Mitigating the Effect of Noise in Iterative Projection Phase Retrieval

Transcription

1 Mitigating the Effect of Noise in Iterative Projection Phase Retrieval & David Hyland Imaging and Applied Optics: Optics and Photonics Congress

2 Introduction Filtering Results Conclusion Outline Introduction Iterative projection phase retrieval 2d projection visualization method Filtering of Modulus Noise Noise model Observed error behavior Theoretical effects New Fourier Domain Projection Resulting Performance 2d projection visualization examples Low SNR example Conclusion

3 Introduction Filtering Results Conclusion Phase Retrieval Problem Statement Objective is to recover the image g x, y which has the discrete Fourier transform G u, v. Given the measured Fourier modulus G u, v, estimate the phase of G u, v so it can inverted to recover g x, y. At each iteration perform several projections onto constrained domains. Historically, there is not much attention paid to noise in the measured Fourier modulus noise. Can it be filtered? R. Trahan and D. Hyland, "Mitigating the effect of noise in the hybrid input-output method of phase retrieval." Applied Optics, Vol. 52, Issue 13, pp (213)

4 Introduction Filtering Results Conclusion Typical Projection Operators Fourier domain projection preserves phase and constrains modulus to F u, v P m g x, y = FT 1 F u, v exp iφ FT g x, y Image domain projection preserves pixel value magnitudes within foreground and zeros others P s g x, y = g x, y, x, y γ x, y, otherwise Reflections are defined as R 2P I.

5 Introduction Filtering Results Conclusion Phase Retrieval Methods Algorithm Error Reduction (ER) Solvent Flipping (SF) Hybrid Input-Output (HIO) Iteration g k+1 = P s P m g k g k+1 = R s P m g k P g k+1 = m g k, I βp m g k, x, y γ x, y otherwise Difference Map (DM) Averaged Successive Reflection (ASR) Hybrid Projection Reflection (HPR) Random Averaged Alternating Reflector (RAAR) Levi-Stark method (LS) g k+1 = I + βp s 1 + γ s P m γ s I βp m 1 + γ m P s γ m I g k g k+1 = 1 2 R sr m + I g k g k+1 = 1 2 R s R m + β 1 P m g k+1 = 1 2 β R sr m + I + 1 β P m G k+1 = 1 λ G k + FT λp s P m g k + I + 1 β P m g k g k

6 Introduction Filtering Results Conclusion Domain Intersection Analog Each projection imposes a constraint. Image resides in the Support Domain (S) Image Fourier modulus resides in the Fourier modulus domain (M) Upon convergence none of the constraints affect the data, i.e. the image is a member of both domains simultaneously. Phase retrieval problem has an analog: find the intersection of two non-convex domains. The analog can be formulated as a two dimensional problem for easy analysis.

7 Introduction Filtering Results Conclusion 2D Visualization Method 2 1 M ER SF HIO DM ASR HPR RAAR g S

8 Noise Model Introduction Filtering Results Conclusion Noise can be modeled as F ±u, ±v = F True u, v 1 + N, σ + N, σ i Here the quantity σ is called the Noise-To-Signal Level. Model is borrowed from intensity interferometry theory. Any other noise model can be considered without changing the rest of this analysis.

9 Introduction Filtering Results Conclusion Effect of Noise on the HIO Oscillation occurs in both image and modulus constraint violations. Oscillations tend to be out of phase. Alternation in semi-satisfied constraint. Mod. Constr. Violation Support Error Image Constraint Violation Modulus Constraint Violation 2.5 x Iteration x 14 Iteration Iteration Iteration

10 Introduction Filtering Results Conclusion Analysis of the Effect of Noise Consider the simple Error-Reduction method, g k+1 = P s P m g k. Let each u, v pixel have a variation Δ from its proper value. Noisy modulus is G u, v = F True u, v + a,b Δ a, b δ u a, v b By linearity of the DFT, gain insight by looking at one noise component, i.e. G u, v = Δδ u a, v b. By going through a phase retrieval iteration, the effect of Δ should be eliminated.

11 Introduction Filtering Results Conclusion Analysis of the Effect of Noise (2) Start G k abs( g' k ) arg( g' k ) u a) Initial Fourier modulus Initial Fourier Modulus x b) Unconstrained Image.25 Unconstrained Image.8.2 Finish G k g k u d) Final Fourier modulus Constrained Fourier Modulus x c) Constrained Image Constrained Image

12 Introduction Filtering Results Conclusion Analysis of the Effect of Noise (3) After one iteration Maximum: G = Δ M 2A Frobeniun norm: Σ M 1 u= G u 2 = Δ M 2A M Filtering is proportional to oversampling. M This leads to the idea that the modulus should be constrained to a mix of the previous iteration s modulus and the measured modulus.

13 Introduction Filtering Results Conclusion New Fourier Modulus Projection Modulus should be constrained to a mix of the measured and previous iteration s modulus. New projection operator: P m g x, y = FT 1 F u, v exp iφ FT g x, y F u, v = 1 λ F u, v Typical relaxation parameter values: To mitigate constraint oscillation:.1 To filter noise:.9 + λ FT g x, y

14 Introduction Filtering Results Conclusion Intersection / No Relaxation ER SF HIO DM ASR HPR RAAR 2 1 S M

15 Introduction Filtering Results Conclusion Relaxation after Iteration ER SF HIO DM ASR HPR RAAR 2 1 S No Spiral M

16 Introduction Filtering Results Conclusion No Intersection / No Relaxation ER SF HIO DM ASR HPR RAAR 2 1 S M

17 Introduction Filtering Results Conclusion Relaxation after Iteration No Divergence ER SF HIO DM ASR HPR RAAR 2 1 S M

18 HIO+Relaxation Introduction Filtering Results Conclusion a) True Image a) True Image a) True Image b) Estimate, Iteration 5 b) Estimate, Iteration 5 b) Estimate, Iteration 5 c) Estimate, Iteration 1 c) Estimate, Iteration 1 c) Estimate, Iteration 1 True Image Iteration 5 No relaxation yet Iteration 1 Relaxed Modulus Constraint

19 Modulus Error Support Error Mod. Constr. Violation Introduction Filtering Results Conclusion HIO+Relaxation Error Metrics 2.5 x 19 5 x 14 Image Constraint Violation Modulus Constraint Violation Iteration Iteration 3.5 x Iteration Iteration Fourier Modulus Error % Reduction Iteration Iteration

20 Conclusion Introduction Filtering Results Conclusion Phase retrieval methods have problem with convergence given low SNR data. Performed an analysis on how the noise affects the phase retrieval operators. Developed a new operator with a filtering rationale. Results show up to 75% noise is filtered in various examples with ~2x oversampling.

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22 Analysis of the Effect of Noise G u, v = Δδ u a, v b g x, y = Δ ax exp i2π + by MN M N Apply P s g x, y = Δ MN 1, A m M 1 A & B n N 1, otherwise G u, v = Δ MN sin M 2A M πu sin πu M sin N 2B N πv sin πv N After one ER iteration, the Fourier modulus has less contribution from Δ.