1
Medida de aberraciones
corneales y oculares
R Monts-Mic
Human Visual Performance
Research Group
University of Valencia, Spain
Profesor Titular ptica
Facultad de Fsica. Universidad de Valencia
Grupo Rendimiento Visual Humano
Miembro del Consejo Editorial:
Journal of Cataract & Refractive Surgery
Journal of Refractive Surgery
2
Lneas de Investigacin:
1.- ptica Visual
2.- Calidad ptica y Visual tras Ciruga Refractiva
3.- Presbicia y Acomodacin.
Produccin Cientfica:
Artculos Internacionales: 91
Patentes: 2
4 Proyectos de Investigacin en marcha (IP)
Outline
1.- Fundamentals of elevation topography
2.- Irregular astigmatism: Fourier analysis
3.- Wavefront
4.- Wavefront sensing
5.- Zernike polynomials
6.- How does wavefront sensing relate to
refractive surgery?
7.- Fourier versus Zernike
3
1.- FUNDAMENTALS OF
ELEVATION MAP
TOPOGRAPHY
Two Types
General: Placido Disc
Orbscan/Pentacam
4
ORBSCAN: MAIN FEATURES
Accurate elevation and
curvature information
Anterior and posterior cornea
surfaces
Full cornea thickness
OPTICAL
ADQUISITION HEAD
Scans the eye using light slits that are
projected at a 45-degree angle.
40 slits in total.
Processing and construction of
elevation maps of the anterior &
posterior cornea.
Pachimetry: Diferences in elevation
between the anterior and posterior
surface
5
HOW THE INFORMATION
DIFFERS TO PLACIDO
BASED SYSTEMS?
Reflective and Slit-scan
One image, one surface.
Angle-dependent
specular reflection.
Measures slope (as a
function of distance).
Multiple images,
multiple surfaces.
Omni-directional diffuse
backscatter.
Triangulates elevation.
Placido reflective systems can only measure the anterior
cornea. ORBSCAN measures the anterior cornea,
posterior cornea, and the anterior lens and iris.
6
Hybrid Technology
of ORBSCAN
3. Unify triangulated and reflective data to obtain
accurate surfaces in elevation, slope, and curvature.
1. Measure surface elevation directly by triangulation
of backscattered
slit-beam.
2. Measure surface
slope directly using
specular reflection,
supplemented with
triangulated elevation.
anterior
cornea
anterior
lens
limbus
anterior
iris
posterior
cornea
fixation
reflex
projector
reflex
Scanning slits measure
several surfaces
7
HOW TO READ
CORNEAL ELEVATION MAPS
Corneal Elevation Topography is viewed
relative to a reference surface
Standardization of the reference surface
high
sea-level
low
color
contour
map
terrain profile
Elevation, whether of the earth or the cornea, is measured
relative to some reference surface. The terrestrial
reference surface is the mean sea level.
Relative Elevation
8
Close-Fitting
Reference Surfaces
For the cornea, a reference surface (typically, a sphere) is
constructed by fitting the reference surface as close as
possible to the data surface.
Fit-zone
Reference surface (sphere)
Data surface
(cornea)
Astigmatism: Elevation
(sphere) Map
This is a relative elevation map (measured from a sphere)
Elevation
9
Elevation Distortion
Spherical
reference surface
Post myopic
Lasik profile
As an example of distortion, consider the corneal surface
following myopic lasik correction. It is centrally flattened by
the surgery.
To see surface features, elevation must be measured with
respect to some reference surface.
This relative elevation peak is NOT the highest point on the
cornea. This apparent central "concavity" does NOT exist.
Relative
elevation
profile
Elevation Topology: Elevation Topology:
Central Hill Central Hill
The normal cornea is prolate, meaning that meridional
curvature decreases from center to periphery. Prolateness of the normal cornea causes it to rise centrally
above the reference sphere. The result is a central hill.
Sharp center
Flat periphery
Immediately surrounding the central hill is an annular sea
where the cornea dips below the reference surface. In the far periphery, the prolate cornea again rises above
the reference surface, producing peripheral highlands.
10
Importance of The Post
Surface of The Cornea
Keratoconus will show as localized
posterior elevation with associated
thinning. Patients with thin corneas
without posterior elevation are
unlikely to be keratoconic.
11
2.- IRREGULAR ASTIGMATISM:
FOURIER ANALYSIS
Astigmatismo regular meridianos
principales perpendiculares entre s, y
correccin con lentes esferoclndricas
Cornea con forma irregular que no puede
describirse con una seccin esfrica,
trica o cnica Astismatismo irregular
Causas comunes: ojo seco,
degeneraciones corneales, traumas,
ciruga de la catarata y refractiva.
ASTIGMATISMO IRREGULAR ASTIGMATISMO IRREGULAR
12
Impossibility to evaluate topographies
without pattern
Problem Problem
Es un procedimiento matemtico que
permite la descomposicin de
cualquier objeto peridico en una
suma de trminos sinusoidales de
frecuencias crecientes y amplitudes
determinadas, lo que se conoce como
espectro de Fourier de dicha funcin.
An An lisis de lisis de Fourier Fourier
13
To apply Fourier Analysis to
videoqueratographic data
Solution Solution
Funtion f(x) periodical
Sum of discrete function f(x)
Sinusoidal terms:
Serie Serie of of Fourier Fourier
f x a a nx p b sin nx p
n
n
n
n
( ) cos( / ) ( / ) = + +
=
0
1 1
2 2
a
p
f x dx
p
0
0
1
= ( ) a
p
f x nx p dx
n
p
=
1
2
0
( ) cos( / ) b
p
f x nx p dx
n
p
=
1
2
0
( ) sen( / )
14
Possibility to apply to non-periodical functions
using the Fourier Transform (FT):
To rebuilt the original function f(x) we apply
the inverse transform to the function F(W):
Fourier Fourier Transform Transform
T F f x F w f x i wx dx . .{ ( )} ( ) ( ) exp( ) = = 2
f x T F F w F w i xw dw ( ) . . { ( )} ( ) exp( ) = =
1
2
Fourier Fourier Analysis Analysis
Topographic image is a matrix
of data M(R) containing radii
as a function of the angle (R)
for each ring of radious .
15
Topography
Software
Data Matrix M(R)
FT
Data Matrix MF(f)
Frequency filtering FT-1
Data Matrix rebuilt M(R)
Fourier Fourier Analysis Analysis
Example Example
1 1 ring ring
First components
0,green; 1,red; 2,blue
3 components
Rest of the
components
sphere
tilt
astig
16
Example Example
Calculation for all topographic rings
3 components
Rest of the
components
Example Example
Regular Part
Irregular Part
17
Irregular Part Regular Part Topography
Spherical
cornea
Keratoconus
Keratoconus
Normal
Astigmatism
Normal
Advanced keratoconus
Early keratoconus
Astigmatism
18
We can divide topographic information
between regular and irregular parts
We can quantify the corneal irregularity
by means two parameters, defined from
the regular and irregular parts.
Conclusions Conclusions
3.- Wavefront
We will describe the wavefront. This is the one of the
most fundamental and useful description of the optical
properties of the eye, from which most of the image
quality metrics can be derived.
19
What is the Wavefront?
converging beam
=
spherical wavefront
parallel beam
=
plane wavefront
What is the Wavefront?
ideal wavefront
parallel beam
=
plane wavefront
defocused wavefront
20
What is the Wavefront?
parallel beam
=
plane wavefront
aberrated beam
=
irregular wavefront
ideal wavefront
What is the Wavefront?
aberrated beam
=
irregular wavefront
diverging beam
=
spherical wavefront
ideal wavefront
21
What is the Wave Aberration?
diverging beam
=
spherical wavefront
wave aberration
Wave aberration is a measure of the difference between the ideal wavefront and the actual
wavefront. You are able to choose whatever ideal wavefront you want, but you commonly
choose the ideal wavefront as one that would focus the light to the image plane
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Wavefront Aberration
mm (right-left)
m
m
(
s
u
p
e
r
i
o
r
-
i
n
f
e
r
i
o
r
)
Wave Aberration of a Surface
22
Adapt i ve Opt i c s Fl at t ens t he Wave Aber r at i on
AO ON
AO OFF
4.- Wavefront Sensing
23
Optical Anatomy of the Eye
Cornea Pupil Retina Lens
Wavefront Sensing Clinical
Utility
Measures integrated function of optical
system
Allows accurate calculation of effective
clinical prescription
Also provides details of higher order
aberrations
Quick measurement easily made in
clinical setting
24
Ideal Vision
Parallel Light Rays
Sharp Focus
on Retina
Ideal Vision
Plane Wavefront
25
Simple Near-Sightedness
(myopia)
Parallel Light Rays
Focus in
Front of
Retina
Simple Near-Sightedness
(myopia)
Diverging Light Rays
Sharp Focus
on Retina
26
Simple Near-Sightedness
(myopia)
Spherical Wavefront
The Reversible Nature of Light
Propagation
A B
A B
27
Wavefront Sensing:
Turn the Rays Around!
Probe Light Beam
Re-Emitted Wavefront for an
Ideal Eye
Plane Wavefront
28
Wavefront Displays for Ideal
Vision
3-D Representation
2-D Color Map
Re-Emitted Wavefront for an
Near-Sighted Eye (myopic)
Spherical Wavefront
29
Wavefront Displays for
Near-Sightedness
3-D Representation
2-D Color Map
How do We Make the
Wavefront Measurement?
Wavefront sensors
30
Usually use ray-tracing methods to reconstruct
the wavefront and are classified into the
following 3 types:
- Outgoing wavefront aberrometry
(Hartmann-Shack)
- Ingoing retinal imaging aberrometry
(cross cylinder, Tscherning aberroscope)
- Ingoing feedback aberrometer
(spatially resolved refractometer, optical path difference)
The Wavefront Sensing Path
Optics...
Lenses CCD Eye
31
Direct CCD Image
Enhanced CCD Image
32
Focussed Spot Associations
Comparison to Ideal Pattern
33
What Are We Comparing With Our
System?
Perfect Wavefront
Perfect Wavefront
Micro -Lenslet Array
Video Sensor
Video Sensor
Ideal Wavefront
Ideal Wavefront
Front Side
Slope
Slope
Front Side
Micro -Lenslet Array
Video Sensor
Video Sensor
Aberrated Wavefront
Slope
Slope
Aberrated Wavefront
Aberrated Wavefront
VS
VS
Ideal & Captured Foci
Ideal & Captured Foci
Wavefront Focusing
Wavefront Focusing
W
a
v
e
f
r
o
n
t
f f
CCD CCD Lenslet Lenslet
x x
How Do Shack-Hartmann
Systems Measure
Aberrations?
How Do Shack-Hartmann
Systems Measure
Aberrations?
34
Principle of Outgoing Wavefront Analyzer Hartmann-Shack
Displacement of spotsfromreferencegrid
indicateslocal slopeof aberratedwavefront
35
Wavefront Analyzer Hartmann-Shack
Montajelaboratorioen bancode ptica
Comercial
Examples of spots position in a Hartmann-Shack
36
Wavefront shape
Examplesof higher-order
aberration maps fromeyes
with four different clinical
conditions.
Zernike orders 0-2
omitted for clarity.
Dry eye Keratoconus
Myopic-LASIK Cataract
5.- Zernike polynomials
37
La aproximacin ms familiar para cuantificar las
aberraciones pticas es la de Seidel, definida para
sistemas rotacionalmente simtricos
Cuando describimos las aberraciones oculares, Seidel
no se utiliza ya que la ptica del ojo no es totalmente
simtrica
Los polinomios de Taylor tambien han sido utilizados
para describir las aberraciones del ojo
Recientemente se han utilizado los polinomios de
Zernike debido a sus propiedades matemticas
adecuadas para pupilas circulares
Introduccin
Polinomios de Zernike: consisten en un
conjunto ortogonal de polinmios que presentan
las aberraciones y adems estn relacionados
con las aberraciones pticas clsicas
Parecen el mtodo ms deseable para
estimaciones precisas del error de frente de
onda, debido a sus propiedades de
ortogonalidad (independencia de los trminos
entre s) y pueden ajustarse por el mtodo de
mnimos cuadrados, que es lineal en parmetros
Introduccin
38
Los topgrafos miden la elevacin corneal slo en un
nmero discreto de puntos y los polinmios de Zernike
no son ortogonales sobre un conjunto discreto de
puntos
La tcnica de ortogonalizacin de Gram-Smith permite
expandir el conjunto discreto de datos de elevacin
corneal, en trminos de polinmios de Zernike y
conseguir las ventajas de una expansin ortogonal.
Una vez completada la expansin, las funciones
ortogonales se transforman en trminos de polinomios
de Zernike, resultando un conjunto nico de coeficientes
de Zernike
Introduccin: Topografa
Definicin y notaciones
Los polinomios de Zernike son un conjunto
infinito de funciones polinmicas, ortogonales
en el circulo de radio unidad.
Son muy tiles para representar la forma del
frente de onda en sistemas pticos. Su uso
est muy extendido y son muy comunes
distintas notaciones, normalizaciones y
criterios en la asignacin de signos.
39
Los polinomios de Zernike pueden expresarse en coordenadas
polares, siendo la coordenada radial (intervalo de variacin
[0,1]) y la componente azimutal (intervalo de variacin es [0,2])
Distinguimos tres componentes:
el factor de normalizacin (N),
la dependencia radial
y la dependencia azimutal.
La dependencia radial es polinmica y la azimutal es armnica.
Se identifica al polinomio con dos ndices n y m, donde n
indica la potencia ms alta (orden) en la componente polinmica
radial y m es la frecuencia azimutal en la componente armnica
Representacin de las aberraciones.
La funcin aberracin de onda W (, ) puede
expresarse como combinacin lineal de los
polinomios de Zernike:
W =
j = 1...N
C
j
Z
j
donde C
j
son los Coeficientes de Zernike que se
expresan en micras y miden el valor de las
distintas aberraciones presentes en el sistema.
40
Para describir las aberraciones oculares se toma como
sistema de referencia un triedro a derechas con origen en
la pupila de entrada del ojo, el semieje positivo Y
apuntando hacia arriba, el X apuntando hacia la izquierda
del sujeto y Z apuntando en direccin emergente al ojo.
Al usar coordenadas polares se mide respecto del
semieje positivo X y es la distancia respecto del origen
medida en unidades normalizadas al radio pupilar
X
Y
Z
Sistema de referencia para la descripcin de la
aberracin ocular en funcin de los polinomios de
Zernike. Se muestra vista frontal del ojo. Los semiejes
positivos se toman de la misma manera en ambos ojos.
O
r
d
e
n
Frecuencia
1 2
3
4 5
6 7 8 9
10 11 12 13 14
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
La siguiente figura muestra la forma del frente de onda
representado por cada polinomio de Zernike, la aberracin
total se expresa como combinacin lineal de esos patrones
caractersticos
Visualizacin los 14 primeros polinomios de Zernike en escala de grises (color claro
para adelanto de fase y oscuro para retraso). Cada patrn se identifica con su ndice
j, cada fila corresponde a un orden n y cada columna a una frecuencia m.
41
Las aberraciones de bajo orden vienen representadas por los
polinomios de ordenes n = 0,1 y 2.
Para n = 0 tenemos un nico polinomio de valor constante unidad
y para n =1 encontramos dos polinomios denominados tilts
stos representan traslaciones y rotaciones del sistema de
referencia
Las aberraciones de 2 orden estn descritas por los 3 polinomios
de Zernike correspondientes a n = 2. Estos polinomios
representan el desenfoque (j=4) y astigmatismo (j=3 y 5)
Las aberraciones de alto orden vienen representadas por los
polinomios de Zernike de orden n 3. Son de tercer orden el
astigmatismo triangular (j = 6 y 9) as como el coma vertical y el
coma horizontal (j =7 y 8) mientras que la aberracin esfrica
(j=12) es de cuarto orden.
Listado de
polinomios de
Zernike hasta 6
orden, notacin
estndar de la
OSA
42
Zernike polynomials
Defocus
Astigmatism(0)
Coma
horizontal
Astigmatism(45)
Coma
vertical
Trefoil
Trefoil
Spherical
aberration
Astigmatism(0)
Astigmatism(45)
4-fold
4-fold
43
Zernike polynomials
Wavefront 3-D Map
Wavefront 2-D Map
44
Zernike Polynomials
1
1
Z
1
1
Z
0
2
Z
2 -
2
Z
2
2
Z
3 -
3
Z
1 -
3
Z
1
3
Z
3
3
Z
The Root-Mean-Square
(RMS) Wavefront Error
The Root Mean Square
Error (RMS) is a
measure of the
difference between the
measured and ideal
wavefronts.
45
Visual Effects of Aberrations
Visual Effects of Aberrations
Visual Acuity Chart Image
Used in Vision Simulation
Reference Dot
46
What Are The Visual Effects of
Under Correcting Aberrations?
1
2
Lower order
Lower order
4
3
Higher Order
Higher Order
Wavefront Error and Simulated
Visual Function
Simulated Chart Image
Simulated Chart Image
2nd Order Defocus
2nd Order Defocus
47
2nd Order
Mixed Astigmatism
2nd Order
Mixed Astigmatism
Simulated Chart Image
Simulated Chart Image
Wavefront Error and Simulated
Visual Function
Wavefront Error and Simulated
Visual Function
3rd Order Coma
3rd Order Coma
Simulated Chart Image
Wavefront Error and Simulated
Visual Function
48
Simulated Chart Image
Simulated Chart Image
4th Order Spherical
Aberration
4th Order Spherical
Aberration
Wavefront Error and Simulated
Visual Function
Wavefront Error and Simulated
Visual Function
4th Order
Secondary Astigmatism
4th Order
Secondary Astigmatism
Simulated Chart Image
Simulated Chart Image
Wavefront Error and Simulated
Visual Function
49
Wavefront Error and
Simulated Visual Function
Flat Wavefront Simulated Chart Image
Wavefront Error and
Simulated Visual Function
Defocus Error Simulated Chart Image
50
Wavefront Error and
Simulated Visual Function
Mixed Astigmatism Simulated Chart Image
Wavefront Error and
Simulated Visual Function
Coma Simulated Chart Image
51
Wavefront Error and
Simulated Visual Function
Spherical Aberration Simulated Chart Image
6.- How Does Wavefront Sensing
Relate to Refractive Surgery?
52
0
0.0001
0.0002
0.0003
0.0004
0.0005
c6/c7 c8/c9 c10 c11/c12 c13/c14
R
M
S
A
m
p
l
i
t
u
d
e
(
m
m
)
Pre-Op
1 Week
1 Month
3 Months
Coma
significantly
worse
Spherical
aberration
significantly
worse
N = 40
Higher-Order Aberrations:
Conventional LASIK Myopes
CustomCornea: Wavefront
Guided Laser Surgery
Measured
Wavefront
53
CustomCornea: Wavefront
Guided Laser Surgery
Desired
Wavefront
CustomCornea: Wavefront
Guided Laser Surgery
Desired
Wavefront
54
CustomCornea: Wavefront
Guided Laser Surgery
Conventional
Treatment
CustomCornea: Wavefront
Guided Laser Surgery
Remove a little
extra here.
Back off a bit
here.
55
Wavefront-Guided Myopic Results
REFRACTION
56
ZERNIKE DATA
TREATMENT
57
VIDEO
-2D
0D
-2D
0D
-2 D 180 Simple Myopic Astigmatism
6 mm OZ, 1.0 mm Blend
-2 D 180 Simple Myopic Astigmatism
6 mm OZ, 1.0 mm Blend
Ablation Profile Ablation Profile Laser Shot Pattern Laser Shot Pattern
Treatment Zone 6 x 8mm Treatment Zone 6 x 8mm
58
0D
+2D
+2D, -2 D 180, Simple Hyperopic
Astigmatism, 6.5 mm OZ, 1.25 mm
Blend
Ablation Profile Laser Shot Pattern
Treatment Zone 9mm
Customized LASIK Example:
Pre-Op Aberrations
Total Aberration Higher Order
RMS 1.51m
UCVA 20/200
RMS 0.31m
BCVA 20/20
4.5m
-2.5m
1.5m
-1.0m
Diameter = 5mm
59
Total Aberration Higher Order
4.5m
-2.5m
1.5m
-1.0m
Diameter = 5mm
RMS 0.14m
UCVA 20/16
RMS 0.09m
BCVA 20/12.5
Customized LASIK Example:
Post-Op Aberrations
STANDARD
Preop: -7 D BCVA 20/15
CUSTOM CORNEA
Preop: -7 D BCVA 20/15
POSTOP Rx -0.25 D BCVA 20/20 POSTOP Rx +0.25 D BCVA 20/15
60
STANDARD
CUSTOM CORNEA
NORMAL
Summary
Wavefront sensing is a powerful tool for
understanding the optical functioning of the
eye.
With the right technology, measurement of
the wavefront can readily be accomplished in
the clinical setting.
Wavefront data has powerful clinical utility,
both in diagnosing visual complaints and in
customizing refractive procedures.
61
7. 7.- - J B Joseph J B Joseph Fourier Fourier
versus versus
Frits Frits Zernike Zernike
Mediante un nmero determinado de ondas
sinusoidales podemos describir una onda cuadrada
62
Crnea
W (r,) Frente de onda
W =ai zi Zernike
W =ai senZi Fourier
Cristalino
Mayor reproduccin con menores ondas
en Fourier que rdenes en Zernike
para una misma superficie de referencia
63
V= i + j Fourier
V= ui Zernike
V
i
j
u1
u2
u3
u4
u5
u6
Reconstrucci Reconstrucci n de la Pupila n de la Pupila
1. 1.- - Zernike necesita de una simetr Zernike necesita de una simetr a de revoluci a de revoluci n n
Pupila Circular Pupila Circular
2. 2.- - Fourier Fourier no necesita de una simetr no necesita de una simetr a de revoluci a de revoluci n n
Pupila no Circular (El Pupila no Circular (El ptica) ptica)
F
Z
Pupila
64
7 mm pupil
Bigger blur
circle
Smaller blur
circle
2 mm pupil
65
Posibles Ventajas Posibles Ventajas Fourier Fourier
1. 1.- - Menos c Menos c lculos de computaci lculos de computaci n n
2. 2.- - Mayor resoluci Mayor resoluci n con menos n con menos rdenes rdenes
(o menor informaci (o menor informaci n) n)
3. 3.- - Aplicable a pupilas m Aplicable a pupilas m s reales s reales
4. 4.- - Reconstrucci Reconstrucci n m n m s real del frente de onda s real del frente de onda
Thank you
robert.montes@uv.es
Human Visual Performance
Research Group
University of Valencia, Spain
