Sample Efficient Fourier Ptychography for Structured Data

Jagatap, Gauri; Chen, Zhengyu; Nayer, Seyedehsara; Hegde, Chinmay; Vaswani, Namrata

doi:10.1109/tci.2019.2948758

Cited by 29 publications

(22 citation statements)

References 54 publications

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“…Clearly, this implies that Another practical point that should be mentioned is that, often, in practice, a very small value of rank r suffices. For example, we used r = 20 in all our experiments on image sequences with n = 32400 in [21]. In follow-up work [30], we show that just r = 5 suffices for the same datasets, as long as a "modeling error correction step" 3 is applied to the output of AltMinLowRaP.…”

Section: A Low Rank Pr (Lrpr) Problem Setting and Notationmentioning

confidence: 98%

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Provable Low Rank Phase Retrieval

Nayer

Narayanamurthy

Vaswani

2020

IEEE Trans. Inform. Theory

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We study the "Low Rank Phase Retrieval (LRPR)" problem defined as follows: recover an n × q matrix X * of rank r from a different and independent set of m phaseless (magnitude-only) linear projections of each of its columns. To be precise, we need to recover X * from y k := |A k x * k |, k = 1, 2, . . . , q when the measurement matrices A k are mutually independent. Here y k is an m length vector and denotes transpose. The question is when can we solve LRPR with m n? Our work introduces the first provably correct solution, Alternating Minimization for Low-Rank Phase Retrieval (AltMinLowRaP), for solving LRPR. We demonstrate its advantage over existing work via extensive simulation, and some partly real data, experiments. Our guarantee for AltMinLowRaP shows that it can solve LRPR to accuracy if mq ≥ Cnr 4 log(1/ ), the matrices A k contain i.i.d. standard Gaussian entries, the condition number of X * is bounded by a numerical constant, and its right singular vectors satisfy the incoherence (denseness) assumption from matrix completion literature. Its time complexity is only Cmqnr log 2 (1/ ). In the regime of small r, our sample complexity is much better than what standard PR methods need; and it is only about r 3 times worse than its order-optimal value of (n + q)r. Moreover, if we replace m by its lower bound for each approach, then the same can be said for the time complexity comparison with standard PR. We also briefly study the dynamic extension of LRPR.The LRPR problem occurs in phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens, where acquiring measurements is expensive. We should point out that LRPR is a very different problem than its A k = A version, or its A k = A and with-phase (linear) version, both of which have been extensively studied in the literature.for each k. If we assume κ is a constant, up to constant factors, (4) also implies (3). Thus, up to constant factors, requiring right incoherence is the same as requiring that the maximum energy of any signal x * k is within constant factors of the average.

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Section: A Low Rank Pr (Lrpr) Problem Setting and Notationmentioning

confidence: 98%

“…The quantity in brackets is the p-th row of (B ). Thus, the n × r reshaped form of SΣ * big U * vec equals U * Σ * B * B and hencê U t+1 satisfies (21).…”

Section: )mentioning

confidence: 99%

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Provable Low Rank Phase Retrieval

Nayer

Narayanamurthy

Vaswani

2020

IEEE Trans. Inform. Theory

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“…Moreover, our analysis is limited to the case of Gaussian measurements schemes, which may or may not be physically realizable. Extending our results to more practical measurement schemes such as Fourier-based sampling or ptychography [42] can be an interesting problem for future study.…”

Section: Discussionmentioning

confidence: 97%

Sparse Signal Recovery from Modulo Observations

Shah

Hegde

2020

Preprint

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We consider the problem of reconstructing a signal from under-determined modulo observations (or measurements). This observation model is inspired by a (relatively) less well-known imaging mechanism called modulo imaging, which can be used to extend the dynamic range of imaging systems; variations of this model have also been studied under the category of phase unwrapping. Signal reconstruction in the under-determined regime with modulo observations is a challenging ill-posed problem, and existing reconstruction methods cannot be used directly. In this paper, we propose a novel approach to solving the inverse problem limited to two modulo periods, inspired by recent advances in algorithms for phase retrieval under sparsity constraints. We show that given a sufficient number of measurements, our algorithm perfectly recovers the underlying signal and provides improved performance over other existing algorithms. We also provide experiments validating our approach on both synthetic and real data to depict its superior performance.

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“…Coded diffraction imaging is used to recover complex-valued wavefront from Fourier measurements [12,40]. In coded diffraction imaging, the signal of interest gets modulated by a sequence of illumination patterns before the K-space measurements were captured [10,29,31]. Ptychography is another related method for capturing high-resolution microscopy images by capturing multiple images of the scene using a sequence of coded illumination patterns [36,41].…”

Section: Related Workmentioning

confidence: 99%

Coded Illumination and Multiplexing for Lensless Imaging

Zheng

Zhang

Asif

2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Mask-based lensless cameras can be flat, thin and lightweight, which makes them suitable for novel designs of computational imaging systems with large surface areas and arbitrary shapes. Despite recent progress in lensless cameras, the quality of images recovered from the lensless cameras is often poor due to the ill-conditioning of the underlying measurement system. In this paper, we propose to use coded illumination to improve the quality of images reconstructed with lensless cameras. In our imaging model, the scene/object is illuminated by multiple coded illumination patterns as the lensless camera records sensor measurements. We designed and tested a number of illumination patterns and observed that shifting dots (and related orthogonal) patterns provide the best overall performance. We propose a fast and low-complexity recovery algorithm that exploits the separability and block-diagonal structure in our system. We present simulation results and hardware experiment results to demonstrate that our proposed method can significantly improve the reconstruction quality.

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Sample Efficient Fourier Ptychography for Structured Data

Cited by 29 publications

References 54 publications

Provable Low Rank Phase Retrieval

Provable Low Rank Phase Retrieval

Sparse Signal Recovery from Modulo Observations

Coded Illumination and Multiplexing for Lensless Imaging

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