Topological derivative methods are used to solve constrained optimization reformulations of inverse scattering problems. The constraints take the form of Helmholtz or elasticity problems with different boundary conditions at the interface between the surrounding medium and the scatterers. Formulae for the topological derivatives are found by first computing shape derivatives and then performing suitable asymptotic expansions in domains with vanishing holes. We discuss integral methods for the numerical approximation of the scatterers using topological derivatives and implement a fast iterative procedure to improve the description of their number, size, location and shape.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. R. Arridge and J. C. Hebden, “Optical Imaging in Medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, pp. 841–853, 1997.
P. J. Michielsen K., DeRaedt H. and G. N, “Computer simulation of time-resolved optical imaging of objects hidden in turbid media,” Phys. Reports 304, pp. 90–144, 1998.
S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Laser. Surg. Med. 26(2), pp. 119–129, 2000.
J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, pp. 6535–6546, 1992.
M. Rowe, E. Pungh, J. Tyo, and N. Engheta, “Polarization-difference imaging: a biologically inspired technique for observation through scattering media,” Opt. Lett. 20, pp. 608–610, 1995.
S. Demos and R. Alfano, “Temporal gating in highly scattering media by the degree of optical polarization,” Opt. Lett. 21, pp. 161–163, 1996.
M. Moscoso, J. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissues,” J. Opt. Soc. Am. A 18, No. 4, pp. 948–960, 2001.
S. Chandrasekhar, Radiative Transfer, Oxford University Press, Cambridge, 1960.
A. Kim and M. Moscoso, “Chebyshev spectral methods for radiative transfer,” SIAM J. Sci. Comput. 23, No.6, pp. 2075–2095, 2002.
B. C. Wilson and S. Jacques, “Optical reflectance and transmittance of tissues: Principles and applications,” IEEE Quant. Electron. 26, pp. 2186–2199, 1990.
J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21, pp. 1310–1312, 1996.
L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, pp. 327–370, 1996.
A. Kim and M. Moscoso, “Influence of the relative refractive index on depolarization of multiply scattered waves,” Physical Review E 64, p. 026612, 2001.
A. Kim and M. Moscoso, “Backscattering of circularly polarized pulses,” Opt. Letters. 27, pp. 1589–1591, 2002.
B. Beauvoit, T. Kitai, and B. Chance, “Contribution to the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, pp. 2501–2510, 1994.
J. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, pp. 3586–3593, 1998.
M. H. Kalos and P. A. Whitlock, Monte Carlo Methods, New York : J. Wiley Sons, 1986.
E. E. Lewis and W. F. Miller Jr., Computational Methods of Neutron Transport, John Wiley and sons, New York, 1984.
G. Bal, G. Papanicolaou, and L. Ryzhik, “Probabilistic Theory of Transport Processes with Polarization,” SIAM Appl. Math. 60, pp. 1639–1666, 2000.
M. Kalos, “On the estimation of flux at a point by Monte Carlo,” Nucl. Sci. Eng. 16, pp. 111–117, 1963.
P. Bruscaglioni, G. Zaccanti, and Q. Wei, “Transmission of a pulse polarized light beam through thick turbid media: numerical results,” Appl. Opt. 32, pp. 6142–6150, 1993.
E. Tinet, S. Avrillier, and M. Tualle, “Fast semianalytical monte carlo simulation for time-resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, pp. 1903–1915, 1996.
S. Chatigny, M. Morin, D. Asselin, Y. Painchaud, and P. Beaudry, “Hybrid monte carlo for photon transport through optically thick scattering media.,” Appl. Opt. 38, pp. 6075–6086, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Carpio, A., Rapún, M.L. (2008). Topological Derivatives for Shape Reconstruction. In: Bonilla, L.L. (eds) Inverse Problems and Imaging. Lecture Notes in Mathematics, vol 1943. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78547-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-78547-7_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78545-3
Online ISBN: 978-3-540-78547-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)