A fast and accurate imaging algorithm in optical/diffusion tomography
- 1 October 1997
- journal article
- Published by IOP Publishing in Inverse Problems
- Vol. 13 (5), 1341-1361
- https://doi.org/10.1088/0266-5611/13/5/015
Abstract
An n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation , , , is considered. The problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors , where is the boundary of . A novel numerical method is derived and tested. Numerical tests are conducted for n = 2 and for ranges of parameters which are realistic for applications to early breast cancer diagnosis and the search for mines in murky shallow water using ultrafast laser pulses. The main innovation of this method lies in a new approach for a novel linearized problem (LP). Such a LP is derived and reduced to a well-posed boundary-value problem for a coupled system of elliptic partial differential equations. A principal advantage of this technique is in its speed and accuracy, since it leads to the factorization of well conditioned, sparse matrices with non-zero entries clustered in a narrow band near the diagonal. The authors call this approach the elliptic systems method (ESM). The ESM can be extended to other imaging modalities.Keywords
This publication has 18 references indexed in Scilit:
- Tomographic image reconstruction from optical projections in light-diffusing mediaApplied Optics, 1997
- Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid mediaProceedings of the National Academy of Sciences of the United States of America, 1996
- A simple method for solving inverse scattering problems in the resonance regionInverse Problems, 1996
- Sensitivity to prior knowledge in optical tomographic reconstructionPublished by SPIE-Intl Soc Optical Eng ,1995
- Issues in electrical impedance imagingIEEE Computational Science and Engineering, 1995
- MRI-guided optical tomography: prospects and computation for a new imaging methodIEEE Computational Science and Engineering, 1995
- The Factorization of Sparse Symmetric Indefinite MatricesIMA Journal of Numerical Analysis, 1991
- A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operatorInternational Journal for Numerical Methods in Engineering, 1979
- Direct Methods for Solving Symmetric Indefinite Systems of Linear EquationsSIAM Journal on Numerical Analysis, 1971
- Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. ICommunications on Pure and Applied Mathematics, 1959