Download APPENDIX Corneal Height Map Reconstruction for the Topcon SL

APPENDIX
Corneal Height Map Reconstruction for the Topcon SL-45 Scheimpflug Instrument
The first commercially available Scheimpflug imaging instruments were equipped with analog
films and were not able to correct its images for the distortion of the tilted film plane and
refraction within the eye. The instruments however evolved over time and the analog film was
replaced by high resolution digital CCD cameras, but adequate image correction software was
still lacking. The Topcon SL-45 Scheimpflug instrument used in this study is originating from
that period and was one of the first to have image correction software. This software was
developed during earlier studies on the shape and thickness of the human cornea and crystalline
lens by Dubbelman et. al.Error! Bookmark not defined.,Error! Bookmark not defined.,54 A measurement protocol
was set up during those studies in which the cornea of a subject was aligned according to its
optical axis with the instrument using a fixation light. This has the advantage that the shape of
the cornea is maximally symmetric, so geometrical models can be fitted with sufficient accuracy.
Series of Scheimpflug images were made in six meridians (90º, 60º, 30º, 0º, 150º, and 120º) in
order to make a three dimensional reconstruction of the cornea. However, vision analysis is more
accurate when performed along the visual axis of the eye and therefore in this study the
measurement protocol has been changed.55 The subjects measured in this study were asked to
look straight into the middle of the illumination slit instead of the fixation light. Consequently,
the geometrical models had to be revised since the maximum symmetry of the cornea in the
image was lost during this procedure. Furthermore, the models had to be extended with higher
order terms to also measure trefoil aberration, since the current models were only applicable up
to coma aberration.
In order to restore the symmetry of the corneal shape in the Scheimpflug images the unknown
translation offset and corneal tilt in the image have to be corrected. This can be done by locating
the corneal apex using the definition of local radius:
1  dz / ds 
r 
2 3/ 2
l
d 2 z / ds 2
(6)
where dz/ds and d2z/ds2 are respectively the first and second derivatives of the corneal surface in
the image. The corneal apex is indicated by the extreme value of the local radius as can be seen
in Figure 5. The first and second derivatives can be calculated out of the corneal image points
using slope calculation by gradient functions in which the effect of noise is reduced by
smoothing algorithms e.g. the Savitzky-Golay filter.56 The translation offset of the corneal
surface in the image can then easily be determined out of the (s,z)-coordinates of the corneal
apex and the corneal tilt can be determined by taking the tangent of this point. After correction
for the translation offset and tilt, the symmetry of the corneal shape is restored.
An extended conic function with a third order term is fitted to the symmetric corneal shape in
order to determine shape and aberration parameters:
z
r  r 2  ks 2
 ms3
k
(7)
where shape parameter r describes the radius of curvature, k the asphericity, m the third order
shapes coma and trefoil. For each of the six meridians, Equation (7) was fitted to a 7.5-mm
corneal zone. The 3D corneal profile is reconstructed by applying the following fit functions to
the measured values of the shape parameters from all six meridians:
r    r0  r cos2  2 
(8)
k    k 0
(9)
m    m0  m1 cos      m2 cos 3  3 
(10)
where r0, k0, and m0 are the mean values for radius of curvature, asphericity and third order shape
respectively; Δr, Δm1, Δm2 are the shape amplitudes for radius of curvature, coma and trefoil,
respectively and; α, β, and γ, are the angles of the meridian where radius, coma amplitude and
trefoil amplitude are maximum. In this system the asphericity, k, is taken as a constant since no
significant evidence for a cosine function was found in earlier studies.Error! Bookmark not defined. The
meridian angles are indicated by θ. The three dimensional corneal shape of any meridian θ can
now be calculated using the determined shape parameters and Equations (7-10), so the whole
three dimensional corneal height map can be created. An example of the fitting for radius of
curvature can be seen in Figure 6.Error! Bookmark not defined.
The previous described model can be easily applied to both anterior and posterior corneal
surfaces, but it lacks the measurement of central corneal thickness. In order to determine the
central corneal thickness Equation (7) is extended to a new model for the posterior corneal
surface specifically with an offset along the z-axis:
z
r  r 2  ks 2
 ms3  z0
k
(11)
while using the same values for translation offset and tilt correction as for the anterior corneal
surface. In this way the relative position of the anterior and posterior corneal surfaces with
respect to each other is conserved. In Equation (11) the central corneal thickness is represented
by z0.