fien up seminar

8/9/2019 Fien Up Seminar

1/53

JRF 2/06-1

Phase RetrievalforImaging and WaveFront Sensing

Jim FienupRobert E. Hopkins Professor of Optics

University of RochesterInstitute of Optics

Presented at EECS Dept., University of Michigan

General Dynamics Distinguished Lecture SeriesFebruary 9, 2006


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JRF 2/06-2

Outline

Image reconstruction for stellar speckle interferometry! Phase retrieval, hybrid input-output algorithm

! Finding bounds on object support

Image reconstruction from far-field diffraction patterns! Optical

! X-ray

Wavefront Sensing for the Hubble Space Telescope! Phase objects (wave fronts) can be reconstructed

! Gradient search algorithms

Wavefront Sensing for James Webb Space Telescope


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JRF 2/06-3

Passive Imaging of Space Objects

Problem: atmospheric turbulence causes phase errors, limits resolution! !1 arc-sec !5*106rad. !!/ rofor != 0.5 microns and ro= 10 cm

Best resolution as though through only 10 cm aperture (vs. Keck 10 m)

Solutions:! Hubble Space Telecope (2.4 m diam.), $2 B

!

Adaptive optics + laser guide star, $10"

s M! Optical interferometry, $10"s M

! Stellar speckle interferometry, < $1 M

} blurredimageobject

aberrated optical system


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JRF 2/06-4

Labeyrie"s Stellar Speckle Interferometry

1. Record blurred images:

where sk(x, y) is kthpoint-spread function due to atmospheric turbulence

2. Fourier transform:

where Sk(u, v) is kthoptical transfer function

3. Magnitude square and average:

4. Measure or determine transfer function

atmospheric model or measure reference star

5.

Amplitude interferometry (Michelson) can also yield Fourier magnitude data

gk(x,y) =f(x, y)! sk(x,y), k = 1, . . ., K

Gk(u,v) = F(u,v)Sk(u,v), k = 1, . . ., K

1K

Gk(u,v)2k=1

K

! = F(u,v)2 1K Sk(u,v)2

k=1

K

!

1

K Sk(u,v)

2

k=1

K

!

Divide by1

K

Sk (u,v)2

k=1

K

! to get F(u,v)2

Reference: A. Labeyrie, "Attainment of Diffraction Limited Resolution in LargeTelescopes by Fourier Analysing Speckle Patterns in Star Images," Astron. andAstrophys. 6, 85-87 (l970).


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JRF 2/06-6

Phase of an Optical Field

Monochromatic EM radiation propagates as a sinusoidal wave Phase is the relative position of the peaks and troughs of the wave Phase also plays an important role in imaging via F.T.

|A1| Re

Im

|A| "

! =2"

#OPD

Field = A2 = A2 ei!

2

Intensity = A22

"2

|A2|


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JRF 2/06-7

Constraints in Phase Retrieval

Nonnegativity constraint: f(x, y) #0! True for ordinary incoherent imaging, x-ray diffraction, MRI, etc.

! Not true for wavefront sensing or coherent imaging

The support of an object is the set of points over which it is nonzero! Meaningful for imaging objects on dark backgrounds

! Wavefront sensing through a known aperture

A good support constraint is essential for complex-valued objects! Coherent imaging or wave front sensing

Atomiticity when have angstrom-level resolution! For crystals -- not applicable for coarser-resolution, single-particle

Object intensity constraint (wish to reconstruct object phase)! E.g., measure wavefront intensity in two planes (Gerchberg-Saxton)

! If available, supercedes support constraint


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JRF 2/06-8

Phase Caries More Informationthan Amplitude

F1[|F[M]|exp[i phase{F[G]}]]

F1[|F[G]|exp[i phase{F[M]}]]G

M

|F[G]|

|F[M]|

phase{FT[G]}

phase{F[M]}


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JRF 2/06-9

WhyPhase is More Important than Amplitude

Wave front= surface of constant phase: Light travels in direction perpendicular to wave front

Where the light is concentrated, after propagation, depends on the phase

!(x,y,z) =const

Light spreads out dimmer

Light is concentrated brighter

!(x,y,z) =const


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JRF 2/06-10

Is Phase Retrieval Possible?


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JRF 2/06-11

First Phase Retrieval Result

(a) Original object, (b) Fourier modulus data, (c) Initial estimate(d) (f) Reconstructed images number of iterations: (d) 20, (e) 230, (f) 600

Reference: J.R. Fienup, Optics Letters, Vol 3., pp. 27-29 (1978).


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JRF 2/06-12

Measured intensity

Iterative Transform Algorithm

F

gk+1 x,y( ) =!gk x,y( ) , !gk x,y( ) satisfies constraints

gk x,y( ) "# !gk x,y( ), !gk x,y( ) violates constraints

$%&

Hybrid Input-Output version

F1


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JRF 2/06-13

Error-Reduction and HIO

Error reduction algorithm

! Satisfy constraints in object domain

! Equivalent to projection onto (nonconvex) sets algorithm

! Proof of convergence (weak sense)

! In practice: slow, prone to stagnation, gets trapped in local minima

Hybrid-input-output algorithm

! Uses negative feedback idea from control theory

#

is feedback constant

! No convergence proof (can increase errors temporarily)

! In practice: much faster than ER, can climb out of local minima

ER: gk+1 x( ) =!gk x( ), x"S & !gk x( )# 0

0 , otherwise

$

%&

HIO: gk+1 x( ) =!gk x( ), x"S & !gk x( )# 0

gk x( )$% !gk x( ), otherwise

&'(


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JRF 2/06-14

Image Reconstruction fromSimulated Speckle Interferometry Data

J.R. Fienup, "Phase Retrieval Algorithms: A Comparison," Appl. Opt. 2l, 2758-2769 (1982).

Labeyrie"sstellar speckleinterferometry

gives this


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JRF 2/06-15

Error Metric versus Iteration Number


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JRF 2/06-16

Autocorrelation Support

Autocorrelation:

rf(x,y) = f( !x , !y )f* ( !x " x, ! y " y)d !x d ! y"##$$ = F

"1 F(u,v)2[ ]

ObjectSupport Forming Autocorrelation Support

Autocorrelation Support

0


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JRF 2/06-17

Bounds on Object Support

Triple-Intersection Rule: [Crimmins, Fienup, & Thelen, JOSA A 7, 3 (1990)]


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JRF 2/06-18

Triple Intersection for Triangle Object

Family of solutions for object support from autocorrelation support

Use upper bound for support constraint in phase retrieval

(a)(b)

(c)

(d)

Object Support Autocorrelation Support

Triple Intersection Support Constraint

Alternative

Object Support


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JRF 2/06-19

PROCLAIM 3-D Imaging ConceptPhase Retrieval with Opacity Constraint LAser IMaging

tunable laser

direct-detectionarray

collecteddata set

,!!,!,1 2 n. . .

reconstructedobject

phaseretrievalalgorithm

3-DFFT

initial

estimatefromlocator

set


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JRF 2/06-20

Object for Laboratory Experiments

ST Object. The three concentric discs forming a pyramid can be seen asdark circles at their edges. The small piece on one of the two lower legswas removed before this photograph was taken.


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JRF 2/06-21

3-D Laser Fourier IntensityLaboratory Data

Data cube:

1024x1024 CCD pixels x 64 wavelengths

Shown at right:128x128x64 sub-cube

(128x128 CCD pixels ateach of 64 wavelengths)


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JRF 2/06-22

Imaging Correlography

Get incoherent-image information from coherent speckle pattern Estimate 3-D Incoherent-object Fourier squared magnitude

! Like Hanbury-Brown Twiss intensity interferometry

Easier phase retrieval: have nonnegativity constraint on incoherentimage

Coarser resolution since correlography SNR lower

FI (u,v,w)2! Dk(u,v,w)" Io[ ]# Dk(u,v,w) " Io[ ] k

(autocovariance of speckle pattern)

References:

P.S. Idell, J.R. Fienup and R.S. Goodman, "Image Synthesis from Nonimaged Laser SpecklePatterns," Opt. Lett. 12, 858-860 (1987).

J.R. Fienup and P.S. Idell, "Imaging Correlography with Sparse Arrays of Detectors," Opt.Engr. 27, 778-784 (1988).

J.R. Fienup, R.G. Paxman, M.F. Reiley, and B.J. Thelen, 3-D Imaging Correlography andCoherent Image Reconstruction, in Proc. SPIE 3815-07, Digital Image Recovery andSynthesis IV, July 1999, Denver, CO., pp. 60-69.


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JRF 2/06-23

Image Autocorrelation from Correlography


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JRF 2/06-24

Thresholded Autocorrelation


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JRF 2/06-25

Triple Intersectionof Autocorrelation Support


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JRF 2/06-26

Locator Set, Slices 50-90


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JRF 2/06-27

Dilated Locator Setused as Support Constraint


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JRF 2/06-28

Fourier Modulus Estimatefrom Correlography


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JRF 2/06-29

Fourier Magnitude, DC Slice

Before Filtering After Filtering


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JRF 2/06-30

Incoherent ImageReconstructed by ITA


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JRF 2/06-31

Support Constraint from ThresholdedIncoherent Image


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JRF 2/06-32

Dilated Support Constraintfrom Thresholded Incoherent Image


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JRF 2/06-33

Coherent Image Reconstructed byITA from One 128x128x64 Sub-Cube


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JRF 2/06-34

PROCLAIM Example System Parameters

Parameter S mbol Microscopic Example MegascopicCenter wavelength ! 0.5 m 0.773 m 0.5 m

Range R 10 cm 89 cm 1,000 km

Detector width N"u 2 cm 1.23 cm 5 m

Number of linear detectors N 1,000 1,024 50

Detector pitch "u 20 m 12 m 10 cm

Lateral resolution #xy 2.5 m 56 m 10 cm

Lateral ambiguity interval N#xy 2.5 mm 57 mm 5 m

Center frequency $ 600x1012Hz 388x1012Hz 600x1012Hz

Bandwidth M"$ 30x1012Hz 3.12x1012Hz 1.5x109Hz

Fractional bandwidth M"$/$ 0.05 0.00828 2.5x106

Number of frequencies M 100 64 100

Frequency interval "$ 0.3x1012Hz 50.2x109Hz 15x106Hz

Range resolution #z 5 m 46.7 m 10 cm

Range ambiguity interval M#z 0.5 mm 3 mm 10 m


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JRF 2/06-35

New Interest in Non-CrystallographicX-Ray Diffraction

Non-crystallographic x-ray diffraction made possible by high-intensity,highly coherent x-ray sources -- synchrotron radiation


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JRF 2/06-36

Image Reconstruction from X-RayDiffraction Intensity

CoherentX-ray beam

Target

Detector

array(CCD)

(electron micrograph)

Collection of gold balls(has complex index of refraction)

Far-field diffraction pattern(Fourier intensity)


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JRF 2/06-37

Example on Real X-Ray Data(Data from M. Howells/LBNL and H. Chapman/LLNL)

(a) X-ray data

(c) Initial Support constraint

computed from (b)

(d) Electron micrograph

of object

(b) Triple Intersection(b) Autocorrelation from (a)


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JRF 2/06-38

Active Conformal Thin Imaging System(ACTIS) without Imaging Optics

Coherent illumination, sensed by a conformal array of detectors in far field! Angle-angle (not range) image

Phase retrieval needed to form an image! Heterodyne detection over large arrays is beyond the state of the art

! Support constraint formed by laser illumination system

With no imaging optics, a wider aperture fits on a platform! Giving finer resolution, wider total field-of-view, thin system

Conformal Array

of DetectorsCoherent

Illuminator

Modification of figure from Brad Tousley (DARPA/TTO)


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JRF 2/06-39

Determine HST Aberrations from PSF

Measurements & Constraints:

Pupil plane: known aperture shape phase error fairly smooth function

Focal plane: measured PSF intensity


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JRF 2/06-40

Nonlinear Optimization AlgorithmsEmploying Gradients

Minimize Error Metric, e.g.:

Repeat three steps:

1. Compute gradient:

!E

!p1,

!E

!p2,

2. Compute direction of

search

3. Perform line search

a

b

c

Parameter 1

Parameter2

Contour Plot of Error Metric

Gradient methods:(Steepest Descent)Conjugate GradientBFGS/Quasi-Newton

E = W(u) G(u) F(u)[ ]2

u

!

g x( ) = g x( )ei! x( ) , G(u) = F g x( )[ ]

pupil model:


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JRF 2/06-41

Analytic Gradientswith Phase Values as Parameters

P[] canbeasingleFFTormultiple-planeFresneltransforms

withphasefactorsandobscurations

Analyticgradientsveryfastcomparedwith

calculationbyfinitedifferences

J.R. Fienup, Phase-Retrieval Algorithms for a Complicated Optical System, Appl. Opt. 32, 1737-1746 (1993).

J.R. Fienup, J.C. Marron, T.J. Schulz and J.H. Seldin, Hubble Space Telescope Characterized by UsingPhase Retrieval Algorithms, Appl. Opt. 32 1747-1768 (1993).

G u( ) = P g x( )[ ]

g(x) =gR x( ) + i gI x( ) =mo x( )ei! x( ) , ! x( ) = ajZj x( )

j=1

J

"Optimizing over

where

GW u( ) = W u( ) F u( ) G(u)

G u( )! G u( )

"

#$

%

&' , and g

W(x) = P GW u( )"# %

&

For point-by-point pixel (complex) value, g(x),!E

!g(x) = 2 Im gW*(x){ }

For point-by-point phase map, $(x), !E

!"(x)

= 2 Im g x( ) gW* x( ){ }

For Zernike polynomial coefficients, !E

!aj = 2 Im g x( ) gW* x( )

x

" Zj x( )#$%

&'(

E = W(u) G(u) F(u)[ ]2

u

!


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JRF 2/06-42

Sources of Obscurations in HST


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JRF 2/06-43

Hubble Telescope Retrieval Approach

Pupil (support constraint) was known imperfectlyPhase was relatively smooth and dominated by low-order Zernike"s

Use boot-strapping approach

1. With initial guess for pupil, fit Zernike polynomial coefficients

(parametric phase retrieval by gradient search)

2. With initial guess for Zernike polynomials, estimate pupil by ITA

(retrieve magnitude, given an estimate of phase)

3. Redo steps 1 and 2 until convergence (2 iterations)

4. Estimate phase map by ITA, starting with Zernike polynomial phase

(nonparametric phase retrieval by G-S or gradient search)

5. Refit Zernike coefficients to phase map

6. Redo steps 2 - 5


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JRF 2/06-44

Pupil Function Reconstruction

Pupil Reconstructed from ITA Inferred Model of Pupil


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JRF 2/06-45

James Webb Space Telescope(Next Generation Space Telescope)

See farther back towards the beginnings of the universeLight is red-shifted into infrared

http://ngst.gsfc.nasa.gov/


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JRF 2/06-46

James Webb Space Telescope (JWST)

See red-shifted light from early universe

! 0.6 m to 28 m

! L2 orbit for passive cooling,avoiding light from sun and earth

! 6 m diameter primary mirror

Deployable, segmented optics

Phase retrieval to align segments


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JRF 2/06-47

Fienup Group Visits JWST


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JRF 2/06-48

Phase Retrieval for JWST

R. Lyon et al., (GSFC)

J. Green (JPL), B. Dean (GSFC) et al.,Proc. SPIE (Glasgow 2004)

D.S Acton et al.( Ball Aerospace),

Proc. SPIE (Glasgow 2004)

NASA has chosen phase retrievalas the fine phasing approach for JWST.


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JRF 2/06-49

JWST at UofRWaveFront Sensing Improvements

Develop improved WFS (phase retrieval) algorithms

! Faster, converge more reliably, less sensitive to noise, 2%jumps

! Work with larger aberrations, broadband illumination, jitter

refining iterative transform, gradient search algorithms

Increase robustness and accuracy

! Extended objects

Phase diversityPhase retrieval performance

Experiments with UofR JWST laboratory simulator! Adaptive optics MEMS deformable mirror

18 hex. segments

! Interferometer measure wavefront independently

! Put in misalignment, reconstruct wavefronts,compare with interferometer truth

Ideal PSF


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JRF 2/06-50

Optical Testing Using Phase Retrieval

Optical wave fronts (phase) can be measured by many forms of interferometry

Novel wave front sensor: a bare CCD detector array, detects reflected intensity

Wave front reconstructed in the computer by phase retrieval algorithm

Approach:

Simulation ResultsmPSF at z=333

PSF1

PSF at z=f=500m

PSF2Actual wrapped phase

TruePhase

Reconstructed wrapped phase

RetrievedPhase

Part

under

test

Display ofmeasured

wavefront

Illumination

Wavefront

CCD Array

measures

intensity ofreflected field

Computer

Part

under

test

Display ofmeasured

wavefront

Illumination

Wavefront

CCD Array

measures


Computer

Part

under

test

Display ofmeasured

wavefront

Illumination

Wavefront

CCD Array

measures


Computer

Part

under

test

Display ofmeasured

wavefront

Illumination

Wavefront

CCD Array

measures


Computer


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JRF 2/06-51

Phase retrieval works when have one measurement and one unknown! Image reconstruction from Fourier magnitude

! Wavefront sensing through a known aperture, from a point source

Some problems have two unknowns: finite source and aberrations! Wavefront sensing from extended object

! Need at least two measurements to determine two unknowns

Phase Diversity

known

defocuslength

}conventional

image

}

unknownextended object

diversity image

aberrated optical system

References:

R.G. Paxman and J.R. Fienup, "Optical Misalignment Sensing and Image Reconstruction Using PhaseDiversity," J. Opt. Soc. Am. A 5, 914-923 (1988).

R.G. Paxman, T.J. Schulz and J.R. Fienup, "Joint Estimation of Object and Aberrations Using PhaseDiversity," J. Opt. Soc. Am. A 9, 1072-85 (1992).

M.R. Bolcar and J.R. Fienup,Method of Phase Diversity in Multi-Aperture Systems Utilizing Individual Sub-Aperture Control, in Unconventional Imaging, Proc. SPIE 5896-14 (July 2005).


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JRF 2/06-52

Summary of Phase Retrieval

Phase retrieval can give diffraction-limited images

! Despite atmospheric turbulence or aberrated optics! Despite lack of Fourier phase measurements

Can work for many imaging modalities

! Passive, incoherent or laser imaging correlography Nonnegativity constraint

Support constraint may be loose and derived from given data! Active, coherent imaging

Complex-valued image, so NOT nonnegative Support constraint: must be tight and of special type or helped by 3-D, or by opaque object, or by correlography

Can be used for wavefront sensing, using simple hardware

! For atmospheric turbulence, telescope aberrations, eye aberrations

Current hot areas: JWST WFSC, Non-crystallographic x-ray imaging,Unconventional laser imaging, Characterization of ultrafast pulses, etc.


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JRF 2/06-53

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