In this paper we consider Sparse Fourier Transform (SFT) algorithms for approximately computing the best s-term approximation of the Discrete Fourier Transform (DFT)f ∈ C N of any given input vector f ∈ C N in just (s log N ) O(1) -time using only a similarly small number of entries of f . In particular, we present a deterministic SFT algorithm which is guaranteed to always recover a near best s-term approximation of the DFT of any given input vector f ∈ C N in O s 2 log 11 2 (N )time. Unlike previous deterministic results of this kind, our deterministic result holds for both arbitrary vectors f ∈ C N and vector lengths N . In addition to these deterministic SFT results, we also develop several new publicly available randomized SFT implementations for approximately computingf from f using the same general techniques. The best of these new implementations is shown to outperform existing discrete sparse Fourier transform methods with respect to both runtime and noise robustness for large vector lengths N . 2 ∩Z, by sampling its associated trigonometric polynomial f (x) = ω∈Bf ω e iωx
We propose a two-step approach for reconstructing a signal $\textbf x\in \mathbb{C}^d$ from subsampled discrete short-time Fourier transform magnitude (spectogram) measurements: first, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix $\widehat{\textbf{x}}\widehat{\textbf{x}}^{*}.$ Secondly, we use angular synchronization to solve for $\widehat{\textbf{x}}$ (and then for $\textbf{x}$ by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems; one which guarantees the recovery of discrete, bandlimited signals $\textbf{x}\in \mathbb{C}^{d}$ from fewer than $d$ short-time Fourier transform magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.
We propose a two-step approach for reconstructing a signal x ∈ C d from subsampled short-time Fourier transform magnitude (spectogram) measurements: First, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix x x * . Second, we use angular syncrhonization to solve for x (and then for x by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems, one which guarantees the recovery of discrete, bandlimited signals x ∈ C d from fewer than d STFT magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.
In this short note, we consider the worst case noise robustness of any phase retrieval algorithm which aims to reconstruct all nonvanishing vectors x ∈ C d (up to a single global phase multiple) from the magnitudes of an arbitrary collection of local correlation measurements. Examples of such measurements include both spectrogram measurements of x using locally supported windows and masked Fourier transform intensity measurements of x using bandlimited masks. As a result, the robustness results considered herein apply to a wide range of both ptychographic and Fourier ptychographic imaging scenarios. In particular, the main results imply that the accurate recovery of high-resolution images of extremely large samples using highly localized probes is likely to require an extremely large number of measurements in order to be robust to worst case measurement noise, independent of the recovery algorithm employed. Furthermore, recent pushes to achieve high-speed and high-resolution ptychographic imaging of integrated circuits for process verification and failure analysis will likely need to carefully balance probe design (e.g., their effective timefrequency support) against the total number of measurements acquired in order for their imaging techniques to be stable to measurement noise, no matter what reconstruction algorithms are applied.
A novel phase retrieval method, motivated by ptychographic imaging, is proposed for the approximate recovery of a compactly supported specimen function f : R → C from its continuous short time Fourier transform (STFT) spectrogram measurements. The method, partially inspired by the well known PhaseLift [4] algorithm, is based on a lifted formulation of the infinite dimensional problem which is then later truncated for the sake of computation. Numerical experiments demonstrate the promise of the proposed approach.
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