VIEWS: 60 PAGES: 53 POSTED ON: 8/13/2011
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