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Evaluating the anterior corneal surface using an improved null-screen system

Manuel Campos-García, Victor de Emanuel Armengol-Cruz, and Arturo Ioan Osorio-Infante

Open Access Open Access

Abstract

In this work, a conical corneal null-screen topographer is presented. The device is designed to adapt to the human morphology of the face and measures the human cornea assuming that its shape is an aspherical surface. The evaluation of the performance of the conical null-screen corneal topographer includes calibration of the device and evaluation and compensation of the distortion introduced by the lens used to acquire the images. For the calibration of the device we perform the evaluation of a spherical reference surface of 7.8 mm radius of curvature and 11 mm in diameter. Here we obtain an rms difference in sagitta between the evaluated surface and the best fitting sphere of about 2.1 μm. We present examples of surface topography measurements on some human corneas. Elevation, sagittal curvature, and meridional curvature maps can be calculated. Other geometrical parameters such as the radius of curvature and the conic constant are obtained. Results of the corneal surface topography were compared with commercial Placido-based corneal topography device.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

It is known in the medical literature that the cornea, including the tear film, provides approximately 70% of the total amount of the refractive power of the eye [1,2]. The shape of the corneal surface is known to be a prolate ellipsoid with radius of curvature between 7.7 to 7.9 mm and conical constant about −0.18 to 0.26 [2]. Many techniques have been developed to make measurements of the cornea (elevation profile or its refractive power) as accurately as possible. Detailed assessment of the corneal shape parameters can be important for several different clinical and research applications such as the diagnosis and monitoring of corneal ectatic disorders, for example, keratoconus and pellucid marginal degeneration, which result in asymmetric corneal steepening.

Among the most common geometric-based techniques, the Placido’s Rings method is a reference for many devices when they are in the calibration process [1,3]. Recently, methods based on optical coherence tomography (OCT) have been used to evaluate the corneal topography [4–6]. One of the most important features of these systems is the ability to provide topographic maps of both anterior and posterior surfaces of the cornea. However, the proposed systems are expensive and have accuracies similar to systems based on Placido’s disk, where the rms error of the height data is in the range of 2 μm to 6 μm [7].

The null-screen testing method [8] is relatively new and has accuracy values close to other geometrical procedures. To perform a quantitative test with the null-screen method, a set of spots (like ellipses) are drawn on the screen in such a way that the image, which is formed by reflection on the test surface, becomes an exact array (square, radial or semiradial) of circular spots if the surface is perfect. Any deviation from this geometry is indicative of defects or misalignments of the surface under test. Moreover, due to its lower costs, it is a very rentable option. Many studies have been performed for testing surfaces with different shapes [9,10] proving both its broad spectrum and its potential. The null-screen method not only has the potential to measure the cornea [11–15], but can also estimate the sagittal and meridional radius of curvature [16]. In particular, the conical null-screen has been under study [12–15] since it can lead to an increase in the image quality and the accuracy of the data when evaluating reference surfaces. However, in order to use the conical null-screen to obtain the topography of human corneas we perform a slight redesign of the proposed topographer; this is due mainly to the morphological characteristics of the human face.

The aim of this work is to present the applicability of the null-screen method for quantitative evaluation of the shape of the human cornea. For this, we describe the proposed test method. Then, we report the new equations used for the design of the conical null-screen for testing human corneas considered as a conic surface. Next, we describe the procedure to evaluate the topography of the surface (elevation and curvature maps). We describe the procedure to calibrate the conical null-screen corneal topographer by testing a reference surface, and finally, results of the evaluation of the topography of some human corneas are shown.

2. Null-screen testing method

The null-screens are a uniform array of spots which by reflection can be seen at the CCD sensor as a perfect array of circular spots [8]; in other words, each point on the screen becomes a point source, P3, see Fig. 1. After reflection on the surface under test at P2, the pencil of rays that pass through a small aperture at P, forms the image of the screen on the CCD plane at P1, as in a camera obscura. The aperture is large enough to avoid important diffraction effects, so this test is in the geometrical optics regime.

 figure: Fig. 1.

Fig. 1. Null-screen testing configuration.

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Here it is important to mention that our experimental setup is not an image-forming optical system, but rather we are interested in the centroid position of the images. In general, the obtained spot images are unfocused. Although ambient illumination may be enough to allow us to see the image, to have better contrast in the image, we illuminate the screen from outside with fluorescent lamps or leds, so the illumination is not necessarily coherent. In practice, a positive lens is used to collect and focus to some degree, these thin pencils of rays onto the CCD sensor in order to increase the luminosity. Certainly, the camera lens introduces a small amount of distortion, which is considered by calibrating it, and which can be compensated.

3. Null-screen design for corneal topography

For the null-screen design we considered the corneal surface as a conic surface with symmetry of revolution, which is described by

$$z = \frac{{c{\rho ^2}}}{{1 + {{({1 - Q{c^2}{\rho^2}} )}^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}}},$$
where z is the sagitta and ρ = (x2 + y2)1/2, is the semidiameter or the distance of each point of the surface to the optical axis z; r = 1/c is the radius of curvature at the vertex, and Q = k + 1 (k is the conic constant of the surface).

The procedure for the design of the null-screen is based on the ray tracing as described in [12]; however, due to morphological characteristics of the human face we have to redesign the conical corneal topographer geometry by placing the conical null-screen base a distance -e from the X-Y plane as is shown in Fig. 2. This makes the mathematical expressions developed here slightly different from the original test [12]. To determine the object points on the conical null-screen that give us a semi-radial array of circular points on the CCD image plane after the reflection from the test surface, we perform an exact ray-tracing calculation, starting with one of the points of the array at the CCD plane P1 = (ρ1, ϕ1, -a-b-e), here P1 is given in cylindrical coordinates (ρ1 > 0; 0 ≤ ϕ1 ≤ 2π), a is the distance from the pinhole to the CCD plane, b is the distance from the pinhole to the cone base, and e is the distance from the vertex of the surface to the cone base (a, b, e > 0), see Fig. 2. A ray passing through the small aperture lens stop at P = (0, 0, b + e) reaches the corneal surface given by the Eq. (1) at the point P2 = (ρ2, ϕ1+π, z2), where

$${\rho _2} = F{\rho _1},\,\,\,\,\,\,{z_2} = Fa - ({b + e} ),$$
here F is given by
$$F = \frac{{a\{{Q({b + e} )+ r} \}- {{[{{{({ar} )}^2} - ({b + e} )\rho_{_1}^2\{{Q({b + e} )+ 2r} \}} ]}^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}}}{{Q{a^2} + \rho _{_1}^2}}.$$

 figure: Fig. 2.

Fig. 2. Null-screen design parameters.

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After reflection on the test surface, the ray hits the surface of the cone at the point P3 = (ρ3, ϕ1+π, z3), given by

$${\rho _3} = \frac{{s\{{\alpha ({{z_2} + e + h} )+ {\rho_2}} \}}}{{\alpha h + s}},\,\,\,\,\,\,{z_3} = \frac{{h({\alpha {z_2} + {\rho_2}} )- s({e + h} )}}{{\alpha h + s}},$$
where
$$\alpha = \frac{{{\rho _1}\rho _2^2 - {\rho _3}{{({Q{z_2} - r} )}^2} + 2a{\rho _2}({Q{z_2} - r} )}}{{a\rho _2^2 - a{{({Q{z_2} - r} )}^2} - 2{\rho _1}{\rho _2}({Q{z_2} - r} )}}.$$
Equations (4) and (5) are the coordinates of the points where the rays coming from the semiradial array of points on the CCD plane, after being reflected by the test surface, hit the cone surface. For general aspherics or free form surfaces, we can numerically design the null-screen following the same ideas.

The distance d on the CCD sensor is given by

$$d = \frac{{aD}}{{b + e + \beta }},$$
where D is the diameter of the test surface, and β is the sagitta at the rim of the surface (Fig. 2), which for a conical surface Eq. (1) is given by
$$\beta = \frac{r}{Q}\left[ {1 - {{\left( {1 - \frac{{Q{D^2}}}{{4{r^2}}}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}} \right]. $$
To build a conical null-screen with a set of custom targets that are located accurately is not an easy task. For a small screen it is easier to draw it on a sheet of paper (with the help of a computer and a laser printer). Here the cone is assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. Then, we must transform the cylindrical coordinates of the targets on the conical null-screen [Eqs. (4) and (5)] into XY Cartesian coordinates of the printed-paper sheet plane; the relationships are given by
$$X = R\,\,Cos({s\theta /l} ),\;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} Y = - R\,\,Sin({s\theta /l} ),$$
where
$$R = {\{{\rho_3^2 + {{({{z_3} - h - e} )}^2}} \}^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}},\,\,\,\,\,\,\,\,\,\,l = {({s + h} )^{1/2}},$$
here l is a generatrix of the lateral surface, see Fig. 2.

4. Corneal topography

The aim of this section is to describe how we evaluate the normals to the test surface and how with this information we can obtain the shape and curvatures of the surface. The term corneal topography refers to the measurement of the corneal surface shape. The main parameters of the corneal surface that can be measured are the elevation and curvature maps. The elevation map refers to the difference between the measured corneal surface shape and a reference surface. The radius of curvature and the conical constant are other relevant geometrical parameters that can be evaluated.

The null-screen testing method gives us information about the normals to the test surface. The evaluation of the normals N to the test surface can be performed with an approximate iterative algorithm described in [12]. In this section we summarize the algorithm for the sake of clarity of this paper. The procedure consists of finding the directions of the rays that join the real positions P1 = (x1, y1, -a-b-e) of the centroids of the spots on the CCD and the corresponding Cartesian coordinates of the objects of the null-screen P3 = (x3, y3, z3), given by Eqs. (4) and (5). According to the reflection law, the approximated normals N to the surface can be evaluated as

$$\textbf{N} = \frac{{\textbf{R} - \textbf{I}}}{{|{\textbf{R} - \textbf{I}} |}} = ({{n_x},{n_y},{n_z}} )$$
where I and R are the directions of the incident and the reflected rays on the surface, respectively; and nx, ny, and nz are the components of the approximated normals (see Fig. 3). The direction of the reflected ray R is known because after the reflection on the surface it passes through the center of the lens stop at P and arrives at the CCD image plane at P1; this direction is given by
$$\textbf{R} = \frac{{({{x_1},{y_1}, - a} )}}{{{{({x_1^2 + y_1^2 + {a^2}} )}^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}}}.$$

 figure: Fig. 3.

Fig. 3. Normal evaluation.

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Next, for the incident ray I we only know the point P3 at the null-screen, so we must approximate a second point to obtain the direction of the incident ray by intersecting the reflected ray with a reference surface at Ps= (xs, ys, zs), the reference surface is given by Eq. (1).

Now, a straight line joining P3 with Ps gives approximately the direction of the incident ray

$${\bf I} = \frac{{({{x_s} - {x_3},\;{y_s} - {y_3},\;{z_s} - {z_3}} )}}{{{{\left[{{{({{x_s} - {x_3}} )}^2} + {{({{y_s} - {y_3}} )}^2} + {{({{z_s} - {z_3}} )}^2}} \right]}^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}}}.$$
Finally, substituting Eqs. (11) and (12) into Eq. (10) the components of the approximated normal to the test surface are calculated. Once the normals are evaluated, the shape of the test surface can be obtained from the deflectometry equation through the formula
$$z - {z_i} = - \int\limits_{{P_i}}^{{P_f}} {\left( {\frac{{{n_x}}}{{{n_z}}}dx + \frac{{{n_y}}}{{{n_z}}}dy} \right)} ,$$
where zi is the sagitta for one point of the surface that must be known in advance. Once the shape of the surface is evaluated for some points according to Eq. (10), the next step is to analyze the details of the resulting evaluation; for this, we propose fitting the experimental data to the aspheric surface given by
$$z = \frac{{r - {{\left\{{{r^2} - Q\left[{{{({x - {x_{\textrm o}}} )}^2} + {{({y - {y_{\textrm o}}} )}^2}}\right]} \right\}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}}}{Q} + {\kern 1pt} A({x - {x_{\textrm o}}} )+ B({y - {y_{\textrm o}}} )+ {z_{\textrm o}}$$
where (xo, yo, zo) are the coordinates of the vertex of the surface; (xo, yo) are the decentering terms, zo is the defocus, and A and B are the terms of tilt in x and y, respectively.

The next step in the iterative normal evaluation procedure is to use the surface given by Eq. (14) as a new reference surface and calculate again the approximated normals to this new reference surface. Next, with these new approximated normals we obtain the new shape of the surface through Eq. (13). This iteration procedure continues until we arrive at a certain tolerance value given in advance. The fitting iteration converges quickly and gives us good accuracy in the determination of the approximated normals and in consequence in the shape of the surface [12].

On the other hand, from the calculated normals [Eq. (10)] we can evaluate the principal curvatures [16]. The axial curvature can be obtained from

$${k_{axi}} = \frac{{x\frac{{{n_x}}}{{{n_z}}} + y\frac{{{n_y}}}{{{n_z}}}}}{{{{({x^2} + {y^2})}^{1/2}}\left[ {{x^2} + {y^2} + {{\left( {x\frac{{{n_x}}}{{{n_z}}} + y\frac{{{n_y}}}{{{n_z}}}} \right)}^2}} \right]}}.$$
And the tangential curvature [16] is calculated from
$${k_{tan}} = {k_{axi}} + \frac{{{k_{fit}}({x^2} + {y^2})}}{{{{[{r_{fit}^2 + {k_{fit}}({x^2} + {y^2})} ]}^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}} \right.}\!\lower0.7ex\hbox{$2$}}}}}},$$
where rfit is the vertex radius of curvature, kfit is the conic constant, and can be obtained by fitting
$$\eta \equiv {\left( {\frac{{{n_x}}}{{{n_z}}}} \right)^2} + {\left( {\frac{{{n_y}}}{{{n_z}}}} \right)^2} = \frac{{{x^2} + {y^2}}}{{{{\left\{{r_{fit}^2 - ({{k_{fit}} + 1} )({{x^2} + {y^2}} )} \right\}}^{1/2}}}},$$
to the components of the approximated normal given by Eq. (10).

5. Results

In this section we report some corneal topography measurements of the corneas of two subjects. Accordingly, following section 2, we designed the conical null-screen for corneal topography with radius of curvature r = 7.8 mm and conical constant of k = -0.2. For the design of the corresponding conical null-screen we consider a cone with height h = 105.9 mm and a radius s = 70.6 mm. To capture the images, we use a CMOS camera (EO-5012M) with a sensitive area of 5.6 mm × 4.2 mm (2560 × 1920 pixels), and a 25-mm focal length lens attached. The rest of the parameters used for designing the conical null-screen are shown in Table 1.

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Table 1. Conical null-screen design parameters.

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For better sampling, the conical null-screen was designed to produce a radial-like array of circular spots on the image plane [12]. Each target was designed in such a way that it had a circular shape of equal size at the CCD (0.02 mm radius); the dot shape on the screen becomes an asymmetrical oval, which we call a drop-shaped target [17]. In this work, it is proposed to use negative null-screens, that is, black background and white dots, since this give us better results in the recovery of centroids due to the shadows produced by the morphology of the human face (nose, eyelids, eyebrows, eyelashes, etc.). The null-screen with black background and white dots is shown in Fig. 4a. Note how the targets on the screen have an almost elliptical shape, and how we increase the density of points as the radial distance increases producing a radial like null-screen. In the radial direction, we consider 10 spots; and the angular separation between spots is 24° for the inner, 12° for the middle, and 6° for the outer section. This give us a total of 560 spots. The flat null-screen was made on a laser printer on bond paper. This null-screen was inserted into an acrylic cone to give it mechanical strength. To have better contrast in the image, we illuminated the screen from outside with commercial leds, see Fig. 4b.

 figure: Fig. 4.

Fig. 4. a) Flat-printed conical null-screen with drop shaped targets, b) Conical corneal topographer targets.

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5.1. Calibration of the conical null-screen corneal topographer

Before performing measurements of human corneas, we make the calibration of the conical corneal topographer. For this, we evaluate the topography of a spherical reference surface of radius of curvature r = 7.8 mm and a diameter of D = 11 mm. The alignment of the surface was performed manually by using an overlay that consists of a reference circle and a cross hair target drawn on the image of the surface. The circular image of the boundary of the surface must be centred at the CMOS sensor and must touch the upper and lower boundaries of the overlay. In addition, the image of the null-screen must show an array of spots. If this condition is not fulfilled, then the screen is misaligned, or the testing surface is different to the design surface. The remaining misalignment can be computed by fitting the experimental data to Eq. (14).

An important feature that can be considered in the calibration of the conical corneal topographer is the evaluation of the distortion of the lens used to collect and focus the rays onto the CMOS sensor. For the distortion evaluation, we use a square grid pattern located in a plane perpendicular to the optical axis; this is considered as the centre of the image array. If the lens system is axially symmetric, the distorted image is still a pattern of spots in another square grid, scaled and distorted. Thus, the positions of the spots in the distorted pattern can be written as

$${x_1} = M{x_o} + E(x_o^2 + y_o^2){x_o},\,\,\,\,\,\,\,\,{y_1} = M{y_o} + E(x_o^2 + y_o^2){y_o},$$
where (x1, y1) are the coordinates of the image of an object point (xo, yo), M is the magnification of the system and E is the distortion coefficient. Then, by a least-squares fitting procedure we can determine the magnification M and the distortion coefficient E. It is worth mentioning that the lens distortion calibration is mandatory for systems where rigorous accuracy is essential. Here we proposed a very simple model; however, choosing the correct model could be a very difficult task as is discussed widely in Ref. [18]. Here the calculated coefficient E = −2.2 × 10−6 mm−2; since E < 0 the lens presents a barrel distortion, and M = 0.24.

The image of the conical null-screen after reflection on the spherical surface is shown in Fig. 5a. The centroids of the image in Fig. 5a were calculated with an image-processing program using custom algorithms. This algorithm obtains the region of interest (roi) of the original image (Fig. 5a), and then we calculate the local maximum of each spot. Finally, we construct a rectangle around each spot and evaluate the coordinates of the centroids using statistical averaging [19] through

$${x_c} = {{\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{x_{i,j}}{I_{i,j}}} } } \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{x_{i,j}}{I_{i,j}}} } } {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{I_{i,j}}} } }}} \right. } {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{I_{i,j}}} } }},\;\;\;\;\;\;{y_c} = {{\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{y_{i,j}}{I_{i,j}}} } } \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{y_{i,j}}{I_{i,j}}} } } {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{I_{i,j}}} } }}} \right. } {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{I_{i,j}}} } }},$$
where xi,j and yi,j are the coordinates of the pixel (i, j) in each subaperture, and Ii,j is the intensity at the pixel (i, j) in each subaperture having N × M pixels, the resultant processed image is shown in Fig. 5b. Finally, all the centroids were corrected for the lens distortion.

 figure: Fig. 5.

Fig. 5. a) The resultant image of the null-screen targets after reflection on the reference surface, b) processed image.

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The next step is calculating the approximated normal [Eq. (10)] to the test surface, as described in Section 4. With the calculated normal, the topography of the surface is obtained using Eq. (13) with the trapezoidal rule as the integration procedure. In Table 2 we show the geometrical parameters resulting from the least square fit to the data obtained from the sagitta of the evaluated reference sphere after 10 iterations. In this case, the peak valley difference in sagitta between the evaluated points and the best fit is Δzpv = 7.6 µm, and the rms difference in sagitta value is Δzrms = 2.1 μm. This final iteration improved the accuracy in the determination of the normal to the test surface and in consequence allows measurement of the shape of the surface with better accuracy. Additionally, we notice that the radius of curvature differs by 5 µm or about 0.06% of the design value of r = 7.80 mm. This result is consistent with the value given by the manufacturer of the reference sphere.

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Table 2. Parameters resulting from least squares fitting of sagitta data.

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In the left plot of Fig. 6, we show the elevation map obtained from the differences in the sagitta between the measured surface and the best fitting sphere obtained by a least-squares fit, here the decentring and tilt have been removed. For comparison, the elevation map obtained with the Placido-based corneal topographer is show in Fig. 6 (right). From the plots we can observe that elevation maps are quite similar. The Placido-based corneal topographer gives a radius of curvature of 7.9 mm that differs slightly from the value of the reference surface too.

 figure: Fig. 6.

Fig. 6. Elevation map for the reference surface.

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5.2. Topography of human corneas

Once the conical corneal null-screen topographer was calibrated the next step is the evaluation of the topography of some human corneas, here we consider both corneas of two different subjects (S1 and S2). The obtained results were compared with those obtained with a Placido-based corneal topographer.

In Fig. 7 we show the images of the conical null-screen after being reflected by the corneas of both subjects. As was explained previously, the null-screen was designed in such a way that the reflected image was a perfect array of circular points; however, from the reflected images of Fig. 7 we can see that most of the reflected points are spots of irregular shape, the irregularities are more evident in the periphery of the corneal surface, this means that the corneal surface is not a symmetric ellipsoidal surface as is assumed in the design of the null-screen. This means that the proposed technique can be applied when the cornea is a highly asymmetric and irregular surface, because according to Eq. (13), that gives the shape of the surface, is an exact expression and is not limited to any model of eye.

 figure: Fig. 7.

Fig. 7. The resultant image of the null-screen targets after reflection, and their corresponding binarized image. S1: left cornea: a) and b), right cornea: c) and d). S2: left cornea: e) and f), right cornea: g) and h).

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From the images of Fig. 7, we can observe that some points of the array are lost due to the shadows produced by the eyelashes and eyelids. The reflected images were binarized and from them we evaluated the centroids (Figs. 7b, d, f and h), and according to Section 4 we evaluated the topography of the corneas. In Table 3 we show the results of the evaluation of the geometrical parameters of the corneas of both subjects with our conical corneal topographer. Here the calculated values of the differences in sagitta were performed between the evaluated surface with the null-screen testing method and the best fitting asphere whose geometrical parameters are listed in Table 3. Then, for S1, the left cornea, the rms difference in sagitta value is Δzrms = 9.5 μm and the the peak valley difference in sagitta is Δzpv = 17.6 µm, for the right cornea Δzrms = 13.9 μm, and Δzpv = 24.2 µm. For S2, the rms difference in sagitta value between the evaluated points and the best asphere is Δzrms = 4.1 μm, and the peak valley difference in sagitta is Δzpv = 6.8 µm for left cornea, and for right cornea we have Δzrms = 9.0 μm and Δzpv = 11.6 μm

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Table 3. Parameters resulting from least squares fitting of corneal surfaces.

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For S1, in Figs. 8a and b the graphs of the elevation maps (differences in the sagitta between the measured surface and the best fitting asphere obtained by a least-squares fit) calculated with the conical null-screen topographer are shown, here the decentring and tilt have been removed. For comparison, the right images of Figs. 8a and b show the elevation map obtained with the Placido-based corneal topographer (the pupil diameter, that for the left cornea is 6.8 mm and for the right cornea is 7.0 mm). Similarly, for S2, Figs. 8c and d show the elevation maps obtained with the conical null-screen topographer, and the right images show the obtained with the Placido-based corneal topographer (here the pupil diameter of the Placido-based corneal topographer is 7.0 mm for the left cornea, and 6.7 mm for the right cornea). From the graphs we can observe that the elevation maps are quite similar. In Table 4 we show that the geometrical parameters, radius of curvature and conic constant, and observe that they are quite similar in both topographers.

 figure: Fig. 8.

Fig. 8. Elevation maps: a) left cornea S1, b) right cornea S1, c) left cornea S2, d) right cornea S2.

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Table 4. Comparison of the geometrical parameters of corneas.

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For the corneas of both subjects, the left plots of Fig. 9, show axial curvature maps calculated from the normal data as proposed in Section 3 [Eq. (15)]. These maps are very important because they give us information of the corneal power. In the axial curvature maps it is evident that there is a normal astigmatism in both corneas because an hourglass shape appears. For comparison the corresponding curvature maps obtained with the Placido-based corneal topographer are shown in the right plots of Fig. 9. As we can see, the axial curvature maps are very similar.

 figure: Fig. 9.

Fig. 9. Axial curvature maps: a) left cornea S1, b) right cornea S1, c) left cornea S2, d) right cornea S2.

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In the left plots of Fig. 10, we show the resulting tangential maps obtained with the null-screen method by fitting the normal data to Eqs. (16) and (17), as is described in Section 4. These maps are important because they give information on the instantaneous curvature changes. The tangential curvature maps obtained with the Placido-based corneal topographer are shown in right of Fig. 10. In all the plots we can observe that the maps obtained with both topographers are quite similar.

 figure: Fig. 10.

Fig. 10. Tangential curvature maps: a) left cornea S1, b) right cornea S1, c) left cornea S2, d) right cornea S2.

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In this study, we have shown that the conical corneal topographer produces accurate topographic results validated with those obtained with a commercial Placido-based corneal topographer.

6. Conclusions

We have proposed a method for measuring the shape of the corneal surface with a redesigned conical null-screen topographer. We have described the null-screen design procedure for conic surfaces and an algorithm for evaluation of the slopes of the surface. The design was performed considering the morphology of the human face and assuming that the cornea shape is aspheric. Additionally, the use of null-screens with a black background and white centroids allows to evaluate more points in the region of the shadows produced by the eyelashes. The results of the evaluation of the reference spherical surface are very close to the design surface; here we found that the variations are approximately 2.1μm rms value measured with respect to the best-fitting sphere, and the radius of curvature calculated differs approximately 0.06% from the design radius of curvature.

With our conical corneal topographer, we obtain the shape of the corneal surface with an accuracy between 4.1 to 13.9 µm for the corneas of two subjects, these values were obtained considering the rms differences in sagitta between the measured corneal surface and the best-fitting aspheric surface. The results obtained in the topographic maps and the geometrical values of the corneal surface (radius of curvature and conic constant) are quite like those obtained by a Placido-based corneal topographer that is used in clinic.

The null-screen test method is a new alternative technique for determining the quality of the corneal surface with high accuracy. The main advantage of the test is that it is a noncontact test and does not require specially designed optics, making it a cheap and easy technique to implement in the development of the conical null-screen corneal topographer. The proposed technique using a single null-screen allows us to have control over the alignment of the measurement system.

Funding

Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (DGAPA, UNAM) (TA100519); Laboratorio Nacional de Óptica de la Visión, Conacyt (293411).

Acknowledgments

The authors of this paper are indebted to Bleps Vision for allowing topographic measurements with their Placido-based corneal topographer. The authors of this paper are indebted to Neil Bruce (Instituto de Ciencias Aplicadas y Tecnología, Universidad Nacional Autónoma de México, México) for his help in revising the manuscript

References

1. M. Kaschke, K. H. Donnerhacke, and M. S. Rill, Optical Devices in Ophthalmology and Optometry. Technology, Design Principles, and Clinical Applications, (Wiley-VCH, 2014).

2. M. Collins, S. Vincent, and S. Read, “The cornea,” in Handbook of visual optics, Vol. 1, P. Artal, ed. (CRC press, 2017).

3. W. D. Furlan, “Basic ophthalmic instruments,” in Handbook of visual optics, Vol. 1, P. Artal, ed. (CRC press, 2017).

4. K. Karnowski, B. J. Kaluzny, M. Szkulmowski, M. Gora, and M. Wojtkowski, “Corneal topography with high-speed swept source OCT in clinical examination,” Biomed. Opt. Express 2(9), 2709–2720 (2011). [CrossRef]  

5. S. Ortiz, D. Siedlecki, P. Pérez-Merino, N. Chia, A. de Castro, M. Szkulmowski, M. Wojtkowski, and S. Marcos, “Corneal topography from spectral optical coherence tomography (sOCT),” Biomed. Opt. Express 2(12), 3232–3247 (2011). [CrossRef]  

6. I. Grulkowski, “Anterior segment OCT,” in Handbook of visual optics, Vol. 2, P. Artal, ed. (CRC press, 2017).

7. W. Tang, M. J. Collins, L. Carney, and B. Davis, “The Accuracy and Precision Performance of Four Videokeratoscopes in Measuring Test Surfaces,” Optom. Vis. Sci. 77(9), 483–491 (2000). [CrossRef]  

8. R. Díaz-Uribe and M. Campos-García, “Null screen testing of fast convex aspheric surfaces,” Appl. Opt. 39(16), 2670–2677 (2000). [CrossRef]  

9. M. Campos- García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing fast aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47(6), 849–859 (2008). [CrossRef]  

10. M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48(5), 1008–1015 (2009). [CrossRef]  

11. M. Campos-García, A. Estrada-Molina, and R. Díaz-Uribe, “New null-screen design for corneal topography,” Proc. SPIE 8011, 801124 (2011). [CrossRef]  

12. M. Campos-García, C. Cossio-Guerrero, V. I. Moreno-Oliva, and O. Huerta-Carranza, “Surface shape evaluation with a corneal topographer based on a conical null-screen with a novel radial point distribution,” Appl. Opt. 54(17), 5411–5419 (2015). [CrossRef]  

13. V. E. Armengol-Cruz, M. Campos-García, and C. Cossio-Guerrero, “Evaluation of a human corneal surface with the null-screen method,” Proc. SPIE 10330, 103300Q (2017). [CrossRef]  

14. A. I. Osorio-Infante, M. Campos-García, and C. Cossio-Guerrero, “Characterization of a conical null-screen corneal topographer,” Proc. SPIE 10330, 103301E (2017). [CrossRef]  

15. M. Campos-García, V. E. Armengol-Cruz, D. Aguirre Aguirre, and C. Camargo-Fierro, “Obtaining the topography of human corneas with the null-screen testing method,” Proc. SPIE 10692, 1069216 (2018). [CrossRef]  

16. A. Estrada-Molina, M. Campos-García, and R. Díaz-Uribe, “Sagittal and meridional radii of curvature for a surface with symmetry of revolution by using a null-screen testing method,” Appl. Opt. 52(4), 625–634 (2013). [CrossRef]  

17. L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

18. C. Ricolfe-Viala and A. J. Sánchez-Salmerón, “Lens distortion models evaluation,” Appl. Opt. 49(30), 5914–5928 (2010). [CrossRef]  

19. P. Arulmozhivarman, L. Praveen Kumar, and A. R. Ganesan, “Measurement of moments for centroid estimation in Shack–Hartmann wavefront sensor—a wavelet-based approach and comparison with other methods,” Optik 117(2), 82–87 (2006). [CrossRef]  

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Figures (10)
Fig. 1.
Fig. 1. Null-screen testing configuration.

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Fig. 2.
Fig. 2. Null-screen design parameters.

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Fig. 3.
Fig. 3. Normal evaluation.

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Fig. 4.
Fig. 4. a) Flat-printed conical null-screen with drop shaped targets, b) Conical corneal topographer targets.

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Fig. 5.
Fig. 5. a) The resultant image of the null-screen targets after reflection on the reference surface, b) processed image.

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Fig. 6.
Fig. 6. Elevation map for the reference surface.

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Fig. 7.
Fig. 7. The resultant image of the null-screen targets after reflection, and their corresponding binarized image. S1: left cornea: a) and b), right cornea: c) and d). S2: left cornea: e) and f), right cornea: g) and h).

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Fig. 8.
Fig. 8. Elevation maps: a) left cornea S1, b) right cornea S1, c) left cornea S2, d) right cornea S2.

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Fig. 9.
Fig. 9. Axial curvature maps: a) left cornea S1, b) right cornea S1, c) left cornea S2, d) right cornea S2.

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Fig. 10.
Fig. 10. Tangential curvature maps: a) left cornea S1, b) right cornea S1, c) left cornea S2, d) right cornea S2.

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Tables (4)

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Table 1. Conical null-screen design parameters.

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Table 2. Parameters resulting from least squares fitting of sagitta data.

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Table 3. Parameters resulting from least squares fitting of corneal surfaces.

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Table 4. Comparison of the geometrical parameters of corneas.

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Equations (19)

Equations on this page are rendered with MathJax. Learn more.

z = c ρ 2 1 + ( 1 − Q c 2 ρ 2 ) 1 / 1 2 2 ,
ρ 2 = F ρ 1 , z 2 = F a − ( b + e ) ,
F = a { Q ( b + e ) + r } − [ ( a r ) 2 − ( b + e ) ρ 1 2 { Q ( b + e ) + 2 r } ] 1 / 1 2 2 Q a 2 + ρ 1 2 .
ρ 3 = s { α ( z 2 + e + h ) + ρ 2 } α h + s , z 3 = h ( α z 2 + ρ 2 ) − s ( e + h ) α h + s ,
α = ρ 1 ρ 2 2 − ρ 3 ( Q z 2 − r ) 2 + 2 a ρ 2 ( Q z 2 − r ) a ρ 2 2 − a ( Q z 2 − r ) 2 − 2 ρ 1 ρ 2 ( Q z 2 − r ) .
d = a D b + e + β ,
β = r Q [ 1 − ( 1 − Q D 2 4 r 2 ) 1 / 1 2 2 ] .
X = R C o s ( s θ / l ) , Y = − R S i n ( s θ / l ) ,
R = { ρ 3 2 + ( z 3 − h − e ) 2 } 1 / 1 2 2 , l = ( s + h ) 1 / 2 ,
N = R − I | R − I | = ( n x , n y , n z )
R = ( x 1 , y 1 , − a ) ( x 1 2 + y 1 2 + a 2 ) 1 / 1 2 2 .
I = ( x s − x 3 , y s − y 3 , z s − z 3 ) [ ( x s − x 3 ) 2 + ( y s − y 3 ) 2 + ( z s − z 3 ) 2 ] 1 / 1 2 2 .
z − z i = − ∫ P i P f ( n x n z d x + n y n z d y ) ,
z = r − { r 2 − Q [ ( x − x o ) 2 + ( y − y o ) 2 ] } 1 / 1 2 2 Q + A ( x − x o ) + B ( y − y o ) + z o
k a x i = x n x n z + y n y n z ( x 2 + y 2 ) 1 / 2 [ x 2 + y 2 + ( x n x n z + y n y n z ) 2 ] .
k t a n = k a x i + k f i t ( x 2 + y 2 ) [ r f i t 2 + k f i t ( x 2 + y 2 ) ] 3 / 3 2 2 ,
η ≡ ( n x n z ) 2 + ( n y n z ) 2 = x 2 + y 2 { r f i t 2 − ( k f i t + 1 ) ( x 2 + y 2 ) } 1 / 2 ,
x 1 = M x o + E ( x o 2 + y o 2 ) x o , y 1 = M y o + E ( x o 2 + y o 2 ) y o ,
x c = ∑ i = 1 N ∑ j = 1 M x i , j I i , j / ∑ i = 1 N ∑ j = 1 M x i , j I i , j ∑ i = 1 N ∑ j = 1 M I i , j ∑ i = 1 N ∑ j = 1 M I i , j , y c = ∑ i = 1 N ∑ j = 1 M y i , j I i , j / ∑ i = 1 N ∑ j = 1 M y i , j I i , j ∑ i = 1 N ∑ j = 1 M I i , j ∑ i = 1 N ∑ j = 1 M I i , j ,
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