A globally accelerated numerical method for optical tomography with continuous wave source

Abstract A new numerical method for an inverse problem for an elliptic equation with unknown potential is proposed. In this problem the point source is running along a straight line and the source-dependent Dirichlet boundary condition is measured as the data for the inverse problem. A rigorous convergence analysis shows that this method converges globally, provided that the so-called tail function is approximated well. This approximation is verified in numerical experiments, so as the global convergence. Applications to medical imaging, imaging of targets on battlefields and to electrical impedance tomography are discussed.

[1]  P. Grinevich.  Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity , 2000 .

[2]  Roman Novikov.  The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator , 1992 .

[3]  E. Cheney.  Introduction to approximation theory , 1966 .

[4]  S. Arridge.  Optical tomography in medical imaging , 1999 .

[5]  David Isaacson,et al.  An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem , 2000 .

[6]  R. Alfano,et al.  Optical spectroscopic diagnosis of cancer and normal breast tissues , 1989 .

[7]  Michael V. Klibanov,et al.  Carleman estimates for coefficient inverse problems and numerical applications , 2004 .

[8]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[9]  M. I. Belishev,et al.  Dynamical variant of the BC-method: theory and numerical testing , 1999 .

[10]  Valentin A. Burov,et al.  Reconstruction of Fine-Scale Structure of Acoustical Scatterer on Large-Scale Contrast Background , 2002 .

[11]  H. Keller,et al.  Analysis of Numerical Methods , 1969 .

[12]  Michael V. Klibanov,et al.  Numerical studies on the globally convergent convexification algorithm in 2D , 2007 .

[13]  P M Schlag,et al.  Development of a time-domain optical mammograph and first in vivo applications. , 1999, Applied optics.

[14]  E. Haber,et al.  On optimization techniques for solving nonlinear inverse problems , 2000 .

[15]  D. Isaacson,et al.  An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem , 2000 .

[16]  R. Novikov,et al.  Multidimensional inverse spectral problem for the equation —Δψ + (v(x) — Eu(x))ψ = 0 , 1988 .

[17]  L. Beilina,et al.  A GLOBALLY CONVERGENT NUMERICAL METHOD FOR SOME COEFFICIENT INVERSE PROBLEMS WITH RESULTING SECOND ORDER ELLIPTIC EQUATIONS , 2007 .

[18]  Jianzhong Su,et al.  Reconstruction method with data from a multiple-site continuous-wave source for three-dimensional optical tomography. , 2006, Journal of the Optical Society of America. A, Optics, image science, and vision.

[19]  Abdigany D. Satybaev,et al.  Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems , 2004 .

[20]  Michael V. Klibanov,et al.  Numerical implementation of the convexification algorithm for an optical diffusion tomograph , 2008 .

[21]  A. Nachman,et al.  Global uniqueness for a two-dimensional inverse boundary value problem , 1996 .

[22]  Inverse problem in optical tomography and its numerical investigation by iteratively regularized methods , 2005 .

[23]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[24]  Mikhail I. Belishev,et al.  TOPICAL REVIEW: Boundary control in reconstruction of manifolds and metrics (the BC method) , 1997 .

[25]  Yuriy A. Gryazin,et al.  Numerical Solution of a Subsurface Imaging Inverse Problem , 2001, SIAM J. Appl. Math..