A projective method for an inverse source problem of the Poisson equation
- 28 February 2003
- journal article
- Published by IOP Publishing in Inverse Problems
- Vol. 19 (2), 355-369
- https://doi.org/10.1088/0266-5611/19/2/307
Abstract
This paper proposes a method for reconstructing the positions, strengths, and number of point sources in a three-dimensional (3D) Poisson field from boundary measurements. Algebraic relations are obtained, base do nm ultipole moments determined by the sources and data on the boundary of a domain. To solve for the source parameters with efficient use of data, we select the necessary number of equations from them in the following two ways: (1) the use of those starting from lower-degree multipole moments; and (2) the use of combined ones involving infinitely higher-degree multipole moments. We show that both methods are based on the projection of 3D sources onto a two- dimensional space: thexy-plane for the first one and the Riemann sphere which is set to contain the domain for the second one. We also show that they share th es ame fundamental equations which can be solved by a procedure proposed by El-Badia and Ha-Duong (2000 Inverse Problems 16 651-63). Numerical simulations show that projection onto the xy-plane is more appropriate for sources scattered in the middle of the domain, whereas projection onto the Riemann sphere is more appropriate for sources concentrated close to the boundary of the domain. We also give an appropriate method of measurement for the Riemann sphere projection.Keywords
This publication has 13 references indexed in Scilit:
- An inverse source problem in potential analysisInverse Problems, 2000
- Identification of dipole sources in a bounded domain for Maxwell's equationsWave Motion, 1998
- A precise estimation method for locations in an inverse logarithmic potential problem for point mass modelsApplied Mathematical Modelling, 1994
- Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brainReviews of Modern Physics, 1993
- Magnetocardiographic functional localization using current multipole modelsIEEE Transactions on Biomedical Engineering, 1991
- Continuous probabilistic solutions to the biomagnetic inverse problemInverse Problems, 1990
- Electric Dipole Tracing in the Brain by Means of the Boundary Element Method and Its AccuracyIEEE Transactions on Biomedical Engineering, 1987
- Application of the single moving dipole inverse solution to the study of the wolff-parkinson-white syndrome in manJournal of Electrocardiology, 1984
- Evaluation of Methods for Three-Dimensional Localization of Electrical Sources in the Human BrainIEEE Transactions on Biomedical Engineering, 1978
- Multipole Representation for an Equivalent Cardiac GeneratorProceedings of the IRE, 1960