January 26-29 2006 Atlantis Hotel, Paradise Island, Bahamasvoi.opt.uh.edu/VOI/WavefrontCongress/2006/... · Atlantis Hotel, Paradise Island, Bahamas. Title: 3-Sarver_Friday_0840.ppt

January 26-29 2006

Atlantis Hotel, Paradise Island, Bahamas

Extracting wavefront errorExtracting wavefront errorfrom S/H images usingfrom S/H images usingSpatial DemodulationSpatial Demodulation

Edwin J. Sarver, PhDEdwin J. Sarver, PhD

OutlineOutline

Traditional Shack-Hartmann (SH)Traditional Shack-Hartmann (SH)processingprocessing

Fourier Transform (FT) processing ofFourier Transform (FT) processing ofSH imagesSH images

Spatial Demodulation (SD) processingSpatial Demodulation (SD) processingof SH imagesof SH images

Simulation and exam examplesSimulation and exam examples DiscussionDiscussion

SH wavefront sensorSH wavefront sensor

Micro lens array Image plane

Wavefront propagation

f Z

Y

SH wavefront sensorSH wavefront sensor

Micro lens array Image plane

Wavefront propagation

f Z

Y

IncidentPlane wavefront

Reference spots in regulargrid on image plane

SH wavefront sensorSH wavefront sensor

Micro lens array Image plane

Wavefront propagation

f Z

Y

SH wavefront sensorSH wavefront sensor

Micro lens array Image plane

Wavefront propagation

f Z

Y

Aberrated wavefrontAberrated spots in irregulargrid on image plane

Traditional SH ProcessingTraditional SH Processing11

Find centroid corresponding to eachFind centroid corresponding to eachmicro lensmicro lens

Deviation Deviation dxdx, , dy dy of detected centroidof detected centroidfrom reference spots yields wavefrontfrom reference spots yields wavefrontgradientgradient

Reconstruct wavefront from gradientReconstruct wavefront from gradient(e.g., Zernike or Fourier)(e.g., Zernike or Fourier)

Traditional SH Processing

1Tyson, Principles of adaptive optics, second edition, Academic Press,New York, 1998.

WF gradient found fromWF gradient found fromspot deviationspot deviation……

Micro lens array Image plane

f

f

delY

dW(x,y) / dy = -delY / f

dW(x,y) / dx = -delX / f

Traditional SH Processing

Fourier SH ProcessingFourier SH Processing11

Compute Fourier Transform of SH imageCompute Fourier Transform of SH image Isolate region of interest (band-pass filter)Isolate region of interest (band-pass filter)

and shift to centerand shift to center Compute inverse Fourier Transform andCompute inverse Fourier Transform and

compute complex angle to yield wrappedcompute complex angle to yield wrappedphasephase

Unwrap phase to yield wavefront gradientUnwrap phase to yield wavefront gradient Reconstruct wavefront from gradients (e.g.,Reconstruct wavefront from gradients (e.g.,

Zernike or Fourier)Zernike or Fourier)

Fourier SH Processing

1Carmon and Ribak, Phase retrieval by demodulation of a Hartmann-Shack Sensor, Optics Comm., 215, 285-8, 2003.

Looking at sameLooking at sameneighborhood as beforeneighborhood as before……

Micro lens array Image plane

ff

delY

Y

Y’

Y

Fourier SH Processing

Warp function is keyWarp function is key……

delY

Y

Y’

Yf

Wavefront warps lens center locations from MLA plane to the image plane.

g0(Y) g1(Y’) = g0(Y’-delY)

( )

( ) ( )

×+=

×−=

fdyydWygyg

fdyydWdelY

'0'1

Fourier SH Processing

If we warp the coordinates of a referencespot pattern, what happens to its FourierTransform?

First, we look at just the reference spots…

Spots and FT of SpotsSpots and FT of Spots

( )

( ) ( ) L

++

−+=

+×

+=

vp

uvp

uvuG

yp

xp

yxg

yx

yx

,1,1810,0

41,

2cos21

212cos

21

21,

0

0

δδδ

ππ

Fourier SH Processing

Reference spots: gReference spots: g00(x,y)(x,y)

Fourier SH Processing

FT of reference spots: GFT of reference spots: G00(u,v)(u,v)

Fourier SH Processing

Now, warped spotsNow, warped spots……

( ) ( ) ( )[ ]

( )( ) ( )( )

++×

++=

++=

yxByp

yxAxp

yxByyxAxgyxg

yx

,2cos21

21,2cos

21

21

,,,, 01

ππ

( ) ( ) ( ) ( ) fdy

yxdWyxBfdx

yxdWyxA ×=×=,,,,, 22

Fourier SH Processing

And FT of warped spotsAnd FT of warped spots……

( ) ( ) ( )

( ) ( ) dydxeexg

dydxeyxAxgvuGvyuxjyxAj

vyuxj

∫∫

+−

+−

=

+=

ππ

π

2),(2

22 ),(,

Our goal is to recover A(x,y) from the low-pass signal exp(j2! A(x,y)) .

Similarily, for B(x,y).

Fourier SH Processing

Sample image of warpedSample image of warpedspotsspots……

Fourier SH Processing

……and its Fourier transformand its Fourier transform

Fourier SH Processing

Regions of interest in FT ofRegions of interest in FT ofspotsspots……

u

v

This region has info about dW/dx

This region has info about dW/dy

Fourier SH Processing

Constraint on WFConstraint on WF……

WF Requirement:WF Requirement:–– Wavefront spectrum must be band limitedWavefront spectrum must be band limited–– In particular, the wavefront slope frequency isIn particular, the wavefront slope frequency is

low compared to MLA spot frequencylow compared to MLA spot frequency

Spot pattern requirement:Spot pattern requirement:–– The width of the spot pattern should be largeThe width of the spot pattern should be large

relative to the spot pitch relative to the spot pitch –– we should have quite we should have quitea few spots across the imagea few spots across the image

Fourier SH Processing

Phase unwrappingPhase unwrapping

Why is it required?Why is it required?–– Because the wavefront slope appears inBecause the wavefront slope appears in

the recovered exponential and the recovered exponential and ““wrapswraps””around every around every 22! . .

How is it done?How is it done?–– If no If no ““residuesresidues””, unwrapping is simple., unwrapping is simple.–– If residues are present, unwrapping canIf residues are present, unwrapping can

be difficult (see be difficult (see 11))

1 Ghiglia and Pritt, Two-Dimensional Phase Unwrapping, John Wiley, 1998.

Fourier SH Processing

Simple unwrappingSimple unwrapping

0

!

2!

Unwrapped function

Wrapped function

Fourier SH Processing

Simple unwrappingSimple unwrapping……

Set U(0) = W(0)Set U(0) = W(0) For n=1 to the end do the followingFor n=1 to the end do the following

D = W(n) D = W(n) – W(n W(n – 1) 1)If D < If D < – Pi then D = D + 2 PiIf D > Pi then D = D – 2 PiU(n) = U(n-1) + D

U = unwrappedW = wrapped

Fourier SH Processing

Test for ResidueTest for Residue

For a continuous surface (wavefront slopes) the sum of the deltasaround a closed curve should be zero.

D(1)

D(2)

D(3)

D(4)

If the sum is non-zero, a residue exists and the simple unwrappingmethod is not valid. Examples given below.

Fourier SH Processing

FT processing steps forFT processing steps forPlane wavePlane wave……

Fourier SH Processing

Contours in input imageContours in input image

Fourier SH Processing

FT of input image (DCFT of input image (DCsuppressed)suppressed)

Fourier SH Processing

Shift spectrum to centerShift spectrum to centerROIROI’’ss……

Fourier SH Processing

Shift to center ROI for dW/dX Shift to center ROI for dW/dY

Shift spectrum to centerShift spectrum to centerROIROI’’ss……

Fourier SH Processing

Shift to center ROI for dW/dX Shift to center ROI for dW/dY

ROIROI’’ss isolated isolated……

Fourier SH Processing

ROI for dW/dX ROI for dW/dY

Wrapped phaseWrapped phase

Fourier SH Processing

Wrapped phase for dW/dX Wrapped phase for dW/dY

Unwrapped phaseUnwrapped phase

Fourier SH Processing

Unwrapped phase for dW/dX Unwrapped phase for dW/dY

Reconstructed wavefrontReconstructed wavefront

Fourier SH Processing

Note zero aberrations since plane wave.

Spatial demodulationSpatial demodulation11

Similar to FT method, but does not requireSimilar to FT method, but does not requireFFTsFFTs

Multiply spots image by complexMultiply spots image by complexexponential to shift the desiredexponential to shift the desiredneighborhood to origin in the frequencyneighborhood to origin in the frequencydomaindomain

Low-pass filter to isolate the desiredLow-pass filter to isolate the desiredfrequency band (box filter is fast)frequency band (box filter is fast)

Unwrap the resultUnwrap the result Reconstruct the wavefrontReconstruct the wavefront

Spatial Demodulation SH Processing

1Talmi and Ribak, Direct demodulation of Hartmann-Shack patterns, JOSA; vol 21, No 4, 632-9, 2004.

FT modulation relationFT modulation relation

( ) ( )asFxfe FTaxj +→←− π2

The FT modulation relation shows that multiplication by a complexexponential in the spatial domain shifts the Fourier Transform in thespectral domain.

Spatial Demodulation SH Processing

Low-pass box filterLow-pass box filter

For spatial demodulation, low-pass filter isFor spatial demodulation, low-pass filter isperformed via convolutionperformed via convolution

To provide efficient calculation, a box filterTo provide efficient calculation, a box filteris used -- (all coefficients equal)is used -- (all coefficients equal)

Using a sliding sum, the filter is computedUsing a sliding sum, the filter is computedusing only 4 adds per output sample (nousing only 4 adds per output sample (nomultiplies)multiplies)

We use two passes to provide a We use two passes to provide a ““cleanercleaner””frequency responsefrequency response

Spatial Demodulation SH Processing

RectRect SincSinc

( )

→←assinc

aaxrect FT

1

Rect x 0.5,( )

x

Sinc x 0.5,( )

x

Spatial Demodulation SH Processing

Triangle Triangle Sinc Sinc22

( ) ( )2

1)(*

→←=assinc

aaxtriangleaxrectaxrect FT

Triangle x 0.5,( )

x

Sinc x 0.5,( )2

x

Spatial Demodulation SH Processing

Comparison of FT domainComparison of FT domainand Spatial domainand Spatial domain Shift spectral region to centerShift spectral region to center

–– FT: Compute FT and shift complex arrayFT: Compute FT and shift complex array–– SD: Multiply spots image by complex exponentialSD: Multiply spots image by complex exponential

Remove unwanted frequenciesRemove unwanted frequencies–– FT: Zero areas outside ROIFT: Zero areas outside ROI–– SD: Apply low-pass filter via convolutionSD: Apply low-pass filter via convolution

Obtain wrapped wavefront derivativesObtain wrapped wavefront derivatives–– FT: Take IFT and calculate complex phaseFT: Take IFT and calculate complex phase–– SD: Calculate complex phaseSD: Calculate complex phase

Rest of processing the sameRest of processing the same……

ExamplesExamples

Simulated spot imagesSimulated spot images–– Astigmatic wavefrontAstigmatic wavefront–– High dynamic range (High dynamic range (±± 20 D) 20 D)–– High resolution (-0.01 D)High resolution (-0.01 D)–– Third-order aberrations (trefoil & coma)Third-order aberrations (trefoil & coma)

Eye imageEye image–– Large amount of background noiseLarge amount of background noise

Simulated Simulated ““badbad”” exam exam–– Not sufficiently band limitedNot sufficiently band limited

Astigmatic wavefrontAstigmatic wavefront-5 S -2 C x 17-5 S -2 C x 17

Examples

Detected contoursDetected contours

Astigmatic wavefront

MTF of input imageMTF of input image

Astigmatic wavefront

Regions of interest in MTFRegions of interest in MTF

Astigmatic wavefront

Wrapped phaseWrapped phase

Astigmatic wavefront

dW/dX dW/dY

Unwrapped phaseUnwrapped phase

Astigmatic wavefront

dW/dX dW/dY

Reconstructed wavefrontReconstructed wavefront

Astigmatic wavefront

High dynamic rangeHigh dynamic range

-20 D + 20 D

Examples

Detected contoursDetected contours

-20 D + 20 D

High dynamic range

MTF of input imageMTF of input image

-20 D + 20 D

High dynamic range

Regions of interest in MTFRegions of interest in MTF

-20 D + 20 D

High dynamic range

Wrapped x-gradientWrapped x-gradient

-20 D + 20 D

High dynamic range

Unwrapped x-gradientUnwrapped x-gradient

-20 D + 20 D

High dynamic range

Wrapped y-gradientWrapped y-gradient

-20 D + 20 D

High dynamic range

Unwrapped y-gradientUnwrapped y-gradient

-20 D + 20 D

High dynamic range

Reconstructed wavefrontReconstructed wavefront

-20 D + 20 D

High dynamic range

Zernike bar graphZernike bar graph

-20 D + 20 D

High dynamic range

High Resolution: -0.01DHigh Resolution: -0.01D

Examples

Detected contoursDetected contours

High resolution

MTF of input imageMTF of input image

High resolution

Regions of interest in MTFRegions of interest in MTF

High resolution

Wrapped phaseWrapped phase

High resolution

dW/dX dW/dY

Unwrapped phaseUnwrapped phase

High resolution

dW/dX dW/dY

Reconstructed wavefrontReconstructed wavefront

High resolution

Third-order aberrationsThird-order aberrationsTrefoil and ComaTrefoil and Coma……

Examples

Trefoil Coma

Detected contoursDetected contours

Third-order aberrations

Trefoil Coma

MTF of input imageMTF of input image

Third-order aberrations

Trefoil Coma

Regions of interest in MTFRegions of interest in MTF

Third-order aberrations

Trefoil Coma

Wrapped phase for Wrapped phase for dxdx

Third-order aberrations

Trefoil Coma

Unwrapped phase for Unwrapped phase for dxdx

Third-order aberrations

Trefoil Coma

Wrapped phase for Wrapped phase for dydy

Third-order aberrations

Trefoil Coma

Unwrapped phase for Unwrapped phase for dydy

Third-order aberrations

Trefoil Coma

Reconstructed wavefrontReconstructed wavefront

Third-order aberrations

Trefoil Coma

Zernike bar graphZernike bar graph

Third-order aberrations

Trefoil Coma

Real eyeReal eye

Examples

Enhanced captured imageEnhanced captured image

Real eye

MTF of input imageMTF of input image

Real eye

Regions of interest in MTFRegions of interest in MTF

Real eye

Wrapped phaseWrapped phase

Real eye

dW/dX dW/dY

Unwrapped phaseUnwrapped phase

Real eye

dW/dX dW/dY

Reconstructed wavefrontReconstructed wavefront

-3.37 – 2.07 x 125 -3.12 – 1.80 x 125

Real eye

SD Reconstruction Spot Centroid Reconstruction

What do bad cases lookWhat do bad cases looklike?like?

C10 = C11 = 2 micronsC10 = C11 = 2 microns–– Not properly band limitedNot properly band limited

C10 = C11 = 2 micronsC10 = C11 = 2 microns

Not band limited

Detected contoursDetected contours

Not band limited

MTF of input imageMTF of input image

Not band limited

Wrapped phaseWrapped phase

Not band limited

dW/dX dW/dY

Unwrapped phaseUnwrapped phase

FAIL TO UNWRAP

Not band limited

dW/dX dW/dY

DiscussionDiscussion

Compared to SH centroid method theCompared to SH centroid method theFT/SD method:FT/SD method:–– Does not require finding spot centroidsDoes not require finding spot centroids–– Easily allows spots to move outside theirEasily allows spots to move outside their

initial aperture regioninitial aperture region–– Requires phase unwrapping processingRequires phase unwrapping processing

The FT technique is especially suitedThe FT technique is especially suitedto large arrays and large aberrationsto large arrays and large aberrations

DiscussionDiscussion……

Compared to FT method the SDCompared to FT method the SDmethod:method:–– Does not require FT of arraysDoes not require FT of arrays–– Allows handling unwrapping problemsAllows handling unwrapping problems

locallylocally

Discussion

SummarySummary

FT/SD method provides another tool forFT/SD method provides another tool forfinding wavefront from a SH imagefinding wavefront from a SH image

For simulated images the FT/SD method:For simulated images the FT/SD method:–– Has a large dynamic rangeHas a large dynamic range–– Has high resolutionHas high resolution

Caution: Errors occur when the bandwidthCaution: Errors occur when the bandwidthof the wavefront exceed the thresholdof the wavefront exceed the thresholdimposed by the MLA lens spacingimposed by the MLA lens spacing

Thank you!Thank you!

January 26-29 2006

Atlantis Hotel, Paradise Island, Bahamas