the gaussian curvature map · the gaussian curvature renzo mattioli 13/09/2007 refractive on line...

$: The Gaussian Curvature Map · The Gaussian Curvature Renzo Mattioli 13/09/2007 Refractive On Line 2007 1 The Gaussian Curvature Map Renzo Mattioli (Optikon 2000, Roma*) *Author is$
The Gaussian CurvatureRenzo Mattioli

13/09/2007

Refractive On Line 2007 1

The The GaussianGaussian Curvature Curvature MapMap

Renzo Renzo MattioliMattioli(Optikon 2000, (Optikon 2000, Roma*Roma*))

*Author*Author isis a full time a full time employeeemployee of Optikon 2000 of Optikon 2000 SpaSpa ((IntInt.: C3).: C3)

HistoricalHistorical background:background:

Theorema Egregium by Carl FriderichGauss


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HistoricalHistorical background:background:C.F. Gauss (1777-1855) waswas professor professor of of mathematicsmathematics in in GGööttingenttingen (D). (D). HasHasdevelopeddeveloped theoriestheories adoptedadopted inin–– AnalysisAnalysis–– NumberNumber theorytheory–– DifferentialDifferential GeometryGeometry–– StatisticStatistic ((GuaussianGuaussian distributiondistribution))–– PhysicsPhysics ((magnetismmagnetism, , electrostaticselectrostatics, , opticsoptics, ,

AstronomyAstronomy))

""AlmostAlmost everythingeverything, , whichwhich the the mathematicsmathematicsof of ourour centurycentury hashas broughtbrought forthforth in the way in the way of of originaloriginal scientificscientific ideasideas, , attachesattaches toto the the namename of Gauss.of Gauss.““

KroneckerKronecker, , L.L.

GaussianGaussian Curvature of a Curvature of a surfacesurface((DefinitionDefinition))

GaussianGaussian curvature curvature isis the the productproduct of of steepeststeepest and and flattestflattest locallocal curvaturescurvatures, at , at eacheach pointpoint..

Negative GC Zero GC Positive GC


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GaussGauss’’ TheoremaTheorema EgregiumEgregium (1828)(1828)

GaussianGaussian curvature (C1 x C2) of a curvature (C1 x C2) of a flexibleflexible surfacesurfaceisis invariantinvariant. (. (nonnon--elasticelastic isometricisometric distortiondistortion) )

C1=0

C2=0

C1=0

C2>0

C1 x C2 = 0C1 x C2 = 0 C1 x C1 x C2C2 = 0= 0

GaussianGaussian curvature curvature isis anan intrinsicintrinsic propertyproperty of of surfacessurfaces

C1C2 C1

C2

Curvature Gaussian Curvature Gaussian

C2

C1 x C2 C1 x C2C1

C2

C1

GaussGauss’’ TheoremaTheorema EgregiumEgregium


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• “A consequence of the Theorema Egregiumis that the Earth cannot be displayed on a map without distortion. The Mercatorprojection… preserves angles but fails topreserve area.” (Wikipedia)

GaussGauss’’ TheoremaTheorema EgregiumEgregium

The The GaussianGaussian MapMap in in CornealCorneal TopographyTopography


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ProposedProposed byby B.BarskyB.Barsky etet al.al. in 1997in 1997

““GaussianGaussian power power withwith cylindercylinder vectorvector fieldfield representationrepresentationforfor cornealcorneal topographytopography mapsmaps””. . Optom.Vis.Sci.Optom.Vis.Sci. 7474--19971997

OtherOther proposalsproposalsT.TurnerT.Turner ((OrbscanOrbscan, 1995) , 1995) –– MeanMean curvature curvature mapsmapsM.TangM.Tang etet alt (AJO alt (AJO DecDec. 2005) . 2005) –– MeanMean curvature curvature mapsmapsR.MattioliR.Mattioli ((KeratronKeratron, , RefractiveRefractive--onon--lineline 2007) 2007) -- GaussianGaussian mapsmaps

MeanMean isis locallocal curvature curvature arithmeticarithmetic averageaverage (D). (D). GaussianGaussian isis curvature curvature geometricgeometric averageaverage (D):(D):

_______ _______ Mean=Mean= (C1+C2) / 2 (C1+C2) / 2 Gaussian=Gaussian=�� C1 x C2C1 x C2

ItIt can can bebe shownshown thatthat::

((GaussianGaussian))²² = = ((MeanMean))²² + |Astigmatism|+ |Astigmatism|²²


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GaussianGaussian mapsmaps: : CancelCancel astigmatismsastigmatisms……butbut preservespreserves ectasiaectasia

Local Curvature Maps:

Gaussian Maps:

WhyWhy ““GaussianGaussian”” differdiffer fromfrom ““CurvatureCurvature””mapsmaps??

Instantaneous Curvatures are measured along sections from the VK axis.

Gaussian are the product of maincurvatures at each point and do notdepend from VK axis


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Curvature Curvature mapsmaps withwith ““MoveMove--AxisAxis””

AdvantagesAdvantages of of GaussianGaussian MapsMaps

Are Are indipendentindipendent fromfrom the VK the VK axisaxisLocate Locate cornealcorneal apexapexPossiblePossible integrationintegration withwith indicesindices (CLMI)(CLMI)


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Are Are indipendentindipendent fromfrom the VK the VK axisaxisLocate Locate cornealcorneal apexapex ((objectivelyobjectively))PossiblePossible integrationintegration withwith indicesindices (CLMI)(CLMI)

ConeCone location, location, withwith 3 3 differentdifferent mapsmaps

Gaussian map (Inst.) Curvature map

Sherical Offset (Best-Fit sphere)

CLMI calculated on Gaussian


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……samesame mapsmaps, , VKVK--axisaxis--movedmoved toto the the GaussianGaussian CLMI CLMI apexapex

Gaussian map Curvature (+ Move Axis )

Sherical Offset (“Tangent to point ”)


Are Are indipendentindipendent fromfrom the VK the VK axisaxisLocate Locate cornealcorneal apexapex objectivelyobjectivelyPossiblePossible integrationintegration withwith indicesindices (CLMI)(CLMI)


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FollowFollow--upup of a of a keratoconuskeratoconusafter 5 after 5 yearsyears

Mc=50.1 2M

CLMI CLMI on on LocalLocal CURVATURECURVATURERC (29y) OSRC (29y) OS

Mc=50.5

Mc=52.5

4y

Mc=53.55M


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Mc=50.1 2M

CLMI CLMI onon GAUSSIANGAUSSIANRC (29y) OSRC (29y) OS

Mc=50.5

Mc=52.5

4y

Mc=53.55M

Mc=49.1 Mc=49.7

Mc=52.3 Mc=53.2

Avg=48.5 2M

Curvature+Curvature+MoveMove--AxisAxis �� SimSim--KK--AvgAvgRC (29y) OSRC (29y) OS

Avg=48.7

Avg=45.4

4y

Avg=46.15M


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FollowFollow up of RC (29y) OD/OSup of RC (29y) OD/OS

45

50

55

60

65

19/04/2001 01/09/2002 14/01/2004 28/05/2005 10/10/2006 22/02/2008

Mc-OS

Avg-OS

Mg-OS

OtherOther clinicalclinical applicationsapplications


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IrregularIrregular astigmatismastigmatism or or keratoconuskeratoconus??Local Curvature Map:

IrregularIrregular astigmatismastigmatism::Local Curvature Maps:


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IrregularIrregular astigmatismastigmatism::Local Curvature Maps:

Gaussian Maps:

IrregularIrregular astigmatismastigmatism::

Gaussian Maps:

Local Curvature Maps:


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SubclinicalSubclinical keratoconuskeratoconusLocal Curvature Maps:

SubclinicalSubclinical keratoconuskeratoconusLocal Curvature Maps:

Gaussian Maps:


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PellucidaPellucidaLocal Curvature Maps:

Gaussian Maps:

PKP PKP –– suture suture removalremovalLocal Curvature Maps:PRE POST


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PKP PKP –– suture suture removalremovalLocal Curvature Maps:PRE POST

Gaussian Maps:PRE POST

ConclusionsConclusions, , GaussianGaussian CurvaturesCurvatures::

RepresentRepresent anan intrinsicintrinsic propertyproperty of of surfacessurfacesAre Are ““AxisAxis independentindependent””PotentiallyPotentially usefuluseful in in clinicalclinical evaluationsevaluations, and , and alongalong withwithCLMI CLMI indicesindicesCAUTIONCAUTION: Do NOT : Do NOT representrepresent shapeshape univoquelyunivoquely ((hidehideastigmatismastigmatism, show , show ectasiaectasia))RecommendedRecommended togethertogether withwith InstantaneousInstantaneous Curvature Curvature and and ““MoveMove--AxisAxis””

THANK YOU !THANK YOU !

the gaussian curvature map · the gaussian curvature renzo mattioli 13/09/2007 refractive on line...

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