A fast and accurate imaging algorithm in optical/diffusion tomography

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.

[1]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[2]  L. Hörmander Linear Partial Differential Operators , 1963 .

[3]  O. Ladyženskaja Linear and Quasilinear Equations of Parabolic Type , 1968 .

[4]  Jacques-Louis Lions,et al.  The method of quasi-reversibility : applications to partial differential equations , 1969 .

[5]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[6]  N. Munksgaard,et al.  A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator , 1979 .

[7]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[8]  I. Duff,et al.  The factorization of sparse symmetric indefinite matrices , 1991 .

[9]  Michael V. Klibanov,et al.  A computational quasi-reversiblility method for Cauchy problems for Laplace's equation , 1991 .

[10]  Michael V. Klibanov,et al.  Inverse Problems and Carleman Estimates , 1992 .

[11]  Michael V. Klibanov,et al.  Numerical solution of a time-like Cauchy problem for the wave equation , 1992 .

[12]  Ivo Babuška,et al.  The method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities , 1993 .

[13]  Harry L. Graber,et al.  MRI-guided optical tomography: prospects and computation for a new imaging method , 1995 .

[14]  F. Natterer,et al.  A propagation-backpropagation method for ultrasound tomography , 1995 .

[15]  David Isaacson,et al.  Issues in electrical impedance imaging , 1995 .

[16]  P. M. van den Berg,et al.  "Blind" shape reconstruction from experimental data , 1995 .

[17]  Simon R. Arridge,et al.  Sensitivity to prior knowledge in optical tomographic reconstruction , 1995, Photonics West.

[18]  M. Gijzen Conjugate Gradient-like solution algorithms for the mixed finite element approximation of the biharmonic equation, applied to plate bending problems , 1995 .

[19]  A. Kirsch,et al.  A simple method for solving inverse scattering problems in the resonance region , 1996 .

[20]  Matthias Orlt,et al.  NIR imaging in random media using time domain data , 1996, European Conference on Biomedical Optics.

[21]  R. Kleinman,et al.  Modified gradient approach to inverse scattering for binary objects in stratified media , 1996 .

[22]  M. Lax,et al.  Time-resolved optical diffusion tomographic image reconstruction in highly scattering turbid media. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[23]  J. Melissen,et al.  Tomographic image reconstruction from optical projections in light-diffusing media. , 1997, Applied optics.