Inverse Source Reconstruction Research Articles

Galerkin-based meshless methods for photon transport in the biological tissue

As an important small animal imaging technique, optical imaging has attracted increasing attention in recent years. However, the photon propagation process is extremely complicated for highly scattering property of the biological tissue. Furthermore, the light transport simulation in tissue has a significant influence on inverse source reconstruction. In this contribution, we present two Galerkin-based meshless methods (GBMM) to determine the light exitance on the surface of the diffusive tissue. The two methods are both based on moving least squares (MLS) approximation which requires only a series of nodes in the region of interest, so complicated meshing task can be avoided compared with the finite element method (FEM). Moreover, MLS shape functions are further modified to satisfy the delta function property in one method, which can simplify the processing of boundary conditions in comparison with the other. Finally, the performance of the proposed methods is demonstrated with numerical and physical phantom experiments.

Exact solutions to the acoustic source reconstruction problem

In this paper, we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed-form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far-field or intermediate regions. Exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain. A stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field. It is found that the degree of ill-posedness of the inverse problem is significantly reduced, when considering a hemi-circular observation arc with respect to a linear array of sensors, by a factor scaling on the small angular aperture subtended by the observation line. Estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace.

Analytic solutions to the acoustic source reconstruction problem

In this paper, we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed-form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far-field or intermediate regions. Exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain. A stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field. It is found that the degree of ill-posedness of the inverse problem is significantly reduced, when considering a hemi-circular observation arc with respect to a linear array of sensors, by a factor scaling on the small angular aperture subtended by the observation line. Estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace.

Influence of tissue conductivity anisotropy on EEG/MEG field and return current computation in a realistic head model: A simulation and visualization study using high-resolution finite element modeling

To achieve a deeper understanding of the brain, scientists, and clinicians use electroencephalography (EEG) and magnetoencephalography (MEG) inverse methods to reconstruct sources in the cortical sheet of the human brain. The influence of structural and electrical anisotropy in both the skull and the white matter on the EEG and MEG source reconstruction is not well understood. In this paper, we report on a study of the sensitivity to tissue anisotropy of the EEG/MEG forward problem for deep and superficial neocortical sources with differing orientation components in an anatomically accurate model of the human head. The goal of the study was to gain insight into the effect of anisotropy of skull and white matter conductivity through the visualization of field distributions, isopotential surfaces, and return current flow and through statistical error measures. One implicit premise of the study is that factors that affect the accuracy of the forward solution will have at least as strong an influence over solutions to the associated inverse problem. Major findings of the study include (1) anisotropic white matter conductivity causes return currents to flow in directions parallel to the white matter fiber tracts; (2) skull anisotropy has a smearing effect on the forward potential computation; and (3) the deeper a source lies and the more it is surrounded by anisotropic tissue, the larger the influence of this anisotropy on the resulting electric and magnetic fields. Therefore, for the EEG, the presence of tissue anisotropy both for the skull and white matter compartment substantially compromises the forward potential computation and as a consequence, the inverse source reconstruction. In contrast, for the MEG, only the anisotropy of the white matter compartment has a significant effect. Finally, return currents with high amplitudes were found in the highly conducting cerebrospinal fluid compartment, underscoring the need for accurate modeling of this space.

Sound source reconstruction using inverse boundary element calculations.

Whereas standard boundary element calculations focus on the forward problem of computing the radiated acoustic field from a vibrating structure, the aim in this work is to reverse the process, i.e., to determine vibration from acoustic field data. This inverse problem is brought on a form suited for solution by means of an inverse boundary element method. Since the numerical treatment of the inverse source reconstruction results in a discrete ill-posed problem, regularization is imposed to avoid unstable solutions dominated by errors. In the present work the emphasis is on Tikhonov regularization and parameter-choice methods not requiring an error-norm estimate for choosing the right amount of regularization. Several parameter-choice strategies have been presented lately, but it still remains to be seen how well these can handle industrial applications with real measurement data. In the present work it is demonstrated that the L-curve criterion is robust with respect to the errors in a real measurement situation. In particular, it is shown that the L-curve criterion is superior to the more conventional generalized cross-validation (GCV) approach for the present tire noise studies.

On inverse source problems for the one-way wave equation

Inverse problems are considered for the linear one-way wave equation or transport equation. The equation is one-dimensional and nonstationary, that is, it has spatial and time-dependent wave speed and dissipation. In particular a number of inverse source reconstruction problems are considered. Both theoretical and numerical results are given for the methods examined. In particular it is shown that the source reconstruction is unique, when the source function is separable as a function of time and space, for the inverse problems discussed. It is shown that the inverse source problems are ill-posed and regularisation is introduced to provide well-posed problems.