Multi-frequency acousto-electromagnetic tomography

Alberti, Giovanni S.; Ammari, Habib; Ruan, Kaixi

doi:10.1090/conm/658/13132

Cited by 7 publications

(13 citation statements)

References 28 publications

(69 reference statements)

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“…As before, we look for illuminations ϕ i and frequencies ω such that the corresponding solutions verify r conditions given by (5) ζ…”

Section: Introductionmentioning

confidence: 99%

Enforcing Local Non-Zero Constraints in PDEs and Applications to Hybrid Imaging Problems

Alberti¹

2015

Communications in Partial Differential Equations

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Abstract. We study the boundary control of solutions of the Helmholtz and Maxwell equations to enforce local non-zero constraints. These constraints may represent the local absence of nodal or critical points, or that certain functionals depending on the solutions of the PDE do not vanish locally inside the domain. Suitable boundary conditions are classically determined by using complex geometric optics solutions. This work focuses on an alternative approach to this issue based on the use of multiple frequencies. Simple boundary conditions and a finite number of frequencies are explicitly constructed independently of the coefficients of the PDE so that the corresponding solutions satisfy the required constraints. This theory finds applications in several hybrid imaging modalities: some examples are discussed.

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“…As before, we look for illuminations ϕ i and frequencies ω such that the corresponding solutions verify r conditions given by (5) ζ…”

Section: Introductionmentioning

confidence: 99%

Enforcing Local Non-Zero Constraints in PDEs and Applications to Hybrid Imaging Problems

Alberti¹

2015

Communications in Partial Differential Equations

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“…However, this is a consequence of special symmetries in Ω and in the parameters: for a generic elliptic operator the eigenvalues are all simple [1,43]. More precisely, under the assumptions a ∈ C 2 (R 3 ; R 3×3 ), q ∈ C 2 (R 3 ; R) and a = a T , for a generic F ∈ Diff(Ω), all the eigenvalues of (5) in Ω F are simple [32, Example 6.3]. Here, Diff(Ω) denotes the open subset of C 2 (Ω; R 3 ) consisting of maps F which are diffeomorphisms to their images Ω F := F (Ω).…”

Section: Propositionmentioning

confidence: 99%

“…This paper represents a generalization of several works [2,3,4,5,6,7,17], in which multiple frequencies are used to enforce several non-zero local constraints for certain frequency-dependent PDE. All these results are based on the fact that the constraint under investigation is satisfied for a particular choice of the frequency, e.g.…”

Section: Introductionmentioning

confidence: 99%

Absence of Critical Points of Solutions to the Helmholtz Equation in 3D

Alberti

2016

Arch Rational Mech Anal

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Abstract. The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain Ω ⊂ R 3 , given by −div(a ∇uWe prove that for an admissible g there exists a finite set of frequencies K in a given interval and an open cover Ω = ∪ ω∈K Ωω such that |∇u g ω (x)| > 0 for every ω ∈ K and x ∈ Ωω. The set K is explicitly constructed. If the spectrum of this problem is simple, which is true for a generic domain Ω, the admissibility condition on g is a generic property.

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“…Second, even if the PDE model is accurate, there may be too many unknowns to have unique reconstruction [19]. Third, even in cases when the reconstruction is unique, this may require the differentiation of the data [13,14,1], which is known to be an unstable process, or may require additional assumptions to be satisfied [13,1,17,14,4].…”

Section: Introductionmentioning

confidence: 99%

Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging

Alberti

Ammari

2017

Applied and Computational Harmonic Analysis

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Abstract. The main focus of this work is the reconstruction of the signals f and g i , i = 1, . . . , N , from the knowledge of their sums h i = f + g i , under the assumption that f and the g i 's can be sparsely represented with respect to two different dictionaries A f and Ag. This generalizes the well-known "morphological component analysis" to a multi-measurement setting. The main result of the paper states that f and the g i 's can be uniquely and stably reconstructed by finding sparse representations of h i for every i with respect to the concatenated dictionary [A f , Ag], provided that enough incoherent measurements g i are available. The incoherence is measured in terms of their mutual disjoint sparsity.This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, we show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.

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Multi-frequency acousto-electromagnetic tomography

Cited by 7 publications

References 28 publications

Enforcing Local Non-Zero Constraints in PDEs and Applications to Hybrid Imaging Problems

Enforcing Local Non-Zero Constraints in PDEs and Applications to Hybrid Imaging Problems

Absence of Critical Points of Solutions to the Helmholtz Equation in 3D

Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging

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