Super-resolution of point sources via convex programming

Fernandez‐Granda, Carlos

doi:10.1109/camsap.2015.7383731

Cited by 38 publications

(86 citation statements)

References 76 publications

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“…(3), referred to as minimum separation factor (MSF). Theorem 1 indicates that MSF≥ 2 guarantees the exact recovery of L 1 minimization with a recent improvement to 1.26 [18] at the cost of an additional constraint that f c ≥ 1000. This line of research was originated from Donoho [12], who showed that MSF > 1 is sufficient if the spikes are on the grid.…”

Section: Definition 1 (Minimum Separation)mentioning

confidence: 93%

“…In addition to these exact recovery results, errors in spike detection and noise robustness are of great interest as well. Fernandez-Granda analyzed error bounds of constrained L 1 minimization in [18], while the unconstrained version was addressed in [34] under a Gaussian noise model as well as in [2] for any sampling scheme. The robustness of spike detection was discussed in [15].…”

Section: Definition 1 (Minimum Separation)mentioning

confidence: 99%

“…This line of research was originated from Donoho [12], who showed that MSF > 1 is sufficient if the spikes are on the grid. Note that both aforementioned works [6,18] are formulated in terms of off-grid spikes. Another article about off-grid spikes was [1] by Aubel et al, who also arrived at MSF > 1 if windowed Fourier (or short-time Fourier transform) measurements are available.…”

Section: Definition 1 (Minimum Separation)mentioning

confidence: 99%

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Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

2016

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We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in (1, 2) of the Rayleigh length (physical resolution limit), L 1 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the L 1 certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two L 1 based nonconvex penalties, the difference of L 1 and L 2 norms (L 1−2 ) and capped L 1 (CL 1 ), subject to the measurement constraints. In one and two dimensional numerical SR examples, the local optimal solutions from difference of convex function algorithms outperform the global L 1 solutions near or below Rayleigh length scales either in the accuracy of ground truth recovery or in finding a sparse solution satisfying the constraints more accurately.

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Section: Definition 1 (Minimum Separation)mentioning

confidence: 93%

Section: Definition 1 (Minimum Separation)mentioning

confidence: 99%

Section: Definition 1 (Minimum Separation)mentioning

confidence: 99%

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Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

2016

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“…In practice, these algorithms may also require postprocessing: for instance thresholding by removing points for which the estimated intensityŵ i is low, or by clustering nearby localizations [45]. Of some interest are the myriad of theoretical results concerning (9): these results stipulate that if the measurement model I is accurate and some additional technical assumptions are satisfied, the solution to (9) is guaranteed to be close (in some sense) to the ground truth [13,40].…”

Section: Maximum-likelihood Methodsmentioning

confidence: 99%

DeepLoco: Fast 3D Localization Microscopy Using Neural Networks

Boyd¹,

Jonas²,

Babcock³

et al. 2018

Preprint

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Single-molecule localization super-resolution microscopy (SMLM) techniques like STORM and PALM have transformed cellular microscopy by substantially increasing spatial resolution. In this paper we introduce a new algorithm for a critical part of the SMLM process: estimating the number and locations of the fluorophores in a single frame. Our algorithm can analyze a 20000-frame experimental 3D SMLM dataset in about one second -substantially faster than real-time and existing algorithms. Our approach is straightforward but very different from existing algorithms: we train a neural network to minimize the Bayes' risk under a generative model for single SMLM frames. The neural network maps a frame directly to a collection of fluorophore locations, which we compare to the ground truth using a novel loss function. While training the neural network takes several hours, it only has to be done once for a given experimental setup. After training, localizing fluorophores in new images is extremely fast -orders of magnitude faster than existing algorithms. Faster recovery opens the door to real-time calibration and accelerated acquisition, and future work could tackle more complicated optical systems and more realistic simulators.

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“…When the temporal and frequency domains are exchanged, this classical problem was reinterpreted as the problem of mathematical super-resolution recently [1,2,3]. This line of work promotes the use of a convex sparse regularizer to solve inverse problems involving spectrally sparse signals, distinguishing them from classical methods based on root finding and singular value decompositions (e.g., Prony's method, MUSIC, ESPIRIT, Matrix Pencil, etc.).…”

Section: Introductionmentioning

confidence: 99%

Approximate support recovery of atomic line spectral estimation: A tale of resolution and precision

Tang

2016

2016 IEEE Global Conference on Signal and Information Processing (GlobalSIP)

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This work investigates the parameter estimation performance of super-resolution line spectral estimation using atomic norm minimization. The focus is on analyzing the algorithm's accuracy of inferring the frequencies and complex magnitudes from noisy observations. When the Signal-to-Noise Ratio is reasonably high and the true frequencies are separated by O( 1 n ), the atomic norm estimator is shown to localize the correct number of frequencies, each within a neighborhood of size O( log n n 3 σ) of one of the true frequencies. Here n is half the number of temporal samples and σ 2 is the Gaussian noise variance. The analysis is based on a primal-dual witness construction procedure. The obtained error bound matches the Cramér-Rao lower bound up to a logarithmic factor. The relationship between resolution (separation of frequencies) and precision or accuracy of the estimator is highlighted. Our analysis also reveals that the atomic norm minimization can be viewed as a convex way to solve a 1-norm regularized, nonlinear and nonconvex least-squares problem to global optimality.

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Super-resolution of point sources via convex programming

Cited by 38 publications

References 76 publications

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

DeepLoco: Fast 3D Localization Microscopy Using Neural Networks

Approximate support recovery of atomic line spectral estimation: A tale of resolution and precision

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