\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A data-driven edge-preserving D-bar method for electrical impedance tomography

Abstract Related Papers Cited by
  • In Electrical Impedance Tomography (EIT), the internal conductivity of a body is recovered via current and voltage measurements taken at its surface. The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise, and requires the use of regularized solution methods, of which D-bar is the only proven method. The resulting EIT images have low spatial resolution due to smoothing caused by low-pass filtered regularization. In many applications, such as medical imaging, it is known a priori that the target contains sharp features such as organ boundaries, as well as approximate ranges for realistic conductivity values. In this paper, we use this information in a new edge-preserving EIT algorithm, based on the original D-bar method coupled with a deblurring flow stopped at a minimal data discrepancy. The method makes heavy use of a novel data fidelity term based on the so-called CGO sinogram. This nonlinear data step provides superior robustness over traditional EIT data formats such as current-to-voltage matrices or Dirichlet-to-Neumann operators, for commonly used current patterns.
    Mathematics Subject Classification: Primary: 65N21; Secondary: 94A08.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    R. Alicandro, A. Braides and J. Shah, Approximation of non-convex functionals in GBV, 1998.

    [2]

    L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence, Communications on Pure and Applied Mathematics, 43 (1990), 999-1036.doi: 10.1002/cpa.3160430805.

    [3]

    K. Astala, J. Mueller, L. Päivärinta, A. Perämäki and S. Siltanen, Direct electrical impedance tomography for nonsmooth conductivities, Inverse Problems and Imaging, 5 (2011), 531-549.doi: 10.3934/ipi.2011.5.531.

    [4]

    K. Astala and L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem, in Proc. 7th Internat. Conference on Harmonic Analysis, Collectanea Mathematica, (2006), 127-139.

    [5]

    K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299.doi: 10.4007/annals.2006.163.265.

    [6]

    J. Bikowski, K. Knudsen and J. L. Mueller, Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27 (2011), 015002, 19 pp.doi: 10.1088/0266-5611/27/1/015002.

    [7]

    R. M. Brown, Global uniqueness in the impedance imaging problem for less regular conductivities, SIAM Journal on Mathematical Analysis, 27 (1996), 1049-1056.doi: 10.1137/S0036141094271132.

    [8]

    A.-P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, (1980), 65-73.

    [9]

    A. Chambolle, Image segmentation by variational methods: Mumford and shah functional and the discrete approximations, SIAM Journal on Applied Mathematics, 55 (1995), 827-863.doi: 10.1137/S0036139993257132.

    [10]

    M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Review, 41 (1999), 85-101.doi: 10.1137/S0036144598333613.

    [11]

    H. Cornean, K. Knudsen and S. Siltanen, Towards a $d$-bar reconstruction method for three-dimensional EIT, Journal of Inverse and Ill-Posed Problems, 14 (2006), 111-134.doi: 10.1515/156939406777571102.

    [12]

    I. N. R. Council, Dielectric properties of body tissues, 2013, http://niremf.ifac.cnr.it/tissprop/htmlclie/htmlclie.htm.

    [13]

    E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Archive for Rational Mechanics and Analysis, 108 (1989), 195-218.doi: 10.1007/BF01052971.

    [14]

    F. Delbary, P. C. Hansen and K. Knudsen, Electrical impedance tomography: 3D reconstructions using scattering transforms, Applicable Analysis, 91 (2012), 737-755.doi: 10.1080/00036811.2011.598863.

    [15]

    E. Erdem and S. Tari, Mumford-shah regularizer with contextual feedback, Journal of Mathematical Imaging and Vision, 33 (2009), 67-84.doi: 10.1007/s10851-008-0109-y.

    [16]

    L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035.

    [17]

    D. E. Finkel, DIRECT Optimization Algorithm User Guide, Technical report, Center for Research in Scientific Computation, North Carolina State University, 2003.

    [18]

    E. Francini, Recovering a complex coefficient in a planar domain from Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107-119.doi: 10.1088/0266-5611/16/1/309.

    [19]

    S. J. Hamilton and J. L. Mueller, Direct EIT reconstructions of complex admittivities on a chest-shaped domain in 2-D, IEEE Transactions on Medical Imaging, 32 (2013), 757-769.doi: 10.1109/TMI.2012.2237389.

    [20]

    S. Hamilton, C. Herrera, J. L. Mueller and A. Von Herrmann, A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D, Inverse Problems, 28 (2012), 095005, 24 pp.doi: 10.1088/0266-5611/28/9/095005.

    [21]

    L. Harhanen, N. Hyvönen, H. Majander and S. Staboulis, Edge-enhancing reconstruction algorithm for three-dimensional electrical impedance tomography, ArXiv e-prints, arXiv:1406.1279.

    [22]

    T. Helin and M. Lassas, Hierarchical models in statistical inverse problems and the mumford-shah functional, Inverse problems, 27 (2011), 015008, 32 pp.doi: 10.1088/0266-5611/27/1/015008.

    [23]

    D. Isaacson, J. Mueller, J. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiological Measurement, 27 (2006), S43-S50.doi: 10.1088/0967-3334/27/5/S04.

    [24]

    D. R. Jones, C. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the lipschitz constant, Journal of Optimization Theory and Applications, 79 (1993), 157-181.doi: 10.1007/BF00941892.

    [25]

    M. Jung, X. Bresson, T. F. Chan and L. A. Vese, Nonlocal Mumford-Shah regularizers for color image restoration, IEEE Trans. Image Process., 20 (2011), 1583-1598.doi: 10.1109/TIP.2010.2092433.

    [26]

    K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiological Measurement, 24 (2003), 391-403.doi: 10.1088/0967-3334/24/2/351.

    [27]

    K. Knudsen, M. Lassas, J. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2009), 599-624.doi: 10.3934/ipi.2009.3.599.

    [28]

    K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Communications in Partial Differential Equations, 29 (2004), 361-381.doi: 10.1081/PDE-120030401.

    [29]

    J. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, vol. 10 of Computational Science and Engineering, SIAM, 2012.doi: 10.1137/1.9781611972344.

    [30]

    J. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM Journal on Scientific Computing, 24 (2003), 1232-1266.doi: 10.1137/S1064827501394568.

    [31]

    D. Mumford and J. Shah, Boundary detection by minimizing functionals, in IEEE Conference on Computer Vision and Pattern Recognition, 1985.

    [32]

    M. Music, P. Perry and S. Siltanen, Exceptional circles of radial potentials, Inverse Problems, 29 (2013), 045004, 25 pp.doi: 10.1088/0266-5611/29/4/045004.

    [33]

    A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96.doi: 10.2307/2118653.

    [34]

    P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639.doi: 10.1109/34.56205.

    [35]

    R. Ramlau and W. Ring, A mumford-shah level-set approach for the inversion and segmentation of x-ray tomography data, Journal of Computational Physics, 221 (2007), 539-557.doi: 10.1016/j.jcp.2006.06.041.

    [36]

    L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional, ESAIM Control Optim. Calc. Var., 6 (2001), 517-538.doi: 10.1051/cocv:2001121.

    [37]

    J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion, in IEEE Conference on Computer Vision and Pattern Recognition, IEEE, (1996), 136-142.doi: 10.1109/CVPR.1996.517065.

    [38]

    S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of a. nachman for the 2-d inverse conductivity problem, Inverse Problems, 16 (2000), 681-699.doi: 10.1088/0266-5611/16/3/310.

    [39]

    J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169.doi: 10.2307/1971291.

    [40]

    J. Weickert, Anisotropic Diffusion in Image Processing, Teubner Stuttgart, 1998.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(85) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return