Phase Retrieval from Local Measurements: Deterministic Measurement Constructions and Efficient Recovery Algorithms

Transcription

1 Phase Retrieval from Local Measurements: Deterministic Measurement Constructions and Efficient Recovery Algorithms Aditya Viswanathan Department of Mathematics and Statistics www-personal.umich.edu/ adityavv 2018 SIAM Conference on Imaging Science (IS18), Bologna, Italy June 8 th, 2018

2 Collaborators Mark Iwen Rayan Saab Brian Preskitt Yang Wang Research supported in part by NSF grant DMS

3 Motivating Application From Huang, Xiaojing, et al. Fly-scan ptychography. Scientific Reports 5 (2015). The Phase Retrieval problem arises in many molecular imaging modalities, including X-ray crystallography Ptychography Other applications can be found in optics, astronomy and speech processing. 1 / 21

4 Motivation Inverse Problem Given y, can we recover x? 2 / 21

5 Motivation Ill-Posed Inverse Problem Which is the correct reconstruction? 3 / 21

6 Mathematical Model where find 1 x C d given y i = a i, x 2 + η i i 1,..., D, y i R denotes the phaseless (or magnitude-only) measurements (D measurements acquired), a i C d are known (by design or estimation) measurement vectors, and η i R is measurement noise. 1 (upto a global phase offset) 4 / 21

7 Existing Computational Approaches Alternating projection methods [Fienup, 1978], [Marchesini et al., 2006], [Fannjiang, Liao, 2012] and many others... Methods based on semidefinite programming PhaseLift [Candes et al., 2012], PhaseCut [Waldspurger et al., 2012],... Others Frame-theoretic, graph based algorithms [Alexeev et al., 2014] (Spectral) initialization + gradient descent (Wirtinger Flow) [Candes et al., 2014] Most methods (with provable recovery guarantees) require impractical (global, random) measurement constructions. 5 / 21

8 Today... We discuss a recently introduced fast (essentially linear-time) phase retrieval algorithm based on realistic (deterministic) 2 local measurement constructions. We provide rigorous theoretical recovery guarantees and present numerical results showing the accuracy, efficiency and robustness of the method. (Time Permitting) extensions to 2D and compressive phase retrieval. 2 for a large class of real-world signals 6 / 21

9 Outline 1 Introduction 2 Solving the Phase Retrieval Problem Measurement Constructions Structured Lifting Obtaining Phase Difference Information Angular Synchronization Solving for the Individual Phases 3 Theoretical Guarantees 4 Numerical Simulations 5 Extensions

10 Local Correlation Measurements From Qian, Jianliang, et al. Efficient algorithms for ptychographic phase retrieval. Inverse Problems Appl., Contemp. Math 615 (2014). Each a i is a shift of a locally-supported vector (mask or window) m (j) C d, supp(m (j) ) = [δ] [d], j = 1,..., K 7 / 21

11 Local Correlation Measurements Each a i is a shift of a locally-supported vector (mask or window) m (j) C d, supp(m (j) ) = [δ] [d], j = 1,..., K Define the discrete circular shift operator S l : C d C d with (S l x) j = x l+j. Our measurements are then (y l ) j = x, S l m (j) 2 + η j,l, (j, l) [K] P, P {0,..., d 1} We will consider K δ and P = [d] 0 := {0,..., d 1} 7 / 21

12 Local Correlation Measurements Each a i is a shift of a locally-supported vector (mask or window) m (j) C d, supp(m (j) ) = [δ] [d], j = 1,..., K Define the discrete circular shift operator S l : C d C d with (S l x) j = x l+j. Our measurements are then (y l ) j = x, S l m (j) 2 + η j,l, (j, l) [K] P, P {0,..., d 1} We will consider K δ and P = [d] 0 := {0,..., d 1} 7 / 21

13 Local Correlation Measurements Each a i is a shift of a locally-supported vector (mask or window) m (j) C d, supp(m (j) ) = [δ] [d], j = 1,..., K Define the discrete circular shift operator S l : C d C d with (S l x) j = x l+j. Our measurements are then (y l ) j = x, S l m (j) 2 + η j,l, (j, l) [K] P, P {0,..., d 1} We will consider K δ and P = [d] 0 := {0,..., d 1} 7 / 21

14 Local Correlation Measurements Each a i is a shift of a locally-supported vector (mask or window) m (j) C d, supp(m (j) ) = [δ] [d], j = 1,..., K Define the discrete circular shift operator S l : C d C d with (S l x) j = x l+j. Our measurements are then (y l ) j = x, S l m (j) 2 + η j,l, (j, l) [K] P, P {0,..., d 1} We will consider K δ and P = [d] 0 := {0,..., d 1} 7 / 21

15 What are we measuring? Lifted System: x, S l m(j) 2 = xx, S l m(j) m (j) S l. Example: (6 6 system, δ = 2, blue denotes non-zero entries) xx, S0m (j) m (j) S 0 = xx, / 21

16 What are we measuring? Lifted System: x, S l m(j) 2 = xx, S l m(j) m (j) S l. Example: (6 6 system, δ = 2, blue denotes non-zero entries) xx, S1m (j) m (j) S 1 = xx, / 21

17 What are we measuring? Lifted System: x, S l m(j) 2 = xx, S l m(j) m (j) S l. Example: (6 6 system, δ = 2, blue denotes non-zero entries) xx, S2m (j) m (j) S 2 = xx, / 21

18 What are we measuring? Lifted System: x, S l m(j) 2 = xx, S l m(j) m (j) S l. Example: (6 6 system, δ = 2, blue denotes non-zero entries) xx, S3m (j) m (j) S 3 = xx, / 21

19 What are we measuring? Lifted System: x, S l m(j) 2 = xx, S l m(j) m (j) S l. Example: (6 6 system, δ = 2, blue denotes non-zero entries) xx, S4m (j) m (j) S 4 = xx, / 21

20 What are we measuring? Lifted System: x, S l m(j) 2 = xx, S l m(j) m (j) S l. Example: (6 6 system, δ = 2, blue denotes non-zero entries) xx, S5m (j) m (j) S 5 = xx, / 21

21 What are we measuring? Lifted System: x, S l m(j) 2 = xx, S l m(j) m (j) S l. Example: (6 6 system, δ = 2, blue denotes non-zero entries) Observation: The only entries of xx we can hope to recover (via linear inversion) are supported on a (circulant) band / 21

22 Useful Observations (I) T δ (C d d ): Let T k : C d d C d d T k (A) ij = { Aij, i j mod d < k 0, otherwise. Lifted System Revisited: x, Sl m(j) 2 = T δ (xx ), Sl m(j) m (j) S l. Bottom Line: If we can find m (j) such that Span { Sl m (j) m (j) S l }l,j = T δ(c d d ), then we can recover T δ (xx ). 9 / 21

23 Useful Observations (I) T δ (C d d ): Let T k : C d d C d d T k (A) ij = { Aij, i j mod d < k 0, otherwise. Lifted System Revisited: x, Sl m(j) 2 = T δ (xx ), Sl m(j) m (j) S l. Bottom Line: If we can find m (j) such that Span { Sl m (j) m (j) S l }l,j = T δ(c d d ), then we can recover T δ (xx ). 9 / 21

24 Useful Observations (I) T δ (C d d ): Let T k : C d d C d d T k (A) ij = { Aij, i j mod d < k 0, otherwise. Lifted System Revisited: x, Sl m(j) 2 = T δ (xx ), Sl m(j) m (j) S l. Bottom Line: If we can find m (j) such that Span { Sl m (j) m (j) S l }l,j = T δ(c d d ), then we can recover T δ (xx ). 9 / 21

25 A Highly Structured Linear System! Example linear system M q = ỹ, where q denotes the vectorized (non-zero) entries of T δ (xx ) ỹ denotes the (interleaved) measurements 10 / 21

26 Useful Observations (II) Why this is useful: (a) Diagonal entries of T δ (xx ) are x i 2. (b) Off-diagonals give the relative phases X := xx xx T δ ( X) (j,k) = e i(arg(xj) arg(x k)), j k mod d < δ Phase Synchronization: (a) The leading eigenvector (appropriately normalized) of ( ) ( ) T δ ( X) x x = diag T δ (11 ) diag x x ( ) ( ) x x = diag F ΛF diag x x Note: is the vector of phases of x. x x = [eiφ 1 e iφ 2... e iφ d] T is the (unknown) phase vector! F C d d is the discrete Fourier transform (DFT) matrix 11 / 21

27 Useful Observations (II) Why this is useful: (a) Diagonal entries of T δ (xx ) are x i 2. (b) Off-diagonals give the relative phases X := xx xx T δ ( X) (j,k) = e i(arg(xj) arg(x k)), j k mod d < δ Phase Synchronization: (a) The leading eigenvector (appropriately normalized) of ( ) ( ) T δ ( X) x x = diag T δ (11 ) diag x x ( ) ( ) x x = diag F ΛF diag x x Note: is the vector of phases of x. x x = [eiφ 1 e iφ 2... e iφ d] T is the (unknown) phase vector! F C d d is the discrete Fourier transform (DFT) matrix 11 / 21

28 Leading Eigenvector Phase Vector [ x1 2 x 1 x 2 x 2 x 1 x 2 2 x 2 x 3 x 3 x 2 x 3 2 x 3 x 4 x 4 x 3 x 4 2 x 4 x 1 x 1 x 4] T x 1 2 x 1 x 2 0 x 1 x 4 x 2 x 1 x 2 2 x 2 x x 3 x 2 x 3 2 x 3 x 4 x 4 x 1 0 x 4 x 3 x 4 2 (re-arrange) (T δ(xx ), 2δ 1 entries in band) (normalize) 1 e i(φ1 φ2) 0 e i(φ1 φ4) e i(φ2 φ1) 1 e i(φ2 φ3) 0 0 e i(φ3 φ2) 1 e i(φ3 φ4) e i(φ4 φ1) 0 e i(φ4 φ3) 1 (top eigenvector) [e iφ1 e iφ2 e iφ3 e iφ4 ] T (Signal Reconstruction) [ x1 e i φ1 x 2 e i φ2 x 3 e i φ3 x 4 e i φ4 ] T 12 / 21

29 Leading Eigenvector Phase Vector [ x1 2 x 1 x 2 x 2 x 1 x 2 2 x 2 x 3 x 3 x 2 x 3 2 x 3 x 4 x 4 x 3 x 4 2 x 4 x 1 x 1 x 4] T x 1 2 x 1 x 2 0 x 1 x 4 x 2 x 1 x 2 2 x 2 x x 3 x 2 x 3 2 x 3 x 4 x 4 x 1 0 x 4 x 3 x 4 2 (re-arrange) (T δ(xx ), 2δ 1 entries in band) (normalize) 1 e i(φ1 φ2) 0 e i(φ1 φ4) e i(φ2 φ1) 1 e i(φ2 φ3) 0 0 e i(φ3 φ2) 1 e i(φ3 φ4) e i(φ4 φ1) 0 e i(φ4 φ3) 1 (top eigenvector) [e iφ1 e iφ2 e iφ3 e iφ4 ] T (Signal Reconstruction) [ x1 e i φ1 x 2 e i φ2 x 3 e i φ3 x 4 e i φ4 ] T 12 / 21

30 Leading Eigenvector Phase Vector [ x1 2 x 1 x 2 x 2 x 1 x 2 2 x 2 x 3 x 3 x 2 x 3 2 x 3 x 4 x 4 x 3 x 4 2 x 4 x 1 x 1 x 4] T x 1 2 x 1 x 2 0 x 1 x 4 x 2 x 1 x 2 2 x 2 x x 3 x 2 x 3 2 x 3 x 4 x 4 x 1 0 x 4 x 3 x 4 2 (re-arrange) (T δ(xx ), 2δ 1 entries in band) (normalize) 1 e i(φ1 φ2) 0 e i(φ1 φ4) e i(φ2 φ1) 1 e i(φ2 φ3) 0 0 e i(φ3 φ2) 1 e i(φ3 φ4) e i(φ4 φ1) 0 e i(φ4 φ3) 1 (top eigenvector) [e iφ1 e iφ2 e iφ3 e iφ4 ] T (Signal Reconstruction) [ x1 e i φ1 x 2 e i φ2 x 3 e i φ3 x 4 e i φ4 ] T 12 / 21

31 Leading Eigenvector Phase Vector [ x1 2 x 1 x 2 x 2 x 1 x 2 2 x 2 x 3 x 3 x 2 x 3 2 x 3 x 4 x 4 x 3 x 4 2 x 4 x 1 x 1 x 4] T x 1 2 x 1 x 2 0 x 1 x 4 x 2 x 1 x 2 2 x 2 x x 3 x 2 x 3 2 x 3 x 4 x 4 x 1 0 x 4 x 3 x 4 2 (re-arrange) (T δ(xx ), 2δ 1 entries in band) (normalize) 1 e i(φ1 φ2) 0 e i(φ1 φ4) e iφ1 e iφ1 e i(φ2 φ1) 1 e i(φ2 φ3) 0 e iφ2 0 e i(φ3 φ2) 1 e i(φ3 φ4) e iφ3 = (2δ 1) e iφ2 e iφ3 e i(φ4 φ1) 0 e i(φ4 φ3) 1 e iφ4 e iφ4 (leading eigenvector) [e iφ1 e iφ2 e iφ3 e iφ4 ] T (Signal Reconstruction) [ x1 e i φ1 x 2 e i φ2 x 3 e i φ3 x 4 e i φ4 ] T 12 / 21

32 Leading Eigenvector Phase Vector [ x1 2 x 1 x 2 x 2 x 1 x 2 2 x 2 x 3 x 3 x 2 x 3 2 x 3 x 4 x 4 x 3 x 4 2 x 4 x 1 x 1 x 4] T x 1 2 x 1 x 2 0 x 1 x 4 x 2 x 1 x 2 2 x 2 x x 3 x 2 x 3 2 x 3 x 4 x 4 x 1 0 x 4 x 3 x 4 2 (re-arrange) (T δ(xx ), 2δ 1 entries in band) (normalize) 1 e i(φ1 φ2) 0 e i(φ1 φ4) e iφ1 e iφ1 e i(φ2 φ1) 1 e i(φ2 φ3) 0 e iφ2 0 e i(φ3 φ2) 1 e i(φ3 φ4) e iφ3 = (2δ 1) e iφ2 e iφ3 e i(φ4 φ1) 0 e i(φ4 φ3) 1 e iφ4 e iφ4 (leading eigenvector) [e iφ1 e iφ2 e iφ3 e iφ4 ] T (Signal Reconstruction) [ x1 e i φ1 x 2 e i φ2 x 3 e i φ3 x 4 e i φ4 ] T 12 / 21

33 Recovery Algorithm Define the map A : C d d C D A(Z) (l,j) = Z, Sl m (j) m (j) S l (l,j). and its restriction A Tδ (C d d ) to our subspace. 13 / 21

34 Recovery Algorithm Define the map A : C d d C D A(Z) (l,j) = Z, Sl m (j) m (j) S l (l,j). and its restriction A Tδ (C d d ) to our subspace. In the noisy setting: Step 1: Step 2: Estimate T δ (xx ) by the banded matrix Z = T δ (Z) := ( A 1 T δ (C d d ) y ) ( + A 1 2 T δ (C d d ) y ). 2 Estimate ( the ) phase by computing the leading eigenvector Z of T δ. Z Step 3: Combine phase with of diagonal entries of T δ (Z) to estimate x. 13 / 21

35 Recovery Algorithm Define the map A : C d d C D A(Z) (l,j) = Z, Sl m (j) m (j) S l (l,j). and its restriction A Tδ (C d d ) to our subspace. In the noisy setting: Step 1: Step 2: Estimate T δ (xx ) by the banded matrix Z = T δ (Z) := ( A 1 T δ (C d d ) y ) ( + A 1 2 T δ (C d d ) y ). 2 Estimate ( the ) phase by computing the leading eigenvector Z of T δ. Z Step 3: Combine phase with of diagonal entries of T δ (Z) to estimate x. 13 / 21

36 Recovery Algorithm Define the map A : C d d C D A(Z) (l,j) = Z, Sl m (j) m (j) S l (l,j). and its restriction A Tδ (C d d ) to our subspace. In the noisy setting: Step 1: Step 2: Estimate T δ (xx ) by the banded matrix Z = T δ (Z) := ( A 1 T δ (C d d ) y ) ( + A 1 2 T δ (C d d ) y ). 2 Estimate ( the ) phase by computing the leading eigenvector Z of T δ. Z Step 3: Combine phase with of diagonal entries of T δ (Z) to estimate x. 13 / 21

37 Recovery Algorithm Define the map A : C d d C D A(Z) (l,j) = Z, Sl m (j) m (j) S l (l,j). and its restriction A Tδ (C d d ) to our subspace. In the noisy setting: Step 1: Estimate T δ (xx ) by Cost: O(d δ 3 + δ d log d) flops Step 2: Z = T δ (Z) := ( A 1 T δ (C d d ) y ) ( + A 1 2 T δ (C d d ) y ). 2 Estimate ( the ) phase by computing the leading eigenvector Z of T δ. Cost: O(δ 2 d log d) flops Z Step 3: Combine phase with of diagonal entries of T δ (Z) to estimate x. Total Cost: O(δ 2 d log d + d δ 3 ) flops 13 / 21

38 Outline 1 Introduction 2 Solving the Phase Retrieval Problem Measurement Constructions Structured Lifting Obtaining Phase Difference Information Angular Synchronization Solving for the Individual Phases 3 Theoretical Guarantees 4 Numerical Simulations 5 Extensions

39 Well-Conditioned Linear Systems Theorem (Iwen, V., Wang 2015) Choose entries of the measurement mask (m (i) ) as follows: { { } (m (i) e 4 l/a ) l = 2δ 1 e 2πi i l 2δ 1, l δ 0, l > δ, a := max 4, δ 1 2, i = 1, 2,..., N. Then, the resulting system matrix for the phase differences (step 1), A Tδ, has condition number } κ(a Tδ ) < max {144e 2, 9e2 (δ 1)2. 4 Deterministic (windowed DFT-type) measurement masks! δ is typically chosen to be c log 2 d with c small (2 3). Extensions: oversampling, random masks / 21

40 Recovery Guarantee Theorem (Iwen, Preskitt, Saab, V. 2016) Let x min := min j x j be the smallest magnitude of any entry in x. Then, the estimate z produced by the proposed algorithm satisfies min θ [0,2π] x e iθ z ( ) ( ) 2 x d 2 C x 2 κ η 2 + Cd 1 4 κ η 2, min δ where C R + is an absolute universal constant. This result yields a deterministic recovery result for any signal x which contains no zero entries. A randomized result can be derived for arbitrary x by right multiplying the signal x with a random flattening matrix. (this is also useful for performing sparse phase retrieval!) 15 / 21

41 Outline 1 Introduction 2 Solving the Phase Retrieval Problem Measurement Constructions Structured Lifting Obtaining Phase Difference Information Angular Synchronization Solving for the Individual Phases 3 Theoretical Guarantees 4 Numerical Simulations 5 Extensions

42 Executime Time (secs.) Efficiency FFT time phase retrieval 10 4 Execution Time (D = d2d log 2 (d)e measurements) 10 2 O(d 3 ) 10 0 O(d log 2 d) 10-2 PhaseLift HIO+ER (300 iters.) BlockPR (w/ 60 iters. HIO+ER) BlockPR (Alg. 1) Problem Size, d 16 / 21

43 Robustness to Measurement Errors / 21

44 Robustness to Measurement Errors / 21

45 Reconstruction Error vs. No. of Measurements / 21

46 Summary and Current/Future Research Directions Today Phase retrieval is an immensely challenging problem seen in important applications such as x-ray crystallography. Proposed mathematical framework: Essentially linear-time robust phase retrieval from deterministic local correlation measurement constructions with rigorous recovery guarantee. Current and Future Directions More robust measurement constructions Compressive phase retrieval Extensions to 2D and Ptychographic datasets Continuous problem formulation 19 / 21

47 Outline 1 Introduction 2 Solving the Phase Retrieval Problem Measurement Constructions Structured Lifting Obtaining Phase Difference Information Angular Synchronization Solving for the Individual Phases 3 Theoretical Guarantees 4 Numerical Simulations 5 Extensions

48 Extension 2D Phase Retrieval Preliminary results for 2D masks with tensor product structure Results from 1D extend to 2D; 2D linear system is a tensor product of the 1D linear system (up to row permutations) Eigenvector-based phase synchronization also works calculation of spectral gap and error analysis pending Test Image ( pixels) Recon. (Rel. error ) 20 / 21

49 Extension Compressive Phase Retrieval Model find x C d given Mx 2 + n = y R D where x is k-sparse, with k d, is entry-wise absolute value, and M is a measurement matrix. Measurement Design Assume M = PC where P C D d is an admissible phase retrieval matrix with an associated recovery algorithm Φ P : R D C d, and C C d d is an admissible compressive sensing matrix with an associated recovery algorithm C : C d C d. Recovery Algorithm (Two-stage) C Φ P : R D C d Performance Metrics No. of measurements required is O(k ln(d/k)) Computational cost (sub-linear) is O(k ln c k ln d) 21 / 21

50 Pubs./Preprints/Code (see www-personal.umich.edu/~adityavv) M. Iwen, B. Preskitt, R. Saab and A. Viswanathan. Phase Retrieval from Local Measurements: Improved Robustness via Eigenvector-Based Angular Synchronization. arxiv: , M. Iwen, A. Viswanathan, and Y. Wang. Fast Phase Retrieval from Local Correlation Measurements. SIAM J. Imag. Sci., Vol. 9(4), pp , Oct Compressive Phase Retrieval M. Iwen, A. Viswanathan, and Y. Wang. Robust Sparse Phase Retrieval Made Easy. Appl. Comput. Harmon. Anal., Vol. 42(1), pp , Jan D Phase Retrieval Mark Iwen, Brian Preskitt, Rayan Saab and A. Viswanathan. Phase Retrieval from Local Measurements in Two Dimensions., Proc. SPIE 10394, Wavelets and Sparsity XVII, X, Aug Code

51 Questions?