A Sampling Framework for Solving Physics-Driven Inverse Source Problems

Murray-Bruce, John; Dragotti, Pier Luigi

doi:10.1109/tsp.2017.2742983

Cited by 21 publications

(12 citation statements)

References 57 publications

(124 reference statements)

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“…The FRI framework has also been applied to arbitrary non-bandlimited convolution kernels [89] and nonuniform sampling patterns [65], but without exact-recovery guarantees. These techniques have been recently extended by Dragotti and Murray-Bruce [60] to physics-driven SNL problems. By approximating complex exponentials with weighted sums of Green's functions, they are able to recast parameter recovery as a related spectral super-resolution problem that approximates the true SNL problem.…”

Section: Other Methodologiesmentioning

confidence: 99%

Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Bernstein

Liu

Papadaniil

et al. 2020

IEEE Trans. Inform. Theory

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Extracting information from nonlinear measurements is a fundamental challenge in data analysis. In this work, we consider separable inverse problems, where the data are modeled as a linear combination of functions that depend nonlinearly on certain parameters of interest. These parameters may represent neuronal activity in a human brain, frequencies of electromagnetic waves, fluorescent probes in a cell, or magnetic relaxation times of biological tissues. Separable nonlinear inverse problems can be reformulated as underdetermined sparse-recovery problems, and solved using convex programming. This approach has had empirical success in a variety of domains, from geophysics to medical imaging, but lacks a theoretical justification. In particular, compressed-sensing theory does not apply, because the measurement operators are deterministic and violate incoherence conditions such as the restricted-isometry property. Our main contribution is a theory for sparse recovery adapted to deterministic settings. We show that convex programming succeeds in recovering the parameters of interest, as long as their values are sufficiently distinct with respect to the correlation structure of the measurement operator. The theoretical results are illustrated through numerical experiments for two applications: heat-source localization and estimation of brain activity from electroencephalography data.

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Section: Other Methodologiesmentioning

confidence: 99%

Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Bernstein

Liu

Papadaniil

et al. 2020

IEEE Trans. Inform. Theory

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“…For the same recovery condition (cf. (37)), our result straight-forwardly generalizes to any arbitrary, bandlimited sampling kernel of the form,…”

Section: Generalization To Arbitrary Bandlimited Sampling Kernelsmentioning

confidence: 61%

“…To this end, (37) guarantees that we can estimate the filter r in (36) which is then used for constructing a polynomial of degree K,…”

Section: B Sparse Signals and Arbitrary Bandlimited Kernelsmentioning

confidence: 99%

“…These are signals which are described by countable degrees of freedom, per unit time and model a broad class of signals. The FRI sampling has been applied to a number of interesting applications such as channel estimation [24], radar [15]- [17], timeresolved imaging [14], [17], [25], sparse recovery on a sphere [26], image feature detection [27]- [29], medical imaging [18], [19], [30], [31], tomography [32], astronomy [33], spectroscopy [34], unlimited sampling architecture [35], [36] and inverse source problems [37], [38].…”

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confidence: 99%

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Sampling and Super Resolution of Sparse Signals Beyond the Fourier Domain

Bhandari

Eldar

2019

IEEE Trans. Signal Process.

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Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, lowpass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, timelimited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution-multiplication property in the SAFT domain.Preliminary ideas leading to this manuscript were presented in parts at ICASSP 2015 [1] and 2016 [2]. A. Bhandari is with the Massachusetts

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“…In addition, certain multidimensional results are stated without proof for brevity. Detailed proofs appear in our paper [22]. , using such algebraic techniques as Prony's method [23].…”

Section: The Inverse Source Problemmentioning

confidence: 99%

Solving inverse source problems for linear PDEs using sparse sensor measurements

Murray-Bruce

Dragotti

2016

2016 50th Asilomar Conference on Signals, Systems and Computers

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Abstract-Many physical phenomena across several applications can be described by partial differential equations (PDEs). In these applications, sensors collect sparse samples of the resulting phenomena with the aim of detecting its cause/source, using some intelligent data analysis tools on the samples. These problems are commonly referred to as inverse source problems. This work presents a novel framework for solving such inverse source problem for linear PDEs by drawing from certain recent results in modern sampling theory. Under the new framework, we study the well-known diffusion PDE and present numerical results that highlight the validity and robustness of the approach.

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A Sampling Framework for Solving Physics-Driven Inverse Source Problems

Cited by 21 publications

References 57 publications

Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

Sampling and Super Resolution of Sparse Signals Beyond the Fourier Domain

Solving inverse source problems for linear PDEs using sparse sensor measurements

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