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 PDEs 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
Copyright
c
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
1
, x
2
) = S(x
1
, x
2
) ˆx ( ˆx = (x
1
, x
2
, 1)
T
)
Here, S(x) ( x = (x
1
, x
2
) ) is the distance measured
from B of the intersection point P defined by the
light ray starting from B in direction ˆx = P
x
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.,
P
y
PP
x
, is described in the mentioned polar coordi-
nate system. By doing so, a mapping is identified be-
tween the points P
y
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
x
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):
1
S(x)
S(x)
x
j
=
v
j
(x) x
j
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
y
P
x
mapping, that is, with map-
ping P
x
P
y
.
Referring to Fig. 4, f(x) = KP
y
, 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
0
, if the P
y
P
x
mapping is known.
A numerical procedure taking discrete values of
the mapping f(x) as input has been devised, firstly, to
calculate the P
x
P
y
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
0
, while it is much less sensitive to er-
rors present in the P
y
P
x
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
i
denote the image of the reflected
measurement pattern taken by i-th camera, and F
i
the
part of the corneal surface actually reflecting the mea-
surement pattern into the i-th camera. The F
i
and F
j
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
i
and C
j
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.
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