Phase Retrieval for Imaging and WaveFront Sensing by dfgh4bnmu

VIEWS: 60 PAGES: 53

									          Phase Retrieval
for Imaging and WaveFront Sensing


                    Jim Fienup
          Robert E. Hopkins Professor of Optics

             University of Rochester
               Institute of Optics
     Presented at EECS Dept., University of Michigan
     General Dynamics Distinguished Lecture Series
                  February 9, 2006
                                                       JRF 2/06-1
                                          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




                                                               JRF 2/06-2
                       Passive Imaging of Space Objects

•   Problem: atmospheric turbulence causes phase errors, limits resolution
     !   ! 1 arc-sec ! 5*10–6rad. ! !/ ro for ! = 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


                                 aberrated optical system



                                        }
                                                                      blurred
      object                                                          image




                                                                                JRF 2/06-3
                      Labeyrie"s Stellar Speckle Interferometry

1. Record blurred images: gk ( x , y ) = f (x, y ) ! sk (x , y ),    k = 1 . . ., K
                                                                          ,
           — where sk(x, y) is kth point-spread function due to atmospheric turbulence
2. Fourier transform: Gk (u, v ) = F (u ,v )Sk (u,v ),       k = 1, . . ., K
           — where Sk(u, v) is kth optical transfer function
                                    K                           K
3. Magnitude square and average: 1                         2 1
                                   ! Gk (u, v ) = F (u ,v ) K ! Sk (u ,v )
                                               2                           2
                                 K k =1                        k =1
                                          1 K
                                            ! Sk (u ,v )
                                                         2
4. Measure or determine transfer function
                                          K k =1
           — atmospheric model or measure reference star
             1 K
                !
                               2
5. Divide by        Sk (u, v )   to get F (u ,v ) 2
             K k =1

Amplitude interferometry (Michelson) can also yield Fourier magnitude data
        Reference: A. Labeyrie, "Attainment of Diffraction Limited Resolution in Large
        Telescopes by Fourier Analysing Speckle Patterns in Star Images," Astron. and
        Astrophys. 6, 85-87 (l970).

                                                                                         JRF 2/06-4
                                               Phase Retrieval Basics
                                                                  #
                Fourier transform: F (u ,v ) =               $$   !#
                                                                     f (x , y )e !i 2" (ux +vy )dx dy

                                                           = F (u ,v ) e         v
                                                                                   = F [f (x , y )]
                                                                           i% (u , )

                                                        #
        Inverse transform: f (x , y ) =            $$   "#
                                                           F (u ,v )e i 2! (ux +vy )dudv    = F "1[F (u ,v )]

                 Phase retrieval problem:
                       Given F (u,v ) and some constraints on f (x , y ),
                       Reconstruct f (x , y ), or equivalently retrieve ! (u,v )

                                       [                               ]       [
    F (u ,v ) = F [f (x , y )] = F e ic f (x ! x o , y ! y o ) = F e ic f * (!x ! x o , ! y ! y o )                  ]
(Inherent ambiguities: phase constant, images shifts, twin image all result in same data)

            Autocorrelation:
            rf (x , y ) =   $$
                                 #
                                 "#
                                    f (x                                                    [
                                           !, y ! )f * (x ! " x , y ! " y )dx !dy ! = F "1 F (u ,v ) 2   ]
•    Patterson function in crystallography is an aliased version of the autocorrelation
•    Simply need Nyquist sampling of the Fourier intensity to avoid aliasing
                                                                                                             JRF 2/06-5
                               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.
                                                                       2"
                                                                    !=    OPD
             "2                                                         #

                                                             Field = A2 = A2 e i!2
|A2|
                                                                               2
                                                             Intensity = A2


                                                                    Im


                                                                         |A|
                                                                                "
|A1|                                                                                        Re




                                                                               JRF 2/06-6
                          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



                                                                                JRF 2/06-7
              Phase Caries More Information
                     than Amplitude




G   |F[G]|      phase{FT[G]}   F –1[| F[G]| exp[i phase{F[M]}]]




M   | F[M]|     phase{F[M]}
                               F –1[| F[M]| exp[i phase{F[G]}]]

                                                        JRF 2/06-8
                                      Why
                      Phase is More Important than Amplitude

•   Wave front = surface of constant phase: ! ( x , y , z ) = const
•   Light travels in direction perpendicular to wave front


                                     Light spreads out – dimmer




                                    Light is concentrated – brighter

          ! ( x , y , z ) = const


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



                                                                       JRF 2/06-9
Is Phase Retrieval Possible?




                               JRF 2/06-10
                           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).
                                                                              JRF 2/06-11
                                 Iterative Transform Algorithm




                                           F




                                                                                Measured intensity

                                          F –1


Hybrid Input-Output version
                  $g ! ( x , y )                 , gk ( x , y ) satisfies constraints
                                                     !
gk +1 ( x , y ) = % k
                  & gk ( x , y ) " # gk ( x , y ) , gk ( x , y ) violates constraints
                                      !               !

                                                                                            JRF 2/06-12
                               Error-Reduction and HIO
                                                      $g k ( x ) , x "S & g k (x ) # 0
                                                         !                   !
•   Error reduction algorithm        ER: g k +1(x ) = %
                                                      &          0 , otherwise
     !   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


                                                    & g ! ( x ) , x "S & gk (x ) # 0
                                                                               !
•   Hybrid-input-output algorithm HIO: g k +1(x ) = ' k
                                                    (g k ( x ) $ %g k ( x ) , otherwise
                                                                    !

     !   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




                                                                                  JRF 2/06-13
                           Image Reconstruction from
                      Simulated Speckle Interferometry Data




     Labeyrie"s
stellar speckle
interferometry
      gives this




J.R. Fienup, "Phase Retrieval Algorithms: A Comparison," Appl. Opt. 2l, 2758-2769 (1982).
                                                                                  JRF 2/06-14
Error Metric versus Iteration Number




                                       JRF 2/06-15
                                            Autocorrelation Support
Autocorrelation:
rf (x , y ) =   $$
                     #
                     "#
                        f (x                                                 [
                               !, y ! )f * (x ! " x , y ! " y )dx !dy ! = F "1 F (u ,v ) 2   ]


                       Object
                       Support                      Forming Autocorrelation Support




                                                     0




                                   Autocorrelation Support
                                                                                                 JRF 2/06-16
                             Bounds on Object Support




Triple-Intersection Rule: [Crimmins, Fienup, & Thelen, JOSA A 7, 3 (1990)]
                                                                             JRF 2/06-17
                 Triple Intersection for Triangle Object




                   Support
            Object (a)           Autocorrelation Support
                                           (b)




                                                     Alternative
                                                          (d)
                                                    Object Support
                            Support Constraint
      Triple Intersection –(c)

$• Family of solutions for object support from autocorrelation support
$•$Use upper bound for support constraint in phase retrieval
                                                                         JRF 2/06-18
                PROCLAIM 3-D Imaging Concept
         Phase Retrieval with Opacity Constraint LAser IMaging




  tunable laser


direct-detection
     array


!1, !2, . . ., ! n
                       initial
                     estimate      phase     3-D
                        from     retrieval
                      locator    algorithm   FFT
                          set

 collected                                     reconstructed
 data set                                          object
                                                                 JRF 2/06-19
                   Object for Laboratory Experiments




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




                                                                       JRF 2/06-20
                          3-D Laser Fourier Intensity
                               Laboratory Data

Data cube:
1024x1024 CCD pixels
$$$x 64 wavelengths
Shown at right:
128x128x64 sub-cube
(128x128 CCD pixels at
each of 64 wavelengths)




                                                        JRF 2/06-21
                                             Imaging Correlography

•    Get incoherent-image information from coherent speckle pattern
•    Estimate 3-D Incoherent-object Fourier squared magnitude
       !   Like Hanbury-Brown Twiss intensity interferometry
                                ! [Dk (u ,v , w ) " I o ] # [Dk (u, v , w ) " Io ]
                            2
           FI (u, v , w )                                                            k
                                    (autocovariance of speckle pattern)

•    Easier phase retrieval: have nonnegativity constraint on incoherent
     image
•    Coarser resolution since correlography SNR lower
    References:

    P.S. Idell, J.R. Fienup and R.S. Goodman, "Image Synthesis from Nonimaged Laser Speckle
    Patterns," 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 and
    Coherent Image Reconstruction,” in Proc. SPIE 3815-07, Digital Image Recovery and
    Synthesis IV, July 1999, Denver, CO., pp. 60-69.
                                                                                                JRF 2/06-22
Image Autocorrelation from Correlography




                                           JRF 2/06-23
Thresholded Autocorrelation




                              JRF 2/06-24
    Triple Intersection
of Autocorrelation Support




                             JRF 2/06-25
Locator Set, Slices 50-90




                            JRF 2/06-26
   Dilated Locator Set
used as Support Constraint




                             JRF 2/06-27
Fourier Modulus Estimate
  from Correlography




                           JRF 2/06-28
          Fourier Magnitude, DC Slice




Before Filtering          After Filtering



                                            JRF 2/06-29
 Incoherent Image
Reconstructed by ITA




                       JRF 2/06-30
Support Constraint from Thresholded
         Incoherent Image




                                      JRF 2/06-31
    Dilated Support Constraint
from Thresholded Incoherent Image




                                    JRF 2/06-32
Coherent Image Reconstructed by
  ITA from One 128x128x64 Sub-Cube




                                     JRF 2/06-33
                 PROCLAIM Example System Parameters

        Parameter          Symbol   Microscopic     Example      Megascopic
   Center wavelength         !         0.5 µm       0.773 µm       0.5 µm
          Range              R          10 cm         89 cm       1,000 km
     Detector width         N"u         2 cm         1.23 cm         5m
Number of linear detectors   N          1,000          1,024          50
      Detector pitch         "u        20 µm          12 µm         10 cm
    Lateral resolution      #xy        2.5 µm         56 µm         10 cm
Lateral ambiguity interval N#xy        2.5 mm         57 mm          5m
    Center frequency          $     600x1012 Hz   388x1012Hz    600x1012 Hz
        Bandwidth           M"$      30x1012 Hz   3.12x1012Hz    1.5x109 Hz
  Fractional bandwidth     M"$/$         0.05        0.00828      2.5x10–6
 Number of frequencies       M           100             64          100
   Frequency interval        "$     0.3x1012 Hz   50.2x109Hz     15x106 Hz
    Range resolution         #z         5 µm         46.7 µm        10 cm
Range ambiguity interval    M#z        0.5 mm         3 mm          10 m


                                                                      JRF 2/06-34
                 New Interest in Non-Crystallographic
                          X-Ray Diffraction




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




                                                                        JRF 2/06-35
                       Image Reconstruction from X-Ray
                             Diffraction Intensity


              Target


Coherent
X-ray beam
                                                                    Detector
                                                                      array
                                                                     (CCD)




     (electron micrograph)
   Collection of gold balls         Far-field diffraction pattern
(has complex index of refraction)       (Fourier intensity)
                                                                         JRF 2/06-36
                        Example on Real X-Ray Data
                 (Data from M. Howells/LBNL and H. Chapman/LLNL)




(a) X-ray data          (b) Autocorrelation from (a)     (b) Triple Intersection




           (c) Initial Support constraint   (d) Electron micrograph
                 computed from (b)                  of object             JRF 2/06-37
                   Active Conformal Thin Imaging System
                       (ACTIS) without Imaging Optics

                                                    Conformal Array
                 Coherent                             of Detectors
                Illuminator




                                    Modification of figure from Brad Tousley (DARPA/TTO)




•   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

                                                                                           JRF 2/06-38
                    Determine HST Aberrations from PSF




Measurements & Constraints:
Pupil plane: known aperture shape
  phase error fairly smooth function
Focal plane: measured PSF intensity



                                                         JRF 2/06-39
                                        Nonlinear Optimization Algorithms
                                              Employing Gradients
                                               i! ( x )
              pupil model: g ( x ) = g ( x ) e          , G(u ) = F [ g ( x )]

                                                      !W (u ) [ G(u ) – F (u ) ]
                                                                                 2
    Minimize Error Metric, e.g.: E =
                                                      u
                            Contour Plot of Error Metric                         Repeat three steps:

                                                                                  1. Compute gradient:
                                                                                      !E !E
                                                                                 !!!!    ,    ,…
                                                                                      !p1 !p2
Parameter 2




                                                              a
                                              c
                                                                                  2. Compute direction of
                                                                                 search
                                                          b                        3. Perform line search


                                        Parameter 1
              Gradient methods:
                       (Steepest Descent)
                       Conjugate Gradient
                       BFGS/Quasi-Newton
                       …                                                                             JRF 2/06-40
                                      Analytic Gradients
                               with Phase Values as Parameters

     !W (u ) [ G(u ) – F (u ) ]             G (u ) = P [ g ( x )]
                                    2
E=
     u                                                                                         J
                                                                       i! ( x )
Optimizing over g (x ) = gR ( x ) + i gI ( x ) = mo ( x ) e                       , ! (x ) =   " a j Z j (x )
                                                                                               j =1
For point-by-point pixel (complex) value, g(x),
                                                                     !E
                                                                    !g (x )
                                                                                   = 2 Im g    {   W*
                                                                                                           }
                                                                                                        (x )

For point-by-point phase map, $(x),
                                                     !E
                                                    !" (x )
                                                                             {
                                                               = 2 Im g ( x ) gW * ( x )           }
                                                    !E       #                             &
For Zernike polynomial coefficients,                 = 2 Im $" g ( x ) gW * ( x )Z j ( x ) '
                                                    !a j     %x                            (
 where                      "                        %
                                     G(u )
      GW      (u ) = W (u ) $ F (u )        ! G (u ) ' , and gW (x ) = P † "GW (u ) %
                                                                           #           &
                            #        G (u )          &

     P [•]$$can$be$a$single$FFT$or                                Analytic$gradients$very$fast
  multiple-plane$Fresnel$transforms                                       compared$with
 with$phase$factors$and$obscurations                             calculation$by$finite$differences
  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 Using
  Phase Retrieval Algorithms,” Appl. Opt. 32 1747-1768 (1993).
                                                                                                               JRF 2/06-41
Sources of Obscurations in HST




                                 JRF 2/06-42
                    Hubble Telescope Retrieval Approach


• Pupil (support constraint) was known imperfectly
• Phase 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


                                                                         JRF 2/06-43
                     Pupil Function Reconstruction




Pupil Reconstructed from ITA           Inferred Model of Pupil

                                                                 JRF 2/06-44
                          James Webb Space Telescope
                        (Next Generation Space Telescope)




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


             See farther back towards the beginnings of the universe
                          Light is red-shifted into infrared
                                                                       JRF 2/06-45
                    James Webb Space Telescope (JWST)

•   See red-shifted light from early universe
     !   0.6 µm to 28 µm
     !   L2 orbit for passive cooling,
         avoiding light from sun and earth
     !   6 m diameter primary mirror
          – Deployable, segmented optics
          – Phase retrieval to align segments




                                                        JRF 2/06-46
Fienup Group Visits “JWST”




                             JRF 2/06-47
                                     Phase Retrieval for JWST




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




            R. Lyon et al., (GSFC)



   NASA has chosen phase retrieval
as the fine phasing approach for JWST.

                                                    D.S Acton et al.( Ball Aerospace),
                                                       Proc. SPIE (Glasgow 2004)
                                                                                   JRF 2/06-48
                                JWST at UofR
                        WaveFront 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
                                                                             Ideal PSF
          – Increase robustness and accuracy
     !   Extended objects
          – Phase diversity
            Phase 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”



                                                                             JRF 2/06-49
                   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 Results
                                                      PSF at z=333µm                      µm
                                                                               PSF at z=f=500

                                Illumination
        Part                    Wavefront
       under
         test
                             CCD Array              PSF1                      PSF2
                             measures              Actual wrapped phase   Reconstructed wrapped phase




 Display of                  intensity of
 measured                    reflected field
 wavefront
                         Computer                   True                  Retrieved
                                                   Phase                   Phase


                                                                                                        JRF 2/06-50
                                                   Phase Diversity
•    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

                                            aberrated optical system               conventional




                                                      }
       unknown                                                                        image
    extended object




                                                                                            known
                                                diversity image
                                                                                        }   defocus
    References:
                                                                                            length
    R.G. Paxman and J.R. Fienup, "Optical Misalignment Sensing and Image Reconstruction Using Phase
    Diversity," 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 Phase
    Diversity," 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).
                                                                                                             JRF 2/06-51
                          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.


                                                                           JRF 2/06-52
JRF 2/06-53

								
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