1

Evaluation for Refractive Surgery

• Want to screen patients prior to performing a surgery that permanently alters the shape of their corneas to ensure they are suitable for the procedure.

• Pupil size is important, especially under dim conditions. Want to ensure the maximum pupil size is not much larger than the optical zone of the treatment.

• Corneal topography is important to ensure cornea irregularities do not exist that might suggest potential disease.

Infrared Pupillometry

2

Locate Purkinje Images & Center

Polar Coordinates

3

Mask Purkinjes

Edge Filter

111111111100000-1-1-1-1-1-1-1-1-1-1

Convolution

4

Find peaks near pupil

Fit points to sinusoid

• Pupil margin is a shifted sinusoidal.• Fit data to truncated Fourier series.• Fit coefficients give center and dimensions of elliptical

pupil.• Iterate (3x) and eliminate outliers.

5

Pupil Margin

Pupil Margin

6

Eyelashes

Hair & Blur

7

Contact Lens

Issues

• Saturation and/or small LCD screen make focusing difficult

• Refresh rate too slow. Currently 10 fps – Need 30 fps• Need “Click” to provide feedback to user that picture was

taken• JPEG compression too high. Defeats Mpix advantage.• Orient LEDs in vertical rectangle to leave sides open.• Use entire imaging chip.

8

Image Compression

KeratometryThe keratometer is a device for measuring the radius of curvature of the anterior corneaalong the flat and steep meridians.

Its primary function today is todetermine corneal curvature for IOL implantation (SRK formula).

9

Keratometry

hy

d

Radius R

hdy2R =

d and h are fixed incommercial systems and Ris obtained from a calibratedmeasurement of the heighty.

Placido DiskThe Placido disk extends the keratometryconcept to examine curvature at differentpoints on the cornea.

10

Corneal Topography

Variations on Placido Concept

11

Variations on Placido Concept

Fringe Projection

12

Stereophotogrammetry

d

Θ1 Θ2

Knowledge of two anglesand side tells where the tipof the triangle lies.

Requires a diffuse reflection!

Stereophotogrammetry

Fluorescein dye is instilledinto the eye. When illuminatedin blue light, the fluoresceinfluoresces in the green.

13

StereophotogrammetryThis system is being linked to excimer lasers in refractive surgery. The ultraviolet laserlight causes the cornea cellsto fluoresce in the blue.

Eliminates the need for dropsand can be done duringsurgery.

Scanning Slit

d

Θ1 Θ2

Knowledge of two anglesand side tells where the tipof the triangle lies.

14

Scanning Slit

Technique uses white lightscatter from cornea and crystalline lens.

By refracting through eachsurface, this technique canmeasure anterior and posterior corneal shape, aswell as anterior crystallinelens shape.

Corneal Shape• Axial Power – (sometimes incorrectly called sagittal

power) gives a map of corneal curvature in terms of dioptric power. Is related to the equivalent sphere with the same slope at a given point.

• Instantaneous Power – (sometimes called tangential power) gives a map of corneal curvature in terms of dioptric power. Is related to the curvature (2nd derivative) of a point on the cornea.

• Sag or Elevation – gives a map of the surface height at a given point. The height can be relative to a reference surface.

15

Total Corneal Power

D 42.356D 6.108-D 48.346

Diopters 5.6

3771.13374.18.7

3771.01000

=Φ=Φ

⎥⎦⎤

⎢⎣⎡ −

+=Φ

Find the equivalent power surface based solely on the anteriorradius of curvature.

3304.1'n1000

356.428.71'n

=

=−

Keratometric Index of Refraction

• Historically, a keratometric index of refraction nk was chosen based on available data on the cornea and for mathematical simplicity.

• The index nk = 1.3375, which gives a power of 45 D for a surface with radius 7.5 mm was chosen.

• The purpose of the index is to account for the power of the posterior cornea based solely on the radius of the front surface of the cornea.

• Still used today in keratometry and corneal topography.

16

Axial Power

Θ

Θ

r

Ra

z

Cornea

Examine each meridian from the axis outward.

( ) a2

22

2

2

a

Rr

dr/dz1

dr/dzsin

drdz1sin

drdz

sin1sin

drdz

drdz tanand

Rrsin

=+

=Θ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+Θ=⎟

⎠⎞

⎜⎝⎛

Θ−

Θ=

=Θ=Θ

( ) ( )( )2

k

a

ka

dr/dz1r

dr/dz1nR

1n

+

−=

−=Φ

Axial Power - Example

constantR

1nrR

rdrdz

rRRz

ka

22

22

=−

=Φ

−=

−−=Sphere:

17

Instantaneous Power

r

RI

Cornea

Examine each meridian from the axis outward.

( )[ ]( )

( )[ ] 2/32

22k

I

kI

2/32

22

I

dr/dz1

dr/zd1nR

1ndr/dz1

dr/zdR1

+

−=

−=Φ

+=z

From Calculus:For a given curve in the r-z plane, thereexists a circle of radius RI which istangent to the curve at (r,z) and has the same curvature 1 / RI as the curve.

Instantaneous Power - Example

[ ]constant

R1n

rRR

drzd

rRr

drdz

rRRz

ki

2/322

2

2

2

22

22

=−

=Φ

−=

−=

−−=Sphere:

18

Axial & Instantaneous Power

( )

( ) ( ) ( )( )

( ) ( ) 'dr'rr1r

drrd

f1df/dr ff

f11

drdf1n

drrd

f1f1nr

drdzf Define

r

0Ia

Ia

2/322ka

2k

a

∫Φ=Φ

Φ=Φ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

++

+−=

Φ

+

−=Φ

=

Elevationr

Cornea

z

rCornea

z

Elevation maps tend to obscuresubtle features on the cornea, unlessa reference surface is subtracted.

19

Astigmatism

RK Incisions

20

Lasik

Central Island

21

KeratoconusLocalized thin spot in the cornea that progressively bulges.

Map Comparison

Axial Instantaneous Elevation Elevation - Ref

www.euclidsys.com

Derivation of Axial and Instantaneous Curvature Jim Schwiegerling

March 5, 2004

Corneal topographic data is typically specified in terms of a dioptric power. Dioptric power is not a true power per se, but instead a description of the surface shape. Two different types of topographic maps are common. The first is typically referred to as the axial power map, or more rarely, the sagittal power map. The second type is called the tangential or instantaneous power map. In this paper, the local curvature of a rotationally symmetric surface is derived and its relationship to axial and instantaneous power is shown. Fundamental Forms of Geometry Differential geometry provides the tools to determine the local curvature on a surface. If a continuous surface is defined as z = f(x,y), then the local curvature can be obtained from the First and Second fundamental forms of geometry. At each point on this surface, there exist two principal curvatures κ1 and κ2. These curvatures represent the maximum and minimum curvature through this point and they are always in orthogonal directions. The surface curvature in other directions is at some value between κ1 and κ2. The First Fundamental quantities are given by

E = 1 + (dz/dx)2 F = (dz/dx)(dzdy) G = 1 + (dz/dy)2

The Second Fundamental quantities are given by

[ ] 2/12

22

FEGdx/zdL

−= [ ] 2/12

2

FEGdxdy/zdM

−= [ ] 2/12

22

FEGdy/zdN

−=

From these, the mean curvature is given by

( ) ( )212 21

FEG2FM2GLENH κ+κ=

−++

= ,

and the Gaussian curvature is given by

212

2

FEGMLNK κκ=

−−

=

The principal curvatures can also be extracted from these quantities as

KHH

KHH2

2

21

−−=κ

−+=κ.

From these relationships, the axial and instantaneous curvature of a surface will be derived.

First Derivatives The display of corneal topography dioptric power maps has an underlying assumption that the surface is rotationally symmetric. In this special case, the axial curvature is given by κ2 and the instantaneous curvature is given by κ1. To calculate these curvatures for a rotationally symmetric surface, it is convenient to describe the surface in polar coordinates. Since all the derivatives in the preceding expression are given in Cartesian coordinates, a transformation is in order. Converting the x derivative to polar form is achieve as follows:

θ⎟⎠⎞

⎜⎝⎛ θ

+⎟⎠⎞

⎜⎝⎛=

dd

dxd

drd

dxdr

dxd ,

where the standard definition of polar coordinates hold (i.e. 22 yxr += and . Calculating the derivatives gives

)x/y(tan 1−=θ

θ=+

= cosyx

x221

dxdr

22 and

rsin

xy

x/y11

dxd

222

θ−=⎟

⎠⎞

⎜⎝⎛ −

+=

θ

since 21

x11)x(tan

dxd

+=− . Thus, the final result is

θθ

−θ=dd

rsin

drdcos

dxd

.

Similarly,

θθ

+θ=dd

rcos

drdsin

dyd

.

Second Derivatives The second derivatives in polar form are given by the product of the first derivates in polar form. Care must be taken when distributing the differentials as they can act on the components they multiply. For the second order derivatives,

⎟⎠⎞

⎜⎝⎛

θθ

−θ⎟⎠⎞

⎜⎝⎛

θθ

−θ=dd

rsin

drdcos

dd

rsin

drdcos

dxd

2

2

, which gives

2

2

2

2

2

2

2

22

2

2

dd

rsin

dd

rcossin2

drd

rsin

drdcos

dxd

θθ

+θ

θθ+

θ+θ= .

Similarly,

2

2

22

22

2

22

dd

rcossin

dd

rsincos

drd

rcossin

drdcossin

dxdyd

θθθ

−θ

θ−θ−

θθ−θθ=

and

2

2

2

2

2

2

2

22

2

2

dd

rcos

dd

rcossin2

drd

rcos

drdsin

dyd

θθ

+θ

θθ−

θ+θ= .

Calculating the First and Second Fundamental Forms For a rotationally symmetric object, the θ derivatives in the preceding expression are all zero. Taking this into account and plugging the derivatives into the expression for the First and Second Fundamental forms gives

2

2

drdzcos1E ⎟

⎠⎞

⎜⎝⎛θ+=

2

drdzcossinF ⎟

⎠⎞

⎜⎝⎛θθ=

22

drdzsin1G ⎟

⎠⎞

⎜⎝⎛θ+=

2/122

2

22

drdz1

drdz

rsin

drzdcosL

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎥

⎦

⎤⎢⎣

⎡ θ+θ=

2/12

2

2

drdz1

drdz

rcossin

drzdcossinM

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎥

⎦

⎤⎢⎣

⎡ θθ−θθ=

2/122

2

22

drdz1

drdz

rcos

drzdsinN

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎥

⎦

⎤⎢⎣

⎡ θ+θ=

Calculating the Gaussian and Mean Curvature The Gaussian and mean curvatures can now be determined for the rotationally symmetric surface.

22

2

2

drdz1r

drdz

drzdK

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎥

⎦

⎤⎢⎣

⎡= (Gaussian Curvature)

2/322

2

2

drdz12

drdz1

drdz

r1

drzdH

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛++= (Mean Curvature)

Calculating the Principal Curvatures Finally, the Principal curvatures can be determined from the Gaussian and mean curvatures such that

2/32

2

2

1 drdz1

drzd

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=κ (Instantaneous Curvature)

2/12

2 drdz1r

drdz

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=κ (Axial Curvature)

The curvatures κ1 and κ2 are typically converted to a dioptric power by multiplying the respective curvatures by (nk - 1), where nk is the keratometric index of refraction. Principal Directions The orientation of the Principal Curvatures are described by a unit vector xix + yiy, where i = 1 for κ1 and i = 2 for κ2. The elements of this vector are given by1

( ) ( )2i

2i

ii

ELFM

FMxκ−+κ−

κ−= and

( ) ( )2i

2i

ii

ELFM

ELyκ−+κ−

κ−−=

For i = 1, and , where θθ=κ− cossinAFM 1 θ−=κ− 2

1 sinAEL

2/32

2

2

2

drdz1

drdz1

drdz

r1

drzd

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+−

=

Plugging these expressions in, the orientation of the Instantaneous Curvature is given by cosθ x + sinθ y, or in other words, the Instantaneous Curvature is in the radial direction. Similarly, for i = 2, θθ=κ− cossinBFM 2 and , where θ=κ− 2

2 cosBEL

2/12

2

2

2

drdz1

drdz1

drdz

r1

drzd

B

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+−

=

Plugging these expressions in, the orientation of the Axial Curvature is given by sinθ x + cosθ y, or in other words, the Axial Curvature is in the azimuthal direction.

References 1. Cohen E, Riesenfeld RF, Elber G. Geometric Modeling with Splines. AK Peters: Natick, MA: 2001. pg. 367.