Recovery of Compactly Supported Functions from Spectrogram Measurements via Lifting Mark Iwen markiwen @ math.msu.edu 2017 Friday, July 7 th , 2017
Joint work with... Sami Merhi (Michigan State University) M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 1 / 24
Joint work with... Aditya Viswanathan (University of Michigan - Dearborn) M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 1 / 24
Introduction Background Motivation Applications: The phase-retrieval problem arises whenever the detectors can only capture intensity measurements. For example, X-ray crystallography Diffraction imaging Ptychographic Imaging ... Our goals: Approaching realistic measurement designs compatible with, e.g., ptychography, coupled with computationally efficient and robust recovery algorithms. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 2 / 24
Introduction Background Motivation Applications: The phase-retrieval problem arises whenever the detectors can only capture intensity measurements. For example, X-ray crystallography Diffraction imaging Ptychographic Imaging ... Our goals: Approaching realistic measurement designs compatible with, e.g., ptychography, coupled with computationally efficient and robust recovery algorithms. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 2 / 24
Introduction Background Motivating Application Figure: Ptychographic Imaging M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 3 / 24
Introduction Background Algorithms for Discrete Phase Retrieval There has been a good deal of work on signal recovery from phaseless STFT measurements in the discrete setting : ◮ First f and g are modeled as vectors ab initio, ◮ Then recovered from discrete STFT magnitude measurements. Recovery techniques include ◮ Iterative methods (Alt. Proj. for STFT) along the lines of Griffin and Lim [8, 12], ◮ Alternating Projections [7], ◮ Graph theoretic methods for Gabor frames based on polarization [11, 9], ◮ Semidefinite relaxation-based methods [5], and others [2, 1, 4, 3]. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 4 / 24
Introduction Problem Statement and Specifications Signal Recovery from STFT Measurements In 1-D ptychography [10, 7], a compactly supported specimen f : R → C , is scanned by a focused beam g : R → C which translates across the specimen in fixed overlapping shifts l 1 ,...,l K ∈ R . At each such shift a phaseless diffraction image is sampled by a detector. The measurements are modeled as STFT magnitude measurements: � � � ∞ 2 � � � f ( t ) g ( t − l k ) e − 2 πiω j t dt � b k,j := , 1 ≤ k ≤ K, 1 ≤ j ≤ N. (1) � � −∞ We aim to approximate f (up to a global phase) using these b k,j measurements. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 5 / 24
Introduction Problem Statement and Specifications Given stacked spectrogram samples from (1), � � T ∈ [0 , ∞ ) NK , � b = b 1 , 1 ,...,b 1 ,N ,b 2 , 1 ,...,b K,N (2) approximately recover a piecewise smooth and compactly supported function f : R → C up to a global phase. WLOG assume that the support of f is contained in [ − 1 , 1] . Motivated by ptychography, we primarily consider the beam function g to also be (effectively) compactly supported within [ − a,a ] � [ − 1 , 1] . Assume also that g is smooth enough that its Fourier transform decays relatively rapidly in frequency space compared to ˆ f . Examples of such g include both Gaussians, as well as compactly supported C ∞ bump functions [6]. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 6 / 24
Signal Recovery Method The Proposed Numerical Approach Using techniques from [4, 3] on discrete PR adapted to continuous PR, recover samples of ˆ f at frequencies in Ω = { ω 1 ,...,ω N } , giving f ∈ C N with f j = � � f ( ω j ) : ◮ First, a truncated lifted linear system is inverted in order to learn a portion of the rank-one matrix � f � f ∗ . ◮ Then, angular synchronization is used to recover � f from the portion of f ∗ above. f � � Reconstruct � f via standard sampling theorems. Invert this approximation in order to learn f . This linear system is banded and Toeplitz, with band size determined by the decay of ˆ g : if g is effectively bandlimited to [ − δ,δ ] the � - essentially FFT-time in � δN (log N + δ 2 ) computational cost is O N for small δ . M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 7 / 24
Signal Recovery Method Our Lifted Formulation Lifted Formulation Lemma (Lifting Lemma) Suppose f : R → C is piecewise smooth and compactly supported in [ − 1 , 1] . Let g ∈ L 2 ([ − a,a ]) be supported in [ − a,a ] ⊂ [ − 1 , 1] for some a < 1 , with � g � L 2 = 1 . Then for all ω ∈ R , � �� � � m � � m � � |F [ f · S l g ]( ω ) | = 1 � � e − πilm ˆ � � f g ˆ 2 − ω � � 2 2 � � m ∈ Z for all shifts l ∈ [ a − 1 , 1 − a ] . M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 8 / 24
Signal Recovery Method Our Lifted Formulation Proof. Plancherel’s Theorem implies that � � � ∞ 2 � � |F [ f · S l g ]( ω ) | 2 = ˆ � g ( − η ) e − 2 πilη dη � f ( ω − η )ˆ . � � −∞ Applying Shannon’s Sampling theorem to ˆ f and recalling that F [ f ⋆g ] = ˆ f ˆ g yield � � 2 � � m �� � � � � � |F [ f · S l g ]( ω ) | 2 = g ( · ) e − 2 πil ( · ) ⋆ sinc π ( m +2( · )) ˆ � � f ˆ ( − ω ) � � 2 � � m ∈ Z � � 2 � � m � � � − l +1 � = 1 � � g ( u ) e − 2 πiu ( m 2 − ω ) du ˆ e − πil ( m − 2 ω ) � � f . � � 4 2 � � − l − 1 m ∈ Z The result follows by noting the support of g and the Fourier type integral in the last equality. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 9 / 24
Signal Recovery Method Our Lifted Formulation Lifted Form We write our measurements in a lifted form |F [ f · S l g ]( ω ) | 2 ≈ 1 � l � Y ω � ω � X ∗ Y ∗ X l 4 X l ∈ C 4 δ +1 and � Y ω ∈ C 4 δ +1 are the vectors where � ˆ f ( ω − δ ) e πil (2 δ ) ˆ g ( − δ ) � � � � ˆ ω − δ + 1 e πil (2 δ − 1) ˆ 1 f 2 − δ g 2 . . . . . . � e πil · 0 ˆ , � ˆ X l = g (0) Y ω = . f ( ω ) . . . . . . � � � � δ − 1 e πil (1 − 2 δ ) ˆ g ˆ ω + δ − 1 f 2 2 e πil ( − 2 δ ) ˆ g ( δ ) ˆ f ( ω + δ ) M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 10 / 24
Signal Recovery Method Our Lifted Formulation Lifted Form We write our measurements in a lifted form |F [ f · S l g ]( ω ) | 2 ≈ 1 � l � Y ω � ω � X ∗ Y ∗ X l 4 Here, � Y ω � Y ∗ ω is the rank-one matrix � � 2 � � � ˆ f ( ω − δ ) ˆ ˆ f ( ω − δ ) ˆ ˆ f ( ω − δ ) ··· f ( ω ) ··· f ( ω + δ ) � . . . . ... . . . . . . . . � � 2 � � f ( ω ) ˆ ˆ � ˆ f ( ω ) ˆ ˆ f ( ω − δ ) ··· f ( ω ) ··· f ( ω + δ ) . � . . . . ... . . . . . . . . � � 2 � � f ( ω + δ ) ˆ ˆ f ( ω + δ ) ˆ ˆ � ˆ f ( ω − δ ) ··· f ( ω ) ··· f ( ω + δ ) � Note the occurrence of the magnitudes of ˆ f on the diagonal, and the relative phase terms elsewhere. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 11 / 24
Signal Recovery Method Our Lifted Formulation Let F ∈ C N × N be defined as � � ˆ � � ˆ j − 2 n − 1 i − 2 n − 1 f f , if | i − j | ≤ 2 δ, , where n = N − 1 2 2 F i,j = . 4 0 , otherwise, F is composed of overlapping segments of matrices � Y ω � Y ∗ ω for ω ∈ {− n,...,n } . Thus, our spectrogram measurements can be written as � b ≈ diag ( GFG ∗ ) , (3) where G ∈ C NK × N is a block Toeplitz matrix encoding the � X l ’s. We consistently vectorize (3) to obtain a linear system which can be inverted to learn � F , a vectorized version of F . M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 12 / 24
Signal Recovery Method Our Lifted Formulation In particular, we have � b ≈ M � F, (4) where M ∈ C NK × N 2 is computed by passing the canonical basis of C N × N through (3). We solve the linear system (4) as a least squares problem. Experiments have shown that M is of rank NK. The process of recovering the Fourier samples of f from � F is known as angular synchronization. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 13 / 24
Signal Recovery Method Our Lifted Formulation Angular Synchronization Angular synchronization is the process recovering d angles φ 1 ,φ 2 ,...,φ d ∈ [0 , 2 π ) given noisy and possibly incomplete difference measurements of the form φ ij := φ i − φ j , ( i,j ) ∈ { 1 , 2 ,...,d }×{ 1 , 2 ,...,d } . We are interested in angular synchronization problems that arise when performing phase retrieval from local correlation measurements [4, 3]. M.A. Iwen (MSU) Continuous Phase Retrieval SampTA 2017 14 / 24
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