Takaaki Nara

An inverse source problem for Helmholtz's equation from the Cauchy data with a single wave number

The inverse source problem of the Helmholtz equation in an interior domain is investigated. We show the uniqueness and local stability, where the source consists of multiple point sources. An algebraic algorithm is proposed to identify the number, locations and intensities of the point sources from boundary measurements. Uniqueness and non-uniqueness results for some distributed sources are also established. The proposed method is verified numerically.

An Exact Direct Method of Sinusoidal Parameter Estimation Derived From Finite Fourier Integral of Differential Equation

In this paper, we propose a novel method for estimating the parameters (frequency, amplitude, and phase) of real sinusoids. To derive the estimator, we start from the characteristic differential equation of a sinusoid. To remove differentials and obtain an algebraic relation for frequency, we introduce finite-period weighted integrals of the differential equation, which become equivalent to the differential equation when a sufficient number of weight functions are applied. As weight functions, we show that Fourier kernels have excellent properties. Terms related to integral boundaries are readily eliminated, observations are provided by Fourier coefficients, and the relation becomes independently accurate for multiple sinusoids if they are sufficiently spaced. We solve the obtained equations in two ways: one is for approaching to the Cramer-Rao lower bound (CRLB), and the other is for enhancing the interference rejection capability. Also, methods are proposed to calculate the weighted integrals from sampled signals with an improved accuracy. Proposed algorithms are examined under noise and sinusoidal interference. Error variances are compared with the CRLB and other fast Fourier transform (FFT)-based methods.

A sensor measuring the Fourier coefficients of the magnetic flux density for pipe crack detection using the magnetic flux leakage method

A simple sensor for magnetic flux leakage methods of detecting cracks on the surfaces of ferromagnetic pipes is proposed. We show that the crack position can be determined by the Fourier coefficients of the leakage magnetic flux density on a circle inside the pipe in the pipe cross-sectional plane. Coils that directly output the Fourier cosine and sine coefficients were made. The experimental results showed that, using only these two coils rather than dozens of magnetic sensors, the center position of a crack on the inside/outside surface of the pipe could be localized.

An algebraic method for identification of dipoles and quadrupoles

This paper proposes an algebraic algorithm for reconstruction of dipole–quadrupole positions and moments in 2D space from the boundary measurements of the potential without providing an initial parameter guess or iterative computing forward solutions. A system of second-order equations is derived for the elementary symmetric polynomials of the pole positions. An algorithm to transform them into a triangular form is shown so that the pole positions are obtained algebraically. Numerical simulations are conducted to verify the algorithm.

Properties of the Linear Equations Derived From Euler's Equation and Its Application to Magnetic Dipole Localization

This paper presents a novel algorithm and sensor for estimating the position of a magnetic dipole. By transforming Eulers equation of degree -3 into an integral form, we have linear equations relating the dipole position to the surface integrals of the magnetic flux densities on a cube. To measure all the quantities required in the linear equations, we develop a cubic sensor with a side length of 50 mm which consists of 18 coils. We show that the coefficient matrix of the linear equations is symmetric and traceless, which can be used to improve localization accuracy. By performing the nonlinear least squares method with the initial solution given by the proposed method, the average and maximum error are 8.3 and 17.5 mm, respectively, in the range of 500 mm.

Direct localization of poles of a meromorphic function from measurements on an incomplete boundary

This paper proposes an algebraic method to reconstruct the positions of multiple poles in a meromorphic function field from measurements on an arbitrary simple arc in it. A novel issue is the exactness of the algorithm depending on whether the arc is open or closed, and whether it encloses or does not enclose the poles. We first obtain a differential equation that can equivalently determine the meromorphic function field. From it, we derive linear equations that relate the elementary symmetric polynomials of the pole positions to weighted integrals of the field along the simple arc and end-point terms of the arc when it is an open one. Eliminating the end-point terms based on an appropriate choice of weighting functions and a combination of the linear equations, we obtain a simple system of linear equations for solving the elementary symmetric polynomials. We also show that our algorithm can be applied to a 2D electric impedance tomography problem. The effects of the proximity of the poles, the number of measurements and noise on the localization accuracy are numerically examined.

Inverse dipole source problem for time-harmonic Maxwell equations: algebraic algorithm and Hölder stability

In this paper, we investigate an inverse source problem of the time harmonic Maxwell equations at a fixed frequency, where the source consists of multiple point dipoles. An algebraic algorithm is proposed to identify the number, locations and moments of the dipoles from boundary measurements of tangential components of the electric and magnetic fields. Also, a H?lder stability result is shown. The proposed algorithm is numerically verified.

Partial differential equation-based localization of a monopole source from a circular array

Wave source localization from a sensor array has long been the most active research topics in both theory and application. In this paper, an explicit and time-domain inversion method for the direction and distance of a monopole source from a circular array is proposed. The approach is based on a mathematical technique, the weighted integral method, for signal/source parameter estimation. It begins with an exact form of the source-constraint partial differential equation that describes the unilateral propagation of wide-band waves from a single source, and leads to exact algebraic equations that include circular Fourier coefficients (phase mode measurements) as their coefficients. From them, nearly closed-form, single-shot and multishot algorithms are obtained that is suitable for use with band-pass/differential filter banks. Numerical evaluation and several experimental results obtained using a 16-element circular microphone array are presented to verify the validity of the proposed method.

Reconstruction of the number and positions of dipoles and quadrupoles using an algebraic method

Localization of dipoles and quadrupoles is important in inverse potential analysis, since they can effectively express spatially extended sources with a small number of parmeters. This paper proposes an algebraic method for reconstruction of pole positions as well as the number of dipole-quadrupoles without providing an initial parameter guess or iterative computing forward solutions. It is also shown that a magnetoencephalography inverse problem with a source model of dipole-quadrupoles in 3D space is reduced into the same problem as in 2D space.

An inverse source problem for quasi-static Maxwell's equations

Abstract In this paper, we show the uniqueness and local stability of an inverse source problem for the quasi-static Maxwell equation in a layered domain, where the source consists of multiple point dipoles. Also, an algebraic algorithm is proposed to identify the number, locations, and moments of the dipoles from boundary measurements of tangential components of the electric and magnetic fields. The proposed algorithm is numerically verified.