Abstract
We study the shape reconstruction of an inclusion from the faraway measurement of the associated electric field. This is an inverse problem of practical importance in biomedical imaging and is known to be notoriously ill-posed. By incorporating Drude’s model of the permittivity parameter, we propose a novel reconstruction scheme by using the plasmon resonance with a significantly enhanced resonant field. We conduct a delicate sensitivity analysis to establish a sharp relationship between the sensitivity of the reconstruction and the plasmon resonance. It is shown that when plasmon resonance occurs, the sensitivity functional blows up and hence ensures a more robust and effective construction. Then we combine the Tikhonov regularization with the Laplace approximation to solve the inverse problem, which is an organic hybridization of the deterministic and stochastic methods and can quickly calculate the minimizer while capture the uncertainty of the solution. We conduct extensive numerical experiments to illustrate the promising features of the proposed reconstruction scheme.
Highlights
Plasmon resonance is the resonant oscillation of conduction electrons at the interface between negative and positive permittivity material stimulated by incident field
We establish the spectral expansion of the shape sensitivity functional, from which we can conclude the sharp relationship between the reconstruction sensitivity and the plasmon resonance
We investigate the inverse problem that utilizing the far-field measurement to reconstruct the shape of of an inclusion
Summary
Plasmon resonance is the resonant oscillation of conduction electrons at the interface between negative and positive permittivity material stimulated by incident field. We study the shape reconstruction of an inclusion from the faraway measurement of the associated electric field This is an inverse problem of practical importance in biomedical imaging and is known to be notoriously ill-posed. We establish the spectral expansion of the shape sensitivity functional, from which we can conclude the sharp relationship between the reconstruction sensitivity and the plasmon resonance. It indicates that the plasmon resonant field can render a more robust and effective reconstruction. Based on the spectral theory of the Neumann-poincare operator, we establish the delicate spectral expansion of the shape sensitivity functional It indicates that when plasmon resonance occurs, the shape sensitivity can be improved dramatically.
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