DEVELOPMENT OF A MULTI-CAMERA CORNEAL

TOPOGRAPHER

Using an Embedded Computing Approach

A. Soumelidis, Z. Fazekas

Computer and Automation Research Institute, Budapest, Hungary

F. Schipp

Department of Numerical Analysis, E¨otv¨os Lor´and University, Budapest, Hungary

A. Edelmayer

CONTWARE Research and Development Ltd., Budapest, Hungary

J. N´emeth, B. Cs´ak´any

Department of Ophthalmology, Semmelweis University, Budapest, Hungary

Keywords:

Corneal topography, stereo vision, specular surface reconstruction, partial differential equations, embedded

computers.

Abstract:

A multi-camera corneal topographer is presented in the paper. Using this topographer, the corneal surface un-

der examination is reconstructed from corneal images taken synchronously by a number of calibrated cameras.

The surface reconstruction is achieved by the joint solution of several partial differential equations (PDE’s),

one PDE for each camera. These PDE’s describe the phenomenon of light-reﬂection for different overlapping

regions of the corneal surface. Both algorithmic and implementation issues are covered in the paper.

1 INTRODUCTION

Due to the high refractivepower of the human cornea,

the knowledge of its detailed topography is of great

diagnostic importance. Examination devices, such

as keratometers, corneal topographers, and examina-

tion methods used in ophthalmology for exploring

and measuring these topographies have a relatively

long history (Jongsma et al., 1999). Nowadays, the

corneal topographers are used in a wide range of oph-

thalmic examinations. They are used in the diagnos-

tics of corneal diseases, in contact lens selection and

ﬁtting, in planning sight-correcting refractive surgi-

cal operations, and in their post-operative check-ups

just to mention a few (Corbett et al., 1999). Also,

dynamic properties – e.g., the average build-up time

– of the pre-corneal tear-ﬁlm can be examined and

measured using fast-operation corneal topographers

(N´emeth et al., 2002).

The majority of the measurement methods applied

in the presently used corneal topographers rely on the

specularity of the pre-corneal tear ﬁlm that is coat-

ing the otherwise non-specular corneal surface. In

these topographers, some bright measurement pattern

of known and well-deﬁned geometry, e.g., a concen-

tric system of bright and dark rings (Placido rings),

is generated and displayed in front of the eye. The

reﬂection of this pattern on the pre-corneal tear ﬁlm

is photographed by one or – in recent topographer ar-

rangements – several cameras. The distorted virtual

image, or images taken by the camera are then anal-

ysed, and the corneal surface is mathematically recon-

structed. Based on this reconstruction, maps showing

the topography of corneal surface and its local optical

properties (e.g., refractive power map) are computed

and displayed.

In case of healthy and regular corneal surfaces,

the presently available corneal topographers generally

produce good quality corneal snapshots, and based

on these, precise and reliable optical power maps are

generated.

However, even for healthy and regular surfaces,

126

Soumelidis A., Fazekas Z., Schipp F., Edelmayer A., Németh J. and Csákány B. (2008).

DEVELOPMENT OF A MULTI-CAMERA CORNEAL TOPOGRAPHER - Using an Embedded Computing Approach.

In Proceedings of the First International Conference on Biomedical Electronics and Devices, pages 126-129

DOI: 10.5220/0001046601260129

 SciTePress

a small impurity, or a tiny discontinuity in the pre-

corneal tear ﬁlm can produce a signiﬁcant and exten-

sive measurement error, if too simplistic measurement

patterns, e.g., a Placido ring-system, is used by the to-

pographer device.

1.1 Reconstruction of Specular Surfaces

The mathematical reconstruction of specular surfaces

has been an active area of research. Savarese and his

co-authors, for example, concentrated on the local re-

construction of specular surfaces. For a given pair

of object-point and image point, there are – in gen-

eral – inﬁnite number of specular surface-patches the

could cause a light-ray emitted from the object-point

to reach the image-point. In order to ﬁnd out which

of these patches is the real one, it is necessary to gain

further information. This information could concern

the global shape of the specular surface (e.g, planar,

spherical, or a general second-order surface). Sufﬁ-

cient conditions for the uniqueness of the local recon-

structions are provided in (Savarese et al., 2004).

Others – such as (Bonfort and Sturm, 2003),

(Fleming et al., 2004), (Kickingereder and Donner,

2004) – published methods for global reconstruction

of specular surfaces. Each of these methods relies on

the smoothness of the surface to be reconstructed and

uses several views – i.e., several cameras – to make

the unique reconstruction possible.

The unit normal vector of a given specular

surface-patch is the same no matter which camera of

a multi-camera arrangement looks at it. Although, a

normal vector itself cannot be seen, it can be calcu-

lated from the reﬂection of a light-ray at the given

surface-patch. For an unknown smooth, convex spec-

ular surface – viewed by several cameras – those

points are located on, or near to the surface for which

the corresponding unit normal vectors – calculated

from two or more views – are approximately the

same. This observation is the basis of the voxel-

carving method suggested by (Bonfort and Sturm,

2003). This method can be used only for those

surface-patches that reﬂect the measurement pattern

into more than one camera.

A mathematically more elegant approach was

proposed by (Kickingereder and Donner, 2004) for

global specular surface recognition. In their approach,

the description of light-reﬂection by a smooth specu-

lar surface takes the form of a total differential equa-

tion. The partial differential equation-based method

proposed in Sect. 2 is to some extent similar to their

approach.

Figure 1: Taking corneal reﬂection images with the pro-

posed multi-camera corneal topographer arrangement. A

special colour-coded measurement pattern is used.

2 THE PROPOSED

TOPOGRAPHER

ARRANGEMENT AND

RECONSTRUCTION METHOD

2.1 The Multi-camera Topographer

Arrangement

The proposed corneal topographer arrangement con-

sists of an embedded computer for handling user in-

teractions and multiple camera inputs, generating var-

ious measurement patterns and computation; a TFT

display that is used for displaying the measurement-

pattern; and up to four colour cameras – mounted

rigidly on the display – aimed at the patient’s eye. A

3-camera arrangement is shown in Fig. 1.

It has been pointed out in the Introduction that a

measurement pattern that is more complex and more

informative than the frequently used Placido ring-

system is required for robust corneal measurements,

and particularly for the proper identiﬁcation of cor-

responding object and image locations. To this end,

the use of various colour-coded measurement patterns

were suggested by (Grifﬁn et al., 1992) and (Sicam

et al., 2007).

Figure 2: A part of the reﬂected colour-coded measurement

pattern after colour segmentation and labeling.

In Fig. 1, a novel colour-coded measurement pat-

tern – displayed in front of the patient’s eye – is

shown. It uses four colours, namely, red, green, blue,

DEVELOPMENT OF A MULTI-CAMERA CORNEAL TOPOGRAPHER - Using an Embedded Computing Approach

127

and yellow, and ensures the unique identiﬁcation of a

3-by-3 ﬁeld-neighbourhood, even if it is rotated and

its squares are distorted. In Fig. 2, a part of such a

reﬂected image is shown after colour segmentation,

morphological ﬁltering and connected component la-

belling.

The measurement pattern itself was generated by a

backtracking algorithm. Presently, this colour-coded

measurement pattern is used in conjunction with a

simple black-and-white one that is shown in Fig. 3.

Each of the white circular spots of the latter is

placed in the centre of a red, green, blue, or yellow

square. The black and white measurement pattern

is used for determining the image grid-points with

a sub-pixel accuracy, while the colour-coded one is

used to ensure robust point-to-point correspondence.

2.2 The Mathematical Reconstruction

of the Corneal Surface

Mathematically, the tear-ﬁlm coated corneal surface

is modelled with a smooth, convex surface F. This

surface is described and sought in preferably chosen

spatial polar-coordinate systems. Each of these polar-

coordinate systems corresponds to one of the cameras

of the topographer arrangement.

In Fig. 4, one of the mentioned polar coordinate

systems is shown. Its origin B is placed in the cam-

era’s optical centre and its axis is the optical axis BB

′

of the camera.

The surface F – that is the corneal surface – is

described in the following form:

F(x

, x

) = S(x

, x

) ˆx ( ˆx = (x

, x

, 1)

)

Here, S(x) ( x = (x

, x

) ) is the distance – measured

from B – of the intersection point P deﬁned by the

light ray starting from B in direction ˆx = P

B on one

hand, and the specular surface F on the other.

The propagation of light from the points of the

measurement pattern to the distorted image, i.e.,

, is described in the mentioned polar coordi-

nate system. By doing so, a mapping is identiﬁed be-

tween the points P

of the measurement pattern and

Figure 3: A simple measurement pattern being reﬂected by

an artiﬁcial cornea.

Figure 4: The spatial polar-coordinate system ﬁxed to one

of the cameras of the arrangement.

the points P

of the camera-image. It follows from the

conditions prescribed for the mathematical surface –

that models the specular corneal surface – that this

mapping is one-to-one.

It follows from the physical law of light-reﬂection,

the two-variable function S(x) describing surface F

satisﬁes the following ﬁrst-order partial differential

equation (PDE):

S(x)

∂S(x)

∂x

(x) − x

h ˆx, ˆx− v(x)i

( j = 1, 2),

where

v(x) = | ˆx|

k+ f (x) − S(x) ˆx

|k+ f(x) −S(x) ˆx|

and function f (x) can be expressed with the inverse

of the mentioned P

→ P

mapping, that is, with map-

ping P

→ P

Referring to Fig. 4, f(x) = KP

, where K is the

origin of the coordinate system chosen in the plane

of the measurement pattern, while k = OK denotes a

vector pointing to point K.

In the above PDE h., .i denotes the scalar product

of the 3D space.

It follows from the mathematical model described

above that surface F can be determined uniquely un-

der the starting condition of S(0, 0) = s

, if the P

→

mapping is known.

A numerical procedure taking discrete values of

the mapping f(x) as input has been devised, ﬁrstly, to

calculate the P

→ P

mapping, and secondly, to solve

the mentioned partial differential equation for a given

Figure 5: The virtual image of a simple chess board pattern

as reﬂected by a living cornea.

BIODEVICES 2008 - International Conference on Biomedical Electronics and Devices

128

Figure 6: Normals of a surface region reconstructed from

one camera view.

camera. In Fig. 6, the reconstructed surface and its

normals are shown from a single camera-view.

In simulations carried out for known surfaces,

good approximations of the original surfaces and their

various curvatures were produced via the mentioned

numerical surface reconstruction procedure. From

these simulations it has turned out that the surface re-

construction procedure is clearly sensitive to the start-

ing condition s

, while it is much less sensitive to er-

rors present in the P

→ P

mapping.

In case of a multi-camera arrangement, the so-

lution of the aforementioned PDE must start from a

surface-point reﬂecting the measurement pattern, or

more precisely, certain parts of it, to two, or more

cameras. Let C

denote the image of the reﬂected

measurement pattern taken by i-th camera, and F

the

part of the corneal surface actually reﬂecting the mea-

surement pattern into the i-th camera. The F

and F

surface-regions corresponding to the i-th and the j-

th cameras of the proposed arrangement usually have

overlapping regions. Nevertheless, in few cases, the

patient’s eye-lids and eye-lashes cover normally over-

lapping areas that would be important for the accurate

surface reconstruction. It can be seen from Fig. 5 that

eye-lashes might cause problems as early as the image

segmentation stage of the measurement.

An algorithm has been devised that determines the

distances of an arbitrarily chosen point of the overlap-

ping surface-region from the i-th and the j-th cameras

based on C

and C

images. This point and these dis-

tances will serve as the starting condition for the i-th

and the j-th PDE (corresponding to the i-th and j-th

cameras, respectively). After appropriate ﬁtting, the

union of the surface-regions will provide the recon-

structed surface. Unit normal vectors, and the various

curvatures used by the ophthalmologistscan be calcu-

lated for any surface points.

3 CONCLUSIONS

The majority of the topographers in use, rely on one

view only, which is theoretically insufﬁcient for the

unique reconstruction of the corneal surface. To over-

come this essential measurement deﬁciency, a multi-

camera arrangement is proposed. Several algorithmic

and technical means were used to improve detection

and surface reconstruction precision. Presently, test

measurements are being carried out on artiﬁcial and

living corneas.

ACKNOWLEDGEMENTS

This research has been partially supported by the Na-

tional Ofﬁce for Research and Technology (NORT),

Hungary, under NKFP-2/020/04 research contract.

Certain parts of the work presented here were car-

ried out for the Advanced Vehicles and Vehicle Con-

trol Knowledge Centre. This Centre is supported by

NORT under OMFB-01418/2004 research contract.

Both supports are gratefully acknowledged.

REFERENCES

Bonfort, T. and Sturm, P. (2003). Voxel carving for specular

surfaces. In 9th IEEE Conference on Computer Vision,

Vol. 2. IEEE.

Corbett, M. C., Rosen, E. S., and O’Brart, D. P. S. (1999).

Corneal topography: principles and practice. Bmj

Publ. Group, London, UK, 1st edition.

Fleming, R. W., Torralba, A., and Adelson, E. H. (2004).

Specular reﬂections and the perception of shape. Jour-

nal of Vision, (4):798–820.

Grifﬁn, P. M., Narasimhan, S., and Yee, S. R. (1992). Gen-

eration of uniquely encoded light patterns for range

data acquisition. Pattern Recognition, (6):609–616.

Jongsma, F., de Brabander, J., and Hendrikse, F. (1999).

Review and classiﬁcation of corneal topographers.

Lasers in Medical Science, 14:2–19.

Kickingereder, R. and Donner, K. (2004). Stereo vision

on specular surfaces. In Proceedings of the Fourth

IASTED International Conference on Visualization,

Imaging, and Image Processing. IASTED.

N´emeth, J., Erd´elyi, B., Cs´ak´any, B., G´asp´ar, P., Soume-

lidis, A., Kahlesz, F., and Lang, Z. (2002). High-speed

videotopographic measurement of tear ﬁlm build-up

time. Investigative Ophthalmology and Vision Sci-

ence, 43(6):1783–1790.

Savarese, S., Chen, M., and Perona, P. (2004). What do

reﬂections tell us about the shape of a mirror? In Ap-

plied Perception in Graphics and Visualization, ACM

SIGGRAPH. ACM.

Sicam, V. A. D. P., Snellenburg, J. J., van der Heijde, R.

G. L., and van Stokkum, I. H. M. (2007). New on-

screen surface validation method in corneal topgra-

phy. In ARVO Annual Meeting, page P#3529. ARVO.

DEVELOPMENT OF A MULTI-CAMERA CORNEAL TOPOGRAPHER - Using an Embedded Computing Approach

129