa and b0 + Δb; for each random value a corresponding null-screen is computed; the new spots are located on different positions as compared with the spots on the design null-screen used in the experimental setup. The sum of the distances between corresponding spots on both null-screens are computed, the a0 and b0 values giving the minimum sum are taken as a best approximation to the real ones. Depending on the optical surface under test, aj, bj are the corresponding distances a0 and b0 which are used for a new evaluation procedure of the j-th surface (j = 1, 2, 3, 4), as described in Section 5.1. The parameters used for the evaluation of the calibration spheres tested with the setup are listed in Table 2.
Table 2. Final parameters aj and bj used in the evaluation of all surfaces derived by the algorithms of the Ref. [16].
(ao, bo, are for designing the null-screen).
[aj, bj are the corresponding distances a0, and b0 for evaluating the j-th surface (j = 1, 2, 3, 4)].
According to the algorithms proposed in Ref. [16], The parameter aj always takes the same value, because in this case, this parameter represents the fixed distance from the lens to the CCD sensor, and bj take different values, depending on the true position of the vertex of each surface; these bj values represent the distances from the object principal point P’ of the lens to the vertex V of each optical surface under test. Also, in the case of the corneal surface under test, aj remains with the same value, but bj changes according to the above definition.
5. Experimental results
5.1 Quantitative evaluation
After the experimental images were obtained, the first step was to calculate the centroid coordinates for every spot on the image; for that, an algorithm for smoothing the background was developed. Each image is independently processed, then, a simple thresholding method is used with the experimental images in order to separate the spot pixels, as is explained in Refs. [24,25]. Figure 9 shows the plots of the spot centroids position for the three calibration spheres and for the anterior corneal surface under test. Also, the distortion and magnification of the camera lens were calibrated as explained in Ref. [24].
Fig. 9. Plot spots centroids position for: a) sphere ball 1, b) sphere ball 2, c) sphere ball 3, and d) the corneal surface as shown in Fig. 8(b).
First, the incident I and reflected R vectors at each incident point on the surface are obtained through an exact ray tracing procedure. The reflected vector R is well defined by line joining the corresponding centroid on the CCD plane and the image principal point P. This line is parallel to the line passing through the object principal point P’ and the incidence point on the surface, but the last is unknown; for small departures of the surface from a spherical one, a good approximation to the incidence point results from intersecting the line starting at P’ with a reference spherical surface 7.8 mm of radius of curvature and its vertex at the origin; after the first evaluation of the surface, this approximation can be improved. The incident vector is defined by the line joining this approximated incidence point and the central point of the corresponding spot at the null-screen, on one of the four planes at the target. With these vectors R and I, the normal vector is defined by the Reflection Law.
The quantitative evaluation of the surface under test is obtained employing the normal components on the shape surface equation originally proposed by Díaz-Uribe [33] in the Eq. (18).
Nx, Ny, and Nz are the Cartesian components of the normal vector, x, y and z, are the Cartesian coordinates for every sampling point on the surface under test, the corresponding axes are oriented as is shown in Fig. 4; z is the coordinate axis along the optical axis of the camera lens. (i.e. z is the sag of the surface); z1 is the sagitta for the initial point on the surface along C, which is the path of integration. The numerical evaluation of the integral is performed using the trapezoidal rule for non-equally spaced data, given by Eq. (19).
In every plot in Fig. 11, for each calibration sphere ball, a color map of the sagitta differences of the evaluation points with reference to the best sphere [Eq. (20)] is shown; in addition, the evaluated points are shown by black dots.
Fig. 11. Difference in sagitta between the measured surface and the best-fitting sphere for three calibration spheres ball: a) R1= 7.70 mm, (b) R2= 9.42 mm, and (c) R3= 6.20 mm.
5.2 Topography maps
To analyze the details of the evaluation, the data of the sagitta were fitted to a spherical model represented by Eq. (20). Figure 11 shows the differences or elevation map between the experimental data and the ideal spherical model.
The P-V differences in sagitta between the evaluating points and the best fit sphere we obtain: δzPV = 0.0563 mm, δzPV = 0.0343 mm, δzPV = 0.0263 mm for each sphere respectively, and their corresponding RMS values are: δzRMS = 0.0062 mm, δzRMS = 0.0042 mm, δzRMS = 0.0035 mm. In addition, for the measured cornea, Fig. 12(a) shows topography maps obtained by null-screen method, while Fig. 12(b) the topography maps obtained by ORBSCAN topographer it is presented. To compare the results, in Fig. 12(c), the corneal topography (elevation map), obtained with a second commercial topographer (ATLAS Topographer of Zeiss), is shown; this shows that there are acceptable differences even between commercial topographers [36–38]. Among several possible causes of such differences, we can list the next: different evaluation methods (reflection, Sheimpflug, scanning, OCT), different evaluation algorithms even for the same technique, and in some cases, even for the same instrument the variation of corneal power throughout the day or different hydration status of the subject, may give significant differences when measuring the same subject at different times.
Fig. 12. Topography plots provided by three different methods correspondingly: a) Elevation and Optical Power Maps. obtained by null-screen method, b) Elevation and Optical Power Map obtained by ORBSCAN Topographer, and c): Elevation Map of the cornea of the same volunteer subject tested by the commercial topographer ATLAS.
In the case of the cornea under test, the P-V differences in sagitta are δzpv = 0.1456 mm and their RMS for the corneal surface under test is δzRMS = 0.0149 mm as compared to the best fitting sphere, for nearly a circular transversal area, 10 mm in dimeter. On the other hand, the value obtained for the dioptric power (DP) in average which is obtained with the best fit sphere was DP = 39.66 Dioptrics (D). The plots of Fig. 11 are made accordingly to the color code obtained from the ORBSCAN topography (Fig. 12); there, each color represents one diopter step. Table 3 shows the numerical values for the parameters obtained for the best fit. In the same Table 3, the numerical values for the radius of curvature recovered for the best fit are shown in the third column. We can see that for each calibration sphere surface; the recovered radius of curvature differs at most 0.02% from the real value respectively.
Table 3. Parameters obtained for the best fit spheric surfaces and for the anterior corneal surface under test.
6. Discussion
The P–V and RMS values are reported in Table 3. The numerical value obtained for the radius of curvature with the best fit is: rcornea = 8.51 mm, while; the same parameter obtained by ORBSCAN topographer is: rcornea ORBSCAN = 8.63 mm. Here we notice that the radius of curvature differs by approximately 0.12 mm, that is approximately 1.39% of the ORBSCAN measurement. On the other hand, the value obtained for the dioptric power (DP) was DP = 39.66 D, while the value obtained by ORBSCAN was DPORBSCAN =39.11 D, [See Fig. 12], this value differs approximately 0.55 diopters (1.41%). In the case of the radius of curvature and optical power measurement, the differences are small compared with the nominal values obtained by the commercial topographer Orbscan, in addition, for the case of the differences in sagitta; it is easy to see (in Fig. 12), that in the evaluation carried out by ORBSCAN topographer, the value obtained for the P-V differences in sagitta of the corneal surface are δzpv = 0.150 mm, which, differs from the value obtained by our proposal in only 0.0044 mm. Currently, the accuracy for clinical topography depends on the instrument within of the diverse range of existing surveyors, and on the other hand, it depends on the position of the surface to be evaluated, especially due to the well-known piston problem. The total error obtained with the method reported here, including alignment of the system and numerical systematic and random errors it is presented. These results do not represent a fundamental limit; they can be improved with better alignment of the system, of the surface under test, or by improving the numerical routines, so that, the uncertainty must be reduced to at least one tenth of the present value, as is the case of the calibration spheres. Recalculation of the parameters of work, helped us to obtain a better topography map.
7. Conclusions
In this paper, the use of four flat-null-screens forming a quadrangular prism setup is employed as a target for an experimental corneal topographer. Experimental results obtained for three calibration spheres and the anterior surface the cornea of an informed subject has been presented. The measured radios of curvature are very close to the design value reported for each calibration sphere; their differences are within 0.02% for the three calibration spheres; for corneal surface the difference value is around of 1.41%. In addition, it was demonstrated that the system can test different type of optical surfaces with a symmetry sampling. Finally, these results are the first report of a different proposal because of the target; we are aware that they can be improved by finer calibration of the quadrangular setup and by introducing better compensation numerical algorithms.
Funding
Universidad Nacional Autónoma de México (FESI-PAPCA 2021-2022-43, PAPIIT IA106823, PAPIIT IT100321, PAPIIT IT101620, PAPIIT IT103823, PAPIME PE109023).
Acknowledgments
The principal author thanks to Alessandra Carmichael-Martins for their useful support in adjusting the setup, and thanks to Angela Abril Suarez-Ajoleza by the evaluation using ORBSCAN Topographer. Thanks to the company Bleeps Vision for the use of the ATLAS Topographer.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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