Fitting ellipsoids to objects by the first order polarization tensor

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DOI:

https://doi.org/10.26637/mjm104/005

Abstract

This article present the manual to determine ellipsoids that has the same first order polarization tensor to any conducting objects included in electrical field. Given the first order polarization tensor for an object at specified conductivity, the analytical formula of the first order polarization tensor for ellipsoid in the integral form is firstly expressed as system of nonlinear equation by the trapezium rule. We will then discuss how the derived equations are simultaneously solved by appropriated numerical method to uniquely compute all semi principal axes of the ellipsoid. Few examples to use the proposed technique in this study are also provided in three different situations. In each case, the first order polarization tensor for the obtained ellipsoid can be calculated back from the analytical formula to examine the effectiveness of the method.

Keywords:

Integral operator, multi-indices, matrices, numerical integration, eigenvalues

Mathematics Subject Classification:

65D30, 34A34, 05B20
  • Pages: 44-53
  • Date Published: 01-10-2013
  • Vol. 1 No. 04 (2013): Malaya Journal of Matematik (MJM)

A. Adler, R. Gaburro and W. Lionheart, Electrical Impedance Tomography, in O. Scherzer (ed) Handbook of Mathematical Methods in Imaging, Springer-Verlag, USA, 2011. DOI: https://doi.org/10.1007/978-0-387-92920-0_14

A. S. N. Alfhaid, Numerical Analysis Part 2, Lecture Note, Mathematics Department, King Abdulaziz University.

H. Ammari and H. Kang, Polarization and Moment Tensors : with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences Series, volume 162, Springer-Verlag, New York, 2007.

D. Bhardwaj, Solution of Nonlinear Equations, Lecture Note, Department of Computer Science and Engineering, Indian Institute of Technology Delhi.

P. J. Davis and P. Rabinowitz, Numerical Integration, Blaisdell Publishing Company, USA, 1967.

S. Gibilisco, Geometry Demystified : A Self Teaching Guide, McGraw-Hill Education, USA, 2003.

T. M. Heath, Scientific Computing : An Introductory Survey, McGraw-Hill, New York, 2002.

D. S. Holder (ed), Electrical Impedance Tomography : Methods, History and Applications, Institute of Physics Publishing, London, 2005. DOI: https://doi.org/10.1201/9781420034462.ch4

D. P. Lerner, Elementary of Integration Theory, Lecture Note, College of Liberal Arts and Sciences, University of Kansas.

E. A. Lord and C. B. Wilson, The Mathematical Description of Shape and Form, Ellis Horwood Limited, England, 1984.

A. Mazer, The Ellipse : A Historical and Mathematical Journey, Wiley, Canada, 2010. DOI: https://doi.org/10.1002/9780470591031

G. Pólya and G. Szeg˝ o, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematical Studies Number 27, Princeton University Press, Princeton, 1951.

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in Fortran 77 Second Edition : The Art of Scientific Computing, Cambridge University Press, USA, 1992.

Taufiq, K. A. K. and W. R. B. Lionheart, Do electro-sensing fish use the first order polarization tensor for object characterization? in 100 years of Electrical Imaging, Presses des Mines, Paris, 2012.

Taufiq, K. A. K. and W. R. B. Lionheart, Some Properties of the First Order Polarization Tensor for 3-D Domains, Matematika UTM, volume 29 issue 1, (2013), 1–18.

G. von der Emde and S. Fetz, Distance, Shape and More : Recognition of Object Features during Active Electrolocation in a weakly Electric Fish, The Journal of Experimental Biology, volume 210, (2007), 3082-3095. DOI: https://doi.org/10.1242/jeb.005694

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Published

01-10-2013

How to Cite

Taufiq K. A. Khairuddin, and William R. B. Lionheart. “Fitting Ellipsoids to Objects by the First Order Polarization Tensor”. Malaya Journal of Matematik, vol. 1, no. 04, Oct. 2013, pp. 44-53, doi:10.26637/mjm104/005.