Low Rank Fourier Ptychography

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

doi:10.1109/icassp.2018.8462480

Cited by 10 publications

(7 citation statements)

References 19 publications

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“…Other recent works include re-weighted amplitude flow [60] and convolutional phase retrieval [61]. Our framework proposed in this paper has also spurred followup work in an optical imaging application called Fourier ptychography with promising results; see [62], [63] for details.…”

Section: B Subsequent Workmentioning

confidence: 83%

“…Moreover, our analysis only applies to the case of Gaussian samples, and extending our results to more realistic measurement schemes such as Fourier samples and coded diffraction patterns [46] will be an interesting direction of future study; indeed, our preliminary experimental results in [62], [63] have shown the empirical benefits of our algorithms for such challenging measurement setups.…”

Section: Discussionmentioning

confidence: 94%

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Sample-Efficient Algorithms for Recovering Structured Signals From Magnitude-Only Measurements

Jagatap

Hegde

2019

IEEE Trans. Inform. Theory

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We consider the problem of recovering a signal x * ∈ R n , from magnitude-only measurements, yi = | ai, x * | for i = {1, 2, . . . , m}. This is a stylized version of the classical phase retrieval problem, and is a fundamental challenge in nano-and bio-imaging systems, astronomical imaging, and speech processing. It is well known that the above problem is ill-posed, and therefore some additional assumptions on the signal and/or the measurements are necessary.In this paper, we consider the case where the underlying signal x * is s-sparse. For this case, we develop a novel recovery algorithm that we call Compressive Phase Retrieval with Alternating Minimization, or CoPRAM. Our algorithm is simple and be obtained via a natural combination of the classical alternating minimization approach for phase retrieval with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that our algorithm achieves a sample complexity of O s 2 log n with Gaussian measurements ai, which matches the best known existing results; moreover, it also demonstrates linear convergence in theory and practice. An appealing feature of our algorithm is that it requires no extra tuning parameters other than the signal sparsity level s. Moreover, we show that our algorithm is robust to noise.The quadratic dependence of sample complexity on the sparsity level is sub-optimal, and we demonstrate how to alleviate this via additional assumptions beyond sparsity. First, we study the (practically) relevant case where the sorted coefficients of the underlying sparse signal exhibit a power law decay. In this scenario, we show that the CoPRAM algorithm achieves a sample complexity of O (s log n), which is close to the information-theoretic limit.We then consider the case where the underlying signal x * arises from structured sparsity models. We specifically examine the case of block-sparse signals with uniform block size of b and block sparsity k = s/b. For this problem, we design a recovery algorithm that we call Block CoPRAM that further reduces the sample complexity to O (ks log n).For sufficiently large block lengths of b = Θ(s), this bound equates to O (s log n).To our knowledge, our approach constitutes the first family of linearly convergent algorithms for signal recovery from magnitude-only Gaussian measurements that exhibit a sub-quadratic dependence on the signal sparsity level.

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Section: B Subsequent Workmentioning

confidence: 83%

Section: Discussionmentioning

confidence: 94%

Sample-Efficient Algorithms for Recovering Structured Signals From Magnitude-Only Measurements

Jagatap

Hegde

2019

IEEE Trans. Inform. Theory

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“…In the description below we use (z) img to refer to the 2D image version of an n-length vector z. As explained in detail in [15,3], for an image, (z) img , the imaging model is…”

Section: Fourier Ptychography As An Approx-lrpr Problemmentioning

confidence: 99%

Undersampled Dynamic Fourier Ptychography via Phaseless PCA

Chen

Nayer

Vaswani

2022

2022 IEEE International Conference on Image Processing (ICIP)

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In recent work, we studied the phaseless PCA (low rank phase retrieval) problem and developed a provably correct and fast alternating minimization (AltMin) solution for it called AltMinLowRaP. In this work, we develop a modification of AltMinLowRaP, called AltMinLowRaP-Ptych, that is designed for reducing the sample complexity (number of measurements required for accurate recovery) for dynamic Fourier ptychographic imaging. Fourier ptychography is a computational imaging technique that enables high-resolution microscopy using multiple low-resolution cameras. Via exhaustive experiments on real image sequences with simulated ptychographic measurements, we show the power of our algorithm for reducing the number of samples required for accurate recovery.

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“…Recently, there has been a surge of interest in solving the low-rank phase retrieval problem [12][13][14][15][16]. This problem can be viewed as a dynamic extension of the standard phase retrieval problem, where the objective is to recover a matrix of vectorized images rather than a single image.…”

Section: Introductionmentioning

confidence: 99%

Low-Rank Phase Retrieval with Structured Tensor Models

Kwon¹,

Li²,

Sarwate³

2022

Preprint

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We study the low-rank phase retrieval problem, where the objective is to recover a sequence of signals (typically images) given the magnitude of linear measurements of those signals. Existing solutions involve recovering a matrix constructed by vectorizing and stacking each image. These algorithms model this matrix to be low-rank and leverage the low-rank property to decrease the sample complexity required for accurate recovery. However, when the number of available measurements is more limited, these low-rank matrix models can often fail. We propose an algorithm called Tucker-Structured Phase Retrieval (TSPR) that models the sequence of images as a tensor rather than a matrix that we factorize using the Tucker decomposition. This factorization reduces the number of parameters that need to be estimated, allowing for a more accurate reconstruction in the under-sampled regime. Interestingly, we observe that this structure also has improved performance in the over-determined setting when the Tucker ranks are chosen appropriately. We demonstrate the effectiveness of our approach on real video datasets under several different measurement models.

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Low Rank Fourier Ptychography

Cited by 10 publications

References 19 publications

Sample-Efficient Algorithms for Recovering Structured Signals From Magnitude-Only Measurements

Sample-Efficient Algorithms for Recovering Structured Signals From Magnitude-Only Measurements

Undersampled Dynamic Fourier Ptychography via Phaseless PCA

Low-Rank Phase Retrieval with Structured Tensor Models

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