On a method of image reconstruction of anisotropic media using applied quasipotential tomographic data

An algorithm for solving the coefficient problems of parameter identification of anisotropic media using applied quasipotential tomographic data is modified for the case of presence of more specific a priori information about the eigendirections of the corresponding conductivity tensor.  Its application is quite common in practice, in particular, in medicine, where the object of such study may be the medium with fibrous or layered areas (which includes muscles, bones, etc.), inside which there are streams of non-spherical particles (e.g. red blood cells).  As in our previous works, the corresponding algorithm is based on alternately solving the quasiconformal mapping and parameter identification problems, but in this work it is supplemented by the procedure of parallelization of calculations and the optimization problem is "accelerated".  The latter is characterized by a significant decrease in the number of intermediate calculations and, when imposing additional restrictions on eigendirections of the conductivity tensor, leads to the possibility of optimal adaptation of the algorithm to specific cases of practice.  The results of numerical experiments of imitative restoration of medium structure are presented.

  1. Holder D.  Electrical Impedance Tomography. Methods, History and Applications.  London, Institute of Physics (2005).
  2. Herwanger J. V., Pain C. C., Binley A., De Oliveira C. R. E., Worthington M. H.  Anisotropic resistivity tomography.  Geophysical Journal International. 158 (2), 409--425 (2004).
  3. Abascal J.-F. P. J., Lionheart W. R. B., Arridge S. R., Schweiger M., Atkinson D., Holder D. S.  Electrical impedance tomography in anisotropic media with known eigenvectors.  Inverse Problems. 27 (6), 1--17 (2011).
  4. Crabb M.  EIT Reconstruction Algorithms for Respiratory Intensive Care. PhD thesis.  Manchester, University of Manchester (2014).
  5. Tallman T. N.  Conductivity-Based Nanocomposite Structural Health Monitoring via Electrical Impedance Tomography. PhD thesis.  Ann Arbor, University of Michigan (2015).
  6. Calderón A. P.  On an inverse boundary value problem.  Computational and Applied Mathematics. 25 (2--3), 133--138 (2006).
  7. Martins T. C., Tsuzuki M. S. G.  Investigating anisotropic EIT with simulated annealing.  IFAC-PapersOnLine. 50 (1), 9961--9966 (2017).
  8. Bomba A., Boichura M.  Numerical complex analysis method for parameters identification of anisotropic media using applied quasipotential tomographic data.  Part 1: Problem statement and its approximation.  Mathematical and Computer Modelling. Series: Physical and Mathematical Sciences. 18 (1), 14--24 (2018).
  9. Bomba A., Safonyk A., Michuta O., Boichura M.  Applied quasipotential method for solving the coefficient problems of parameter identification of anisotropic media.  Informatics, Control, Measurement in Economy and Environment Protection. 9 (1), 33--36 (2019).
  10. Bomba A., Kashtan S.  On one method for constructing a dynamical mesh of nonlinear quasiperfect processes in deformable anisotropic media.  Journal of Applied Computer Science. 12 (2), 7--21  (2004).
  11. Astala K., Päivärinta L., Lassas M.  Calderón's inverse problem for anisotropic conductivity in the plane.  Communications in Partial Differential Equations. 30 (1--2), 207--224 (2005).
  12. Lionheart W. R. B.  Conformal uniqueness results in anisotropic electrical impedance imaging.  Inverse Problems. 13 (1), 125--134 (1997).
  13. Tikhonov A. N., Arsenin V. Y.  Solution of Ill-posed Problems.  New York, Wiley (1977).
  14. http://www.alglib.net/
  15. https://developer.nvidia.com/cuda-zone/
  16. Konstantinidis E., Cotronis Y.  Graphics processing unit acceleration of the red/black SOR method.  Concurrency and Computation: Practice and Experience. 25 (8), 1107--1120 (2013).
  17. Martyniuk P. M., Michuta O. R., Ulianchuk--Martyniuk O. V., Kuzlo M. T.  Numerical investigation of pressure head jump values on a thin inclusion in one-dimensional non-linear soil mousture transport problem.  International Journal of Applied Mathematics. 31 (4), 649--660 (2018).
  18. Murugananthi C., Ramyachitra D.  Life science applications in grid environment.  Proceedings on International Conference on Research Trends in Computer Technologies. Coimbatore, India. 1--5 (2013).
Mathematical Modeling and Computing, Vol. 6, No. 2, pp. 211–219 (2019)