Advances in corneal topography measurements with ...

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Universitaria S/N, C.P. 70760, Tehuantepec, Oaxaca, México. ABSTRACT. In this work we report the design of a null-screen for corneal topography. To avoid the ...
Advances in corneal topography measurements with conical nullscreens

Manuel Campos-Garcíal*a, Cesar Cossio-Guerreroa, Oliver HuertaCarranzaa, Víctor Iván Moreno-Olivab a Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Apdo. Postal 70-186, México, 04510, D.F. México; bUniversidad del Istmo, Ciudad Universitaria S/N, C.P. 70760, Tehuantepec, Oaxaca, México. ABSTRACT In this work we report the design of a null-screen for corneal topography. To avoid the difficulties in the alignment of the test system due to the face contour (eyebrows, nose, or eyelids), we design a conical null-screen with a novel radial points distribution 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. Additionally, an algorithm to compute the sagittal and meridional radii of curvature for the corneal surface is presented. The sagittal radius is obtained from the surface normal, and the meridional radius is calculated from a function fitted to the derivative of the sagittal curvature by using the surfacenormals raw data. Experimental results for the testing a calibration spherical surface are shown. Also, we perform some corneal topography measurements. Keywords: Null-screen, Corneal Topography, Aspherics, Surface Measurements, Optical Metrology

1. INTRODUCTION 1-4

In previous works , we proposed the null-screen method to test fast aspheric convex, concave and off-axis surfaces. The method consists of drawing a set of 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. 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 cylinder5,6. 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 works7-9, we proposed measuring the corneal topography using a conical null-screen. The aim of the work is to present the applicability of this null-screen method for quantitative evaluation of the shape of the cornea. For this, we describe the proposed test method. Then, we report the equations used for the design of the conical null-screen for testing conic surfaces. Next, we describe the procedure to evaluate the topography of the surface. Finally, results of the test of a spherical reference surface and a human cornea are shown.

2. CONICAL NULL-SCREEN DESIGN METHOD 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. 1. 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, z2), given by

ρ2 =

{

}

a ( Qb + r ) − ( ar ) − bρ21 ( Qb + 2r ) 2

Qa + ρ1 2

2

12

ρ1 ,

z2 =

ρ2 a − b, , ρ1

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(1)

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, here Q = k + 1 (k is the conic constant).

L "UP

Figure 1. Conical null-screen configuration.

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

s ( z3 + h ) , h

h ( αz 2 + ρ 2 − s )

z3 =

αh + s

(2)

,

with ρ1ρ 22 − ρ3 ( Qz2 − r ) + 2aρ 2 ( Qz2 − r ) 2

α=

aρ 22 − a ( Qz2 − r ) − 2ρ1ρ 2 ( Qz 2 − r ) 2

.

(3)

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. Eqs. (2) give us the coordinates of the points where the rays coming from the custom array of points on the CCD plane, after been reflected by the test surface, hits the cone surface. 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). Then, we have to transform the cylindrical coordinates of the targets on the conical null-screen [Eqs. (2)] into XY Cartesian coordinates of the printedpaper sheet plane; the relationships are given by X = R Cos ( s θ / l ) ,

Y = − R Sin ( s θ / l ) ,

(4)

, l = ( s + h ) , θ = 2πs l .

(5)

where

{

R = ρ32 + ( z3 − h )

}

2 12

1/ 2

here l is the generatrix of the lateral surface of the cone and θ is the angle between both generatrices. Finally, to support the null-screen we place the sheet of paper inside an acrylic cone of the proper dimensions.

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3. CORNEAL TOPOGRAPHY 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 Pf

ny ⎞ ⎛n z − zi = − ∫ ⎜ x dx + dy ⎟, n nz ⎠ Pi ⎝ z

(6)

where nx, ny, and nz are the Cartesian components of the normal vector N to the test surface, and zi 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 evaluation of the normals N to the test surface can be performed with an approximate algorithm9. The proposed algorithm involves three-dimensional ray-tracing. The procedure consists of finding the directions of the rays that join the actual positions P1 = (x1, 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 = (x3, y3, z3). According to the reflection law, the approximated normals N to the surface can be evaluated as R −I , R −I

N=

(7)

where I and R are the directions of the incident and the reflected rays on the surface, respectively (see Fig. 1). In reference to Fig. 1, 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

R=

(x

( x1 , y1 , −a )

2 1

+ y12 + a 2 )

12

.

(8)

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 at Ps = (xs, ys, zs)9, given by

{

}

2 2 r − r 2 − Q ⎡( x − xo ) + ( y − yo ) ⎤ ⎣ ⎦ z= Q

12

+ A ( x − xo ) + B ( y − yo ) + zo

(9)

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. Then a straight line joining P3 with Ps gives approximately the direction of the incident ray

I=

( xs − x3 , ys − y3 , zs − z3 ) . 12 ⎡( xs − x3 )2 + ( ys − y3 )2 + ( zs − z3 )2 ⎤ ⎣ ⎦

From the normals measurements the sagittal radius of curvature can be obtained from10

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(10)

1/ 2

rsag = − ( x + y 2

)

2 1/ 2

2 ⎧⎪ n y ⎞ ⎫⎪ ⎛ nx 2 2 x + y + x + y ⎨ ⎜ ⎟ ⎬ nz ⎠ ⎪ ⎝ nz ⎪⎩ ⎭

x

ny nx +y , nz nz

(11)

and the meridional radius of curvature is calculated from

rmer = rsag

{r +

2 fit

}

− k fit ( x 2 + y 2 ) k fit ( x 2 + y 2 )

3/ 2

,

(12)

where rfit is the vertex radius of curvature, kfi is the conic constant, and can be obtained by fitting 2

2

⎛ n ⎞ ⎛ ny ⎞ x2 + y2 η≡ ⎜ x ⎟ +⎜ ⎟ = ⎝ nz ⎠ ⎝ nz ⎠ rfit2 − ( k fit + 1) ( x 2 + y 2 )

{

}

1/ 2

.

(13)

Eqs. (11) and (12) shows that from measurements of the normal to the test surface we can calculate the principal radius curvature10.

4. CORNEAL TOPOGRAPHY Testing a spherical surface We evaluate the topography of a 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. 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 x 3.69 mm (1280 x 1024 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. Table 1. Conical null-screen design parameters. Element

Symbol

Size (mm)

Surface radii of curvature

r

7.8

Surface diameter

D

11

Camera lens focal length

f

25

CCD length

d

2.8

Vertex – sensor distance

a

30

Stop aperture – surface vertex distance

b

113.4

Cone height

h

105.9

Cone radius

s

70.6

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

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dot shape on the screen becomes an asymmetrical oval, which we call a drop-shaped target. Figure 2.a shows how the screen looks before it is put in the experimental setup (flat screen).

.

fr :4V 11 Y Ya

c

!

I,:

[

'

Y

r A 1Y . ! . ¡ e ! r4 M1L 4, f . a +! ' , ! v ! w

: t'.

s

6 a

1

°

y ±

'

a

p

°

s

?

w

.

y P°; a

e 4. «

,,

.

.

.

:

. .

.

Y

,

R

....

o

. .

.

_

. .d

.

. . . .

.

1

: .a {iM11 i P

+

!

i

.

10

4 v.Ai wa! .

.

ry1

P

f

F t4

4

'

Y

fywt

: ; ,«4

A

r

.4

!

f w

{

(a)f

i{

(b)

+*? . .i .wJ b) The resultant image of the null-screen targets after reflection on the test surface. F 4. Figure 2. a) Flat-printed null-screen, +conical . 4.1~1` .1..y 1 4. y~ 4 fy f. 4.

wi

f

°

k

.

1

yA.1

t

001

0.135

0 008 0.12

-0.015

0.105

Ñ 0.09

-0022 0.075

-0,029 -2

(a)

0

(b)

Y(mm)

X(mm)

X(mm)

(d)

(c)

Figure 3. Topography of the spherical surface: a) Elevation map, b) Meridional curvature, c) Sagittal curvature, d) Dioptric Power.

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In Fig. 3.a graph 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 Δzpv = 12.5 µm, and the rms difference in sagitta value is Δzrms = 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. Figs. 3.b, and 3.c, shows, respectively, the meridional and sagittal curvature calculated as it is proposed in Section 3. Here, departures from the perfect shape can be clearly observed. 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. Finally, Figure 3.d shows a dioptric power map calculated from D=

(1 − 1.3375) rsag

(10)

Here we found that the accuracy of the refractive power is less than 0.10 D. Design of a conical null-screen for human cornea evaluation As proof of principle, we design a conical null-screen for corneal topography measurements. Accordingly with section 2 and data of table 1 we design the conical null-screen for a human cornea with radius of curvature r = 7.8 mm and conical constant of k = -0.2. In Figure 4, we observe the resultant image of the null-screen targets after reflection on the corneal surface and the centroids obtained.

,. .. ...' . :: ... ' . ,' , ., .

.'

-

,

'

,

.' ,'

,.

.

.

' .

.

.

.

.

.

,

.

.., . 'a



R

Figure 4. The resultant image of the null-screen targets after reflection on the corneal surface, b) centroids.

In Fig. 5.a graph of the differences in the sagitta between the measured surface and the best fitting sphere obtained by a least-squares fit are shown, as before, the decentering and tilt have been removed. Here, the P-V difference in sagitta between the evaluated points and the best fit is Δzpv = 42.6 µm, and the rms difference in sagitta value is Δzrms = 7 μm. Figs. 5.b, and 5.c, shows, respectively, the meridional and sagittal curvature calculated as it is proposed in Section 3. Here, departures from the perfect shape can be clearly observed. Figure 5.d shows a dioptric power map. Additionally,

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we obtain a radius of curvature of 7.9 mm and a conical constant of -0.78; both values differs from the design values (r = 7.8 mm, k = -0.2).

-o.'

-0.2

-0.3

-0.4

0

(a)

(b)

X(mm)

0.132

44.5

0.124

41.5

0.116 8.

0.100 3

0.1

-2

(c)

0

2

4

0

X(mm)

6

X(mm)

(d)

Figure 5. Corneal Topography: a) Elevation map, b) Meridional curvature, c) Sagittal curvature, d) Dioptric Power.

5. CONCLUSIONS 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 and human cornea. For qualitative analysis, we designed a novel radial-like conical null-screen. 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 literature11,12.

ACKNOWLEDGMENTS This research was supported by the Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (DGAPA-UNAM) under project Programa de Apoyo a Proyectos de Investigación e Inovación Tecnológica (PAPIIT) No: IT101414.

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REFERENCES [1] R. Díaz-Uribe and M. Campos-García, "Null screen testing of fast convex aspheric surfaces," Appl. Opt. 39(16), 2670-2677 (2000). [2] 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). [3] M. Avendaño-Alejo and R. Díaz-Uribe, "Testing a fast off-axis parabolic mirror using tilted null-screens," Appl. Opt. 45(12), 2607-2614 (2006). [4] M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, R. Díaz-Uribe, "Quantitative evaluation of an offaxis parabolic mirror by using a tilted null screen," Appl. Opt. 48, 1008-1015 (2009). [5] Y. Mejía-Barbosa, D. Malacara-Hernández, "Object surface for applying a modified Hartmann test to measure corneal topography," Appl. Opt. 40, 5778-5786 (2001). [6] I. E. Funes-Maderey, "Videoqueratometría de campo plano" ("Flat field videokeratometry"), B.A. Thesis Universidad Nacional Autónoma de México, (México 1998). [7] M. Campos-García, A. Estrada-Molina, and R. Díaz-Uribe, “New null-screen design for corneal topography,” Proc. SPIE 8011, 801124 (2011). [8] M. Campos-García, C. Cossio-Guerrero, O. Huerta-Carranza, A. Estrada-Molina, and V. I. Moreno-Oliva, “Characterizing a conical null-screen by using a reference spherical surface,” in Latin America Optics and Photonics Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper LTh3B.7. [9] M. Campos-García, C. Cossio-Guerrero, V.I. Moreno-Oliva, 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, 5411-5419 (2015). [10] A. Estrada-Molina, M. Campos-García, 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, 625-634 (2013). [11] Kaschke, M., Donnerhacke, K. H., and Rill, M. S., [Optical Devices in Ophthalmology and Optometry. Technology, Design Principles, and Clinical Applications], Wiley-VCH, 187–200 (2014). [12] J. H. Massig, E. Lingelbach, and B. Lingelbach, “Videokeratoscope for accurate and detailed measurement of the cornea surface,” Appl. Opt. 44, 2281–2287 (2005).

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