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Manabu Machida

Publications and source records attributed to Manabu Machida.

At least 37 records · Page 2Linked to original sources
arXiv2021

Diffuse optical tomography by simulated annealing via a spin Hamiltonian

Diffuse optical tomography (DOT) is an imaging modality which uses near-infrared light. Although iterative numerical schemes are commonly used for its inverse problem, correct solutions are not obtained unless good initial guesses are chosen. We propose a numerical scheme of DOT which works even when good initial guesses of optical parameters are not available. We use simulated annealing (SA) which is a method of the Markov-chain Monte Carlo. To implement SA for DOT, a spin Hamiltonian is introduced in the cost function, and the Metropolis algorithm or single-component Metropolis-Hastings algorithm is used. By numerical experiments, it is shown that an initial random spin configuration is brought to a converged configuration by SA and targets in the medium are reconstructed. The proposed numerical method solves the inverse problem for DOT by finding the ground state of a spin Hamiltonian with SA.

math.NA↗
arXiv2020

Global Lipschitz stability for inverse problems for radiative transport equations

We consider inverse problems of determining coefficients or time independent factors of source terms in radiative transport equations by means of Carleman estimate. We establish global Lipschitz stability results with an additional condition which requires some strict positivity for initial value or given factor of source, but we need not any extra conditions on domains of velocities, which is the main achievement of this article compared with the existing work by Machida and Yamamoto ({\it Inverse Problems} {\bf 30} 035010, 2014). The proof relies on a Carleman estimate with a piecewise linear weight function according to the partition of the velocity domain.

math.AP↗
arXiv2020

Chandrasekhar polynomials -- A brief review

A review on the Chandrasekhar polynomials is given. The polynomials often appear in transport theory. The relation to the method of rotated reference frames for the three-dimensional radiative transport equation is clarified.

math-ph↗
arXiv2020

Linear transport in porous media

The linear transport theory is developed to describe the time dependence of the number density of tracer particles in porous media. The advection is taken into account. The transport equation is numerically solved by the analytical discrete ordinates method. For the inverse Laplace transform, the double-exponential formula is employed.

physics.comp-ph↗
arXiv2020

Numerical algorithms of the radiative transport equation using rotated reference frames for optical tomography with structured illumination

We consider optical tomography with structured illumination in spatial-frequency domain using the three-dimensional radiative transport equation. Without the diffusion approximation, the radiative transport equation is solved by the technique of rotated reference frames. In addition to the method of rotated reference frames (spherical-harmonic expansion), the three dimensional FN method is applied to this optical tomography.

physics.optics↗
arXiv2020

Born series for the photon diffusion equation perturbing the Robin boundary condition

The photon diffusion equation is solved making use of the Born series for the Robin boundary condition. We develop a general theory for arbitrary domains with smooth enough boundaries and explore the convergence. The proposed Born series is validated by numerical calculation in the three-dimensional half space. It is shown that in this case the Born series converges regardless the value of the impedance term in the Robin boundary condition.

math-ph↗
arXiv2019

A hybrid inversion scheme combining Markov chain Monte Carlo and iterative methods for determining optical properties of random media

Near-infrared spectroscopy (NIRS) including diffuse optical tomography is an imaging modality which makes use of diffuse light propagation in random media. When optical properties of a random medium is investigated from boundary measurements of reflected or transmitted light, iterative inversion schemes such as the Levenberg-Marquardt algorithm are known to fail when initial guesses are not close to the true value of the coefficient to be reconstructed. In this paper, we investigate how this weakness of iterative schemes is overcome by the use of Markov chain Monte Carlo. Using time-resolved measurements performed against a polyurethane-based phantom, we present a case that the Levenberg-Marquardt algorithm fails to work but the proposed hybrid method works well. Then with a toy model of diffuse optical tomography we illustrate that the evenberg-Marquardt method fails when it is trapped by a local minimum but the hybrid method can escape from local minima by using the Metropolis-Hastings Markov chain Monte Carlo algorithm until it reaches the valley of the global minimum. The proposed hybrid scheme can be applied to different inverse problems in NIRS which are solved iteratively. We find that for both numerical and phantom experiments optical properties such as the absorption and reduced scattering coefficients can be retrieved without being trapped by a local minimum when Monte Carlo simulation is run only about $100$ steps before switching to an iterative method. The hybrid method is compared with simulated annealing. Although the Metropolis-Hastings MCMC arrives at the steady state at about $10000$ Monte Carlo steps, in the hybrid method the Monte Carlo simulation can be stopped way before the burn-in time.

physics.comp-ph↗
arXiv2019

The linear Boltzmann equation in column experiments of porous media

The use of the linear Boltzmann equation is proposed for transport in porous media in a column. By column experiments, we show that the breakthrough curve is reproduced by the linear Boltzmann equation. The advection-diffusion equation is derived from the linear Boltzmann equation in the asymptotic limit of large propagation distance and long time.

physics.geo-ph↗
arXiv2016

The time-fractional radiative transport equation -- Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion

We consider the radiative transport equation in which the time derivative is replaced by the Caputo derivative. Such fractional-order derivatives are related to anomalous transport and anomalous diffusion. In this paper we describe how the time-fractional radiative transport equation is obtained from continuous-time random walk and see how the equation is related to the time-fractional diffusion equation in the asymptotic limit. Then we solve the equation with Legendre-polynomial expansion.

math-ph↗
arXiv2016

The radiative transport equation in flatland with separation of variables

The linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in three-dimensional space with planar symmetry. In this method, solutions are given by superpositions of eigenmodes which are sometimes called singular eigenfunctions. In this paper, we explore the singular-eigenfunction approach in flatland or two-dimensional space.

math-ph↗
arXiv2015

The Green's function for the three-dimensional linear Boltzmann equation via Fourier transform

The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering.

math-ph↗
arXiv2015

An FN method for the radiative transport equation in three dimensions

The FN method is an accurate and efficient numerical method for the one-dimensional radiative transport equation. In this paper the FN method is extended to three dimensions using rotated reference frames. To demonstrate the method, the exiting flux from structured illumination reflected by a medium occupying the half space is calculated.

math.NA↗
arXiv2015

Polarization oscillations of near-field thermal emission

We consider the polarization of thermal emission in the near-field of various materials including dielectrics and metallic systems with resonant surface modes. We find that at thermal equilibrium, the degree of polarization exhibits spatial oscillations with a period of approximately half the optical wavelength, independent of material composition. This result contrasts with that of Setala, Kaivola and Friberg [Phys. Rev. Lett. 88, 123902 (2002)], who find monotonic decay of the degree of polarization for systems in local thermal equilibrium.

physics.optics↗
arXiv2013

Singular eigenfunctions for the three-dimensional radiative transport equation

Case's method obtains solutions to the radiative transport equation as superpositions of elementary solutions when the specific intensity depends on one spatial variable. In this paper, we find elementary solutions when the specific intensity depends on three spatial variables in three-dimensional space. By using the reference frame whose z-axis lies in the direction of the wave vector, the angular part of each elementary solution becomes the singular eigenfunction for the one-dimensional radiative transport equation. Thus Case's method is generalized.

math-ph↗