Fourier phase retrieval with random phase illumination

Transcription

1 Fourier phase retrieval with random phase illumination Wenjing Liao School of Mathematics Georgia Institute of Technology Albert Fannjiang Department of Mathematics, UC Davis. IMA August 5, 27

2 Classical phase retrieval Fourier Transform: Let {f (n)} be a discrete array with n = (n,..., n d ) Z d. We assume n N and d 2. F (ω) = N f (n)e 2πiω n n= ω = (ω,..., ω d ) [, ] d Phase retrieval: recover f from the Fourier intensity measurements Y = Φf, where Φ represents Fourier transform. Chanllenge: non-uniqueness non-convexity 2 / 26

3 Non-uniqueness Sampling: L = F (ω) 2 = N n= N f (m + n)f (m) m }{{} auto-correlation function of f { (ω,..., ω d ) : ω k =, 2N k +,..., Oversampling ratio = 2 d e 2πin ω } 2N k 2N k +, k =,..., d Trivial ambiguity: the auto-correlation function is invariant under: spatial translation: f ( ) f ( + t) conjugate reflection: f ( ) f (N ) constant global phase change: f ( ) e iθ f ( ) 2 Nontrivial ambiguity: exists for few images when d 2 [Bruck & Sodin 979, Bates 982, Hayes 982] 3 / 26

4 Non-convexity Standard phasing algorithms: [Fienup 982, Marchesini 27] Find f such that f n (n)e 2πiω n = n f (n)e 2πiω n, ω L 2 f satisfies the same constraint as f, such as the support or positivity constraint (a) cameraman (b) phantom Figure : Images with a loose support 4 / 26

5 relative residual at each iteration iteration relative residual at each iteration iteration relative residual at each iteration iteration relative residual at each iteration iteration Numerical stagnation [Fienup and Wackerman 986] Residual: r(f k ) = Φf k Y Algorithms: Error Reduction (ER) [Gerchberg & Saxton 972, Fienup 982] and Hybrid Input Output (HIO) [Fienup 982] relative residual relative residual (a) ER (b) Residual (c) HIO + ER (d) Residual relative residual relative residual (e) ER (f) Residual (g) HIO + ER (h) Residual 5 / 26

6 Outline Our approach: random phase illumination 2 Recovery with exact mask information Uniqueness Numerical results 3 Recovery with phase-uncertain mask Uniqueness Phasing algorithms Numerical results 6 / 26

7 Random phase illumination Structured illuminations for phase retrieval: Ptychography [Thibault et al. 28, Dierolf et al. 2, Maiden and Rodenburg 29, Marchesini et al ] Masking [Marchesini et al. 28, Candès et al. 2,... ] Inverse Problems 28 (22) 758 Random phase mask: Inverse Problems 28 (22) 758 A Fannjiang A Fannjiang (a) (a) (b) (b) Figure. Illumination Figureof. aillumination partially transparent of a partiallyobject transparent (the blue object oval) (thewith blue aoval) deterministic with a deterministic (a) or (a) or random field λ created randomby field a diffuser λ created(b) byfollowed a diffuser by (b) followed intensity by an measurement intensity measurement of the diffraction of the diffraction pattern. In the case pattern. of wavefront In the casereconstruction, of wavefront reconstruction, therandom modulator theillumination randomis modulator placed at isthe placed exitatpupil the exit pupil instead of the entrance insteadpupil of theas entrance in (b). pupil as in (b). deterministic illumination λ(n) = λ(n) = e iφ(n) : φ(n) random, known or roughly known problem of stagnation problem of may stagnation be duemay to the be possibility due to the possibility of the iterative of the process iterative process to approach to approach the object and the its object twin or and shifted its twin image, or shifted the support image, the not support tight enough not tight or enough the boundary or the boundary not not sharp enough [3, sharp 4, enough 8]. [3, Besides 4, 8]. the Besides uniqueness the uniqueness issue, phase issue, retrieval phase retrieval is also inherently is also inherently nonconvex and many nonconvex and many researchers have researchers have believed the lack of convexity in the Fourier magnitude believed the lack of convexity in the Fourier magnitude constraint to be a main, constraint to be a main, if not the if not the dominant, source of numerical problems with the standard dominant, source of numerical problems with the standard phasing algorithms [4, 2, 3]. While there have been dazzling advances in applications of phasing algorithms [4, 2, 3]. While there f (n)λ(n)e 2πin ω have been dazzling advances in applications of phase retrieval in the past decades [22], we still do not know just how much of the error and phase retrieval in the past decades [22], we still do not know just how much of the error and stagnation problems is attributable to to the lack of uniqueness or convexity. stagnation problems is attributable to to the lack of uniqueness or convexity. We propose here ton refocus on the issue of uniqueness as uniqueness is undoubtedly We propose here to refocus on the issue of uniqueness as uniqueness is undoubtedly the first foundational issue of any inverse problem, including phase retrieval. Specifically the first foundational issue of any inverse problem, including phase retrieval. Specifically we will first establish uniqueness in the absolute sense with random illumination under we will first establish general, physically uniqueness reasonable in the absolute object constraints sense with (figure random ) and illumination secondlydemonstrate under that7 / 26 Measurement: Y (ω) =

8 How does the problem change? f physical process Φλf Classical phase retrieval Φf λ(n) = phase retrieval f Phase retrieval with random phase mask } Φλf phase retrieval f known or roughly known λ(n) 8 / 26

9 Outline Our approach: random phase illumination 2 Recovery with exact mask information Uniqueness Numerical results 3 Recovery with phase-uncertain mask Uniqueness Phasing algorithms Numerical results 9 / 26

10 Uniqueness with exact mask information Real-valued images: Theorem [Fannjiang 22] Suppose f is real-valued and its support has rank 2. Then with probability one, f is determined by one random phase illumination up to a ± sign. Complex-valued images: Theorem [Fannjiang 22] Suppose the support of f has rank 2. Then with probability one, f is determined by two random phase illuminations up to a global phase. The rank of a support set is the dimension of its convex hull. / 26

11 relative residual at each iteration iteration relative residual at each iteration iteration relative residual at each iteration iteration relative residual at each iteration iteration Numerical results relative residual relative residual (a) ER (b) Residual (c) HIO + ER (d) Residual relative residual relative residual (e) ER (f) Residual (g) HIO + ER (h) Residual We observe global convergence as the residual decreases to. / 26

12 Stability to noise Error versus noise-to-signal ratio average relative error in 5 trials relative error versus noise level high resolution & gaussian noise low resolution & gaussian noise high resolution & poisson noise low resolution & poisson noise high resolution & illuminator noise low resolution & illuminator noise average relative error in 5 trials relative error versus noise level one uniform + one high resolution & gaussian noise one uniform + one low resolution & gaussian noise one uniform + one high resolution & poisson noise one uniform + one low resolution & poisson noise one uniform + one high resolution & illuminator noise one uniform + one low resolution & illuminator noise noise percentage noise percentage Left: non-negative phantom with one random phase mask and oversampling rate = 2 Right: complex-valued image whose intensity is the phantom with two random phase masks and oversampling rate = 3 2 / 26

13 Outline Our approach: random phase illumination 2 Recovery with exact mask information Uniqueness Numerical results 3 Recovery with phase-uncertain mask Uniqueness Phasing algorithms Numerical results 3 / 26

14 Phase-uncertain mask Exact mask: λ(n) = e iφ(n) Mask estimate: λ (n) = e iφ (n) φ(n) φ (n) δπ, φ (n) + δπ φ (n) ± δπ, n, φ (n) φ(n) δπ Fourier magnitude data: Y = Φλf Fourier phase retrieval with a roughly known mask: } Φλf phase retrieval f λ 4 / 26

15 Uniqueness for real-valued images with one phase-uncertain mask Theorem [Fannjiang and L. 23] Suppose f is a real-valued image with sparsity S and its support has rank 2. The exact mask phases {φ(n)} are i.i.d. samples in [, 2π) and δ < /2. If we can find real-valued f and λ(n) = exp (i φ(n)) such that then Φ λ f = Φλf ; λ satisfies mask constraint: φ(n) φ (n) ± δπ, f (n) = ±f (n) n λ(n) = e iθ λ(n) on n where f (n) with probability no less than (2δ) S/2 d j= (N j + ). 5 / 26

16 Uniqueness for complex-valued images with two phase-uncertain masks Theorem [Fannjiang and L. 23] Suppose f is a complex-valued image with sparsity S and its support has rank 2. The first mask λ () = λ has i.i.d. random phases and δ <. The second mask λ (2) is exactly known. If we can find f and λ(n) = exp (i φ(n)) such that then Φ λ f = Φλf and Φλ (2) f = Φλ (2) f ; λ satisfies mask constraint: φ(n) φ (n) ± δπ, f (n) = e iα f (n) n λ(n) = e iα2 λ(n) on n where f (n), with probability no less than δ S/2 d j= (N j + ). 6 / 26

17 Algorithm: alternating updates between image and mask Example: recovery of real-valued images with one phase-uncertain mask Objective: Find f and λ such that Φ λ f = Φλf ; f is real-valued; λ satisfies mask constraint. Alternating updates: (f k, λ k ) standard phasing method of (f k+, λ k ) (f k+, λ k+ ). algorithms alternating projection Image update: (f k, λ k ) (f k+, λ k ) by standard phasing algorithms [Fienup 982, Marchesini 27] Mask update: (f k+, λ k ) (f k+, λ k+ ) by alternating projection 7 / 26

18 standard phasing Image update (f k, λ k ) (f k+, λ k ) algorithms Two operators: Object-constraint fitting operator P o Fourier intensity data fitting operator P f,k = λ k Φ T Φλ k where { Y (ω) exp (i G(ω)) if G(ω) > T G(ω) = Y (ω) if G(ω) =. where Y = Φλf. Two phasing algorithms: Error Reduction (ER) [Gerchberg & Saxton 972, Fienup 982]: f k+ = P o P f,k f k Douglas-Rachford (DR) [Douglas & Rachford 956]: f k+ = I + (2P o I )(2P f,k I ) f k. 2 8 / 26

19 method of Mask udpate (f k+, λ k ) (f k+, λ k+ ) alternating projection Two operators: Fourier intensity data fitting operator Q f,k { ] [Φ λ k = Q f,kλ k (n) = T ΦΛ k f k+ (n)/f k+ (n) if f k+ (n). λ k (n) else. Mask-constraint fitting operator Q m φ (n) δπ { Q m λ exp(i λ k (n) = k (n)) orthogonal projection λ k (n) φ (n) ± δπ else.. Mask update by alternating projection: λ k+ = Q m Q f,k λ k 9 / 26

20 Alternating updates between image and mask (f k, λ k ) standard phasing method of (f k+, λ k ) (f k+, λ k+ ) algorithms alternating projection Alternating Error Reduction (AER): (f k+, λ k+ ) = (P o P f,k f k, Q m Q f,k λ k ) (AER) Theorem Let r(f k, λ k ) = ΦΛ k f k Y. ER iteration satisfies r(f k+, λ k+ ) r(f k, λ k ). Alternating Douglas Rachford and Error Reduction (DRER): ( ) (f k+, λ k+ ) = 2 [I + (2P o I )(2P f,k I )]f k, Q m Q f,k λ k (DRER) 2 / 26

21 Test images and masks Test images: cameraman mandrill 2 2 phantom High-Resolution Mask (HRM) and Low-Resolution Mask (LRM): HRM LRM 2 / 26

22 Non-negative images and one phase-uncertain mask Illumination: One low-resolution phase-uncertain mask with uncertainty δ =.3 Algorithm: DRER + AER recovered images mask error residual.9 Relative residual of each iterate iteration.9 Relative residual of each iterate iteration.9 Relative residual of each iterate iteration 22 / 26

23 How large can the mask uncertainty δ be? Image error versus δ with % gaussian or poisson noise average relative error in 5 trials relative error versus mask uncertainty HRM, % gaussian nosie LRM, % gaussian nosie HRM, % poisson noise LRM, % poisson noise average relative error in 5 trials relative error versus mask uncertainty HRM, % gaussian nosie LRM, % gaussian nosie HRM, % poisson noise LRM, % poisson noise uncertainty level (i) non-negative cameraman with one phase-uncertain mask uncertainty level (j) complex-valued image whose intensity is cameraman with one uniform illumination and one phaseuncertain mask 23 / 26

24 Summary Phase retrieval with random phase illumination uniqueness global convergence and stability of the standard phasing algorithms 2 Phase-uncertain mask uniqueness of the image and the mask algorithm: alternating update between the image and the mask good reconstruction of the image and the mask 3 Convergence analysis for non-convex problems Wirtinger flow [Candès, Li and Soltanolkotabi 25] Geometric analysis of phase retrieval [Sun, Qu and Wright 26] 24 / 26

25 Continuing: sensor calibration in imaging N sensors s point sources location: ω j [, ) amplitudes: x j C Measurements: {y n : n =,..., N } where Unknowns: y n = g n s j= x je 2πinω j + e n Calibration parameters {g n } N n= Source locations {ω j } s j= and source amplitudes {x j} s j= Connection to phase retrieval: equivalent to phase retrieval if g n = Y. C. Eldar, W. Liao and S. Tang, Sensor calibration for off-the-grid spectral estimation, arxiv: / 26

26 Thank you for your attention! Wenjing Liao Georgia Institute of Technology 26 / 26