A fast and accurate imaging algorithm in optical/diffusion tomography

Abstract

An n-dimensional (n = 2, 3) inverse problem for the parabolic/diffusion equationThe problem consists of determining the function a(x) inside of a bounded domain ⊂ R n given the values of the solution u(x, t) for a single source location x 0 ∈ ∂ on a set of detectors {x i } m i=1 ⊂ ∂ , 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.

Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy

“…This is the basis of image reconstruction algorithms in which an image of Dl a (i.e., the vector x) can be obtained from a set of measurements (i.e., the vector y) through inversion of the matrix A (i.e., the effective pathlengths L i,j ). While several advanced imaging algorithms have been developed-including analytic diffraction tomography approaches (Cheng and Boas, 1998;Li et al, 1997;Matson and Liu, 1999;Schotland, 1997), perturbation approaches (Arridge and Schweiger, 1995;Barbour et al, 1995;O'Leary et al, 1995;Schotland et al, 1993;Yao et al, 1997), the Taylor series expansion approach (Jiang et al, 1996;Paulsen and Jiang, 1995), gradient-based iterative techniques (Arridge and Schweiger, 1998), elliptic systems method (ESM) (Gryazin et al, 1999;Klibanov et al, 1997), and Bayesian conditioning (Barnett et al, 2003;Eppstein et al, 1999) -the most widely used methods for diffuse optical functional brain imaging incorporate a semiinfinite forward model (Kienle and Patterson, 1997a;Patterson et al, 1989) and either backprojection (Colak et al, 1997;Franceschini et al, 2000;Maki et al, 1995, Walker et al, 1997 or perturbation approaches (Arridge, 1999).…”

Three-dimensional diffuse optical mammography with ultrasound localization in a human subject

“…A variety of methods have been developed for DOT. These include fits to analytic solutions, [1][2][3] backprojection methods, [4][5][6][7] diffraction tomography in k-space, [8][9][10][11][12][13][14] perturbation approaches, [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] elliptic systems method ͑ESM͒, [31][32][33] and a direct method. 34 All of these approaches have various advantages and disadvantages.…”

A Novel Principle for Data Processing from Hand-Held Ground Penetrating Radars

“…This issue was addressed in this project. Specifically, a new method for the corresponding inverse problem was developed, which is a second generation of the so-called Elliptic Systems Method (ESM), being previously developed by the Co-PIs for Diffusion Tomography with medical applications [10,[12][13][14]. (In this report references in bold face letters refer to Bibliography section.…”

“…Hence, linearization leads to dropping the h(x)v term in (4.13). This approach was used previously in publications about the ESM [10,[12][13][14]. The assumption about the linearization can actually be relaxed if using Newton-like updates [10,[12][13][14].…”

“…In the first generation of this ESM, which was aimed on medical applications and was applied to the time dependent diffusion equations, integrals were eliminated via truncated generalized Fourier series with respect to time [10][11][12][13][14]. As a result, a BVP for an elliptic system was obtained (the latter led to the name of this method).…”