Reconstruction from Digital Holograms by Statistical Methods

Transcription

1 Reconstruction from Digital Holograms by Statistical Methods Saowapak Sotthivirat Jeffrey A. Fessler EECS Department The University of Michigan 2003 Asilomar Nov. 12, 2003 Acknowledgements: Brian Athey, Emmett Leith, Kurt Mills 0

2 Outline Background on holography Image model Conventional numerical reconstruction Statistical holographic reconstruction Simulation results Conclusions and future work 1

3 Conventional Holography Record both amplitude and phase of a wave field Hologram = recorded interference pattern between an object beam and a reference beam object object beam recording medium Recording reference beam primary image hologram conjugate image Reconstructing reference beam 2

4 Image Plane Holography object P 1 P 2 incoherent source. object beam x 1 x 2.. hologram x 3. reference beam x 1 x 2. P 1 P 1 Optical sectioning property as in confocal microscopy No xy scanning Larger source decreases coherence and improves spatial resolution 3

5 Challenges Finite source size limits spatial resolution Optical reconstruction inconvenient Use statistical image reconstruction Hologram Numerical reconstruction on a computer Statistical image restoration Hologram Statistical image reconstruction 4

6 Measurement Model Source Object f( r )= Σ χ ( r ) j x j j Optical system h Object beam u o Reference beam u ref Record samples of + u + 2 o u ref on a digital detector Object beam, superposition: u o ( r )= h( r, r) f ( r)d r Discretized object model u o ( r i )=[Ax] i, Hologram intensity at recording plane I( r) = u o ( r)+u ref ( r) 2 a ij = h( r i, r)χ j ( r)d r = u o ( r) 2 + u ref ( r) 2 + u o ( r)u ref( r)+u o( r)u ref ( r) Mean of ith measured sample E[Y i ]=I( r i )+b i background dark current = u o ( r i )+u ref ( r i ) 2 + b i = [Ax] i + u i 2 + b i, i = 1,...,N 5

7 Conventional Reconstruction Method Plane-wave assumption: u ref ( r)=u ref e ı2π r α Hologram intensity I( r)= u o ( r) 2 +U 2 ref +U ref u o ( r)e ı2π r α +U ref u o( r)e ı2π r α Fourier transform of hologram intensity I ( ν)=i o ( ν)+u 2 refδ( ν)+u ref U o ( ν α)+u ref U o( ν α) r 2 ν 2 r 2 Zero order spectrum r 1 IFFT ν 1 r 1 Recorded Hologram First order spectra α α Magnitude of Fourier Transform Reconstructed Holographic Image 6

8 Statistical Model Poisson model of the hologram measurement data: } Y i Poisson { [Ax] i + u i 2 + b i, i = 1,...,N [Ax] i = P j=1 a ij x j Known: imaging system A, data {Y i }, reference beam {u i }, dark current offset {b i } Cost function: Ψ(x)=L(x)+V (x) Negative likelihood function: L(x)= N i=1 h i ([Ax] i ) h i (z)=( z + u i 2 + b i ) y i log( z + u i 2 + b i ) Penalty: V (x)=β r i=1 ψ([cx] i ) Penalized-likelihood (aka MAP) reconstruction: x = argminψ(x) x No closed-form solution need an iterative algorithm 7

9 Optimization Transfer Illustration 8

10 Optimization Transfer Principle S-step: find a majorizing surrogate function φ (n) 1. φ (n)( x (n)) = Ψ M-step: minimize φ (n) instead of Ψ (x (n)) 2. φ (n) (x) Ψ(x), x C P x (n+1) = argminφ (n) (x) x Fact: algorithm monotonically decreases cost function (cf. Newton) Ψ(x (n+1) ) Ψ(x (n) ) φ(x;x n) Φ(x) x ^ x n+1 x n x 9

11 Log-likelihood Surrogate Functions h (l R ;0) h R (l R ;l n ) q R (l R ;l n ) z (n) i N L(x)= h i ([Ax] i ) (independence) i=1 h i (z) =( z + u i 2 + b i ) y i log( z + u i 2 + b i ) q i (z;z (n) i )=h i (z (n) i )+ q i (z (n) i )(z z (n) i )+ 1 2 c(n) i z z (n) i l R =[Ax (n) ] i. Curvature: c (n) i = c(y i,u i,b i,[ax (n) ] i ) (JOSAA, in review) l n 2 10

12 S-step: L(x)= Quadratic Surrogate Function N h i ([Ax] i ) i=1 N q i ([Ax] i ;[Ax (n) ] i ) = Q(x; x (n) ) i=1 Q(x; x (n) ) = N q i ([Ax] i ;[Ax (n) ] i ) i=1 = L(x (n) )+ L(x (n) )(x x (n) )+ 1 2 (x x(n) ) A DA(x x (n) ) { } D = diag M-step: x (n+1) = argminq(x; x (n) )+V(x) x C P Surrogate Q is quadratic, so minimize by PCG algorithm. Quadratic surrogates for regularizing penalty function also available. (cf. half quadratic methods) c (n) i 11

13 Separable Quadratic Surrogate Algorithm Convexity of q i allows separable surrogate (De Pierro, 1995): ( P [ q i ([Ax] i ;z (n) [aij i )=q i π (x j x (n) ]) j )] ij +[Ax (n) ] i ; z (n) i j=1 π ij ) π ij 0 and P j=1 π ij = 1 P j=1 ( [aij π ij q (x j x (n) j )] i +[Ax (n) ] i ; z (n) i π ij Separable quadratic surrogate function: Q (x; x (n) P )= Q j (x j ; x (n) ) j=1 Q j (x j ; x (n) )= N i=1 Parallelizable update: x (n+1) j for j = 1,...,P ( π ij q [aij (x j x (n) j )] i +[Ax (n) ] i ; z (n) i π ij = argmin x j C ) Q j (x j ; x (n) )+V j (x j ; x (n) ), 12

14 How Many Holograms? Number of measurement elements N is fixed by the recorder (e.g., CCD camera pixels) Number of reconstructed pixels P is chosen by algorithm designer. Natural choice is P = N, but this is under-determined! (Measured data values are real, reconstructed field is complex.) Need P N/2 to avoid an under-determined problem. In conventional digital holography, FFT zero-padding is used. Possible strategies: 1 recorded hologram; reconstruct half-size image: P = N/2 1 recorded hologram; reconstruct full-size image: P = N (Must rely on regularization.) 2 recorded holograms with different reference beams; reconstruct full-size holographic image P = N/2 13

15 Simulation Data: Complex Object Original Object Magnitude Phase x 10 4 x Holograms Data #1 Data #2 0 14

16 Reconstruction Results: Complex Object Filtering Method 0.2 One Data Set, P=N/2 0.2 One Data Set, P=N 0.2 Two Data Sets, P=N/2 (NMSE = 0.16) (NMSE = 0.03) (NMSE = 0.03) (NMSE = 0.02) 0.2 Nonquadratic, edge-preserving regularization ψ(t)= t/δ log(1 + t/δ ), with δ chosen empirically 200 iterations of separable quadratic surrogate (SQS) algorithm 15

17 Contours of Marginal Objective Functions 300 x x x x x I x I x R One Data Set x R Two Data Sets h i (z) for z C, and h i ([Ax (n) ] i ) vs n (Unregularized.) 16

18 Real-Valued Object (Constrained) Hologram Data #1 Filtering Method 1 Data Set, P=N (NMSE = 0.19) (NMSE = 0.05) Original image Hologram Data #2 1 Data Set, P=N/2 (NMSE = 0.04) 2 Data Sets, P=N/2 (NMSE = 0.03) 17

19 Summary Statistical method for reconstructing a complex-valued object field from real-valued hologram intensity data Poisson statistical model. Extendable to others, e.g., Poisson+Gaussian (Snyder et al.) Optimization transfer monotonically decreases the cost function Preliminary simulations suggest improved image quality compared with conventional digital holography reconstruction method. Does not assume the reference beam is a plane-wave! But requires reasonably accurate reference beam model... Can incorporate constraints such as real-valued object field Accommodates hologram reference beam diversity, reducing the problem of multiple minima 18

20 Future Work? The usual story Use a space-variant model Regularization parameter selection Faster converging algorithms Test with real data Specific to this topic Different types of digital holography (e.g., Fresnel, Fourier) Phase retrieval problems 19