Conjugate Phase Retrieval in the Paley-Wiener Space Eric Weber Iowa - PowerPoint PPT Presentation

Conjugate Phase Retrieval in the Paley-Wiener Space Eric Weber Iowa State University CodEx Seminar September 22, 2020 Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 1 / 26

Acknowledgements ChunKit Lai Friedrich Littmann San Francisco State University North Dakota State University Support from NSF Award #1830254. Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 2 / 26

Phase Retrieval Problem (Phase Retrieval) Can a signal f be reconstructed from the magnitudes of linear measurements of f , up to the ambiguity of uniform phase factor α ? Formally, we define an equivalence class on the signal space H by: f ∼ g if f = α g for some | α | = 1. then ask whether the mapping A : H / ∼→ ℓ 2 ( I ) : f �→ ( | φ n ( f ) | ) n is injective, where φ n are linear functionals on H . If so, the next question is: how to invert A ? Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 3 / 26

Conjugate Phase Retrieval Suppose the signal space H has the property that if f ∈ H , then f ∈ H (e.g. C d or the Paley-Wiener space). We formulate a weaker variation: Problem (Conjugate Phase Retrieval–Evans & Lai [EL17]) Can a signal f be reconstructed from the magnitudes of linear measurements of f , up to the ambiguity of uniform phase factor α and the ambiguity of conjugation? Formally, we define an equivalence class on the signal space H by: f ∼ g if f = α g , or f = α g for some | α | = 1. then ask whether the mapping A : H / ∼→ ℓ 2 ( I ) : f �→ ( | φ n ( f ) | ) n is injective, where φ n are linear functionals on H . If so, the next question is: how to invert A ? Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 4 / 26

Paley-Wiener Space Definition For β > 0 , we denote PW β = { f ∈ L 2 ( R ) | ˆ f ( ξ ) = 0 a . e . | ξ | > β } . Definition If f ∈ PW β , then f ♯ ( z ) = f ( z ) ∈ PW β Note that for real z , f ♯ = f . From here on, our equivalence relation is on PW β : f ∼ g if f = α g , or f = α g ♯ for some | α | = 1. Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 5 / 26

The Problem with Phase Retrieval in the PW β Goal Design a sampling regime { φ n } on PW β that: does conjugate phase retrieval; 1 admits a numerical reconstruction method. 2 Preferably, the sampling regime occurs on the real axis. Problem If f ∈ PW β , | f ( x ) | does not determine f ( x ) up to unimodular scalar, or conjugation either (here, x ∈ R ). If ˆ f is supported in an interval smaller than 2 β , ˆ f can be shifted to remain in ( − β, β ), which modulates f ( x ). Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 6 / 26

Phase Retrieval: A Short History Optics: Gerchberg-Saxton [GS72], Fienup [Fie78], Rosenblatt [Ros84], 1 Levi-Stark [LS84] Inverse Spectral Theory [KS92, KST95] 2 Frames: Balan et. al. [BCE06, BBCE09] Bandeira et. al. [BCMN14] 3 Reconstructions: Alternating projections; Wirtinger Flow; PhaseLift; 4 PhaseMax; AltMinPhase; Kaczmarz; etc Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 7 / 26

Phase Retrieval: A Short History (cont’d) Paley-Wiener space: Thakur [Tha11], Pohl-Yang-Boche [PYB14]. 1 [Tha11] considers the case of real phase retrieval in PW π . The reconstruction 1 occurs off of the real axis, where there are no zeros of the function. [PYB14] considers the case of (complex) phase retrieval in PW π by designing 2 a sampling scheme that occurs off of the real axis. In particular, the sampling scheme as presented in [PYB14] takes the form � φ n ( f ) = c j , n f ( z n − b j , n ) (1) j for complex scalars c j , n , z n , b j , n . Sampling schemes such as this are referred to as structured modulations in [PYB14] because the authors there consider the reconstruction in the Fourier domain, where the shifts become modulations. Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 8 / 26

Conjugate Phase Retrieval: A Short History Proposition (McDonald [McD04]) Suppose f , g ∈ PW β . If b < β/π , and for all x ∈ R , | f ( x ) | = | g ( x ) | and 1 | f ( x + b ) − f ( x ) | = | g ( x + b ) − g ( x ) | , then f ∼ g. If for all x ∈ R , | f ( x ) | = | g ( x ) | and | f ′ ( x ) | = | g ′ ( x ) | , then f ∼ g. 2 Proposition (Evans-Lai [EL17]) � a 1 � � � � b 1 c 1 v 3 ∈ R 2 is written as If � v 1 , � v 2 , � v 1 � v 2 � v 3 = then � v 1 , � v 2 , � v 3 does a 2 b 2 c 2 conjugate phase retrieval in C 2 if and only if   a 2 a 2 2 a 1 a 2 1 2   � = 0 . b 2 b 2 det 2 b 1 b 2 (2) 1 2 c 2 c 2 2 c 1 c 2 1 2 Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 9 / 26

Conjugate Phase Retrieval in C d CPR is weaker than PR: there exist vectors in C d that do CPR but not PR. 1 Q: Can CPR be done with 3 d vectors? 2 No proven reconstruction method exists for Conjugate Phase Retrieval in C d . 3 We will use Gerchberg-Saxton for reconstruction. We show experimentally 4 that it works well. Q: Can CPR be made robust to noise? 5 Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 10 / 26

Gerchberg-Saxton for Reconstruction Given A that does conjugate phase retrieval, | A T � v | , reconstruct � w ∈ [ � v ]. Choose phases λ 1 , . . . , λ n ; 1 Apply ( A T ) † to 2 v n �| ) T ( λ 1 |� � v , � v 1 �| , . . . , λ n |� � v , � to obtain estimate � w ; w with ( A T ) † applied to Replace � 3 � � � � T w , � v 1 � v 1 �| , . . . , � � w , � v n � v 1 �||� � v , � v n �||� � v , � v n �| ; |� � w , � |� � w , � Repeat step 3. 4 Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 11 / 26

Gerchberg-Saxton Method Results Observed behavior: the alternating projections method converges (Levi-Stark) 1 the method may not converge to a solution (i.e. Traps) 2 Traps/Tunnels were observed in numerical simulations (reconstruction failed; 3 however, reseeding succeeds) for the 3 × 6 matrix A used in the PW example, reconstruction error was 4 < 10 − 4 after 400 iterations for approximately 80% of instances (performed 1000 random examples) Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 12 / 26

A Corollary of McDonald’s Theorem Lemma If f ∈ PW β , then: f ′ ∈ PW β ; ff ♯ ∈ PW 2 β ; f ′ ( f ′ ) ♯ ∈ PW 2 β . Theorem Suppose { t n } ⊂ R is a set of sampling for PW 2 β . Then the mapping A : PW β / ∼→ ℓ 2 ( Z ) ⊕ ℓ 2 ( Z ) : f �→ ( | f ( t n ) | , | f ( t n + b ) − f ( t n ) | ) n is one-to-one whenever b < β π . Similarly, the mapping A : PW β / ∼→ ℓ 2 ( Z ) ⊕ ℓ 2 ( Z ) : f �→ ( | f ( t n ) | , | f ′ ( t n ) | ) n � is one-to-one. Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 13 / 26

Inversion Fundamental Question: How to invert these mappings? A : PW β / ∼→ ℓ 2 ( Z ) ⊕ ℓ 2 ( Z ) : f �→ ( | f ( t n ) | , | f ( t n + b ) − f ( t n ) | ) n � A : PW β / ∼→ ℓ 2 ( Z ) ⊕ ℓ 2 ( Z ) : f �→ ( | f ( t n ) | , | f ′ ( t n ) | ) n We can reconstruct | f ( x ) | 2 and | f ( x + b ) − f ( x ) | 2 (or | f ′ ( x ) | 2 ) easily. 1 McDonald’s Theorem guarantees injectivity of the mappings using 2 Weierstrass factorizations of functions of finite order. Reconstruction from McDonald’s Theorem requires knowledge of the zeros of 3 f . Inversion is unstable: observed by Mallat-Waldspurger [MW15], proven by 4 Cahill-Casazza-Daubechies [CCD16]. We propose a sampling and reconstruction method following Pohl-Yang-Boche [PYB14] using “structured convolutions”. Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 14 / 26

Injectivity via Structured Convolutions Theorem (Lai, Littmann & W.) The following sampling scheme does conjugate phase retrieval on PW π : {| α m ∗ f ( t n ) | : m = 0 , 1 , . . . , M − 1; n ∈ Z } where K − 1 � α m ∗ f = a km f ( · − b k ) (3) k =0 provided: A = ( a km ) be a K × M matrix which does conjugate phase retrieval on C K 1 { t n } n ∈ Z ⊂ R is a set of sampling for the space PW 2 π 2 { b j } K − 1 j =0 ⊂ R be such that the group Z ( { b 0 , b 1 , . . . , b K − 1 } ) has finite upper 3 Beurling density and lower Beurling density greater than one. Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 15 / 26

Reconstruction via Structured Convolutions For f ∈ PW π we sample {| α m ∗ f ( t n ) | : m = 0 , 1 , . . . , M − 1; n ∈ Z } (4) where { t n } and α m satisfy the hypotheses of Theorem 3. Choose b j = j / B for some integer B. Reconstruction Algorithm From the samples in Equation (4) , reconstruct the functions 1 | α m ∗ f ( x ) | 2 , m = 0 , 1 , . . . , M − 1 , using the Shannon sampling theorem. Choose β at random † . 2 By Lemma 4, with probability 1, f ( n B − b j − β ) � = 0 for all j = 0 , . . . , K − 1 , n ∈ Z . Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 16 / 26

Reconstruction via Structured Convolutions Reconstruction Algorithm (Continued) Calculate the following samples using Step 1: 3 | α m ∗ f ( n B − β ) | 2 , m = 0 , 1 , . . . , M − 1 , n ∈ Z . Use the fact that the matrix A does conjugate phase retrieval to calculate for 4 each n ∈ Z the vector   f ( n B − b 0 − β ) F n := λ ( n  .  � . B − β ) (5)   . f ( n B − b K − 1 − β ) up to the unknown phase λ ( n B − β ) and unknown conjugation. Eric Weber Conjugate Phase Retrieval in the Paley-Wiener Space 17 / 26