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Surface shape evaluation with a corneal topographer based on a conical null-screen with a novel radial point distribution MANUEL CAMPOS-GARCÍA,1,* CESAR COSSIO-GUERRERO,1 VÍCTOR IVÁN MORENO-OLIVA,2 OLIVER HUERTA-CARRANZA1

AND

1

Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Apdo. Postal 70-186, Coyoacán, México, D. F. 04510, Mexico 2 Universidad del Istmo, Ciudad Universitaria s/n, C.P. 70760, Tehuantepec, Oax., Mexico *Corresponding author: [email protected] Received 3 March 2015; revised 4 May 2015; accepted 12 May 2015; posted 13 May 2015 (Doc. ID 235538); published 5 June 2015

In order to measure the shape of fast convex aspherics, such as the corneal surface of the human eye, we propose the design of a conical null-screen with a radial point distribution (spots similar to ellipses) drawn on it in such a way that its image, which is formed by reflection on the test surface, becomes an exact array of circular spots if the surface is perfect. Any departure from this geometry is indicative of defects on the evaluated surface. We present the target array design and the surface evaluation algorithm. The precision of the test is increased by performing an iterative process to calculate the surface normals, reducing the numerical errors during the integration. We show the applicability of the null-screen based topographer by testing a spherical calibration 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 less than 1 μm. © 2015 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.6650) Surface measurements, figure; (220.4840) Testing; (330.7325) Visual optics, metrology. http://dx.doi.org/10.1364/AO.54.005411

1. INTRODUCTION The cornea, considering the tear film, is the most anterior optical surface of the eye, and is responsible for approximately two thirds of the eye’s refractive power. In recent years, there has been increased interest in quantitative measurements of the shape of the surface of the human cornea. Detailed assessment of the corneal shape parameters can be important for a number of different clinical and research applications, including the diagnosis and monitoring of corneal ectatic disorders (keratoconus and pellucid marginal degeneration, resulting in asymmetric corneal steepening), the monitoring of corneal dystrophies, rigid and soft contact-lens fitting and design, the detection and monitoring of contact lens induced corneal changes, the preoperative and postoperative screening of refractive surgery candidates, and for customized refractive corrections. These procedures demand accurate measurement of the corneal surface in order to correct properly the refractive state of the eye; thus, topographic measurements of the corneal surface are important in planning, performing, and assessing the effects of these procedures. The current methods for determining the shape of the corneal surface are mainly based on Placido’s disk, are inadequate 1559-128X/15/175411-09$15/0$15.00 © 2015 Optical Society of America

in terms of accuracy, and their limitations have recently been recognized [1–6]. In principle, this disk is a plate with concentric bright and dark rings. A virtual image of these rings is formed by the reflection from the tear film on the cornea. Regularly, in the commercially available videokeratoscopes the rings are attached to a concave surface in order to reduce the field curvature of the image. The image is captured by a CCD camera that is placed behind a central bore in the plate. The numerical evaluation of the image proceeds along meridians. For this, the center of the innermost ring or the intersection of the instrument axis with the cornea is taken as the center of these meridians. An algorithm finds the intersections of the rings and the meridian, and a local radius of curvature can be calculated using the distances of the intersections of neighboring rings. In this way, a number of tangential radii are found for each meridian. The curvature is related to the first and the second derivative of the corneal height. Finally, this height can be calculated by a numerical integration along the meridians. With the purpose of flattening the image produced by the target, several geometries have been considered. The Placido disk systems use cones, hemispheres, cylindrical or even ellipsoidal surfaces as targets [5,6].

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The Placido disk-based corneal topographers allow a very accurate determination of the refractive power distribution and the radii of curvature of the patient’s cornea. The accuracy and reproducibility of the refractive power lies in the range of
0.10D and the corresponding radii of curvature have a tolerance of
0.02 mm [6]. The accuracy of the Placido disk-based videokeratoscopes in measuring test surfaces has been studied extensively. In [2], the accuracy and precision performance of four commercially available systems in measuring the surface elevations of a variety of test surface shapes was examined. Three of the four instruments investigated were Placido disk-based videokeratoscopes (the Keratron, Medmont, and TMS), and the fourth system (the PARCTS) uses the raster-stereogrammetry technique; the rms error of the height data was in the range of 2–6 μm. Other studies gives rms errors less than 3 μm in such systems [7]. Other commercial Placido disk-based corneal topographers, such as the Topcon CM1000, have accuracies lower than 0.05 mm in radii of curvature measurements of spherical reference surfaces [8]. Recently, methods based on optical coherence tomography (OCT) have been used to evaluate the corneal topography [9,10]. 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. In previous works [11–14], we proposed the null-screen method to test fast aspheric convex, concave, and off-axis surfaces. The method consists of drawing a set of curved lines or spots (similar to ellipses) on a cylinder or a plane in such a way that by reflection on the test surface, the image consists of a perfect grid or an array of circular spots. Any departure from this geometry is indicative of defects on the surface. Here the whole surface is tested at once. The proposal of using a cylinder (for testing convex surfaces) as the object, comes from the fact that, from paraxial calculations it can be shown that for a convex spherical reflecting surface, the real object having a plane image, is a highly eccentric ellipsoidal surface. However, building ellipsoidal surfaces is an involved task, so a good approach is to build a cylinder [5,15]. Nevertheless, to test the corneal surface a cylindrical screen would be impractical due to the face contours (eyebrows, nose, or eyelids), and because the cornea must be located inside the cylinder. In order to avoid these difficulties, in previous works [16,17], we proposed measuring the corneal topography using a conical null-screen. In [16] we performed a statistical error analysis to determine the characteristics of the conical null-screen that reduces the error in the evaluation of the topography, and in [17] we performed a first trial evaluation of the shape of the surface. However, in order to increase the accuracy of the technique, in this paper we propose a new null-screen design with a novel radial target pattern drawn on it; as we will show later, this new null-screen increases the data sampling and the accuracy of the method. Additionally, we present a custom algorithm that allows an improvement of the evaluation of the normals on the test surface. The aim of the paper is to present the applicability of this null-screen method for quantitative evaluation of the shape of the cornea. For this, in Section 2 we describe the proposed test

method. Then in Section 3, we report the equations used for the design of the conical null-screen for testing conic surfaces. Next, in Section 4 we describe the procedure to evaluate the shape of the surface, and develop the iterative algorithm that allows the evaluation of the normals on the test surface with accuracy. In Section 5, results of the test of a spherical reference surface are shown. Finally, a discussion of the results and the conclusions are given at the end of the paper. 2. NULL-SCREEN METHOD In this work, to measure the shape of the corneal surface of the human eye we propose the use of a conical null-screen placed along the optical axis, as shown in Fig. 1. The conical nullscreen is a uniform array of targets which, through reflection we can see at the CCD sensor as an ordered array of points. After reflection on the surface under test, the pencil of rays that pass through a small aperture at the center of the screen (without taking into account diffraction effects), forms the image of the null-screen, as in a camera obscura. The aperture is large enough to avoid important diffraction effects, so this test is in the geometrical optics regime. A positive lens is used to collect and focus these thin pencils of rays onto the CCD sensor. The effects of the aberrations introduced by the positive lens of the camera have been well explained in [18] in which coma and spherical aberrations are considered; the distortion is taken into account by calibrating the camera lens, with a method described below. Departures from a regular array of points on the CCD or image are indicative of deformations of the surface under test. For the determination of the points on the null-screen belonging to a custom array of spots we performed an exact ray-tracing calculation, similar to that developed for the calculations of the cylindrical screens to test fast surfaces (convex or concave). The expressions are quite similar; they only differ in some signs [11,12]. To collect the light reflected from the surface, in principle, a pinhole can be used before the CCD sensor; then, all the calculations are made with this simple setup. However, in order to increase the image intensity for the experiments, a combination of lens and aperture stop is very useful. An important restriction for the CCD camera lens is that it must be able to produce an image of the whole surface at the CCD. A schematic of the proposed setup is shown in Fig. 1.

y Conical null-screen

P1

x

P

z P2

Diaphragm

P3 Cornea

Fig. 1. Layout of the testing configuration.

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P1 is the point where a ray starting at P3 in the null-screen reaches the image plane, after being reflected by the surface at P2 .

where

3. CONICAL NULL-SCREEN DESIGN

Here we consider that the conical null-screen has radii s, height h, is oriented along the z axis, and its base is located at z  0. Equations (4) and (5) are the coordinates of the points where the rays coming from the custom 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 design numerically the null-screen following the same ideas. The distance d on the CCD sensor is given by

As the system has symmetry of revolution, all the calculations can be performed on a meridional plane (x–z plane, Fig. 2), there are no skew rays for the case when the surface is perfectly aligned along the cone axis and only one ray passes through the pinhole. In order to keep the expressions short, we will restrict ourselves to conic surfaces described by z

cρ2 ; 1  f1 − Qc 2 ρ2 g1∕2

(1)

where z is the sagitta and ρ  x 2  y 2 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). To determine the object points on the conical null-screen that give us a custom array of points on the CCD image plane after the reflection with 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. Here P1 is given in cylindrical coordinates (ρ1 > 0; 0 ≤ ϕ1 ≤ 2π; a, b > 0), see Fig. 2. A ray passing through the small aperture lens stop at P  0; 0; b reaches the corneal surface at the point P2  ρ2 ; ϕ1  π; z 2 , where ρ2 

aQb  r −

far2

− bρ21 Qb Qa2  ρ21



2rg1∕2

ρ1 ;

ρ2 a − b; ρ1

(3)

where a is the distance from the aperture stop to the CCD plane, and b is the distance from the aperture stop to the vertex of the surface. After reflection on the test surface, the ray hits the surface at the point P3  ρ3 ; ϕ1  π; z 3 , given by s ρ3  z 3  h; (4) h hαz 2  ρ2 − s z3  ; αh  s

(5)

ρ1 ρ22 − ρ3 Qz 2 − r2  2aρ2 Qz 2 − r : aρ22 − aQz 2 − r2 − 2ρ1 ρ2 Qz 2 − r

d

aD ; bβ

(6)

(7)

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 (6) is given by 8 < D2 for Q  0; 8r 8 β  r h  QD2 1∕2 i : Q 1 − 1 − 4r 2 for Q ≠ 0: For a full hemisphere Q  1, the marginal ray is tangent to the surface; it defines the boundary of the image. For such a case, D

(2)

and z2 

α

81∕2 ra ; d 2  2a2 1∕2

(9)

must be used. To build a conical null-screen with a set of custom targets that are located accurately on it is not an easy task. For a small screen (i.e., less than 50 mm in diameter), 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 center of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. The perimeter of the base of a cone is called the directrix, and each of the line segments between the directrix and apex is a generatrix of the lateral surface, which is given by l  s  h1∕2

10

and the angle θ between both generatrices is given by θ  2πs∕l :

(11)

Then, we have to transform the cylindrical coordinates of the targets on the conical null-screen [Eqs. (4) and (5)] into X Y Cartesian coordinates of the printed-paper sheet plane; the relationships are given by X  R Cossθ∕l ;

Y  −R Sinsθ∕l ;

(12)

where R  fρ23  z 3 − h2 g1∕2 : Fig. 2. Setup for testing the corneal surface using a conical nullscreen.

(13)

Finally, to support the null-screen we place the sheet of paper inside an acrylic cone of the proper dimensions.

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4. SURFACE SHAPE EVALUATION METHOD The shape of the test surface can be obtained from measurements of the positions of the incident points on the CCD plane through the formula [19]  Z P  ny f nx z − zi  − dx  dy ; (14) nz nz Pi where nx , ny , and nz are the Cartesian components of the normal vector N to the test surface, and z i is the sagitta for one point of the surface that must be known in advance. This expression is exact; evaluating the normals and performing the numerical integration, however, are approximate, so they introduce some errors that must be reduced. The next step is the numerical evaluation of Eq. (14). A simple method used for the discrete evaluation of the integral is the trapezoidal rule for nonequally spaced data [20],  m−1  X nx l nx l 1 x l 1 − x i  zm  −  nz l nz l 1 2 l 1    nyl nyl 1 y l 1 − yl   zi;   (15) nz l nz l 1 2 where m is the number of points along some integration path. To analyze the details of the evaluation, we propose fitting the experimental data to an aspheric surface given by z

Fig. 3. Approximated normal evaluation.

r − fr 2 − Qx − x o 2  y − y o 2 g1∕2  Ax − x o  Q  By − y o   z o ;

R (16)

where x o ; yo ; z o  are the coordinates of the vertex of the surface, x o ; yo  are the decentering terms, z o is the defocus, and A and B are the terms of tilt in x and y, respectively. Note that the first term of Eq. (16) is the same as that of Eq. (1). This new representation is more convenient for numerical evaluation. The proposed fit can be generalized to more general equations (aspherics with deformation coefficients or other freeform expressions). Thus, the misalignment can be computed by fitting the experimental data to Eq. (16). The fit of Eq. (16) was performed by using the Levenberg–Marquart method [21] for nonlinear least-squares fitting that is suitable for this task. The evaluation of the normals N to the test surface can be performed with an approximate algorithm. The proposed algorithm involves three-dimensional ray-tracing. The procedure consists of finding the directions of the rays that join the actual positions P1  x 1 ; y1 ; −a − b of the centroids of the spots on the CCD and the corresponding Cartesian coordinates of the objects of the null-screen P3  x 3 ; y 3 ; z 3 . According to the reflection law, the approximated normals N to the surface can be evaluated as R−I N ; (17) jR − Ij where I and R are the directions of the incident and the reflected rays on the surface, respectively (see Fig. 3). In reference to 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

x 21

x 1 ; y1 ; −a :  y 21  a2 1∕2

(18)

On the other hand, for the incident ray I we only know the point P3 at the null-screen, so we have to approximate a second point to obtain the direction of the incident ray by intersecting the reflected ray with a reference surface. In general, for the reference surface we consider an aspheric surface described by Eq. (16). Then the reflected ray, whose direction is given by Eq. (18), intersects the reference surface [Eq. (16)] at Ps  x s ; y s ; z s  (see Fig. 3), where −B 1 − B 21 − 4A1 C 1 1∕2 ; 2A1 x y y s  − 1 z s  b: x s  − 1 z s  b; a a zs 

(19)

Here, A1  Qη1  12 

x 21  y 21 ; a2

B 1  2η1  1Qbη1  η2 − Qz o − r 

η3 x 1  η4 y 2 ; a2

C 1  Qz 2o  b2 η21  2bη1 η2  η22  − 2Qz o − rbη1  η2  

η23  η24 ; a2

(20)

with Ax 1  By 1 ; a η3  bx 1  ax o ; η1 

η2  Ax o  Byo ; η4  by 1  ay o :

(21)

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

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x s − x 3 ; ys − y 3 ; z s − z 3  : x s − x 3 2  y s − y 3 2  z s − z 3 2 1∕2

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(22)

Finally, substituting Eqs. (18) and (22) into Eq. (17) the approximated normals to the test surface are calculated by N

N x ; N y ; N z  N 2x  N 2y  N 2z 1∕2

;

(23)

with

x1 x3 − xs  ; x 21  y 21  a2 1∕2 x s − x 3 2  y s − y 3 2  z s − z 3 2 1∕2 y3 − ys y1  ; Ny  2 2 2 1∕2 2 x 1  y 1  a  x s − x 3   y s − y 3 2  z s − z 3 2 1∕2 −a z3 − zs Nz  2  : 2 2 1∕2 2 x 1  y 1  a  x s − x 3   y s − y 3 2  z s − z 3 2 1∕2

Nx 

(24) It is important to mention that, as a first trial, in practice we choose the design surface [Eq. (1)] of the null-screen as the reference surface, thus the errors in the determination of the normals are minimal [12]. However, if the reference surface differs from the test surface, we have to perform an iterative procedure by using the fitting surface given by Eq. (16) as a new reference surface and calculating again the approximated normals to this new reference surface [Eq. (23)]. Next, with these new approximated normals we obtain the new shape of the surface through Eq. (15). 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, as we show below. 5. TESTING A SPHERICAL SURFACE As proof of principle, we performed a quantitative test of a convex spherical reference surface. In this case, the test surface was mounted on a stage that allowed transverse x and y movements for easy centering, and on a lab jack to put it at the correct height position. The test surface was a calibration spherical surface for corneal topography with a curvature radius of 7.8 mm and a diameter of 11 mm, placed in a cylinder for easy handling. In Fig. 4, we show the experimental setup used for testing the spherical surface. 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 (DCC1645C) with a sensitive area of 4.61 mm × 3.69 mm (1280 × 1024 pixels), and a 25 mm focal length lens attached. The lens is a Tamron format 2∕3} and an aperture f ∕# from f ∕1.6 to f ∕16. The lens diaphragm was used as an aperture stop. The CCD camera was located in such a position that the entire surface could be observed, so that the whole surface could be evaluated at once [see Eq. (7)]. The rest of the parameters used for designing the conical null-screen are shown in Table 1. For better sampling, the conical null-screen was designed to produce a radial-like array of circular spots on the image plane. 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

Fig. 4. Experimental setup.

on the screen becomes an asymmetrical oval, which we call a drop-shaped target [22]. Figure 5 shows how the screen looks before it is put in the experimental setup (flat screen). 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 12 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 465 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. Although ambient illumination may have been enough to allow us to see the image, to have better contrast in the image, we illuminated the screen from outside with a commercial circular fluorescent lamp, see Fig. 4. The conical null-screen was supported on an x-y-z stage for locating the screen in the correct position (see Fig. 4). 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 centered at the CCD 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 Table 1. Conical Null-Screen Design Parameters Element Surface radii of curvature Surface diameter Camera lens focal length CCD length Vertex—sensor distance Stop aperture—surface vertex distance Cone height Cone radius

Symbol

Size (mm)

r D f d a b h s

7.8 11 25 2.8 30 113.4 105.9 70.6

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Research Article x 1  M x o  Ex 2o  y 2o x o ; y 1  M y o  Ex 2o  y2o y o ;

Fig. 5. Flat-printed conical null-screen with drop-shaped targets for quantitatively testing a spherical surface.

experimental data to Eq. (16). The image of the conical nullscreen (see Fig. 5) after reflection on the spherical surface is shown in Fig. 6(a). The centroids of the image in Fig. 6(a) were calculated with an image-processing program using custom algorithms. This algorithm obtain the region of interest (roi) of the original image [Fig. 6(a)], and then we calculate the local minimum of each spot [see Fig. 6(b)]. Finally, we construct a rectangle around each spot and evaluate the coordinates of the centroids using statistical averaging [23] through PN PM PN PM j1 y i;j I i;j j1 x i;j I i;j i1 i1 x c  PN PM ; y c  PN PM ; (25) j1 I i;j j1 I i;j i1 i1 where x i;j and yi;j are the coordinates position of the pixel i; j in each subaperture, and I i;j is the intensity at the pixel i; j in each subaperture having N × M pixels. All the centroids were corrected for the lens distortion. For the distortion evaluation, we use a square grid pattern located in a plane perpendicular to the optical axis; this is considered as the center of the image array. Assuming that the lens system is axially symmetric, the distorted image is still a pattern of spots in another square grid, scaled and distorted. This procedure is similar to the one employed in [24], where the pattern is displayed on an LCD screen. Thus, the positions of the spots in the distorted pattern can be written as [25]

Fig. 6. (a) Resultant image of the null-screen targets after reflection on the test surface and (b) centroids.

(26)

where x 1 ; y1  are the coordinates of the image of an object point x o ; y o , 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 propose a very simple model; however, choosing the correct model could be a very difficult task as is discussed widely in [26]. Here the calculated coefficient E  −3.3 × 10−6 mm−2 ; since E < 0 the lens presents a barrel distortion, and M  0.19. The coordinates of the positions of all the centroid spots are shown in Fig. 7. The next step is calculating the approximated normals [Eq. (23)] to the test surface, as described in Section 4. With the calculated normals, the shape of the surface is obtained using Eq. (14) with the trapezoidal rule as the integration procedure [Eq. (15)]. In Fig. 7, we show the selected integration paths for all the evaluation points and the initial starting integration point for each trajectory, Pi . The integration paths were obtained by using the Dijkstra algorithm [27,28]. This algorithm was implemented to give the shortest integration path [29]. As is described in Section 4, in order to analyze the details of the evaluation, we fit the experimental data to a spherical surface k  0 given by Eq. (16). In the first evaluation, we take as reference surface the design sphere [Eq. (1) with k  0]. The differences in the sagitta between the measured surface and the best-fitting sphere obtained by a least-squares fit are shown in Fig. 8. In this case, the P-V differences in sagitta between the evaluated points and the best fit is Δz pv  15.3 μm, and the rms difference in sagitta value is Δz rms  4.7 μm. In Table 2, we observe the details of this first evaluation (first row). According to the algorithm described in Section 4, we use this fitted spherical surface as the new reference surface in order to evaluate the approximated normals and obtain the shape of the surface; the procedure is performed iteratively until the algorithm converges, in our particular case, we only need four iteration steps. In Table 2, we show the

Fig. 7. Coordinates of the positions of all the centroid spots. Integration paths in the x–y plane. All paths start at the same point Pi; the paths were obtained with the Dijkstra algorithm.

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Fig. 8. Difference in the sagitta between the measured surface and the best-fitting sphere (first evaluation).

parameters resulting from the least-square fit. Here we note how the P-V and rms difference in sagitta diminish in each iteration. Note how in this first evaluation the tilt is evident (see Fig. 8). In Fig. 9 graphs of the final iteration of the differences in the sagitta between the measured surface and the best-fitting sphere obtained by a least-squares fit are shown; here, the decentering and tilt have been removed. In this case, the P-V difference in sagitta between the evaluated points and the best fit is Δz pv  12.5 μm, and the rms difference in sagitta value is Δz rms  0.7 μm. This final iteration improved the accuracy in the determination of the normals to the test surface and in consequence allows measurement of the shape of the surface with better accuracy. Here, departures from the perfect shape can be clearly observed. At a finer detail, the large shape variations observed in Fig. 9 are very unlikely to be real deformations of the surface. Instead, they seem to be a result of the accumulation of errors during the numerical integration procedure. Additionally, we notice that the radius of curvature differs by 5.5 μm or about 0.07% of the design value of r  7.80 mm. This result is consistent with the value given by the manufacturer of the reference sphere. 6. DISCUSSION In this work, we describe a procedure that improves the measure of the shape of an aspherical convex surface with the nullscreen testing method. In particular, the improvements (the new null-screen design and the iterative evaluation algorithm) can allow corneal topography measurements. It is important to mention that the proposed technique takes into account the fact that the cornea is a highly asymmetric and irregular surface, because according to Eq. (14), that gives the shape of the surface, is an exact expression, and is not limited to any model of

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eye. As pointed out, the conical null-screen is designed in such way that it gives us an ordered array of points if the surface is a perfect conic, any departure from this geometry is indicative of defects on the evaluated surface. In a previous work [16] we performed an error analysis of our proposal. In that proposal, we performed a numerical simulation introducing Gaussian random errors in the coordinates of the centroids of the spots on the image plane, and in the coordinates of the sources (spots on the null-screen) in order to obtain the conical null-screen that reduces the error in the evaluation of the topography. However, the analysis only referred to the statistical random errors due to the determination of the CCD centroids and in the construction of the nullscreen. (In that case we found that in order to have differences in sagitta lower than 3.5 μm, the error in the measurement of the centroid coordinates must be lower than 0.5 pixel. With our algorithms we can attain measurements of the centroids lower than 0.1 pixel, in consequence the accuracy of our proposal can be increased.) For the case of the determination of the CCD centroids, the main source of error is due to processing of the reflected images. Here, we have taken into account that the illumination of the null-screen is performed with fluorescent light producing variations of the imaged centroids. For the case of the null-screen we have to take into account the points of the null-screen drawn by the plotter have inaccuracies. In addition, another source of random errors could be the macroscopic roughness of the paper and the inaccuracy of the pasting procedure of the paper null-screen on the acrylic cone used to give it mechanical strength. In summary, we can say that this proposed procedure to determine the statistical errors involved in the measurements allows us to know the size of the error in the calculation of the normals to the test surface. Of course, accurate positioning is also mandatory for better testing. The systematic errors to be considered are due to translations and rotations of the components of the proposed optical testing [30]; however, we think that this analysis is outside the scope of this work where we proposed the use of a conical null-screen to perform the test. In a future publication, we plan to present a detailed analysis of the systematic errors in order to obtain appropriate information of the errors in the approximated normals. A decision must be made by considering all these facts. However, in order to known how small decentering in the z-axis coordinate affects the results of the measurements we perform an experimental analysis that consists of decentering the test surface each 10 μm along the z axis, and evaluate the surface [31,32]. Results of the evaluation are shown in Fig. 10, where differences in radii of curvature Δr and in rms difference in sagitta Δz rms against defocus z o shows a linear dependence.

Table 2. Parameters Resulting from Least-Squares Fitting of Sagitta Data Iteration 0 1 2 3 4

x o
μm

y o
μm

z o
μm

A × 10−4

B × 10−6

Δr
μm

Δz pv
μm

Δz rms
μm

−0.04 −0.06 −0.06 −0.06 −0.06

0.04 0.03 0.03 0.03 0.03

−0.6 −0.5 −0.3 −0.3 −0.3

−10.4 −2.7 −2.7 −2.7 −2.7

21.1 5.3 4.5 4.5 4.5

11 5.6 5.5 5.5 5.5

15.3 12.7 12.5 12.5 12.5

4.5 2.3 0.8 0.7 0.7

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and of the CCD camera. However, it is possible to increase the density of points of evaluation by considering the point shifting procedure by using a dynamic null-screen [33], this nullscreen could be implemented on an LCD monitor [34] and one can use chromatic [35] and/or monochromatic arrays of points [36]. 7. CONCLUSIONS Fig. 9. Difference in the sagitta between the measured surface and the best-fitting sphere (final iteration).

Fig. 10. (a) Difference in radii of curvature against defocus and (b) rms difference in sagitta against defocus.

From the plots in Fig. 10, we can observe that for small defocus z o the values of the differences in radii of curvature and rms difference in sagitta are inside the values reported in the literature [2,6–8]. The linear dependence of Δr with the defocus z o is evident. These results are in accordance with one obtained in [32], where a method for measuring conic constant and vertex radius of fast conic surfaces from Hartmann patterns are developed. In accordance with [32], from the plots it is possible to obtain the value of the defocus z o by putting Δr equals zero (or Δz rms  0) in the line equation of Fig. 10. In our particular case we obtain a defocus value of z o  31 μm. On the other hand, it is important to point out that the density of the spots of the null-screen depends on several issues: first, a greater density allows detection of smaller details of the surface, i.e., the lateral resolution is increased. Second, the number of evaluation points is also increased, as is the evaluation time. Finally, in [33] it is shown that with a higher density of points the numerical errors introduced by the integration algorithm are reduced. The density, however, cannot be increased indefinitely because of the resolution of the laser printer

We have proposed a method for measuring the shape of the corneal surface. We have described the screen-design procedure for conic surfaces and an algorithm for evaluation of the slopes of the surface. We have shown the feasibility of such a proposal by the testing of a spherical surface of 7.8 mm of radius of curvature. For qualitative analysis, we designed a novel radial-like conical null-screen. For the quantitative evaluation, we found that the measurement performed with the trapezoidal method gives a radius of curvature that differs approximately 0.07% from the design radius of curvature. The result of the least-square fit algorithm shows that the tested surface is very close to the design surface. Here, we found that the variations are approximately 0.7-μm rms value measured with respect to the best-fitting sphere. This variant of 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. The proposed technique using a single null-screen allows us to have control over the alignment of the measurement system. Furthermore, the accuracy of the proposed testing method is comparable with some other works reported in the literature [2,6–8]. Dirección General Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IT101414). The authors of this paper are indebted to Neil Bruce (Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, México) for revising the manuscript. REFERENCES 1. J. E. Greivenkamp, M. D. Mellinger, R. W. Snyder, J. T. Schwiergerling, A. E. Lowman, and J. M. Miller, “Comparison of three videokeratoscopes in measurement of toric test surfaces,” J. Refract. Surg. 12, 229–239 (1996). 2. 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, 483–491 (2000). 3. G. A. Kounis, M. K. Tsilimbaris, G. D. Kymionis, H. S. Ginis, and I. G. Pallikaris, “Estimating variability in placido-based topographic systems,” Optom. Vis. Sci. 84, E962–E968 (2007). 4. Y. Mejía-Barbosa and D. Malacara-Hernandez, “A review of methods for measuring corneal topography,” Optom. Vis. Sci. 78, 240–253 (2001). 5. Y. Mejía-Barbosa and D. Malacara-Hernandez, “Object surface for applying a modified Hartmann test to measure corneal topography,” Appl. Opt. 40, 5778–5786 (2001). 6. M. Kaschke, K. H. Donnerhacke, and M. S. Rill, Optical Devices in Ophthalmology and Optometry. Technology, Design Principles, and Clinical Applications, 1st ed. (Wiley-VCH, 2014), pp. 187–200.

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22. 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). 23. 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, 82–87 (2006). 24. S. Lee, R. Parks, and J. H. Burge, “Self-consistent way to determine relative distortion of axial symmetric lens systems,” Appl. Opt. 51, 588–593 (2012). 25. A. K. Ghatak and K. Tahyagarajan, Contemporary Optics (Plenum, 1978), pp. 31–49. 26. C. Ricolfe-Viala and A. J. Sánchez-Salmerón, “Lens distortion models evaluation,” Appl. Opt. 49, 5914–5928 (2010). 27. E. W. Dijkstra, “A note on two problems in connexion with graphs,” Numer. Math. 1, 269–271 (1959). 28. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, “Section 24.3: Dijkstra’s algorithm,” in Introduction to Algorithms, 2nd ed. (McGraw-Hill Science/Engineering/Math. 2001), pp. 595–601. 29. V. I. Moreno-Oliva, A. Castañeda-Mendoza, M. Campos-García, and R. Díaz-Uribe, “Improving the quantitative testing of fast aspherics surfaces with null screen using Dijkstra algorithm,” Proc. SPIE 8011, 801125 (2011). 30. J. Pfund, N. Lindlein, and J. Schwider, “Misalignment effects of the Shack–Hartmann sensor,” Appl. Opt. 37, 22–27 (1998). 31. A. Estrada-Molina, “Topografo corneal portatil basado en pantallas nulas (Portable corneal topographer based on null screens),” Ph.D. thesis (Universidad Nacional Autonoma de Mexico, 2014). 32. Y. Mejía-Barbosa, R. Díaz-Uribe, A. L. Pacheco, A. Estrada-Molina, and F. Spors, “Measuring conic constant and vertex radius of fast convex conic surfaces from a set of Hartmann patterns,” submitted to Appl. Opt., paper number 233513. 33. V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point-shifting in the optical testing of fast aspheric concave surfaces by a cylindrical null-screen,” Appl. Opt. 47, 644–651 (2008). 34. M. Campos-García, V. I. Moreno-Oliva, R. Díaz-Uribe, F. GranadosAgustín, and A. Santiago-Alvarado, “Improving fast aspheric convex surface test with dynamic null screens using LCDs,” Appl. Opt. 50, 3101–3109 (2011). 35. M. I. Rodríguez-Rodríguez, A. Jaramillo-Nuñez, and R. Díaz-Uribe, “Dynamic point shifting with null-screens using three LCDs as targets for corneal topography,” submitted to Appl. Opt. paper number 236222. 36. J. Beltran-Madrigal and R. Díaz-Uribe, “Progress in the design of chromatic null screens to test cylindrical parabolic concentrators,” Proc. SPIE 8011, 80111R (2011).