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Register a new userNumerical reconstruction of radiative sources in an absorbing and non-diffusing scattering medium in two dimensions
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Added by
Hiroshi Fujiwara
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We consider the two dimensional quantitative imaging problem of recovering a radiative source inside an absorbing and scattering medium from knowledge of the outgoing radiation measured at the boundary. The medium has an anisotropic scattering property that is neither negligible nor large enough for the diffusion approximation to hold. We present the numerical realization of the authors recently proposed reconstruction method. For scattering kernels of finite Fourier content in the angular variable, the solution is exact. The feasibility of the proposed algorithms is demonstrated in several numerical experiments, including simulated scenarios for parameters meaningful in optical molecular imaging.
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We study the efficient approximation of integrals involving Hankel functions of the first kind which arise in wave scattering problems on straight or convex polygonal boundaries. Filon methods have proved to be an effective way to approximate many types of highly oscillatory integrals, however finding such methods for integrals that involve non-linear oscillators and frequency-dependent singularities is subject to a significant amount of ongoing research. In this work, we demonstrate how Filon methods can be constructed for a class of integrals involving a Hankel function of the first kind. These methods allow the numerical approximation of the integral at uniform cost even when the frequency $omega$ is large. In constructing these Filon methods we also provide a stable algorithm for computing the Chebyshev moments of the integral based on duality to spectral methods applied to a version of Bessels equation. Our design for this algorithm has significant potential for further generalisations that would allow Filon methods to be constructed for a wide range of integrals involving special functions. These new extended Filon methods combine many favourable properties, including robustness in regard to the regularity of the integrand and fast approximation for large frequencies. As a consequence, they are of specific relevance to applications in wave scattering, and we show how they may be used in practice to assemble collocation matrices for wavelet-based collocation methods and for hybrid oscillatory approximation spaces in high-frequency wave scattering problems on convex polygonal shapes.
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