Support Recovery for Sparse Super-Resolution of Positive Measures

Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel

doi:10.1007/s00041-016-9502-x

Cited by 89 publications

(105 citation statements)

References 24 publications

(59 reference statements)

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“…In this section, we describe the main properties of the positive Beurling LASSO (Blasso + ) that are needed in the paper. These are mainly minor adaptations of the properties stated in [17,13] for the Blasso, and we state them here without proof.…”

Section: The Blasso With Nonnegativity Constraintsmentioning

confidence: 87%

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A characterization of the Non-Degenerate Source Condition in super-resolution

Duval

2019

Information and Inference: A Journal of the IMA

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In a recent article, Schiebinger et al. provided sufficient conditions for the noiseless recovery of a signal made of M Dirac masses given 2M + 1 observations of, e.g. , its convolution with a Gaussian filter, using the Basis Pursuit for measures. In the present work, we show that a variant of their criterion provides a necessary and sufficient condition for the Non-Degenerate Source Condition (NDSC) which was introduced by Duval and Peyré to ensure support stability in super-resolution. We provide sufficient conditions which, for instance, hold unconditionally for the Laplace kernel provided one has at least 2M measurements. For the Gaussian filter, we show that those conditions are fulfilled in two very different configurations: samples which approximate the uniform Lebesgue measure or, more surprisingly, samples which are all confined in a sufficiently small interval.

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Section: The Blasso With Nonnegativity Constraintsmentioning

confidence: 87%

“…The (2M −1)-Nondegeneracy of a point entails the support stability of finite measures which are clustered around x 0 . Theorem 2 ( [13]). Assume that ϕ ∈ KER (2M +1) and that the point is (2M − 1) Non Degenerate.…”

Section: Low Noise Behavior and Exact Support Recoverymentioning

confidence: 99%

A characterization of the Non-Degenerate Source Condition in super-resolution

Duval

2019

Information and Inference: A Journal of the IMA

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“…The above quantity is well-defined (as a Bochner integral) as soon as ϕ is continuous and bounded. In order to apply some results of [28], we add the hypotheses that are summarized below.…”

Section: Notations and Definitionsmentioning

confidence: 99%

The sliding Frank–Wolfe algorithm and its application to super-resolution microscopy

Denoyelle¹,

Duval²,

Peyré³

et al. 2019

Inverse Problems

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167

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This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem.The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

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“…Often in these application much is known about the point spread function of the sensor, or can be estimated and, given such model information, it is possible to identify point source locations with accuracy substantially below the essential width of the sensor point spread function. Recently there has been substantial interest from the mathematical community in posing algorithms and proving super-resolution guarantees in this setting, see for instance [16,17,18,19,20,21,22,23]. Typically these approaches borrow notions from compressed sensing [24,25,26].…”

Section: Introductionmentioning

confidence: 99%

Sparse non-negative super-resolution — simplified and stabilised

Eftekhari¹,

Tanner

Thompson

et al. 2021

Applied and Computational Harmonic Analysis

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The convolution of a discrete measure, x " ř k i"1 aiδt i , with a local window function, φps´tq, is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources tai, tiu k i"1 with an accuracy beyond the essential support of φps´tq, typically from m samples ypsjq " ř k i"1 aiφpsj´tiq`δj , where δj indicates an inexactness in the sample value. We consider the setting of x being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that x is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. δj " 0, m ě 2k`1 samples are available, and φps´tq generates a Chebyshev system. This is independent of how close the sample locations are and does not rely on any regulariser beyond non-negativity; as such, it extends and clarifies the work by Schiebinger et al. in [1] and De Castro et al. in [2], who achieve the same results but require a total variation regulariser, which we show is unnecessary.Moreover, we characterise non-negative solutionsx consistent with the samples within the bound ř m j"1 δ 2 j ď δ 2 . Any such non-negative measure is within Opδ 1{7 q of the discrete measure x generating the samples in the generalised Wasserstein distance. Similarly, we show using somewhat different techniques that the integrals ofx and x over pti´ǫ, ti`ǫq are similarly close, converging to one another as ǫ and δ approach zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of φps´tq being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution and that, while regularisers such as total variation might be particularly effective, they are not required in the non-negative setting.

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Support Recovery for Sparse Super-Resolution of Positive Measures

Cited by 89 publications

References 24 publications

A characterization of the Non-Degenerate Source Condition in super-resolution

A characterization of the Non-Degenerate Source Condition in super-resolution

The sliding Frank–Wolfe algorithm and its application to super-resolution microscopy

Sparse non-negative super-resolution — simplified and stabilised

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